One of our main conclusions is that for oscillatory elds associated with wave vectors on curved ..... 0, the solutions u"(t) and v"(t) diverge to in nity in Hs when s is.
DIFFRACTIVE NONLINEAR GEOMETRIC OPTICS WITH RECTIFICATION * Jean-Luc JOLY
MAB, Universite de Bordeaux I 33405 Talence, FRANCE
Guy METIVIER
IRMAR, Universite de Rennes I 35042 Rennes, FRANCE
Je rey RAUCH
Department of Mathematics, University of Michigan Ann Arbor 48109 MI, USA
Abstract. This paper studies high frequency solutions of nonlinear hyperbolic equations for time scales at which di ractive e ects and nonlinear e ects are both present in the leading term of approximate solutions. The key innovation is the analysis of recti cation e ects, that is the interaction of the nonoscillatory local mean eld with the rapidly oscillating elds. The main results prove that in the limit of frequency tending to in nity, the relative error in our approximate solutions tends to zero. One of our main conclusions is that for oscillatory elds associated with wave vectors on curved parts of the characteristic variety, the interaction is negligible to leading order. For wave vectors on at parts of the variety, the interaction is spelled out in detail.
Outline. x1. Introduction. x2. The ansatz and the rst pro le equations. x3. Large time asymptotics for linear symmetric hyperbolic systems. x4. Pro le equations, continuation. x5. Solvability of the pro le equations. x6. Convergence and stability. x7. Applications, examples and extensions. References
x1. Introduction.
This paper continues the study, initiated in [DJMR], [D], of the behavior of high frequency solutions of nonlinear hyperbolic equations for time scales at which di ractive e ects and nonlinear e ects are both present in the leading term of approximate solutions. By di ractive e ects we mean that the leading term in the asymptotic expansion has support which extends beyond the region reached by the rays of geometric optics. Our expansions are for problems where the time scale for nonlinear interaction is comparable to the time scale for the onset of di ractive e ects. The key innovation is the analysis of recti cation e ects, that is the interaction of the nonoscillatory local mean eld with the rapidly oscillating elds. On the long time scales associated with di raction, these nonoscillatory elds tend to behave very di erently from the oscillating elds. One of our main conclusions is that for oscillatory elds associated with wave vectors on curved parts of the characteristic variety, the interaction is negligible to leading order. For wave vector on at parts of the variety, and in particular problems for problems in one dimensional space, the interaction
* Research partially supported by the U.S. National Science Foundation, U.S. Oce of Naval Research, and the NSF-CNRS cooperation program under grants number NSF-DMS-9203413 and OD-G-N0014-92-J-1245 NSF-INT-9314095 respectively, and the CNRS through the Groupe de Recherche G1180 POAN.
1
cannot be ignored and is spelled out in detail. In all cases the leading term in an approximation is rigorously justi ed, the correctors are o(1) as the wavelength " tends to zero. The error is not O(" ) with > 0. The point of departure are the papers [DJMR], [D] where in nitely accurate asymptotic expansions of the form X j " aj ("x; x; x: =") ; (1:1) u" "p j 2Np
are justi ed. Here x = (t; y) 2 R1+d , = ( ; ) 2 R1+d and the pro les aj (X; x; ) are periodic in . The exponent p is a critical exponent for which the time scales for di ractive and nonlinear e ects are equal. Note the three scales, a wavelength of order ", variations on the scale 1 coming from the dependence of the pro les on x and variations on the long and slow scales 1=" from the dependence on "x. This three scale structure is typical of di ractive geometric optics. In [DJMR] the following key hypotheses guaranteed that nonlinear interaction did not create nonoscillatory terms, the nonlinear terms were odd functions of u, and the pro les aj (X; x; ) in the approximations (1.1) had spectrum contained in Zodd. The rst hypothesis excludes quadratic nonlinearities and the second implies that the pro les have vanishing mean value with respect to . The main achievement of this paper is the analysis of solutions when these hypotheses are not made. Note that quadratic interaction of oscillatory terms tends to create nonoscillatory sources as the simple identities eik e?ik = 1 and sin2 k +cos2 k = 1 illustrate. The creation of nonoscillatory waves from highly oscillatory sources is called recti cation. The spatial Fourier Transform of an initial datum of the form a(y) ei:y=" is roughly localized at wave numbers 2 Rd with j ? ="j = O(1). In the linear case solutions would be superpositions of plane waves moving with velocities with angular spread O("). For times small compared with 1=" this angular dispersion is not important. For times O(1=") the angular dispersion becomes important since at those times the accumulated e ects are O(1). This scale t 1=" is the time scale of di ractive geometric optics.
1. Example of creation of nonoscillating parts in the principal pro le. Consider the semilinear initial value problem for for u = (u1 ; u2 )
@u + 1 0 @u + u21 = 0 ; 0 @t 0 ?1 @y
u" (0; y) = " (f (y); g(y)) sin y=" ;
with f; g 2 C01 (R; R ). The equations for the uj are decoupled. The linear part is a rst order system which corresponds to the D'Alembert operator 1+1 on R1+1 . The characteristic variety is the union of the two lines = . The exact solutions with g = 0 satis es u"2 = 0 and (y ? t)=" u"1 (t; y) = 1 +" f"t(yf (?y t?) tsin ) sin (y ? t)="
? =" f (y ? t) sin (y ? t)=" ? "t f (y ? t) sin (y ? t)=" 2 + O("3t) :
They have p = 1 and pro le
a(T; t; y; ) = f (t ? y) sin ? T f (t ? y)2 sin2 ; 0 : 2
For t 1=" the solution has nonoscillatory terms " "tf (t ? y)2 of the same order as the oscillating part. The factor is the mean value of sin2 .
2. Example of the principal of nonoverlapping waves. The preceding example should be contrasted to the behavior of the system where the nonlinear term (u21 ; 0) is replaced by the nonlinear term (u22 ; 0). To see any e ect one must take g = 6 0. The equation for u1 then reads @t u1 + @y u1 + "2 g(t + y)2 = 0 : The source term "2 g(t + y)2 travels at speed minus one so crosses the integral curves of @t + @y transversally. The u1 wave and the u2 wave overlap for only a nite period of time so the in uence of the source term is O("2 ) therefore negligible compared to the leading O(") term even for times t 1=". If g were not integrable in y this argument would not be correct. In this example and for the remainder of the introduction the principal pro les will depend only on (T; t; y; ). There is no dependence on Y .
3. Examples where recti cation e ects must be analysed. 1. The compressible Euler equations in uid dynamics and all more complicated models containing variants of the Euler equations, have quadratic terms. 2. From nonlinear optics the important phenomenon of second harmonic generation is a quadratic phenomenon. 3. Even for odd nonlinearities initial pro les which contain even harmonics are not included in the previous analysis. When recti cation e ects are present, correctors to the leading approximation are required which do not have the structure of modulated high frequency wave trains. The correctors can look like typical nonoscillatory solutions of nonlinear or linear wave equations. In particular they have no reason to follow the rays of the oscillatory parts. They sometimes spread in all directions so are correspondingly smaller than the leading term in the approximation.
4. Example of the principal of spreading waves. Consider the two dimensional analogue of
the equations in Examples 1 and 2,
1 0 @u + 0 1 @u + (u) = 0 ; + L u + (u) := @u @t 0 ?1 @y1 1 0 @y2 where is quadratic and u" has oscillations of wavelength ". For this problem the critical exponent p = 1 so the critical initial value problem is
?
u" (0; y) = " b(y; y1 ="); 0 ;
b 2 C01(R2 T) :
Theorem 1.1 shows that the solution is given for t C=" by
?
u" (t; y) = " a(t; x) + " a(T; t ? y1 ; y2 ; : : : ; ) ; 0) + o(") : where the oscillatory pro le a(T; t; y; ) has mean zero with respect to and is determined by the initial value problem ? @T a + 21 y ;:::;yd @?1 a + (a; 0) = 0 ; a(0; y; ) = b(y; ) ; 3 2
where denontes oscillatory part, that is the function less its mean with respect to . The nonoscillatory part of the pro le, a, is determined by the linear hyperbolic dynamics
Z 2 1 a(0; x) = 2 b(0; y; ) d : 0
@a + 1 0 @a + 0 1 @a = 0 ; @t 0 ?1 @y1 1 0 @y2
The function a(t; x) decays to zero in L1 (R2 ) as t ! 1 but has L2 (R2 ) norm independent of time. The conservation is maintained while the waves spread. The L1 decay is responsible for the fact that the mean eld does not in uence the leading term of the oscillatory part. The oscillatory and nonosillatory part do not interact to leading order. If ajt=0 = 0 as in Examples 1 and 2, it remains zero, in contrast to Example 1. However, even though the principal term in the expansion has mean zero in , to prove the accuracy of the leading term requires correctors which have nonoscillatory parts. The approximate solutions which we construct are not in nitely accurate and the proof of accuracy requires a fundamentally di erent analysis than our earlier papers on di ractive nonlinear geometric optics. Analogous diculties arose when studying multiphase nonlinear geometric optics for example in [JMR 2]. For the longer time scales of di ractive geometric optics, the diculty occurs for one phase problems. That one does not have a simple expansion like (1.1) is clear in the next example.
5. Example of a large corrector. Consider the coupled system 1+2 v = jrt;y uj
2:
1+2 u = 0 ;
With
0 0 @=@y1 @=@y2 1 B C @ B C L := @t ? B @=@y 0 0 C 1 @ A; @=@y2
0 0 this yields the triangular system of equations for u := rt;y u and v := rt;y v,
Lu = 0;
?
L v = juj2 ; 0; 0 :
Take Cauchy data for v" to be equal to zero. The critical exponent is p = 1. For t 1=" one can take u" = O(") to be a linearly di ractive solution and the computation in x7.5 shows that v" = O("3=2 ) is a corrector but is large compared to the O("2) correctors in (1.1). Three scale expansions like those discussed in this paper arise in a variety of applied problems. Several from nonlinear optics are presented in [DR]. Others with origins in uid dynamics including a variety of problems in shock di raction where rigorous error estimates have yet to be proven can be found in [H]. The latter reference also contains an interesting historical discussion. We present examples in x7. We now describe a typical result from our analysis. One must distinguish between hyperplanes in the characteristic variety and curved sheets since that determines the extent of the interaction between the mean eld and the oscillatory parts. 4
Consider the constant coecient semilinear equation
L(@x ) u + (u) = 0 ;
L :=
d X =0
A @x@
(1:2)
where u is a smooth C N or RN valued function and is a polynomial of degree J with variables (Re u; Im u). Quasilinear problems are treated in the body of the paper. The system is assumed to be symmetric hyperbolic, that is
A = A ;
A0 = I :
(1:3)
In the introduction we suppose that the characteristic variety consists of a nite number ` of smooth nonintersecting sheets = j (), j = 1; : : : ; `. This is the case for strictly hyperbolic equations, Maxwell's equations and the linearized Euler equations. For J th order semilinear problems (1:4) p := J ?1 1
is the critical exponent for which the time of nonlinear interaction is O(1="), which is the time for the onset of di ractive e ects. In this introduction we consider only pro les a0 which are independent of Y . For 2 Char L R 1+d n f0g we construct approximate solutions
u" (x) = "p a0 ("t; t; y; x: =") ;
(1:5)
where a0 (T; t; y; ) is smooth, periodic in and has derivatives square integrable on [0; T ] R dy T uniformly for t 2 R. Hyperplanes which are contained in the characteristic variety play a crucial role in describing di ractive nonlinear geometric optics. Decompose the index set
f1; : : : ; `g = Af [ Ac
(1:6)
where Ac are the indices of curved sheets and Af those of at sheets. Then Af is empty when there are no hyperplanes. De ne a switch by = 1 when belongs to a hyperplane and = 0 otherwise. When = 1 relabel the sheets so that belongs to the rst hyperplane = 1 (). For 2 Char L let ( ) denote the orthogonal projection of C N onto the kernel of L( ). De ne
?
E () := ( ); ) ;
2 A:
(1:7)
The Fourier multiplication operators E (Dy ) are orthogonal projections on H s (Rd ) for all s. One has = ( ( ); ) for a unique 1 ` and we associate two di erential operators
V (@x ) := @t + v:@y ; and
v := ?r () ;
d @ 2 () @ 2 X 1 : R(@y ) := ? 2 j;k=1 @j @k @yj @yk
5
(1:8) (1:9)
For a periodic function a, the mean value in is denoted by either a or hai. The oscillating part is a := a ? a. With the above de nitions, we can write the equations for a0 . The rst equation is
L(@x ) a0 = 0 :
which allows us to decompose
a0 =
X 2A
a0; ;
(1:10)
a0; := E (Dy ) a0 ;
(1:11)
and the elements of this modal decomposition satisfy ? @ + (D ) a = 0 : (1:12) t y 0; De ne a switch : A ! f0; 1g by ( ) = 1 if and only if 2 Af . The dynamics of the mean values is given by the equations
D
E (Dy ) @T a0; + ( ) ( 1; a0 + a0; )
E
= 0;
where Kronecker's 1; has value 1 when = 1 and vanishes otherwise. The oscillating part is polarized and has simple dynamics with respect to t, ( ) a0 = a0 ; V (@x ) a0 = 0 : These equations are equivalent to a0 (T; t; y; ) := a(T; y ? vt; ) ; a = ( ) a : The reduced pro le, a(T; y; ), has dynamics given by
@T a0 ? R(@y ) @?1 a0 + ( ) (a0 + a0;1 )
= 0:
(1:13)
(1:14) (1:15) (1:16)
The on the last term indicates that the nonoscillating term has been removed so the constraint a = a from (1.15) is satis ed as soon as it is satis ed in fT = 0g. This paper is devoted to the derivation of more general versions of equations (1.10)-(1.16), the proof of their solvability, and the accuracy of the approximate solution in the limit " ! 0. The nature of equation (1.16) can be understood as follows. The linear part given by the rst two terms acts as @T ? iR(@y )=n on the Fourier components ^a(T; y; n). The operator R is symmetric so iR=n is antisymmetric and therefore R generates unitary evolutions in H s . The semilinear term is roughly of the form aJ . The next results are a specialization of Theorems 5.1 and 6.1 to the semilinear case and pro les which do not depend on Y .
Theorem 1.1. If g(y; ) 2 \s H s (Rd T) satis es g = ( )g, then there is T 2]0; 1] and unique maximal solutions (a0 ; a) to the pro le equations (1.10-1.16) satisfying
? a0 (T; t; y; ) ; a(T; t; y)
L ([0;T ]RdT) < 1 ; 8 8 T 2]0; T [ ; sup
@T;y;
and the initial condition
t2R
2
a0 (0; 0; y) + a(0; y; ) = g(y; ) : 6
y
(1:17)
Theorem 1.2. With pro les fa0 ; ag from Theorem 1.1, de ne a0 = a0 + a0 with a0 given by (1.15) and a family of approximate solution u" by (1.5). Let v" 2 C 1 ([0; t (")[Rd ) be the maximal
solutions of the initial value problems L(@x ) v" + (v" ) = 0 ; v" jt=0 = u"jt=0 : Then, for any T 2]0; T [, there is an "0 > 0 so that if " < "0 , then t (") > T=" and v" (t) ? u" (t) = o("p) : sup 0tT="; y2Rd
(1:18) (1:19)
Remarks. 1. The discussion before the theorem suggests that a0 = O(1) for 0 t T so that in
(1.19) neither of the terms in the di erence is o("p). 2. Typically for xed t and " ! 0, the solutions u" (t) and v" (t) diverge to in nity in H s when s is large. This prevents the application of elementary continuous dependence arguments. In addition, the residuals L(@ ) u" + (u" ) , with u" := "p a0 , are O("1+p) so the accumulated e ect of the residual for times of order 1=" is crudely estimated to be O("p) and therefore nonnegligible. For the proof of Theorem 1.2 we construct a corrector of size O("p+1) which reduces the residual to size o("p+1 ). 3. The equations determining the pro le a0 are found so that such correctors exist. 4. To derive the equations for a0 requires detailed information about the large time asymptotic behavior of solutions of the linear equation (1.10), and related inhomogeneous equations. For example, all solutions of a linear symmetric hyperbolic equation with C01 initial data tend to zero in sup norm if and only if the characteristic variety contains no hyperplane. The proof of such results in x3 is by a nonstationary phase arguments with attention paid to possible singular points in the characteristic variety. 5. Qualitative information can be gleaned from the pro le equations. For example, if does not lie on a hyperplane in the characteristic variety then the oscillatory part of the solution does not in uence the nonoscillatory part to leading order. In particular, if a0 = 0 at ft = T = 0g, then it vanishes identically, and the pro le equations simplify to the single equation (1.16). These reductions, applications, and extensions are discussed in x7.
Acknowledgement. Thanks to J.-M. Delort, P. Donnat and J. Hunter for instructive discussions which were crucially important in carrying out this research. In addition, the suggestions of David Lannes and Eric Dumas concerning an earlier version of the text were very helpful.
x2. Formulating the ansatz and the rst pro le equations. x2.1. The ansatz and the pro le equations.
In this section approximate solutions of the nonlinear symmetric hyperbolic equation L(u; @x ) u + F (u) = 0 : are constructed. The principal part of (2.1) is the quasilinear operator
L(u; @x ) u :=
d X
=0
A (u) @ u
(2:1) (2:2)
Symmetric hyperbolicity assumption. The coecients A are smooth hermitian symmetric valued functions of u on a neighborhood of 0 2 C N , and, for each u, A0 (u) is positive de nite. 7
Smoothness means that A (u) = A (Re u; Im u) with A 2 C 1 (RN RN ; Hom (C N )).
Order of nonlinearity. The quasilinear terms are of order 2 K 2 N in the sense that ? j j K ? 2 =) @Re (2:3) u;Im u A (u) ? A (0) u=0 = 0 : Then A (u) ? A (0) = O(jujK ?1 ) and so the quasilinear term, (A (u) ? A (0))@u is of order K . The semilinear term F is smooth on a neighborhood of 0 2 C N , and is of order J 2 in the sense that
j j J ? 1 =) @Re (2:4) u;Im u F (0) = 0 : Then F (u) = O(jujJ ). Making the change of dependent variable to A0 (0)?1=2 u and multiplying the resulting equations by A0 (0)?1=2 u preserves these hypotheses and reduces to the case A0 (0) = I .
Convention. A0 (0) = I . If the solutions have amplitude of order "p with derivatives of order "p?1 , then the size of the quasilinear terms is "p(K ?1) "p?1 . The accumulated e ects after time T are crudely estimated to be of order T"pK ?1 . Setting this equal to the order of magnitude of the solutions, "p , yields the following estimate for the time of nonlinear interaction T 1 : quasilinear
"pK?p?1
Similarly, the semilinear terms are of order "pJ , with accumulation T"pJ and time of interaction estimated by setting this equal to "p , T 1 : semilinear
"pJ ?p
The goal is that the time of nonlinear interaction is of order "?1 which is the time for the onset of di ractive e ects in linear problems with linear phases and wavelengths " (see [DJMR]). Thus one wants pK ? p ? 1 1 and pJ ? p 1 with equality in at least one of the two.
The standard normalization. Consider waves of wavelength of order " and amplitude of order
"p satisfying
p = max K 2? 1 ; J ?1 1 :
(2:5)
Examples. The classic case is quadratic nonlinearities. For quadratic quasilinear e ects one has
K = 2, and p = 2. A semilinear term with J 2 will not a ect the leading behavior. For quadratic semilinear e ects one has J = 2, p = 1, and K 3. For odd order quasilinear terms, the time of interaction for a semilinear term of order J = (K +1)=2 is of the same order as that of the quasilinear term. For even order quasilinear terms, there is never such agreement. If K > 3 and J > 2, then p is a fraction with 0 < p < 1. 8
De nition of leading order nonlinear terms. If p = 2=(K ? 1), the degree K ? 1 Taylor polynomial of A (u) ? A (0) at u = 0 is denoted (u). If p > 2=(K ? 1) set := 0 for all . If p = 1=(J ? 1), let (u) denote the order J Taylor polynomial of F at u = 0. If p > 1=(J ? 1) set := 0.
With this de nition, the contribution of the quasilinear and semilinear terms to the dynamics of the leading term in the approximate solution is given in terms of and . Note that and are homogeneous polynomials of degree 2=p and (p + 1)=p respectively with the convention that if either degree is not an integer, the corresponding polynomial vanishes identically. The basic ansatz has three scales
u" (x) = "p a("; "x; x; x: =") ;
(2:6)
where
a("; X; x; ) = a0 (X; x; ) + " a1 (X; x; ) + "2 a2 (X; x; ) : (2:7) This expansion has terms which are chosen so that the leading terms in L u" + F (u" ) vanish.
Example of nonuniqueness of pro les. Di erent pro les can lead to the same approximate solution when multiple scale expansions are used. For example if a(T; t) 2 C01 (R2 ) de ne g(T; t) 2 C01(R2 ) by a(T; t) ? a(0; t) = Tg(T; t) :
Then the pro les a(T; t) and a(0; t) + "tg(T; t) de ne the same function when T = "t is injected. This lack of uniqueness means the aj are not determined so that in deriving equations we are obliged to make choices. We have tried to take a path where these choices are as natural as possible. Since the expansion (2.7) is used for times t 1=" one must control the growth of the pro les in t. In order for the corrector a1 in (2.7) to be smaller than the principal term for such times, it must satisfy " a ("x; x; ) = lim sup " a (X; X="; ) = 0 : lim sup (2:8) 1 1 "!0 "!0 [0;T="]RdT
[0;T ]RdT
Our pro les satisfy the stronger condition of sublinear growth 1 sup a (X; t; y; ) = 0 : lim 1 t!1 t X;y;
(2:9)
This condition plays a central role in the analysis to follow. The pro les aj (T; Y; x; ) are smooth on [0; T [Rd R1+d T.
x2.2. Equations for the pro les.
The chain rule implies that
h i L(u"; @x ) u" + F (u" ) = L("pa" ; "@X + @x + " @ ) "p a + F ("p a) X ="x; =x: =" :
(2:10)
The strategy is to expand the function of (X; x; ) in brackets in powers of " and to choose a so that the leading terms vanish.
9
Lemma 2.1. Suppose that a0 ; a1 ; a2 are smooth functions on [0; T ] Rd R1+d T and that a is given by (2.7). Then
L("p a" ; "@X + @x + " @ ) "p a + F ("p a) = "p?1 r0 (X; x; ) + "r1(X; x; ) + "2 r2 (X; x; ) + "2+min f1;pg h("; X; x; ) ;
(2:11)
where the rj (X; x; ) are given by
r0 = L(0; ) @ a0 ;
r1 = L(0; ) @ a1 + L(0; @x ) a0 ;
r2 = L(0; ) @ a2 + L(0; @x ) a1 + L(0; @X ) a0 + (a0 ) +
d X =0
(2:12)
(a0 ) @ a0 :
(2:13)
The function h("; X; x; ) depends on the aj , is periodic in and each of its derivatives with respect to X; x; is continuous on [?1; 1] R1+d R1+d T .
Remark. For the aj and h which we construct precise estimates are given in Theorem 5.1 and Proposition 5.4. Proof. Compute
L(0; "@X + @x + " @ ) "p a = "p?1 L(0; ) @ a + " L(0; @x ) a + "2 L(0; @X ) a :
(2:14)
The leading term in the Taylor expansion of A ("p a) ? A (0) is ("p a0 ) and is homogeneous of degree 2=p. Thus,
A ("p a) ? A (0) = "2 (a0 (X; x; )) + "min f1;pg h1("; X; x; ) : Similarly, is homogeneous of degree (p + 1)=p so,
(2:15)
F ("p a) = "p+1 (a0(X; x; )) + "min f1;pg h2 ("; X; x; ) : This together with equations (2.14), and (2.15) proves the Lemma. The pro les in (2.7) are chosen so that in (2.11) one has rj = 0 for j = 0; 1; 2. Equations (2.12-2.13) show that r0 ; r1 ; r2 vanish if and only if
L(0; ) @ a0 = 0 ;
(2:16)
L(0; ) @ a1 + L(0; @x ) a0 = 0 :
(2:17)
L(0; ) @ a2 + L(0; @x ) a1 + L(0; @X ) a0 + (a0 ) +
d X
=0
(a0 ) @ a0 = 0 :
(2:18)
The analysis of these equations is as in [DJMR], with the important exception that the mean value or zero mode of the periodic functions must be treated. This is more subtle than meets the eye, and is the essential diculty of the present paper. 10
Notation. Introduce two notations for the mean value with respect to a := hai := 21 The oscillating part is denoted a := a ? a :
Z 2 0
a d :
2.3. Analysis of equation 2.16.
Equation (2.16) holds if and only if a takes values in the kernel of L(0; ). This kernel is nontrivial if and only if is characteristic for the linear di erential operator L(0; @ ) . The set of such is denoted Char L(0; @ ) := f 2 Rd+1 n f0g : det L(0; ) = 0g : There are two naturally de ned matrices which play a central role.
De nition. For 2 R1+d , ( ) denotes the linear projection on the kernel of L(0; ) along the
range of L(0; ). Q( ) is the partial inverse de ned by
Q( ) ( ) = 0 ;
Q( ) L(0; ) = I ? ( ) :
and
(2:19)
Symmetric hyperbolicity implies that both ( ) and Q( ) are hermitian symmetric. In particular ( ) is an orthogonal projector. With this notation, (2.16) is equivalent to
( ) a0 = a0 :
(2:20)
This asserts that the oscillating part of the principal pro le is polarized along the kernel of L(0; ). Equation (2.16) imposes no constraints on the nonoscillating part a.
x2.4. Analysis of equation 2.17.
Equation (2.17) involves both a0 and a1 . Taking the mean value with respect to eliminates a0 and yields a fundamental evolution equation for the nonoscillating part a0 (X; x),
L(0; @x ) a0 = 0 :
(2:21)
Multiplying the oscillating part of (2.17) by ( ) annihilates L(0; ) so eliminates the a1 term to give ( ) L(@x ) ( ) a0 = 0 : (2:22) A vector w 2 C N vanishes if and only if ( )w = 0 and Q( )w = 0: Thus the information in (2.17) complementary to (2.21-2.22) is obtained by multiplying the oscillatory part of (2.17) by Q( ). This yields (2:23) (I ? ( )) a1 = ?Q( ) L1 (@x ) @?1 a0 : Equations (2.20) and (2.22) are the fundamental equations of linear geometric optics (see [R]). They determine the dynamics of a0 = ( )a0 with respect to the time t. The following hypothesis guarantees that the linear geometric optics is simple. It excludes for example along the optic axis of conical refraction. 11
Smooth characteristic variety hypothesis. = ( ; ) 2 Char L(0; @ ) and there is a neighborhood ! of in Rd (resp. O of in R1+d ) so that for each 2 ! there is exactly one point ( (); ) 2 O \ Char L(0; @ ). The next proposition is virtually identical to Proposition 3.1 in [DJMR]. The proof proceeds by P di erentiating the identity ( Aj j )( ) = ( )( ) with respect to k . The formula (2.24) is simpler than in [DJMR] thanks to the convention that A0 (0) = I . The traditional presentation of this identity uses left and right eigenvectors as in xVI.3.11 of [C]. The eigenvectors are not uniquely determined but the projector ( ) is.
Proposition 2.2. If the smooth characteristic variety hypothesis is satis ed then the functions,
(), ( (); ) and Q( (); ) are real analytic on !. If A0 (0) = I then
@ X d @ ( ) @ ( ) L(0; @x ) ( ) = ( ) @t ? @j @yj : j =1
(2:24)
De nitions. If ; belongs to the characteristic variety and satis es the smoothness assumption, de ne the transport operator V and group velocity v by d @ ( ) @ @ ?X V ( ; ; @x ) := @t := @t + v:@y : j =1 @j @xj
(2:25)
The ; dependence of ; Q; V , v will often be omitted when there is little risk of confusion. Equations (2.24) and (2.25) show that (2.22) is equivalent to
V (@x ) a0 = 0 :
(2:26)
Corollary 2.3 If the smooth characteristic variety hypothesis is satis ed at = ( ; ) then Char V (@x ) is the tangent plane to Char L(0; @x ) at ( ; ).
Proof. Near ( ; ), the characteristic variety of L(0; @x ) has equation = (). Therefore its tangent plane has equation ? = r ( ) ( ? ).
Formula (2.25) shows that the characteristic variety of V (@x ) is the codimension 1 linear subspace with equation ? r ( ) = 0. Thus ( ; ) + Char V is the tangent plane to Char L at ( ; ) . To complete the proof it suces to show that ( ; ) 2 Char V . Since the function ? () is homogeneous of degree one in ; , Euler's identity implies that ?r (): = ? (). At ( ; ) , the right hand side vanishes which proves that ( ; ) 2 Char V . 12
x2.5. A rst look at equation 2.18.
Equation (2.18) involves three pro les because there are three scales in the ansatz. Taking the mean value with respect to eliminates the a2 part and yields an equation for the nonoscillating part of a1 E D X L(0; @x ) a1 = ?L(0; @X ) a0 ? (a0 ) + (a0 )@ a0 : (2:27) The last term on the right may be nonzero, even if a0 = 0. This possibility of creating nonoscillating contributions from oscillating terms is called recti cation. The analysis of equation (2.27) is the main diculty in the derivation of the pro le equations. The idea is that since a0 must satisfy (2.21) and (2.26) its asymptotics as t ! 1 are quite structured so the form of the source terms on the right of (2.27) can be described. Requiring that a1 satisfy the sublinear growth condition (2.9) places constraints on a0 . To understand these we must study the asymptotic behavior of solutions of the linear symmetric hyperbolic system (2.21) and also the asymptotics of the inhomogeneous equation L(0; @x )u = f where the source term f is constructed from solutions of the homogeneous equation as in the right hand side of (2.27). The next section is devoted to the study of these linear problems. In x4 we return to the study of (2.18) and in particular (2.27).
x3. Large time asymptotics for linear symmetric hyperbolic systems.
In this section we suppose that
L(@x ) =
d X =0
A @x@ ;
A = A ;
A0 = I
(3:1)
is a general constant coecient symmetric hyperbolic operator with coecient of @t equal to the identity matrix. In our applications, L will be taken to be L(0; @x ). Proposition 3.2 decomposes solutions of Lu = 0 into modes which rigidly translate and modes which spread out in space. The latter decay in sup norm. The analysis is by stationary and nonstationary phase. Care is needed because the characteristic variety may have singular points. The basic strati cation theorem of real algebraic geometry implies that these singularities form a lower dimensional variety and this implies that the contributions of the singular points can be treated as error terms.
Examples. The solutions of the equation in dimension d and the equation
@v + @v = 0 @t @y1
@v + 1 0 @v = 0 0 ?1 @y @t
(3:2) (3:3)
in dimension 1 have only purely translating modes. In dimension 2, the rst order analogue of D'Alembert's wave equation @u + 1 0 @u + 0 1 @u = 0 : (3:4) 0 ?1 @y1 1 0 @y2 @t has solutions which spread in time and whose L1 norm decays like t?1=2 while the L2 norm is conserved. 13
The characteristic variety for (3.2) and (3.3) consist of hyperplanes while for (3.4) the variety is curved. This dichotomy is crucial as indicated by the following heuristic argument about the group velocities ?r (). If the variety is at with equation = ?v: these velocities do not depend on and wave packets will translate at this xed velocity. If the variety is not at, the variation of the speed spreads wavepackets leading to their decay in L1 .
x3.1. Hyperplanes and singularities in the characteristic variety. The characteristic variety is de ned by
Char L := f( ; ) 2 R1+d n f0g : det L( ; ) = 0 g :
(3:5)
For each point ( ; ) in the variety and a 2 ker L( ; ), u = ei( t+:y)a is a plane wave solution of Lu = 0. Since ( ; 0) is noncharacteristic for L, any hyperplane fa + b: = 0g contained in the characteristic variety must have a 6= 0 so the hyperplane is necessarily a graph f = ?v:g. Since over each 2 Rd there are at most N points in the characteristic variety, the variety contains at most N distinct hyperplanes H1 ; : : : ; HM ,
Hj = f ( ; ) : = ?vj : g ;
j = 1; : : : ; M N :
(3:6)
Examples. 1. When d = 1 the characteristic variety is a union of lines so consists only of
hyperplanes. There are no curved sheets. 2. The operator from (3.4) has characteristic variety is given by 2 = jj2 so the variety is the classical light cone, and there are no hyperplanes. 3. The characteristic varieties of Mawell's Equations and the the linearization at u = 0 of the compressible Euler equations are the union of a curved light cone and a single horizontal hyperplane = 0.
De nition. A wave number ! 2 Rd n f0g is good when there is a neighborhood of ! and a nite of smooth real valued functions 1 () < 2 () < < m () so that the spectrum of Pd number j =1 Aj j is f1 (); : : : ; m ()g for 2 . The complementary set consists of bad wave numbers. The set of bad wave numbers is denoted ?(L).
Over a good , the characteristic variety of L is m smooth nonintersecting sheets = ?j (). The bad points are points where eigenvalues cross and therefore multiplicities change. The examples above have no bad points.
Examples. Consider the characteristic equation ( 2 ? jj2 )( ? c1) = 0 with c 2 R. If jcj < 1 then the variety is a cone and a hyperplane intersecting only at the origin and all points are good. If jcj > 1 the plane and cone intersect in a hyperbola whose projection on space is the set of bad points
? = : (c2 ? 1)12 = 22 + : : : + d2 : When jcj = 1 the hyperbola degenerates to a line of tangency. 14
Proposition 3.1. i. ?(L) is a closed conic set of measure zero in Rd n f0g. ii. The complementary set, Rd n (? [f0g), is the disjoint union of a nite family of conic connected open sets g Rd n f0g, g 2 G . iii. The multiplicity of = ?vj : as a root of det L( ; ) = 0 is independent of 2 Rd n (? [ f0g). iv. If () 2 C ! ( g ) is an eigenvalue of P Aj j depending real analytically on , then either there is j 2 f1; : : : ; M g such that () = ?vj or r2 = 6 0 almost everywhere on g . Proof. i. Use the basic strati cation theorem of real algebraic geometry (see [BR], [CR]). The characteristic variety is a conic real algebraic variety in R1+d n f0g. Since over each it contains at least 1 and at most N points the dimension is d. The singular points are therefore a stratum of dimension at most d ? 1. The bad frequencies are exactly the projection of this singular locus and so is a subvariety of Rd of dimension at most d ? 1 and (i) follows.
ii. That there are at most a nite number of components in the complementary set is a classical theorem of Whitney (see [BR], [CR]). iii. Denote by m the multiplicity on g and m0 the multiplicity on g0 . By de nition of multiplicity,
k 2 g and k < m =) @ det@ Lk( ; ) = 0: =?vj :
(3:7)
Then @ k L(?vj :; ) is a polynomial in which vanishes on the nonempty open set g , so must vanish identically. Thus it vanishes on g0 and it follows that m0 m. By symmetry one has m m0 . iv. If is a linear function = ?v: on g , then det L(?v:; ) = 0 for 2 g so by analytic continuation, must vanish for all . It follows that the hyperplane = ?v: lies in the characteristic variety and therefore that = ?vj : for some j . If is not a linear function, then the matrix r2 is a real analytic function on g which is not identically zero and therefore vanishes at most on a set of measure zero in g .
De nitions. Enumerate the roots of det L( ; ) = 0 as follows. Let Af := f1; : : : ; M g denote the indices of the at parts, and for 2 Af , () := ?v :. For g 2 G and 2 g , number the roots other than the f : 2 Af g in order g;1 () < g;2 () < < g;M (g). Multiple roots are not repeated in this list. Let Ac denote the indices of the curved sheets
Ac := (g; j ) : g 2 G and 1 j M (g) :
(3:8)
Let A := Af [ Ac . For 2 Af and 2 Rd de ne E () := (?vj :; ). For 2 Ac de ne
8 ? (); < E () := : 0
for 2 g for 2= g :
(3:9)
The next proposition decomposes an arbitrary solution of Lu = 0 as a nite sum of simpler waves. 15
Proposition 3.2. 1. For each 2 A , E () 2 C 1(Rd n (? [ f0g)) is an orthogonal projection valued function positive homogeneous of degree zero. 2. For each 2 Rd n (? [ f0g), C N is equal to the orthogonal direct sum CN
= 2A Image E () :
(3:10)
3. The operators E (Dy ) := F E ()F are orthogonal projectors on H s (Rd ), and for each s 2 R,
H s (Rd ) is equal to the orthogonal direct sum, H s (Rd ) = 2A Image E (Dy ) :
(3:12)
4. The solution of the initial value problem ujt=0 = f
L(@x ) u = 0 ; is given by the formula
u^(t; ) =
X 2A
u^ (t; ) :=
X 2A
(3:13)
eit () E () f^() :
(3:14)
Remarks. 1. The last decomposition is also written u :=
X
2A
u :=
X
2A
eit (Dy ) E (Dy )f :
2. Since is real valued on the support of E () the operator eit (Dy ) E (Dy ) is a contraction on H s (Rd ) for all s. 3. If 2 Af then ?i (Dy ) = v :@y . For = (g; j ) 2 Ac, since j ()j C jj the operator (Dy )f is continuous from H s to H s?1 . The mode u = eit (Dy ) E (Dy )f satis es @t u = i (Dy )u .
x3.2. Large time asymptotics for the homogeneous equation. Theorem 3.3. L1 Asymptotics for symmetric systems. Suppose that f^ 2 L1 (Rd ) and u is the solution of the initial value problem L(@x )u = 0, ujt=0 = f . Then with the notation introduced in the preceding de nition,
tlim !1 u(t) ?
X 2Af
(E (Dy )f ) (y ? v t)
L1 (Rd)
= 0:
(3:15)
Remarks. 1. This result shows that a general solution of the Cauchy problem is the sum of M rigidly translating waves, one for each hyperplane in the characteristic variety, plus a term which tends to zero in sup norm. The last part decays because of the spreading of waves. 2. The? Theoremdoes not extend to f whose Fourier Transform is a bounded measure. For example, u := ei(y ?t) ; 0 is a solution of (3.4) with f^ equal to a point mass. The characteristic variety contains no hyperplanes so (3.15) asserts that solutions with f^ 2 L1 tend to zero in L1 (Rd ) while u(t) has sup norm equal to 1 for all t. 1
16
Proof. Step 1. Approximation-decomposition. De ne a Banach space A and norm as the set of tempered distributions whose Fourier transform is in L1 (Rd ) with
kf kA
:= (2)?d=2
Z
Rd
jf^()j d :
(3:16)
The Fourier Inversion Formula implies that
kf kL1(Rd) kf kA :
(3:17)
? P
Symmetric hyperbolicity implies that exp it Aj j is unitary on C N so the evolution operator ? P S (t) := exp ? t j Aj @j is isometric on A . Since the family of linear maps
f 7?! S (t)f ?
X
2Af
(E (Dy )f ) (y ? v t)
is uniformly bounded from A to L1 (Rd ), it suces to prove (3.15) for a set of f dense in A . For 2 Ac , Propostion 3.1.iv shows that the matrix of second derivatives, r2 can vanish at most on a set of measure zero. The set of f we choose is those with
[ ? f^ 2 C01 Rd n ? [ f0g [ f 2 g : r2 () = 0 g ; 2Ac
which is dense since the set removed from Rd has measure zero. To prove (3.15) for such f decompose
f=
X
2A
f :=
X
2A
E (Dy ) f ;
u(t) = S (t)f =
X
u (t) :=
X
S (t) f :
(3:18)
For 2 Af , u (t) = (E (Dy )f ) (y ? v t) which recovers the summands in (3.15). To prove (3.15) it suces to show that for 2 Ac lim ku (t)kL1 (Rd) = 0 :
(3:19)
t!1
Step 2. Stationary and nonstationary phase. Proposition 3.2.4 shows that for 2 Ac, u (t; y) =
Z
g
ei( ()t+y:) f^ () d ;
f^ 2 C01( g ) :
(3:20)
For each in the support of f^ , there is a vector r 2 Rd so that hr2 () r; r i 6= 0 on a neighborhood of . Using a partition of unity we can write f^ as a nite sum of functions h^ 2 C01 ( g ) so that for each there is a r 2 C N so that on an open ball containing the support of h^ , hr2 () r ; r i 6= 0. It suces to show that for each tlim !1
Z i( ()t+y:) ^ e h ( ) d
L1 (Rd)
17
= 0:
(3:22)
To prove (3.22) rst make a linear change of variables in so that r = (1; 0; : : : ; 0) and therefore
@ 2 6= 0 ; @ 2 1 For
on supp h^ :
Z
K (t; y; 2 ; : : : ; d ) :=
one has the simple estimate
Z
jK (t; y1; 2 ; : : : ; d )j
(3:23)
ei( ()t+y : ) h^ () d1 1
1
jh^ (1 ; 2 ; : : : ; d ) j d1 2 L1 (Rd?1 ) :
Lebegue's Dominated Convergence Theorem implies that to prove (3.22) it suces to show that
8 2 ; : : : ; d 2 Rd?1 ;
lim
t!1
Z it( ()+z: ) sup e h^ () d1 = 0 : z2R 1
(3:24)
The one dimensional oscillatory integral in (3.24) is analysed by the method of stationary phase. The derivative of the phase with respect to 1 is equal to @1 + z . Thus if
jzj 1 +
sup
2supp F h
j@1 j
the principle of nonstationary phase implies that the integral in (3.24) is o(t?1 ). Thus it suces to prove (3.24) with z restricted to a compact set. For 2 ; : : : ; d xed, the family of smooth phases is compactly parametrized by z . By construction @12 6= 0 on a ball containing the support of h^ so for each 2 ; : : : ; d there is at most one stationary point for the phase. The stationary phase theorem then implies that
Z
eit( ()+z:) h^ () d1 = O(t?1=2 )
uniformly for z belonging to the compact set. This proves (3.24) and therefore Theorem 3.3.
Corollary 3.4.
If L(@x ) is a constant coecient symmetric hyperbolic operator, then the following are equivalent. 1. The characteristic variety of L contains no hyperplanes. 2. For every smooth solution of Lu = 0 with u t=0 2 C01 (Rd ), lim ku(t)kL1(Rd) ! 0 :
t!1
(3:25)
3. For every f 2 A the solution of the Cauchy problem
ujt=0 = f
Lu = 0 ;
(3:26)
satis es (3.25). 4. If () is a C 1 solution of det L( ; ) = 0 de ned on a open set of 2 Rd then for every v 2 Rd , f 2 Rd : r = ?vg has measure zero. 18
Proof. That (1) =) (4) is immediate. On the other hand if (4) is violated there is a smooth solution so that r = ?v on a set of positive measure. It follows from the Fundamental Strati cation Theorem (see [BR],[CR]) that r = ?v on a conic open real algebraic set of dimension d in Rd n 0. Then = ?v: on this set and we conclude that the polynomial det L(?v:; ) vanishes on this set and therefore everywhere. Thus the hyperplane f = ?v:g is contained in the characteristic variety and (1) is violated. Thus (1) and (4) are equivalent. That (3) implies (2) is immediate and the converse follows from the fact that C01 (Rd ) is dense in A . Thus (2) and (3) are equivalent. Theorem 3.3 shows that (1) is equivalent to (3).
Remark. Part four of this Corollary shows that for any velocity v the group velocity ?r associated to a curved sheet of the characteristic variety takes the value v for at most a set of frequencies of measure zero. x3.3 Asymptotics for the inhomogeneous equation.
Equation (2.27) is a linear inhomogeneous equation for a1 with source term de ned in terms of a0 . Since a0 satis es (2.21) It can be decomposed using Proposition 3.2 which expresses the rst term on the right of (2.27) as a sum of terms of the form eit (Dy ) f with 2 A. For large time we will see that the second term in (2.27) is well approximated by a sum of rigidly translating waves. These can come from terms in a0 or from recti cation of the oscillating parts a0 which thanks to (2.26) translates with velocity ?r (). If this velocity is equal to one of the vj then the characteristic variety of V is a hyperplane inside Char L and the source terms are of the form eit (Dy ) f . Otherwise, the characteristic variety of the transport operator V is a hyperplane which is not contained in Char L.
Main Lemma 3.5. Consider the solution of the initial value problem Lu = f ; ujt=0 = 0 ; (3:27) where s 2 R, and f 2 C ([0; 1[ ; H s (Rd )). 1. If the spatial Fourier Transform of f vanishes outside g , () de nes one of the smooth sheets
of Char L over g , and @t f = i (Dy )f , then
lim 1t k u(t) ? tE (Dy )f (t) kH s(Rd) = 0 :
t!1
(3:28)
2. If V (@x ) = @t + v:@y is a constant coecient vector eld whose characteristic variety is not contained in the characteristic variety of L, and f satis es V (@x ) f = 0, then 1 ku(t)k s d = 0 : lim H (R ) t!1 t
(3:29)
Proof. 1. One has f = exp(i (Dy )) f0 with f0 := f jt=0 . Since the family of operators
f0 7?! 1t u(t) ? tE (Dy )f (t) ; 19
1t d=2. It follows that kw(t)kH s cs t. The corrector, ?"w, is small compared to the principal term, v0 for times t = o(1=") which veri es again that the nonlinear e ects are small for t = o(1="). We nd approximate solutions of (4.4) by decomposing v0 into modes and observing that the nonlinear interaction is simpli ed using two fundamental principles.
Example illustrating the principle of nonoverlapping waves. Consider the system of two equations in space of dimension one,
@v + 1 0 @v + " (v) = 0 : 0 ?1 @y @t The linear approximation
?
(4:5)
v0 = v10 (y ? t) ; v20 (y + t) :
(4:6) contains two waves, a v10 component which moves to the right with speed one and a v20 component moving to the left. In the source term in (4.4), products of the waves moving leftward with waves moving rightward are essentially zero except for a time interval of order 1 so make a contribution O(") to w and are therefore negligible. In this way one shows that the corrector w is given by w = (w1 ; w2 ) + O(") where w is determined by the uncoupled equations (@t + @y ) w1 + 1 (v10 (y ? t); 0) = 0 ;
(@t ? @y ) w2 + 2 (0; v20 (y + t)) = 0 :
(4:7)
with initial condition w(0; y) = 0. Then w = O(t) so "w is an appreciable corrector for times t 1=". In the nonlinear term only products of terms propagating in the same way through space time need be retained. 22
Example illustrating the principle of ignoring spreading waves. Consider in two dimensions the rst order analogue of a semilinear wave equation @u + 1 0 @u + 0 1 @u + (u) = 0 ; u(0; y) = "p g(y) 2 C 1 (R2 ) : 0 0 ?1 @y1 1 0 @y2 @t Rescaling yields @v + 1 0 @v + 0 1 @v + " (v) = 0 : 0 ?1 @y 1 0 @y @t 1
2
(4:8) (4:9)
This case is strikingly di erent from the preceding example. The reason is that the solution v0 of the linear equation is O(t?1=2 ) in L1 (R2 ) and has H s (R2 ) norm independent of time. Thus the L2 norm of (v0 ) is at most O(t?(J ?1)=2 ). More generally the H s norm is o(1) as t ! 1. It follows that the corrector w = o(t) so "w makes a small correction even for times 1=". For modes which decay because they spread over larger and larger regions of space time, the nonlinear term can be neglected. The next result is a quantitative result incorporating the two principles.
Proposition 4.1. Suppose that (u) is a smooth function which vanishes along with its partial derivatives of order 1 when u = 0. If fk P2 \s H s (Rd ) is a nite family of functions, vk 2 Rd a family of pairwise distinct velocities, u = k fk (y ? vk t) + (t; y), and 8 s; lim k @y (t; y) kL1 (Rdy) = 0 ;
sup k (t; y) kH s(Rdy) < 1 ;
t!1
t2[0;1[
then for all s
lim (u) ? t!1
X ? fk (y ? vk t) H s (Rdy) = 0 : k
(4:10)
(4:11)
Proof. Let u1 := Pk fk (y ? vk t) : Since the rst derivatives of vanish at the origin, Taylor's Theorem expresses
(u) ? (u1 ) =
X
(u1 ; ) ` ()
(4:12)
where the are smooth function which vanish when u1 = = 0 and the ` are real linear functions. Leibniz' rule expresses derivatives of (u) ? (u1 ) as a nite sum of terms ` (@ ) @ ( ). Since vanishes at the origin and u1 and are bounded in H s as t tends to in nity, the L2 norm of the second factor is bounded as t ! 1. At the same time, the L1 norm of the rst factor tends to zero, so the product tends to zero in L2 . Thus k (u) ? (u1 )kH s(Rd) ! 0. For s > d=2, the mapping w 7! (w) is uniformly lipshitzean from bounded sets in H s to H s . Given a challenge number ", one can choose hk 2 C01 (Rd ) so that fk ? hk is as small as one likes in H s , and therefore for all t
X
? ? hk (y ? vk t) ? fk (y ? vk t) H s < " ; k
and
(u ) ? X h (y ? v t)
< " : 1 k k Hs k
23
Since the hk have compact support and the speeds vk are distinct it follows that for t large,
X k
X ? hk (y ? vk t) :
hk (y ? vk t) =
k
The triangle inequality yields X
lim sup (u1 ) ? (fk (y ? vk t)) H s 2 " t!1 k
and the result follows.
x4.2. Pro le equations, endgame.
In this section we complete the description of the equations determining the pro les aj which guarantee that r0 ; r1 ; r2 vanish. It remains to analyse r2 . The equation r2 = 0 is split into three pieces, r2 = 0, ( ) r2 = 0, and Q( ) r2 = 0. x4.2.1. The equation r2 = 0. Use (2.21) and Theorem 3.3 to decompose a0 into modes of L(0; @x ), X a0; ; a0; = E (Dy ) a0 (4:13) a0 = a0 + a0 = a0 + 2A
with ka0; (t)kL1 ! 0 as t ! 1 if 2 Ac and (@t + v :@x )a0; = 0 if 2 Af . This decomposition is injected into equation (2.27). The constraint (2.9) that a1 (T; Y; t; y) must grow sublinearly in t imposes conditions on a0 . The key step is to simplify the nonlinear term
D
(a0 ) +
d X
=0
(a0 ) @ a0
E
using the principles of the last sections. The principle of ignoring spreading terms suggests that the terms a0; with 2 Ac can be dropped. The other terms involve a0 and a0; with 2 Af each of which satis es a transport equation. The principle of nonoverlapping waves suggests that we ignore the interaction of summands which move at di erent speeds. For this it is important to know whether the speed of V is equal to one of the vj . The speed is one of the vj if and only if the hyperplane Char V (@x ) is contained in Char L(0; @x ). Since Char V (@x ) is the tangent plane to the variety at by Corollary 2.3, Char V (@x ) is contained in Char L(0; @x ) if and only if lies on a hyperplane in the variety. This dichotomy strongly a ects the nature of the pro le equation since if belongs to a hyperplane in the variety, the oscillations travel at exactly the same speed as one of the nonspreading modes for L. This encourages interaction between the mean eld and the oscillations. For belonging to a curved sheet in the variety, the interaction is weaker.
De nitions. Suppose that = ( ; ) satis es the simple characteristic variety hypothesis. De ne 8 1 if Char V (@ ) Char L(0; @ ) < x x =: (4:14) 0 if Char V (@x ) 6 Char L(0; @x ) : When Char V Char L, enumerate the projections E ; 2 f1; : : : ; M g = Af from the de nition before Proposition 3.2 so that V corresponds to = 1, that is V = @t + v1 :@y .
a switch by
24
When Char V 6 Char L the term a0 translates at a speed di erent than the vj . Then the principle of ignoring spreading waves and the interaction of nonoverlapping waves suggest the replacement (a0 ) 7?! (a0 ) +
M X j =1
(a0;j ) :
When Char V (@x ) Char L, the term a0 translates at the same speed as the term a0;1 and the principles suggest the replacement (a0 ) 7?! (a0 + a0;1 ) +
M X j =2
(a0;j ) :
This dichotomy is summarized by (a0 ) 7?! (1 ? )(a0 ) + ( a0 + a0;1 ) +
M X
(a0;j ) :
j =2
For the quasilinear term, reasoning as above suggests the replacement (a0 ) @ a0 7?!
(4:15)
X (1 ? ) (a0 ) + (a0 + a0;1 ) + (a0;j ) @ a0 : M
j =2
The factor @ a0 = @ a moves with the speed of the vector eld V , and each of the summands in the other factor also translates at constant velocity. The principle of nonoverlapping waves suggests that only the summands annihilated by V need be retained. When = 0 this is (a0 )@ a0 . while when = 1 it is (a0 + a0;1 ). These choices are summarized by the simpli cation (a0 ) @ a0 7?! (a0 + a0;1 ) @ a0 ;
which generates the quasilinear term d X =0
(a0 + a0;1 ) @ a0 ;
(4:16)
Adding (4.15) and (4.16) yields
N (a0 ; @ a0 ) := (1 ? )(a0 ) + ( a0 + a0;1 ) +
M X j =2
(a0;j ) +
d X =0
(a0 + a0;1 )@ a0 : (4:17)
With these de nitions one has for a0 satisfying (2.21) (a0 ) +
X
(a0 )@ a0 = N (a0 ; @ a0 ) + o(1)
as t ! 1 :
(4:18)
Here and in succeeding estimates the terms o(1) tend to zero in L1 (Rd ) as t ! 1. This follow from Proposition 4.1 and Theorem 3.3. Estimate (5.26) is a more precise version. 25
Inserting (4.18) in (2.27) yields
D
E
L(0; @x ) a1 = ? L(0; @X ) a0 ? N (a0 ; @ a0 ) + o(1) :
Solving with initial data a1 t=0 = 0 yields
a1 = L(0; @x )?1
X ?
2A
D
L(0; @X ) a0; ? N (a0 ; @ a0 )
E
(4:19)
+ o(t) :
(4:20)
Each source term on the right hand side of (4.20) satis es a di erential equation which allows us to use Main Lemma 3.5. Introduce the switch 8 1 if 2 A < f ( ) := : (4:21) 0 if 2 Ac Decompose the source term
S :=
X
2A
D
E
L(0; @X ) a0; + N (a0 ; @ a0 ) = S + S1 +
where S for = 1 is de ned by
D
X
2Anf1g
S ;
(4:22)
E
S1 := L(0; @X )a0;1 + (1) (a0 + a0;1 ) ; while
S := L(0; @X )a0; + ( ) (a0; )
D
(4:23)
for 6= 1 ;
(4:24)
E
X S := (1 ? )(a0 ) + (a0 + a0;1 )@ a0 : d
(4:25)
=0
They satisfy
V (@x ) S = 0 ;
? @ ? i (D ) S = 0 : t y
The rst part of Main Lemma 3.5 yields L(0; @x )?1 S = t E (Dy ) S + o(t) ;
(4:26) L(0; @x )?1 S = t E1 (Dy ) S + o(t) : Note that since (1 ? ) = 0 the (1 ? )(a0 ) term in S does not contribute to the right hand side
of the second equation in (4.26). ?S + is sublinear in time if and only if E ( D ) Equations (4.20), (4.22), and (4.26) show that a 1 y 1 1 S = 0 and for 6= 0, 0 = E (Dy ) S , that is
D
X E1 (Dy ) L(0; @X ) a0;1 + (1) ( a0 + a0;1 ) + (a0 + a0;1 ) @ a0
and
E
E (Dy ) L(0; @X ) a0; + ( ) (a0; ) = 0 ; for 2 A n f1g : Equations (4.27), (4.28) determine the dynamics in T of the mean values. 26
= 0;
(4:27) (4:28)
x4.2.2. The equation ( ) r2 = 0
Multiplying the expression for r2 on the left of (2.18) by ( ) eliminates the L(0; )@ term. Taking the oscillatory part yields
L(0; @X ) a0 + L(0; @x )a1 + (a0 ) +
X
(a0 ) @ a0
= 0:
(4:29)
Using (2.23) write
a1 = ( ) a1 + (1 ? ( )) a1 = ( ) a1 ? Q( ) L(0; @x) @?1 a0 : Inserting this into (4.29) yields
L(0; @X ) a0 ? L(0; @x ) Q( ) L(0; @x) @?1 a0 X + (a0 ) + (a0 ) @ a0 = ? L(0; @x ) a1 : Use Proposition 2.2 two times to nd
V (@X ) a0 ? L(0; @x ) Q( ) L(0; @x ) @?1 a0 X + (a0 ) + (a0 ) @ a0 = ?V (@x ) a1 :
(4:30)
The constraint that a1 (T; Y; t; y; ) grow sublinearly in t places restrictions on the left hand side of (4.30) which determines our next pro le equation. Equation (2.26) shows that the rst two summands on the left are annihilated by V (@x ). Thus,
V (@x )?1 V (@X ) a0 ? L(0; @x ) Q( ) L(0; @x) @?1 a0 = t V (@X ) a0 ? L(0; @x ) Q( ) L(0; @x ) @?1 a0 : Equation (4.18) implies that
V (@x )?1 (a0 ) +
X
(a0 ) @ a0
= V (@x )?1 N (a0 ; @ a0 ) + o(t) :
(4:31)
(4:32)
We next consider the terms from the right hand side of (4.32). Since (a0 ) and (a0 + a0;1 )@ a0 are annihilated by V (@x ) one has
d X ? 1 V (@x ) (1 ? ) (a0 ) + (a0 + a0;1 )@ a0
=0
d X = t (1 ? ) (a0 ) + (a0 + a0;1 )@ a0 :
(4:33)
=0
Similarly if = 1 one has
V (@x )?1 ( a1 + a0;1 ) = t ( a1 + a0;1 ) :
If = 0, ( a1 + a0;1 ) is nonoscillatory so the contribution vanishes. 27
(4:34)
The terms (a0;j ) in N are nonoscillating so make no contribution to (4.32). Combined with (4.30-34) this shows that a1 grows sublinearly in t if and only if V (@X ) a0 ? L(0;@x ) Q( ) L(0; @x) @?1 a0 + (1 ? ) (a0 ) d X (4:35) + (a1 + a0;1 ) + (a0 + a0;1 )@ a0 = 0 : =0
When the smooth characteristic variety hypothesis is satis ed, the operator L Q L appearing in (4.35) is essentially scalar. ThatPis the content of the next proposition whose proof proceeds by di erentiating twice the identity ( Aj j )( ) = ( )( ) . The details can be found in [DJMR]. The expression here is simpler than in that reference thanks to the convention that A0 (0) = I .
Proposition 4.2. If the smooth characteristic variety hypothesis is satis ed at = ( (); ) and
A0 (0) = I , then
0 1 2 2 X 1 ( ) @ @ () @ A : 2 @j @k @yj @yk
( ) L(0; @x ) Q( ) L1 (@x ) ( ) =
(4:36)
j;k
De ne the scalar di erential operator
d @ 2 () @ 2 X 1 : R(@y ) := ? 2 j;k=1 @j @k @yj @yk
(4:37)
The dynamics (4.35) of a0 = a0 with respect to T simpli es to V (@X ) a0 ? R(@y ) @?1 a0 + (1 ? ) ( ) (a0 )
+ ( ) (a0 + a0;1 ) + ( )
d X
=0
(a0 + a0;1 )@ a0
(4:38)
= 0:
Remarks. 1. In all cases V (@x ) a0 = 0 and the operator V (@X ) in (4.38) is converted to @T by writing a0 = a(T; Y ? vT; t; y ? vt; ). Then (4.38) is equivalent to d X
@T a ? R(@y ) @?1 a + (1 ? ) (a) + (a + a0;1 ) +
=0
(a + a0;1 )@ a
= 0:
2. In the semilinear case, the terms are absent, yielding (1.16) of the introduction. x4.2.3. The equation Q( ) r2 = 0. The equation in the title is satis ed if and only if (I ?) a2 =
X ? 1 ?Q( ) @ L(0; @x )a1 + L(0; @X ) a0 + (a0 ) + (a0 ) @ a0 :
(4:39)
Summary. The principal pro le a0 is determined from its initial data at t = T = 0 from equations (2.20), (2.21), (2.26), (4.27), (4.28) and (4.38). The parts a1 and ( ) a1 of the second pro le are determined by solving (2.27) and (4.30) with initial data vanishing at t = 0 . Finally, (I ? ( ))a1 and a2 = (I ? )a2 are given by (2.23) and (4.39) respectively. 28
In the next section we verify that the initial value problems referred to in this summary are in fact well posed locally in time, and the terms expected to be sublinear are in fact sublinear. That the solutions provide accurate approximate solutions is proved in x6.
x5. Solvability of the pro le equations. x5.1. Solvability of the equations for the leading pro le a0 .
Theorem 5.1. Suppose that g(Y; y; ) belongs to \s H s (Rd Rd T) and satis es the polarization
condition ( ) g = g . Then there is a T > 0 and a unique a0 (T; Y; t; y; ) so that for all 0 < T < T and all 2 N 2d+3 sup k @T;Y;t;y; a0 (T; :; t; :; :) kL (RdY RdyT) < 1 ;
sup
(5:1)
2
0T T t2R
and a0 satis es (2.20), (2.21), (2.26), (4.27), (4.28) and (4.38) together with the initial condition
a0 (0; Y; 0; y; ) = g(Y; y; ) :
(5:2)
Proof. The strategy is the following. Equations (4.27), (4.28), and (4.38) involve @T;Y;y; but not @t . In the plane ft = 0g one solves the quasilinear initial value problem de ned by (4.27), (4.28), (4.38) with time variable T . Then the Cauchy problem de ned by equations (2.21), (2.26) with initial values determined by a0 jt=0 extends the solution to all t. The validity of (4.27), (4.28),
(4.38) for all t is proved using a commutation argument together with uniqueness for hyperbolic initial value problems. The next lemma performs the rst step.
Lemma 5.2. Solvability in ft = 0g. There is a T > 0 and unique solution w(T; Y; y; ) 2 \s C s ([0; T ] ; H s (R2Y;yd T)) to equations (4.27), (4.28), (4.38) and satisfying the initial condition w(0; Y; y; ) = g. In addition, ( ) w = w .
Proof. Step 1. Existence follows from a di erential inequality. We construct solutions as limits as h ! 0+ of the nonlinear ordinary di erential equations obtained
by replacing @Y;y; in (4.27), (4.28), and (4.38) by the associated symmetric nite di erence operh . The replacement is not made in the expressions E (Dy ) and @ ?1 which are bounded ators, Y;y; operators. With w := E (Dy ) w, the resulting equations are
E1 (Dy ) L(0; @T ; Yh )wh1 +
D
X + (1) (wh1 + wh; ) + (wh; + wh1 ) h wh;
E (Dy ) L(0; @T ; Yh )wh + ( ) (wh ) = 0 ;
d X
=0
29
= 0;
for 2 A n f1g ;
V (@T ;Yh ) wh; ? R(yh ) @?1 wh; + (1 ? ) ( ) (wh; ) + ( ) (wh; + wh1 ) + ( )
E
(wh; + wh1 )h wh;
= 0:
(5:3) (5:4) (5:5)
These three equations together express
@T wh = E1 (Dy ) @T wh1 +
X 6=1
E (Dy ) @T wh + @T wh;
as a function Gh (wh ). If s > (2d +1)=2 and h are xed, Schauder's Lemma implies that the nonlinear map Gh is uniformly lipshitzean from bounded subsets of H s (R2d T) to H s (R2d T). Picard's Fundamental Existence Theorem for ordinary di erential equations implies that there is a T (s; h) > 0 and a unique local solution ? wh 2 C 1 [0; T (s; h)[ ; H s (R2d T) (5:6) and that if T (s; h) < 1 then
lim wh (T; ; ; ) H s (RdRdT)) = 1 : t%T (s;h)
(5:7)
Equations (5.3)-(5.5) imply that @T (I ? ( ))wh; = 0, so the polarization ( )wh; = wh; holds as soon as it holds at T = 0. Lemma 5.2 follows from the following a priori estimate. Fix s > (2d + 3)=2. For each s s, there is a continuous function Ks : R+ ! R+ independent of h so that the local solutions satisfy for T < T (s; h)
d kwh (T )k2H s(R dT) ? k wh (T ) k s d kwh (T )k2 (5:8) K s H (R T) H s(R dT) : dT To see that this suces rst take s = s. Let (t) 2 C ([0; T1 [ ; R) be the maximal solution of d = K ( (T )) (T ) ; (0) = jjgjj s d : 2 dT s H (R T) Then inequality (5.8) implies that for T < T1 kwh (T )kH s(R dT) (T ) : (5:9) 2
2
2
2
2
This proves that T (s; h) T1 which gives an h independent domain of existence for wh . For any T 2 [0; T1 [ and s s let
C1 = C1 (T ) := sup j (T )j ; 0T T
C2 = C2 (s; T ) := sup jKs ()j : jjC1 (T )
Inequality (5.8) implies that for T T ,
kwh(T )k2H s(R dT) eC T kgk2H s(R dT) : 2
2
(5:10)
2
Using the di erential equation to express time derivatives in terms of space derivatives implies corresponding estimates
k@Tk wh (T )kH s(R dT) C (s; k; T ) kgkH s 2
k (R2dT) ;
+2
0T T:
(5:11)
With ?these bounds uniform in h it is routine to pass to the limit constructing a solution in s 2 d s \s C [0; T ] ; H (R T) . Passing to the limit also yields the polarization identity. 30
Step 2. Proof of the di erential inequality (5.8).
On T use the measure d=2 of total mass equal to one. To prove (5.8) add the results of taking the real part of the L2 (R2d T) scalar product of (5:3) with
(1 ? Y;y )s wh1
(5:4) with
(1 ? Y;y )s wh
(5:12)
(5:5) with (1 ? Y;y; )s wh; : The symmetry of A(0) and the antisymmetry of Yh show that from (5.3) and (5.4) one has for all 1 j d and 2 A
Re (1 ? Y;y )s wh ; E (Dy ) Aj (0) Yhj wh1
= 0:
L2 (R2d)
Similarly from (5.5)
= 0: Re (1 ? Y;y; )s wh; ; v:Yh wh; ? R(yh ) @?1 wh; L (R dT) 2
2
Thus summing the real parts from (5.12) yields d 1 d kwh (T )k2H s(R dT) = Re I + X Re I2; ; 1 2 dT =0
(5:13)
2
where I1 contains the terms
I1 := (1 ? Y;y )swh1 ; (1) (wh1 + wh; ) L (R d) +
M X j =2
(1 ? Y;y )s whj ; (wj )
2
2
L2 (R2d)
+ (1 ? Y;y; )s wh; ; (1 ? ) (wh; ) + ( wh; + w1 )
L2 (R2dT)
(5:14)
:
The more troublesome terms are the quasilinear terms I2;
I2; := (1?Y;y )s wh1 ; (w0 + w1 ) h wh; +
L2 (R2d)
? (1 ? Y;y; )s wh; ; ( wh1 + wh; ) h wh; : L (R dT) 2
(5:15)
2
Moser's inequality shows that
jI1 j Ks ( fjjw h (T )jjL1 : 2 Ag ) jjwh jj2H s r:h:s: of (5:8) : The crucial point for estimating the real part of I2; is the identity
I2; = (1 ? Y;y; )s ( wh1 + wh; ) ; ( wh1 + wh ) h ( wh1 + wh; ) L (R dT) ; 2
31
2
(5:16) (5:17)
which is proved in the next lines. First consider the case = 1. In the scalar product in the rst summand on the right in (5.15), the second factor is a mean value so is orthogonal to purely oscillating terms. Thus in this summand w1 can be replaced by w1 + wh; . Similarly in the second summand on the right of (5.15) the second factor has mean zero so that in the rst factor wh; can be replaced by w1 + wh; . Finally use that h wh; = h (w1 + wh; ) to nd that when = 1,
L (R d) ? : (1 ? Y;y; )s (wh1 + wh; ) ; (wh1 + wh; ) h (wh1 + wh; ) L (R dT)
I2; := (1 ? Y;y; )s (wh1 + wh; ) ; (w0 + w1 ) h (wh1 + wh; ) +
2
2
2
2
The rst factors in the two scalar products are equal so that the sum simpli es and one has
I2; = (1 ? Y;y; )s (wh1 + wh; ) ; (wh1 + wh ) h (wh1 + wh; ) L (R dT) : 2
This proves (5.17) when = 1. When = 0, use the fact that wh; has mean zero to nd that
2
(5:18)
= 0 =) I2; = (1 ? Y;y; )s wh; ; (wh ) h(wh; ) L (R dT) : 2
2
This proves (5.17) when = 0. The expression (5.17) is like those encountered in the energy method for quasilinear symmetric hyperbolic systems. The crucial fact is that the smooth matrix valued functions have hermitian symmetric values. The symmetry of (1 ? )s and together with the antisymmetry of of h shows that the symmetric part of the operator (1 ? )s (wh; )h is of order 2s uniformly in h. Estimating the commutator [ ; h ], the Lipshitz norm of is needed which explains the lower bound (2d + 3)=2 on s. In total, the Gagliardo-Nirenberg inequalities are used to prove Re I2; r:h:s: (5:8) . This together with (5.16) completes the proof of (5.8) and therefore the existence part of Lemma 5.2.
Step 3. Sketch of uniqueness proof. It suces to show that the L2 (R2d T) norm of the di erence of two solutions satis es a di erential inequality d kw1 ? w2 k2L (R dT) K (k (w1; w2 )(T ) kH s ) kw1 ? w2 k2L (R dT) : dt 2
2
2
2
This is done by subtracting the two equations and using multipliers analogous to (5.12) but without the operator 1 ? . The details are left to the reader.
End of proof of Theorem 5.1. Once w is constructed on [0; T [ R dY Rdy T, a0 is uniquely determined in [0; T [ R dY Rt Rdy T by solving the linear initial value problems determined from (2.21) and (2.26), namely
V (@x ) a0 = 0 ;
a0 jt=0 = w ;
(5:19)
L(0; @x ) a0 = 0 ;
a0 jt=0 = w ;
(5:20)
32
The resulting function a0 satis es (5.1) and (5.2) as well as the constraints a0 = a0 . This proves uniqueness of the solution a0 , since the values of w were uniquely determined. To complete the proof of Theorem 5.1, it suces to show that equations (4.27), (4.28), and (4.38) are satis ed when t 6= 0. To verify these three equations it is sucient to show that
L(0; @x ) l:h:s: of (4:27) and (4:28) = 0 ;
V (@x ) l:h:s: of (4:38) = 0 :
(5:21)
To see that this suces note that for each T 2 [0; T [ the restriction of the left hand side of (4.27) (resp (4.28), and (4.38)) to fT = T g is annihilated by a rst order symmetric hyperbolic operator with t timelike, and has vanishing initial data at t = 0. It follows that they vanish at all points of fT = T g. The veri cations of the three equations in (5.21) rely on the identities
?
L(0; @x ) E (Dy ) = E (Dy ) L(0; @x ) = E (Dy ) @t ? i (Dy ) :
(5:22)
For (4.27) compute
? L(0; @x ) l:h:s: of (4:27) =L(0; @x ) E1 (Dy ) L(0; @X )a0;1 D X + (1)(a0;1 + a0 ) +
(a0 + a0;1 ) @ a0
E
(5:23)
:
The right hand side of (5.23) is equal to
E X ? E1 (Dy ) @t ? i 1 (Dy ) L(0; @X ) a0;1 + (1)(a0;1 + a0 ) + (a0 + a0;1 ) @ a0 :
When = 1, @t ? i 1 (Dy ) = @t + v1 :@y = V (@x ) and so (@t + v1 :@y ) a0 = (@t + v1 :@y ) a0;1 = 0. It follows that the right hand side of (5.23) vanishes. When = 0, the right hand side simpli es to
L(0; @x ) E1 (Dy ) L(0; @X )a0;1 + (1)h(a0;1 )i : For (4.28) one has 6= 1 and
?
(5:241)
L(0; @x ) l:h:s: of (4:28) = L(0; @x ) E1 (Dy ) L(0; @X )a0; + ( )(a0; ) :
(5:242)
The rst term on the right of (5.24) is equal to
E1 (Dy ) (@t ? i (Dy )) L(0; @X ) a0; = E1 (Dy ) L(0; @X ) (@t ? i (Dy )) a0; = 0 : The second term can be nonzero only if 2 Af , in which case @t ? i (Dy ) = @t + v :@y , and therefore (@t ?i (Dy ))a0; = 0 implies that (@t ?i (Dy ))(a0; ) = 0 proving that (5.24) vanishes. Reasoning as above shows that V (@x ) annihilates every term on the left hand side of (4.38). This veri es the nal assertion in (5.21) and completes the proof of Theorem 5.1. 33
x5.2. The correctors a1 and a2 .
Recall from the summary at the end of x4, that (I ? )a1 is given by (2.23) while a1 and a1 are de ned by solving the hyperbolic initial value problems (2.27) and (4.30) with timelike variable t and initial values equal to zero at t = 0. The corrector a2 = (I ? ) a2 is given by formula (2.28).
Proposition 5.3. The rst corrector a1 grows sublinearly in the sense that for all T 2]0; T [ and
2 N 2d+3 ,
a (T; Y; t; y; ) lim 1t @X;x; 1 L ([0;T ]RdY RdyT) = 0 :
t!1
(5:25)
2
Proof. We must prove estimate (5.25) for each of the three parts, a1 , ( ) a1 , and (I ? ( ))a1 . Formula (2.23) together with (5.1) immediately yield the estimate for (I ? ( ))a1 . It suces to
treat the other two.
Step 1. Estimate (5.25) for a1 . Recall that a0; := E (Dy ) a0 and de ne by a0 =
X
2Af
a0; (x ? v t) + (T; Y; t; y) :
Estimate (5.1) for a0 shows that a0; and satisfy estimates analogous to (5.1). Theorem 3.3 implies that lim k (T; Y; t; y) kL1 ([0;T ]RdY Rdy) = 0 : t!1
Using (2.26) for a0 one can apply the proof of Proposition 4.1 to nd the following quantitative version of (4.18). For all T < T , and 2 N 2d+3
tlim !1 @X;x; (a0 ) +
X
(a0 )@ a0 ? N (a0 ; @ a0 )
L ([0;T ]Rd RdT) = 0 : 2
Y
y
(5:26)
The quantity estimated to be small here, serves as an error term below so denote
Err := (a0 ) + Equation (4.20) reads
a1 = L(0; @x )?1
X ?
2A
X
(a0)@ a0 ? N (a0 ; @ a0 ) :
D
L(0; @X ) a0; ? N (a0 ; @ a0 )
E
+ L(0; @x )?1 Err :
With (5.26) this yields the quantitative version of (4.20)
1
@ a lim t!1 t X;x; 1 + L(0; @x )?1
X 2A
L(0; @X ) a0; + N (a0 ; @ a0 )
L ([0;T ]Rd RdyT) = 0 :
Using the notation (4.22-4.25), decompose
L?1 S = L?1 S +
X
L?1 S := v (T; Y; y; ) + 34
2
X
Y
v (T; Y; t; y) :
(5:27)
For each T xed, S ; S are continuous functions of t with values in H s (RdY Rdy ) and satisfy (4.26). The pro le equations guarantee that E (Dy )S = 0, and E1 (Dy )S = 0. Viewing L(0; @x ) and @t ? i (Dy ) as di erential operators in t; Y; y , and applying Main Lemma 3.5 yields a quantitative version of (4.27)
lim 1 v (T; Y; t; y; ) ; v (T; Y; t; y) = 0: t!1
t
L2 (RdY RdyT)
fS ; S g) showing that Nearly the same reasoning applies to L?1 (@X;x 1
L?1 (@ fS ; S g)
lim X;x L (RdY RdyT) = 0 : t!1 t
(5:28)
2
In this derivation, the Main Lemma is applied with source terms f whose initial values at t = 0 are
v (T; Y; 0; y) ; @ v (T; Y; 0; y)g : f0 := f@X;x (5:29) X;x These are continuous functions of T with values in L2 (R2d ) so their values lie in a compact subset of L2 for 0 T T . It follows that the convergence in (5.28) is uniform in T . Thus,
= 0 (5:30) lim 1 L?1 (@ fS ; S g)
d T) X;x L ([0;T [RY;y t
fv ; v g? L?1 (@ fS ; S g) is the solution of Lw = 0 whose initial data The di erence w := @X;x X;x
agrees with those of @X;x fv ; v g. Thus the L2 ([0; T ] R2d ) norm of w is bounded independent of t. Together with (5.30) this shows that
t!1
2
2
1
@ fv (T; Y; t; y) ; v (T; Y; t; y)g
lim
t!1 t X;x
d T) L2 ([0;T ]R2Y;y
= 0:
(5:31)
This together with (5.27) yields (5.25) for a1 . Step 2. Estimate (5.25) for ( ) a1. The pro le equation (4.38) for a0 is derived exactly so that X ? 1 ( ) a1 = V (@x ) ( ) (a0 ) + (a0 ) @ a0 ? ( ) N (a0 ; @ a0 ) : That this is sublinear follows from the quantitative version of (4.18) as in the proof of (5.31).
x5.3. Estimate for the residual.
Once pro les are de ned, the approximate solution is given by (2.6) and (2.7). The residual is computed in (2.10)-(2.11). In the present case, r0 = r1 = r2 = 0. Our estimates for the pro les a0 ; a1 ; a2 yield estimates for the residual.
Proposition 5.4. With the pro les constructed in this section the residual
? k("; X; x; ) := L "p a ; "@X + @x + " @ "pa + F ("pa)
satis es the following estimates. For all 2 N 2d+3 and T < T ,
k("; X; x; )
p+1 sup @X;x; d T) = o(" ) L ([0;T ]RY;y as " ! 0.
0t 1 + (d + m)=2. The standard local existence theory for quasilinear symmetric hyperbolic systems shows that there is a unique maximal local solution
W " 2 \k C k ([0; T (")[ ; H k (Rd Tm )) and if the time of existence T (") < 1 then lim k W "(t) kH s(RdTm) = 1 :
(6:24)
t%T(")
To prove that T T=" it suces to bound the H s norm of W " (t) for 0 t T=". The next Lemma is the key element in the stability proof. The important fact is that the ampli cation factor C is uniform for times t 1=".
Lemma 6.3. Hs(Rd Tm ) estimate. For s ? 1 > (d + m)=2, there is a constant C so that if ? " 2]0; 1], t 2 [0; T="], and W 2 C 1 [0; t] ; H 1+s (Rd Tm ) satis es sup
0tt="
then for 0 t t
W (t); @ W (t); "@ W (t)
z t L1 (RdTm) 1 ;
kW (t)kH s(RdTm) C kW (0)kH s(RdTm) +
Zt 0
(6:26)
kL"(U " + W ) W (t)kH s(RdTm) dt :
(6:27)
Proof of Theorem 6.2 assuming Lemma 6.3. Equations (6.18), (6.22), and (6.23), imply that
there is a 0 < "2 1 so that for 0 < " < "2 ,
W "(0); @ W "(0); "@ W "(0)
z t L1 (RdTm) 1=2 ; 41
For 0 < " < "2 de ne 0 < t (") by n t (") = sup t < minfT ("); T="g : sup
0tt
o
W "(t); @ W "(t); "@ W "(t)
1 : z t L1 (RdTm)
Fix N 3 s 1 + (d + m)=2. Inequality (6.27) then implies that for 0 t t ("), kW "(t)kH s(RdTm)
C
Zt
kW "(0)kH s(RdTm) +
0
kK "(t) ? " G("; U "; "@t U "; @z U "; W " ) W "kH s(RdTm) dt
(6:28)
:
(6:29)
Hypothesis (6.17) bounds U " and "@t U " in H s (Rd Tm ). Then Moser's inequality shows that for
times no larger than t (") kG("; U "; "@t U "; @z U " ; W ") W " kH s(RdTm) C k W " kH s(RdTm) : Then (6.29) yields kW "(t)kH s(RdTm) C kW "(0)kH s(RdTm) +
Zt
Zt
kW "(t)kH s(RdTm) dt + kK "(t)kH s(RdTm) dt : 0 0 Hypothesis (6.18) shows that as " ! 0, Zt kW "(0)kH s(RdTm) + kK "(t)kH s(RdTm) dt "
0
kg"kH s(RdTm) + Inserting this in (6.30) yields
Z T=" 0
(6:30)
k K "(t) kH s(RdTm) dt = o(1) :
Zt " kW (t)kH s(RdTm) C o(1) + " kW "(t)kH s(RdTm) dt : 0 " C"t Gronwall's lemma yields kW (t)kH s(RdTm) e o(1) : Since "t T , this implies sup kW " (t)kH s(RdTm) = o(1) : 0tt (")
(6:31) (6:32)
Then Sobolev's Lemma shows that kW "(t); @z; W "(t)kL1(RdTm) = o(1) : Using equation (6.22) to express the time derivative in terms of z derivatives and using hypothesis (6.17) then shows that k"@t W (t)kL1(RdTm) = o(1). Thus, one can choose 0 < "3 < "2 so that for 0 < " < "3 and 0 < t t ("),
sup W " (t); @z W " (t); "@t W " (t) L1 (RdTm) 1=3 : 0tt (")
It follows from the de nition (6.28) that for these ", t = T=" , and therefore that (6.32) holds for all 0 t T=". In particular T (") > t (") = T=" and sup k W " (t); @z; W " (t) kL1 (RdT) 1 : 0tT="
Next consider larger integer values s. Given the sup norm bound for W " ; @z; W " one can repeat the argument leading to (6.32) to show that (6.32) holds for the new s and 0 t T=" which proves the desired conclusion (6.21) 42
Proof of Lemma 6.3. Use the energy method for symmetric hyperbolic systems. There are
three key ingredients in the computation. First is that one estimates z = y; derivatives and not t derivatives. Second, the singular terms are dissipative and have constant coecients, so commute with @z . Third the scales in " must be carefully followed. They work but with no margin for error because p is a critical exponent. Write dX +m " L(U + W ) = A" (x) @ + L0 ; =0
where
?
and for j = 1; : : : ; d + m
A"0 (x) := A0 + "2 B0 "; U " (x) + W (x) ;
? A"j (x) := Aj + 1" A~j + " Bj "; U " (x) + W (x) :
Step 1. L2 (Rd Tm ) energy estimate. The standard energy identity for the linear operator L(U " + W ) applied to Z 2 C 1 ([0; t] ; H 1 (Rd Tm ) ) reads @t (A" Z; Z ) 0
L2 (Rd) +
Z
Rd Tm
?
X L0 + L0 + 2 @ A" Z ; Z dz = d+m =0
2 Re
Z
Rd Tm
Z ; L(U " + W ) Z
dz :
(6:33)
The term on the right is estimated using the Schwartz inequality to give
Z " Z ; L(U + W ) Z dz k Z (t) kL k L Z (t) kL : RdTm
Hypothesis (6.13) implies that
Z
Rd Tm
?
(6:34)
2
2
L0 + L0 )Z ; Z dz 0 :
The explicit form of the coecients A"0 shows that
(6:35)
k @0 A"0 (t; z) kL1 (z) " C (kU " + W kL1 ) "@0 (U "(t) + W ) L1 C" ;
thanks to (6.17) and (6.26). Similarly
k @j A"j(t; z) kL1 (z) " C (kU " + W kL1 ) @j (U " + W )(t) L1 C" :
(6:36) (6:37)
It is important that in (6.36) one has "@0 derivatives on the right hand side. The extra power of " compared to (6.37) comes from the "2 in the B0 term in (6.14) as compared to the " in the Bj terms. Combining the last ve equations yields a di erential inequality
@t (A"0 Z; Z )L (Rd) C "k Z (t) k2L + k Z (t) kL k L(U " + W )Z (t) kL : 2
2
43
2
2
(6:38)
De ne
?
(6:39) (t) := A"0 Z (t) ; Z (t) L1=2(RdTm) : The symmetric hyperbolicity assumption guarantees that so long as U " +W is pointwise bounded, is equivalent to the L2 norm in the sense that 1 k Z (t) k d m (t) C k Z (t) k d m : (6:40) 2
L2(R T )
C
L2 (R T )
Using this in (6.38) yields the di erential inequality
d 2 (t) " C (t)2 + C (t) k L(U " + W ) Z (t) k : L dt
(6:41)
2
As usual this yields
(t)
e"Ct (0) +
Zt
e"C(t?) k L(U " + W ) Z () kL d : 2
0
(6:42)
Since "t T in our domain this proves the crucial L2 estimate
kZ (t)kL C kZ (0)kL + C
Zt
2
2
0
kL(U " + W )Z ()kL d : 2
(6:43)
The constants in this equation depend on the sup norms of U " + W , @j (U " + W ), and "@t (U " + W ). Step 2. A commutator estimate. The strategy is to apply estimate (6.43) to Z (t) := @z W with j j s. Toward that end write
?
L(U " + W ) @z W = @z L(U " + W )W + L(U " + W ) ; @z W :
(6:44)
The next step is to estimate the commutator.
Commutator estimate. There is a constant C (s; U "; L" ) so that for all t T , j j s and W satisfying (6.26) one has for 0 t t=",
L(U " + W ) ; @ W (t)
L (RdTm) z (6:45) ? " C k W (t) kH s(RdTm) + " k L" U " (t) + W (t) W (t) kH s(RdTm) : 2
Proof of the commutator estimate. The commutator on the left of (6.45) is a sum of
"2 B0 (U " + W ) @0 ; @z and the terms
" Bj (U " + W ) @j ; @z ;
(6:46)
j = 1; 2; : : : ; d + m :
First consider (6.47). Leibniz' rule for di erentiating a product shows that
@z Bj (U " + W )@j W = Bj (U " + W )@j @z W + 44
X
06= = ?
c; (@z Bj ) @z @j W :
(6:47) (6:48)
Thus (6.47) is a linear combination of terms " (@z Bj ) @z @j W . Leibniz' rule expresses @z Bj (U " +W ) as a sum of terms of the form
" G(U " + W )
Y
@zk (U " + W ) ;
X
k = :
The G factor and all of @zk U " factors are bounded thanks to (6.17) and (6.26). It remains to estimate the L2 (Rd Tm ) norm of
"
Y
@zk W @j @z W ;
jk j 1 ; 1 j j s ? 1 ; j j +
X
P
k
jk j s :
(6:49)
Note the total order 1 + j j + k jk j could be as large as s + 1. We use the Gagliardo-Nirenberg inequalities together with Holder's inequality. If jk j = 1 then (6.26) bounds the corresponding factor in L1 so those factors can be eliminated. Thus it suces to consider expressions (6.49) with all jk j 2. For such expressions, let X X r := j j + (jk j ? 1) s ? 1; so j2 rj + jk2j r? 1 = 21 : (6:50) k k
Interpolating between @j W 2 L1 and j@z jr @j W 2 L2 one has
k @z @j W kL r=j j C k @j W k1L?1j j=r k j@z jr @j W kjL j=r :
(6:51)
k @zk W kL r= jkj? C k rz W kL1?1(jkj?1)=r k j@z jr+1 W kL(jkj?1)j=r :
(6:52)
2
2
Similarly writing @zk as the product of a rst order partial derivative and a partial derivative of order jk j ? 1 yields 2
(
1)
2
Applying Holder's inequality yields
Y
"
@zk W @j @z W
L (RdTm) " F ( kW ; rz W kL1 ) k j@z jr+1 W kL (RdTm) : 2
2
(6:53)
This suces to estimate the terms (6.47) by the the rst summand on the right hand side of (6.45). The analysis of the terms (6.46) follows the same lines with @j replaced by "@0 leading to
Y k
" @z W "@0 @z W L (RdTm) " F ( kW ; "@0 W ; rz W kL1 ) k "@0 j@z jr W kL : 2
2
Expressing the time derivative
dX +m " " ? 1 "Aj + A~j @W "@0 W = A0 " L (U + W )W ? j =1
yields
(6:54)
@zj ? "L0 W :
k "@0W (t) kH s? (RdTm) C k W (t) kH s(RdTm) + "k L"(U " + W )W kH s(RdTm) : (6:55) 1
Estimates (6.54) and (6.55) suce to estimate the terms (6.46) by the right hand side of (6.45). This completes the proof of the commutator estimate. 45
Step 3. Endgame. Applying (6.43) to Z := @z W with j j s and using the commutator estimate (6.45) yields with norms on Rd Tm ,
kZ (t)kL C kZ (0)kL + 2
Zt
2
0
? k@z L(U " + W )W ()kL2
d + "
Zt 0
kW ()kH s
d :
Summing over all j j s yields
Zt Zt ? " kW (t)kH s C kW (0)kH s + k L(U + W )W ()kH s d + " kW ()kH s d :
0
0
Gronwall's inequality implies that
kW (t)kH s C e"CtkW (0)kH s +
Zt 0
?
e"C(t?) k L(U " + W )W ()kH s d :
Since 0 t=" T=" on the domain in consideration, this proves (6.27) and completes the proof of Lemma 6.3, and therefore Stability Theorem 6.2.
x7. Applications, examples, and extensions. Once the construction of the approximate solution and the proof of its validity is complete, it remains to study the structure of the approximate solutions. In this section we present a selection of qualitative results concerning the dynamics de ned by the pro le equations. Several examples show the forms these equations can take. In addition, we describe several extensions of the results whose proof does not require essentially new ideas. In particular we discuss nonoscillatory solutions and pro les periodic in Y .
x7.1. Nonoscillatory solutions.
The dynamics of the oscillatory part a0 is determined by equations (2.26) and (4.38). These equations always have the solution a0 = 0. The solutions with a0 = 0 describe the behavior for times t 1=" of hyperbolic systems with nonoscillatory initial data u(0; y) = "p a(0; "y; 0; y). We record some features of the result as a Proposition.
Proposition 7.1. If the oscillatory part a0 (T; Y; y; ) vanishes for T = t = 0 then it vanishes for all time, and the nonoscillatory part is determined by the uncoupled system of equations for a0; = E (Dy ) a0; , 2 A ? (7:1) @t ? i (Dy ) a0; (T; Y; t; y) = 0 :
E (Dy ) L(0; @X )a0; + ( ) (a0; ) = 0 :
(7:2)
Proof. Equations, (2.21), (2.26), (4.27), (4.28) and (4.38) with initial data a0 (0; Y; y; ) =
a0 (0; Y; 0; y) are solved if one takes a0 = 0 and a0; to be the unique solution of (7.1), (7.2) with initial value a0; (0; Y; 0; y) = E (Dy ) a0 (0; Y; 0; y). By uniqueness this yields the solution. 46
Remarks. 1. The modes are uncoupled. 2. The nonlinear parts of the quasilinear terms do not contribute to the principal term. 3. For curved parts of the characteristic variety, ( ) = 0 and the equation for a is linear. If all
sheets of the variety are curved the dynamics is linear. 4. In the one dimensional case, one can diagonalize A1 (0)
A1 (0) = diag (1 Im ; 2 Im ; : : : ; N ImN ) ; 1
2
where 1 < 2 < : : : < N , and Imj denotes the mj mj identity matrix. In this basis we have components a0 = a0 = (a1 ; : : : ; aN ) ; = (1 ; : : : ; N ) : The pro le equations are uncoupled nonlinear transport equations
?@ + @ a = 0; t n y n
?@
T + n @Y an + n (an ) = 0 ;
n = 1; : : : ; N :
Writing an = bn (T; Y ? n T; y ? n t; ) with bn = bn (T; Y; y; ) these equations reduce to
@T bn + n (bn ) = 0 ;
n = 1; : : : ; N :
x7.2. Pro les independent of or periodic in Y.
The equations for the pro le a0 (X; x; ) are translation invariant in the sense that for any X; x; 2 R 2d+3 , a0 (X + X; x + x; + ) is a solution if and only if a0 is a solution. This suggests the following argument. If the initial data a0 (0; Y; t; y; ) is independent of Y , then the solution a0 is also independent of Y since a0 (T; Y + Y ; t; y; ) is a solution with the same initial data, so must be equal to a0 . The error in the argument is that we have only studied the initial value problem for data belonging to \s H s (RdY Rdy T) which excludes functions independent of Y . However, the arguments which work for Y 2 Rd work exactly the same way for Y 2 Td . In this way a theory exactly analogous to that we have developed works for modulations on the slow scale which are periodic rather than square integrable in Y . One nds exactly the same pro le equations and perfectly analogous stability results where H s (RdY ) is systematically replaced by H s (TdY ). Once this is remarked the preceding uniqueness argument is correct and we have the following important case of pro les independent of Y for which the pro le equations simplify.
Proposition 7.2. If the initial data g(y; ) 2 \s H s (Rd T) with ( ) g = g is independent of
Y then there is a unique pro le a0 (T; t; y; ) independent of Y determined as the solution of (2.21), (2.26) and the T dynamic equations
E (Dy ) @T a0; + ( ) (a0; ) = 0 ;
D
for 2 A n f1g ;
X E1 (Dy ) @T a0;1 + (1) ( a0 + a0;1 ) + (a0 + a0;1 ) @ a0
E
@T a0 ? R(@y ) @?1 a0 + (1 ? ) (a0 ) d X
+ (a0 + a0;1 ) +
=0
(a0 + a0;1 )@ a0
= 0;
(7:3) = 0;
(7:4) (7:5)
which are formally identical to equations (4.28), (4.27) and (4.38) for Y independent functions. 47
Combining the simpli cations of x7.1 and x7.2 yields the situation of the warmup problem in x4.1. Pro les independent of Y arise in another natural way. Whenever one has a principal pro le which is rapidly decreasing in y, the asymptotic description can be achieved with a principal pro le which is independent of Y . In particular this is the case of the pro les constructed in the published (as opposed to internet) version of [DJMR]. The argument is the following. Write
a0 (T; Y; t; y; ) = a0 (T; 0; y; ) + Y:b(T; Y; t; y; ) ;
b :=
Z1 0
rY a(T; Y; t; y; ) d :
Let c(T; Y; t; y; ) := y:b(T; Y; t; y; ). When pro les are used one substitutes Y = "y so that the pro le a0 (T; 0; t; y; ) + "c(T; Y; t; y; ) describes the same asymptotic behavior as a0 and has principal pro le a0 (T; 0; t; y; ) which is independent of Y . In order that c be suitably square integrable in y requires that yrY a0 be square integrable in y which is a decay condition on rY a0 . If the initial values a0 (0; Y; 0; y; ) are rapidly decaying in the sense that for all y a0 (0; Y; 0; y; ) 2 \s H s (Rd Rd T) then the process can be repeated yielding an expansion
a0 (0; Y; 0; y; ) a0 (0; 0; 0; y; ) +
1 X j =1
"j cj (y; )
all of whose pro les are independent of Y . Solving the pro le equations with the initial data for a1 ; ( ) a1 changed from zero to a1 t=T =0 = c1 (y) ; ( ) a1 t=T =0 = ( ) c1 yields an asymptotic description with pro les independent of Y .
Summary. If a0 is rapidly decaying in y, the above recipe constructs an asymptotic description of the solution with pro les independent of Y .
x7.3. Curved sheet implies that a0 does not in uence a0 . Proposition 7.3. If satis es the smooth variety hypothesis and belongs to a sheet of the
characteristic variety which is not a hyperplane, then the evolution of the nonoscillating part a0 is not in uenced by the oscillating part a0 . For such if a0 vanishes at T = t = 0 then it vanishes identically. For solutions with a0 = 0, the pro le equations simplify appreciably which makes this Proposition particularly interesting.
Proof. The condition on is equivalent to = 0. In that case the pro le equations for a0 are ? @ ? i (D ) a = 0 ; (7:6) t y
?
E (Dy ) L(0; @X ) a + ( )(a0;j ) = 0 ; (7:7) These equations do not involve a0 proving the rst assertion of the Proposition. If the initial data vanishes at T = t = 0 then (7.6) implies that a vanishes in fT = 0g. Then the second equation implies that it vanishes for all T . 48
This result shows that creation of nonoscillating contributions from oscillating terms is restricted to the case of belonging to hyperplanes in the characteristic variety.
Examples of hyperplanes in the variety. 1. When d = 1 all sheets of the characteristic variety
are hyperplanes. For this reason the one dimensional case is not a good guide to the general case. 2. The characteristic varieties of both the compressible Euler equations and Maxwell's Equations are the union of a light cone and the horizontal plane = 0. For the Euler equation the hyperplane corresponds to entropy waves. For the Maxwell equations, the hyperplane in the characteristic variety correspond to unphysical solutions which are eliminated by the constraints div E = div B = 0. 3. It is possible that in a conic neighborhood of the characteristic variety contains a curved sheet and its tangent plane. An example is the characteristic equation ( 2 ? jj2 )( + 1 ) = 0
= (1; ?1; 0; : : : ; 0) :
with
In such cases the simple characteristic variety hypothesis is violated and our construction of approximate solutions does not apply.
x7.4. Two examples with a0 = 0.
We compute the form taken by the pro le equations for two examples satisfying the hypotheses of Proposition 7.3 and for which the mean value a0 vanishes identically.
Example 7.1. Consider the semilinear system
@u + 1 0 @u + 0 1 @u + (u) = 0 ; @t 0 ?1 @y1 1 0 @y2
(7:8)
where (u1 ; u2 ) = (1 ; 2 ) is homogeneous of degree J . The standard normalization is then p = 1=(J ? 1). For quadratic (resp. cubic) nonlinearity one has p = 1 (resp. p = 1=2). The characteristic variety is given by 2 = jj2 so the simple characteristic variety and curved characteristic variety hypotheses are satis ed at all . Take = (?1; 1; 0; 0) then
L( ) = 00 ?02 ; and;
( ) = 10 00 :
The transport operator is V (@x ) = @t + @1 . The principal pro le satis es ( )a0 = a0 so has vanishing second component. Risking confusion write (a(T; Y1 ? T; Y2 ; y1 ? t; y2 ; ); 0) for the leading pro le. Then (4.38) shows that the scalar valued function a(T; Y; y; ) is determined from its values at T = 0 by the equation
@ a + 1 @ ?1 a + (a; 0) = 0 : 1 @T 2 y
(7:9)
If J is an odd integer and 1 (a; 0) = (jajJ ?1 )a, there are monochromatic solutions a(T; Y; y; ) = eik v(T; Y; y) whose pro le v is determined as a solution of the classical nonlinear Schrodinger equation @ v + 1 v + (jvjJ ?1 ) v = 0 : @T 2ik y 49
Example 7.2. The inviscid compressible Euler equations.
The isentropic inviscid 2d compressible Euler equations describe ows with negligible heat conduction.* The unknowns are the velocity v = (v1 (t; y); v2 (t; y)) and the density (t; y). The dynamics is governed by the system of equations
@
0
p () @ @ @ @t + v1 @y1 + v2 @y2 v1 + @y1 = 0 : @ 0 + v @ + v @ v + p () @ = 0 :
(7:10) (7:11)
@t 1 @y1 2 @y2 2 @y2 @ @ + v @ + @v1 + @v2 = 0 ; + v @t 1 @y1 2 @y2 @y1 @y2
(7:12)
where the pressure as a function of the density. This system is of the form P B (v;p =)@ p((v;))gives = 0 with coecients
B0 = I ;
0 v 0 p0 ()= 1 1 B 1 = @ 0 v1 0 A; 0
v1
0 v 0 p0 ()= 1 2 B 2 = @ 0 v2 0 A: 0
v2
(7:13)
The system is symmetrized by multiplying by B~0 yielding coecients
0 2 =p0() B~0 := @ 0 0
0 2 v =p0() 0 1 j B~j = @ 0 vj 0 A :
1
0 0 1 0A ; 0 1
0 vj
(7:14)
The background state is v = 0; = > 0 with constant density. Introduce
u := B~01=2 (0; ) (v; ? ) =
v1
c ; v2 ; ? ;
q
c := p0 ()
to nd an equivalent system with coecients
A (u) = B~0?1=2 (0; ) B~ c u1 ; u2 ; u3 + B~0?1=2 (0; ) :
satisfying the conventions that the background state is u = 0 and the coecient of @t is equal to I when u = 0. This is usually a simple zero so the nonlinearity is of order K = 2 and the critical exponent for di ractive nonlinear geometric optics is p = 2=(K ? 1) = 2. The system is strictly hyperbolic with characteristic equation
?
2 ? c2 jj2 = 0 : For = (c; ?1; 0) = ( ; ) the basic operators are
@ +c @ ; V (@x ) = @t @y 1
R(@y ) = y ;
(7:15)
0 1=p2 ( ) = @ 0p
* The nonisentropic, and 3d equations can be analysed in the same manner. 50
p 1
0 1= 2 0 0p A : 1= 2 0 1= 2
(7:16)
The last is orthogonal projection on R(1; 0; 1). Since the variety is curved, Proposition 7.3 yields pro les with vanishing nonoscillatory part. In that case, the principal pro les
p
? p
a0 = a(T; Y1 ? cT; Y2 ; y1 ? ct; y2 ; ) 1= 2; 0; 1= 2
with scalar valued amplitude a(T; Y; y; ) are determined as solutions of the quasilinear evolution equations @T a + y @?1 a + a @ a = 0 ; with 2 R determined from the identity
( )
X
A0 (0) ( ) = ( ) c A00 (0) ? A01 (0) ( ) = ( ) :
x7.5. An example of a large corrector.
We present the computations for Example 5 of the introduction. In the notations of that example u" and v" are solutions of a pair of coupled scalar wave equations. The gradients, u" and v" satisfy coupled rst order systems. Either by analysing the exact solution with oscillatory initial data as in [DJMR], or by applying Theorem 1.1 in the absence of nonlinear terms, one nds solutions u" := rt;y u" with 1 X u" " ei(y ?t)=" "j aj ("t; y1 ? t; y2 ) : 1
j =0
The leading pro le a0 (T; y) satis es the Schrodinger equation 2i @T a0 + y a0 = 0 : Take Cauchy data for v" to be equal to zero. Then for t 1=" v" "2 ?1 ja0 ("t; y1 ? t; y2 )j2 ;
v" := rt;y v" = (v0" ; v1" ; v2" ) :
Lemma 7.4. If the initial data for a0 are a0 (0; y) := e?y , then there are positive constants 0 < c < C so that as " ! 0 sup kv" (t)kL1(R ) C "3=2 and ? v1"(1="; 1=") c "3=2 : 2
0t1="
2
The upper bound veri es that v" is a corrector, that is small compared to u" = O("). The more interesting part is the lower bound which shows that this corrector is large compared to the O("2) correctors in (1.1).
Proof of the lower bound. The upper bound is proved by a similar argument. The function ja0 (T; y)j2 is a spreading gaussian with maximum at the origin. For suciently small c > 0 and 0 T 1,
?(sgn y1 ) @y@ a0 (T; y)j2 c jy1j e?y =c := h(y) : 2
1
51
Choose g 2 C01 (R2 ) so that 0 g h : 1 imply that in fy tg, Finite speed and positivity of ?1+2 1 " y1 ? t; y2 )j2 "2 w : ?v1" = ? @@yv = "2 ?1 @ ja0 ("t; @y 1
1
(7:17)
where w is the solution of the initial value problem w" jt=0 = wt" jt=0 = 0 ; 1+2 w = g(y1 ? t; y2 ) ; whose source term is a pulse moving at speed 1. De ne G(t; y) to be the solution of Gjt=0 = 0; Gtjt=0 = g(y) : 1+2 G = 0 ; Then Zt w(t; y) = G(t ? s; y1 ? s; y2 ) ds ; 0 so Zt w(t; t; 0) = G(s; s; 0) ds ; 0 For t > 0, Z 1 pt2 ? [(yg(y?) t)2 + y2 ] dy G(t; t; 0) = 2 D(t;(t;0)) 1 2 where D(r; z ) denotes the disc of center z and radius r. Let d2 := (y1 ? t)2 + y22 so in the intersection of the disc and supp g, t2 ? (y1 ? t)2 ? y22 = t2 ? d2 = (t + d)(t ? d) = 2t(t ? d) ? (t ? d)2 = 2t dist (y; @D(t; (t; 0)) ? dist (y; @D(t; (t; 0))2 = 2t dist (y; @D(t; (t; 0)) + O(1) ; as t ! 1. For t large, D(t; (t; 0)) \ supp g approaches the intersection supp g \ fy1 0g . In particular dist (y; @D(t; (t; 0)) = y1 + O(1=t) on D(t; (t; 0)) \ supp g. Therefore, Z g(y) 1 G(t; t; 0) = p p dy + O(t?1) 2 2t y >0 y1 and consequently p Z g(y) dy + O(log t) : w(t; t; 0) = pt p 2 y >0 y1 Together with (7.17), this proves the desired lower bound. 1
1
x7.6. A scalar higher order equation.
An analysis like that for rst order systems applies to equations of higher order. One has the option of reducing to a rst order system or to start from scratch deriving pro le equations. The equations for the pro les for scalar equations are a little easier to nd because there are no polarizations and projectors . There are two changes which should be born in mind for higher order equations. For an equation of order m, nonlinearities can be present in derivatives of all orders between 0 and m. Each nonlinear term must have a time of interaction no smaller than O(1=") For a linear mth order equation with source term oscillating with wavelength " the response will be of size "m if the phase has noncharacteristic di erential. The response is O("m?1) in the characteristic case, that is when the phase satis es the eikonal equation. Our phases fall in the latter category. In the applied literature phases of this sort are sometimes called phase matched. 52
Example 7.3. Consider the equation t;y u + (@t;y u) = 0 ;
(7:18)
where is a homogeneous polynomial of degree J . The characteristic variety is 2 = jj2 so is everywhere simple and curved. The equation can be converted to the semilinear symmetric hyperbolic system
@t wj ? @j w0 = 0 ; @t w0 ?
d X j =1
j = 1; : : : ; d ;
@j wj ? (w) = 0 ;
for w = rt;y u. Since is of order J the critical exponent is q = 1=(J ? 1) and we have solutions
w" = "q w0("x; x; :x=") + h:o:t: The characteristic variety is given by the equation d?1 ( 2 ? jj2 ) = 0 with the extraneous hyperplane f = 0g of multiplicity d ? 1. These roots are easily understood since solutions of the system satisfy ? @t d w1 dx1 + : : : + wddxd = 0 ; and the solutions of interest are those in the invariant subspace de ned by d(w1 dx1 + : : : + wddxd ) = 0. The L2 (Rd )-orthogonal complement of these solutions are stationary solutions satisfying div (w1 ; : : : ; wd ) = 0. These correspond to the extraneous roots. If one chooses = (1; ?1; 0; : : : ; 0), then
( ) = j(1; 1; 0 : : : ; 0)ih2 (1; 1; 0 : : : ; 0)j ;
V (@ ) = @0 + @1 ;
R(@y ) = 12 y ;
and for pro les whose mean value is initially zero and independent of Y it remains so for positive time by Propositions 7.2 and 7.3. Pro les with vanishing mean and independent of Y are given
a0 (T; t; y; ) = a0 (T; t; y; ) = a(T; t ? y1 ; y2 ; : : : ; yd ; ) p12 ; p12 ; 0 : : : ; 0
with scalar valued a satisfying
? @T a + 21 y @?1 a + p1 ; p1 ; 0 : : : ; 0 a p1 ; p1 ; 0 : : : ; 0 = 0 : 2
2
2
2
An alternate approach is to reason directly on the scalar equation seeking approximate solution of the form (2.6-2.7). To compute the critical exponent p one must estimate the accumulated e ect of the nonlinear term (@u). For u of wavelength " and amplitude "p as in (2.6), @u "p?1 so (@u) "(p?1)J . The phase will satisfy the eikonal equation, and then linear geometric optics shows that the e ect of the oscillatory part is ?1 ((@u) ) t" "(p?1)J . Setting this equal to "p for times t 1=" yields p(J ? 1) = J so p = J=(J ? 1). Note that p + 1 = q is the critical power for a semilinear rst order system with nonlinearity of degree J . 53
Inject the ansatz (2.6-2.7) into the di erential equation and insist that the nonlinear term contribute at the same power in " that yields linear di ractive e ects. A rst computation yields
? 2 u" = "p?2 02 ? 12 ? : : : ? d2 @@ 2a0 + O("p?1) : This forces the eikonal equation 02 = 12 + : : : + d2 . Then one has u" = 2"p?1 which yields
@a @ X d @ 0 @t ? j @y @0 + O("p) ; j j =1
0 0 @a @t ?
X @a0 j = 0 : j
(7:19)
@yj
The nonlinear term should appear in the next term which gives the laws of di ractive geometric optics. The next term is of order "p and the nonlinear term is of order "(p?1)J . Equating these orders gives again the critical exponent p = J=(J ? 1). Choosing = (1; ?1; 0; : : : ; 0), yields the vector eld V (@x ) = @t + @1 . With the choice of p above one nds
0 + a + u" + (rt;y u" ) = "p 2V (@X ) @a x 0 @
@a0 @a 1 + 2V (@x ) + O("p+1) : @
@
(7:20)
This suggests imposing the equation
0 + a + @a0 + 2V (@ ) @a1 = 0 : 2V (@X ) @a x @ x 0 @ @
(7:21)
0 + a + @a0 = 0 : 2V (@X ) @a x 0 @ @
(7:22)
Thanks to (7.19), applying V (@x ) to the oscillating part of (7.21) yields V (@x )2 a1 = 0. In order that a1 grow sublinearly in x one must have V (@x )a1 = 0 and one nds the pro le equation Equations (7.19) and (7.22) suce to determine the oscillating part a0 from its initial values at ft = T = 0g. Note that the mean values of a0 do not enter in this determination, and that the pro le equation just derived is equivalent to the equation found from reducing to a rst order system with the identi cation w0 = @ a0 =@ = @ a0 =@. Di erentiating one has
ru" = "p?1 @ a0 ("x;@x; :x=") + O("p) ; and one recovers the answer from the rst order system computation.
References
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