STABILITY OF A RESONANT SYSTEM OF CONSERVATION LAWS MODELING POLYMER FLOW WITH GRAVITATION CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO Abstract. We prove L1 uniqueness and stability for a resonant 2 2 system of conservation laws that arise as a model for two phase polymer ow in porous media. The analysis uses the equivalence of the Eulerian and Lagrangian formulation of this system, and the results are rst established for an auxiliary scalar equation. Our methods are based on front tracking approximations for the auxiliary equation, and the Kruzkov entropy condition for scalar conservation laws.
1. Introduction We study the initial value problem for scalar conservation laws of type (1.1) ut + f (a(x); u)x = 0; where the coecient a is a suitable function of x. Using the transformation given by Wagner in [23], for u 6= 0, one nds the equivalent 2 2 system: 1 ? uf a; 1 =0 u t u x (1.2) a 1 ? auf a; u = 0: u t x
If 1=u is interpreted as the saturation of water, and a the concentration of polymer dissolved in the water, then (1.2) is a model of polymer ow in a porous two phase environment. This model is one of the prime motivations for the present paper. Furthermore, conservation laws of the type (1.1) are related to models of transonic ow of gas in a variable area duct. If the coecient a is smooth, and a0 is of bounded variation, one can use nite di erences to show existence of a weak solution to (1.1). This method was rst used by Oleinik in [19], where she used the Lax-Friedrichs scheme. As long as a is smooth, we can also show uniqueness using the Kruzkov entropy condition, and the classical \doubling of the variables" technique, see Kruzkov [15] or Kuznetsov [16]. However, if a is allowed to be discontinuous, none of the above techniques work. In this case, it is more pro table to view (1.1) as a system, by adding the trivial conservation law at = 0: a This system has characteristic speeds = 0 and u = fu . If these two speeds coincide, the system is non-strictly hyperbolic, and is called resonant. The solution of the Riemann problem for resonant conservation laws is more complicated and interesting than the solution in the strictly hyperbolic case, see Isaacson and Temple [8]. This paper ([8]) studies a general system of resonant conservation laws, and solves the Riemann problem locally around a state where two wave speeds coincide. Date : November 1, 1999. 1991 Mathematics Subject Classi cation. 65M10,35L65,35L67. Key words and phrases. conservation laws,stability, non-strictly hyperbolic. Nils Henrik Risebro is grateful to the University of Heidelberg and to SFB359 for support during the completion of this work. The authors are grateful to John Towers for spotting a mistake in an earlier version of this work. 1
2
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
For the 2 2 systems (1.2) or (1.1), the global Riemann solution was studied by Gimse and Risebro in [4], where it was shown that for a large class of ux functions f , a unique solution exists, subject to an additional entropy condition. However, for certain f s, some Riemann problems do not have a solution. Also for the special case of the polymer model without gravitation, the solution of the Riemann problem was reported in Isaacson [7], and the solution of the Riemann problem with gravitation was investigated in [3] Utilizing the solution of the Riemann problem, Temple [21] showed existence of a weak solution for the polymer model. This was done by generating approximate solutions by the Glimm scheme, and then showing compactness using a nonlinear functional. This functional was subsequently used in [5], to show existence for a model of two phase ow in a heterogeneous porous medium, including gravitational e ects. The Temple functional was also used in [14] where uniqueness and existence was shown for a model problem where f (a; u) = ag(u) for a convex g. Here, uniqueness was obtained using a wave entropy condition. This model problem was also studied by Klausen and Risebro [13, 12], where it was shown that there is a unique solution, which is the limit of solutions with smooth coecients, and in addition a stability estimate was established, both with respect to a and u. Moreover, the functional introduced in [21] was used to prove convergence of the Godunov scheme for (1.1) by Longwai, Temple and Wang in [17], and for an inhomogeneous balance law by Isaacson and Temple in [9]. In these papers, the coecient a was assumed to be Lipschitz continuous with the total variation of a0 bounded. In this case, the solution generated by Godunov's method satis es Kruzkov's entropy condition, so that one has L1 stability with respect to u, see [18]. For discontinuous a, L1 contraction of the solution operator with respect to u can be proved by using front tracking and the semigroup approach by Bressan, [2, 10], and in the special case where fu 6= 0 L1 stability was shown by Baiti and Jenssen in [1]. Note that by Wagner's fundamental result [23], corresponding existence and stability results as the ones alluded to above, hold for (1.2) as well. Without using this equivalence, Tveito and Winther [22] proved existence and L1 stability for the polymer system (1.10), (1.11), in the case without gravitational e ects. In this paper we incorporate gravitational e ects (i.e., f in (1.1) need not be convex in u), and then proceed to prove existence and L1 stability without using smoothness assumptions on either the initial data or on the coecients. The rest of this paper is organized as follows: In the remainder of this introductory section we motivate the polymer model. In section 2 we de ne a \canonical" auxiliary conservation law. When viewed in the proper coordinates, the Riemann solution for the auxiliary model is identical to the Riemann solution for (1.10), (1.11). We therefore carry out the bulk of our analysis for the auxiliary model, and defer the discussion of the corresponding results for the polymer model to the next section. Based on the solution of the Riemann problem, we then de ne approximate solutions by a front tracking scheme. Using a Temple functional, we proceed to show compactness, and hence existence of a weak solution. For the moment assuming that the coecient a is smooth, we next show that the limit satis es Kruzkov's entropy condition, and hence in this case, the solution operator is L1 contractive. Furthermore, in the case that a0 is bounded, we obtain a convergence rate for the front tracking approximation. Then, using both the front tracking approximations and the Temple functional, we show a stability estimate for non-smooth coecients. Finally in section 3 we establish corresponding results for the polymer system with gravitation. 1.1. The polymer model. In this section we will motivate the polymer model for for a one dimensional reservoir with constant geological properties, i.e., porosity and permeability. Assume that we have two phases present in the reservoir, oil and water. Let s denote the saturation of the water. The saturation of a phase is de ned as the percentage of the available pore volume occupied by that phase. Hence the saturation of the other phase, oil, is 1 ? s. Let x denote the position in the reservoir and let t denote the time. Partial derivatives will be denoted by subscripts.
STABILITY OF A RESONANT SYSTEM
3
The velocity of each phase, denoted by subscript i, is assumed to obey (the experimentally veri ed) Darcy's law (1.3) vi = ?i ((Pi )x + i gDx ) where the mobility of each phase is de ned as (1.4)
k i = K i : i
Here K denotes the absolute permeability of the rock, ki denotes the relative permeability of phase i, and i the viscosity of the phase. Furthermore, i denotes the density of the phase, g the gravitational acceleration, and D measures vertical distance in the reservoir. The relative permeabilities are convex functions from [0; 1] to [0; 1]. In subsequent equations the index i will be o(denoting the oleic phase) and w(denoting the aqueous phase). We will ignore the capillary pressure and set Po = Pw = P . Conservation of mass for the aqueous phase now reads ? ( w vw )x + qw = (w sw )t ; where denotes the cross section of the reservoir, denotes the porosity, i.e., the fraction of the total volume available for one of the phases. The term qw denotes sources or sinks present in the reservoir. We will assume that the cross section is constant, and that the densities and the porosity are independent of the pressure. Hence, formally, the conservation equation reads qw (1.5) = (sw )t ; ? (vw )x + w and similarly for the oleic phase. Adding the conservation equation for oil and for water, and using that so + sw = 1, we are left with
qo qw + (vo + vw )x = ? (1.6) w o (vtot )x = Qtot where the so-called total velocity is denoted by vtot , similarly Qtot denotes the total volumetric
injection or production rate. Thus we see that in a one-dimensional reservoir, the total velocity is constant if Qtot is zero. Using this in the equation for mass conservation (1.5), we nd (1.7) st + (f0 (s) (vtot ? (w ? o ) gDx Ko (s)))x = 0; where we write s for sw (so = 1 ? s), and the function f0 is the so called fractional ow function, given by w (s) f0 (s) = (1.8) : w (s) + o (s) In oil reservoir applications, injection of water is done in order to help prevent pressure loss in the reservoir, thereby forcing more oil out. However, since the viscosity of oil is larger than that of water, \ ngers" may form, thereby rendering this process less e ective than desirable. In order to prevent this ngering, a polymer is sometimes added to the water to increase its viscosity. This polymer is passively transported with the water, yielding a conservation equation (1.9) (sc)t + (cf (s; c))x = 0; where f (s; c) is given by f (s; c) = f0 (s; c) (vtot ? Go (s)) ; and G = (w ? o ) gDx K: So summing up, we have the following system of conservation laws (1.10) st + f (s; c)x = 0 (1.11) (sc)t + (cf (s; c))x = 0 In the gure below we show how f typically looks for two di erent c values. In the rest of this
4
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO The flux function as a function of s for c=1 and c=0 1
0.8
0.6
f(s,c)
0.4
0.2
0
c=0 c=1
-0.2
-0.4
0
0.1
0.2
0.3
0.4
0.5 s
0.6
0.7
0.8
0.9
1
Figure 1. The ux function f (s; c) as a function of s for two di erent c values.
paper we will use the following properties of the ux function f . It is a di erentiable function of s and c for (s; c) in [0; 1] [0; 1]. There is an s^ such that f (^s; c) = 0 for all c, and f (s; c) < 0 for c in (0; s^), and 1 > f (s; c) > 0 for s in (^s; 1). Furthermore f (0; c) = 0 and f (1; c) = 1 for all c. Also @f > 0; for c in (0; s^) @c @f < 0; for c in (^s; 1). @c
2. The auxiliary model It turns out that viewed in the proper coordinates, the Riemann problem for (1.10){(1.11) is identical to the Riemann solution for a scalar equation with x dependent coecients. This \system" is the following t + (r sin )x = 0 (2.1) rt = 0 where the unknown is (x; t). We are interested in the initial value problem for (2.1) and assume that r(x; t) = r(x) and (x; 0) = 0 (x) are known. Furthermore, we shall assume that r(x) is a positive function of bounded variation, and that the initial function 0 (x) takes values in the interval [?; ]. We let h(; r) = r sin . 2.1. The Riemann problem. The Riemann problem for (2.1) is the initial value problem where ( ( l ; for x 0, r; for x 0, 0 (x) = r(x) = l (2.2) r ; otherwise, rr ; otherwise, where l;r and rl;r are constants. Such Riemann problems have been studied and solved by Gimse and Risebro in [4], where it was demonstrated that subject to an extra entropy condition, there existed a unique solution. The solution Riemann problem for (2.1) consists of two types of waves, r waves, over which h is constant, but r varies, and waves, over which r is constant. Note that viewed as a system (2.1), has two characteristic speeds r = 0 and = r cos . It is linearly degenerate in the rst family, and for = =2, the two eigenvalues coincide.
STABILITY OF A RESONANT SYSTEM
5
It turns out that the (r; ) coordinates are awkward to work with when de ning the front tracking approximations, and we use instead the following coordinates: v = sign (cos()) r (1 ? jsin()j) (2.3) w = r sin We denote this coordinate mapping by r , i.e., (v; w) = r (; r). We see that in the (v; w) plane, the r waves will be lines with constant w, and the waves will be on diamonds of constant r. A Riemann problem is said to have a solution if the left state is connected with the right state via a wave, similarly the solution the two states are connected with an r wave the solution is labeled r. Hence we have essentially three types of solutions to the Riemann problem for (1.10), (1.11): r, r and r. In the gure below, we show the solution in each of the cases ? < ?=2, ?=2 < 0, 0 < =2 and =2 . These cases appear counterclockwise, starting in the lower left corner. This gure is interpreted by following the dashed lines from, representing w
w
L
rθ
L
θr
θr
θr θ
θr θ
v
v
θr θ
w
w
rθ
θr θ θr θ
θr θ v
θr L
v
θr
L
Figure 2. The solution of the Riemann problem
either waves or r waves, from the left state, denoted L, to any right state. In each case, the (v:w) plane is divided into three regions, separated by thick lines, and in each region the wave con guration is constant, e.g., r. Note that r waves cannot cross the lines = =2, where the two eigenvalues coincide. Hence to solve a Riemann problem, we rst nd the wave con guration from this gure 2. If a wave appears in the solution, the left and right states have the same r value, and the solution is found by solving a scalar Riemann problem. This is done by nding the lower (if l < r ) or upper (if l > r ) envelope of h(; r). Below we show how this works on a speci c example. The diagrams show how the solution will look in (v; w) coordinates, in (; h)
6
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
coordinates, and then we show the waves in (x; t) space and nally as a function of x=t. For a more detailed explanation of the solution such Riemann solutions, we refer the reader to [4]. w
h
R R
θ
v
L
L
Solution in (v; w) coordinates.
Solution in (; h) coordinates. 3.1
3.1
1.0
1.0
-1.0
-1.0
t
-3.1
-3.1
x
-0.80
-0.27
0.27
0.80
Solution in (x; t) coordinates.
Solution in (x=t; ) coordinates. 2.2. The front tracking scheme. Now we de ne the front tracking approximations which will be our primary tool. The scheme we will use is an adaptation of the schemes used in [5] and in [14]. Fix a (small) positive number . For i 2 Z, let vi = i and wi = i This will be our grid in phase space. For ri = jij , we make a piecewise linear approximation to h(; r), interpolating between the points on the grid, i.e., let
ij = sign (j ) cos?1 1 ?
jj j ; for ?2i j 2i: i
Let h be the piecewise linear approximation to h, h (r; ij ) (2.4) h (; r) = h (ri; ij ) + ( ? ij ) h (r; i;j +1) ? ; for r = ri and 2 [ij ; i;j +1] ? i;j
+1
ij
For a xed i, the initial value problem for (2.5) t + h ( ; ri)x = 0; (x; 0) 2 fij g2j =i ?2i ; can be solved exactly using Dafermos' method, see [6]. Furthermore, the unique weak solution (x; t) will be constant on a nite number of polygons in the (x; t) plane, and take values in the set fij g2j =i ?2i. Let r (x) be an approximation to r(x), such that r takes values in the set fri g, and
r ? r ! 0 (2.6) 1 as ! 0. Similarly 0 (x) be an approximation to 0 taking values in the set fij g, and
? 0 ! 0 (2.7) 0 1
STABILITY OF A RESONANT SYSTEM
7
as ! 0. We shall use front tracking to de ne a weak solution to ? t + h ; r x = 0: (2.8) rt = 0 Firstly, note that each initial Riemann problem de ned by r and 0d will have solution which are piecewise constant, and that takes values on the grid (vi ; wi). Furthermore, the discontinuities emanating from each initial discontinuity, all have nite speed. Therefore, by solving the initial Riemann problems, we have de ned a weak solution initial two discontinuities collide. At the collision point, we solve the Riemann problem de ned by the states to the left and right of the collision point. Since these states are on our grid, also the resulting states will be on the grid. Now we have a solution de ned until the next collision point. Therefore, we have a weak solution up to any collision point. We call this construction front tracking. It remains, of course, to show that this process is well de ned, we have to check whether we can reach any predetermined time, and whether the ?number of discontinuities stays nite for any time. To accomplish this we de ne a functional on ; r . This functional is chosen so that it dominates the total in (v; w), and such that it is non-increasing in t. ? variation Let now u = v ; w , we have that u de nes a path on the (v; w) grid. This path consists of waves and r waves. We call any nite connected sequence of and r waves an I curve, and say that I connects uL to uR if the left state of the rst segment is uL and the right state of the last segment is uR. An I curve can then be written r11 r2 2 : : :rN . This terminology and the subsequent techniques are ultimately borrowed from Temple [21], but see also [5] and [14]. We rst de ne F on simple wave segments, for a wave, let (2.9) F () = T:V: (w()) ; i.e., F () is times the number of horizontal grid lines crossed by the wave. For an r wave connecting points (vL ; w) and (vR ; w), let r = vR ? vL . Then (2.10) F (r) = (3 ? sign(r)) jrj : For general I curves, I = b1 b2 bN , F is de ned additively, i.e., F (I ) = ?
N X
=1
i
F (bi ) :
For later use, note that F u can also be written ?
F u = (2.11) and we then clearly have that
Now we can show that
T:V:
Z
?
?
3 ? sign vx
R
?
?
v + w x
x
?
dx;
u F u 4T:V: u :
Lemma 2.1. Let I be any I curve connecting uL to uR , and let [uL; uR] be the I curve de ned by the solution of the Riemann problem with left state uL and right state uR. Then (2.12) F ([uL; uR]) F (I ): Proof. First we show the lemma in case where sign(I ) is constant, i.e, I does not cross the v axis. Then it suces to observe that 2 [?; 0] and 2 [0; ] are both invariant regions with respect to
the solution of the Riemann problem, cf. gure 2. In this case the proof of this lemma is identical to the proof of the corresponding lemma in [14] or [21]. Next assume that wL = wR = 0,and vL ; vR > 0, in this case the [uL; uR ] = r, and by the above remarks F ([uL ; uR]) F (J ) for any J curve connecting the states and not crossing the v axis. Assume next that wL 0 and wR 0, and I crosses the v axis. Let J be the I curve from uL to uR consisting of the part of I with non-negative w coordinate, together with parts on the v axis. Then J = J1J10 J2 J20 : : : J`0J`+1
8
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
where Jk denotes the parts of I with non-negative w and Jk0 denotes the parts of J with zero w. Corresponding to Jk0 we let Ik denote the part of I connecting the same points (on the v axis) as Jk0 . Then we have F ([uL; uR]) F (J ) =
` X k
=1
` X k
=1
?
F Jk0 + F (Ik ) +
`+1 X
=1 +1
k ` X k
=1
F (Jk )
F ( Jk )
= F (I ): An identical argument covers the case where wL ; wR < 0. Assume now that sign (wL) 6= sign (wR ). From gure 2 we see that [uL; uR] = [uL; uM ][uM ; uR] ; where wM = 0. Furthermore F ([uL; uR]) = F ([uL ; uM ]) + F ([uM ; uR ]) : First we show the lemma in case I crosses the v axis once at u1, the general case will then follow by induction. Now we have that [u1; uR] = r = [u1; uM ][uM ; uR] : Let I1 denote the part of I connecting uL and u1 and let I2 denote the part connecting u1 and uR . Then F (I ) = F (I1) + F (I2 ) F (I1) + F ([u1 ; uM ]) + F ([uM ; uR]) = F (I1 [u1; uM ]) + F ([uM ; uR]) F ([uL; uM ]) + F ([uM ; uR]) = F ([uL; uR ]) : Regarding the general case, assuming that I crosses the v axis k times. Using the above arguments, we can nd another I curve I~, with F (I ) F I~ connecting uL and uR and crossing the v axis k ? 1 times. This concludes the proof of Lemma 2.1. From this lemma the following is immediate:
Lemma 2.2. F ?u is non-increasing in time.
Proof. It is clear that F only changes value when two discontinuities in u collide. At collision points, a section of the I curve traced out by u connecting uL and uR is replaced by [uL; uR]. Hence, F is non-increasing.
Now we can also use this to show that front tracking is well de ned. We have that the following types of collisions can occur 0 ! r ! 0 r0 r ! r0 0 r ! 10 r0 20 r ! 10 r0 20
meaning that the two waves to the left of the arrow collides, producing a solution indicated by the waves to the right of the arrows. Each of the 0 waves to the right may consist of several discontinuities. This will be the case if the 0 wave is approximating a rarefaction. To aid the further analysis we use the next lemma
Lemma 2.3. If we have a collision of type r ! 10 r0 20 or r ! 10 r0 20 , then F ? F 0 .
STABILITY OF A RESONANT SYSTEM
9
Proof. The proof of this is a study of cases. We prove the lemma in the case where we have a collision of type r, the other case being entirely similar. Let separate uL and uM , and r separate uM and uR . The point uL has coordinates (L ; rL). Similarly for the points uM and uR. To show the lemma we consider four cases, depending on which quadrant in the (v; w) plane uL is in. With a slight abuse of notation, let L be the coordinate of the left state. The reader is urged to consult the Riemann solution, g. 2 as a tool to understand our arguments.
Case 1: ? L < ?=2: In this case ?=2 R , and rR rL, since this is the set where [uL; uR] = r . Now rM = rL since uL and uM are connected via a wave. Furthermore, this wave has positive speed. Now there are no single waves of positive speed having L as its left state, which follows from the Rankine-Hugoniot condition. Hence in case ? L < ?=2 there are no such collisions. Case 2 ?=2 L < ?0: In this case the set of states where [uL ; uR] = r is given by ?=2 R , and rR jwLj ? , where wL is the w coordinate of uL. Since uM and uR are connected by an r wave, uM must be such that jwM j jwLj ? , and since uL and uM are connected by a wave, rM = rL. Now it is a simple exercise to control that F 0 ? F . Case 3 0 L < =2: Now ? R =2 and jwR j wL ? . Also in this case it is straightforward to check that F decreases by at least . Case 4 0 L < =2: This case is similar to the rst case.
This concludes the proof of the lemma. A consequence of this lemma is that for some xed , \re ections" of waves can only occur a nite number of times. Hence after some nite time, all waves of nonzero speed will have passed into the region to the left or right where r is constant. Recall that the initial data, and in particular r, is constant outside some bounded interval. Regarding collisions, these always result in a single discontinuity if F is constant, and if more than one discontinuity results, then F ? F 0 , see [6] for details. Hence after some nite time, any collisions will result in a single discontinuity. Also the speed of this discontinuity will be between the speeds of the colliding discontinuities. In particular, two colliding discontinuities of negative speed, will result in a single discontinuity moving with negative speed. Combining the above remarks, we see that after some nite time T , there will be no further collision of fronts. Hence the front tracking method is well de ned, and for a xed , requires only a nite number of operations. 2.3. Compactness and convergence. Now we are in a position to use standard arguments to show that there is a subsequence of u that converges to some u, such that (; r) def = ?r1u is a weak solution to (2.1). For the auxiliary system the relevant theorem is:
Theorem 2.1. Let u0 = r (0 ; r) be such that T:V: (u) is nite, and assume that u0 is in L1. Let u denote the function de ned by the front tracking construction. Then for any sequence f g such that ! 0, there exists a subsequence fj g, such that for any nite time t 0, j (; t) ? converges uniformly in L1loc , where ; r = ?r1u . Furthermore the limit of j is a weak solution of (2.1).
10
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
Proof. The compactness of the sequence fug follows by standard arguments, see e.g., [20], for u
satis es
u 1 M;
u (; t) M; 1 T:V: (u) M;
u (; s) ? u (; t) M jt ? sj ; 1
(2.13)
where M is a generic constant. The proof of the the fourth of these inequalities follows from nite speed of propagation and the third inequality. Then the convergence of a subsequence of fug in L1loc follows by applying Helly's theorem and using a further diagonal argument, see e.g., [20]. Since r is injective, the corresponding convergence for a subsequence of follows. For simplicity we now denote the subsequence fj g by f g. For a test function ' we de ne the functional W () =
(2.14)
Z Z
R0
1
't + h(; r)'x dxdt + ?
Z
R(x; 0)'(x; 0) dx:
By the front tracking construction, ; r is a weak solution to the approximate problem ?
?
t + h ; r x = 0:
Let = lim , then we nd Z Z ?
?
?
Z
?
jW ()j = ? 't + h(; r) ? h ; r 'x dxdt + 0 ? 0 '(x; 0) dx
? ? M k ? k1 + h(; r) ? h r ; 1 + 0 ? 0 1 ;
?
where M is a bound on j'j, j'x j and j't j, and as ; r takes values at precisely those points where h = h, we have replaced h by h in the middle term. Since converges to and r to r in L1 , the terms on the right hand side can be made arbitrarily small by making small. Hence is a weak solution to (2.1). 2.4. Stability for the scalar equation. We rst study stability for the scalar auxiliary equation with respect to perturbations in the initial value . If r is twice di erentiable, we can use the classical results of Oleinik [19] and Kruzkov [15] to show that
1
(
; t) ? 2 (; t) 1 01 ? 02 1 ;
(2.15)
where i is the entropy solution of (2.1) with initial data 0i for i = 1; 2. Now, our rst goal is to show that this also holds for the solutions produced by front tracking. To do this, we must rst show that for smooth r, the front tracking method produces the correct entropy solution. Lemma 2.4. Assume that r(x) is uniformly Lipschitz continuous and constant outside some bounded interval. Let = lim!0 . Then we have that Z
(2.16)
0
T
Z
Rj ? kj ' + sign ( ? k) [(h(; r) ? h(k; r)) ' ? j(x; T ) ? kj '(x; T ) dx + R t
Z
x
Z
? (h(k; r)x ) '] dxdt
R
j0 ? kj '(x; 0) dx 0;
for all non-negative test functions ' and all constants k, i.e., satis es the Kruzkov entropy condition.
STABILITY OF A RESONANT SYSTEM
11
Proof. Let the discontinuities of r be located at xi, i = 1; : : : ; N . Since is a weak solution in each interval hxi ; xi+1i, we have that
(2.17) Z
T
0
Z
xi+1
xi
+
Thus
Z
T
0
?
?
?
? k 't + sign ? k h ; r ? h k; r 'x dxdt ?
?
?
?
T
Z
0
(2.18)
?
?' (xi ; t)sign (xi; t) ? k h (xi ; t) ; r (xi) ? h k; r (xi) dt Z xi+1 Z xi+1 ? k '(x; 0) dx 0: (x; T ) ? k '(x; T ) dx + ? 0 xi
xi
Z
?
' (xi+1 ; t)sign (xi+1 ; t) ? k h (xi+1 ; t) ; r (xi+1) ? h k; r (xi+1)
R
?
?
?
? k 't + sign ? k h ; r ? h k; r 'x dxdt ?
Z
(x; T )
R ?
Z
N T X
?
k '(x; T ) dx + ?
Z
R
0 ? k '(x; 0) dx
?
?
?
'i+1 sign i+1;? ? k h i+1;? ; ri+1;? ? h k; ri+1;?
0 i=1
?
?
?'i sign i; + ? k h i; + ; ri; + ? h k; ri; +
dt 0
where ri; = limy!xi r (y), similarly for i; , and 'i = ' (xi ; t). Consider now the last term in (2.18), since r is smooth, we can de ne ri; + = r (xi ) =: ri: Furthermore, by the solution of the Riemann problem, ? ? h i+1;? ; ri = h i+1;+ ; ri+1 =: hi: Let now
P = xj max i? ; i+ > =2 ; and S = xj max i? ; i+ =2 :
Then we write the sum under the integral in (2.18) as X ? ? (2.19) 'i hi ? h (k; ri?1) sign i; ? ? k ? sign i; + ? k (2.20) (2.21)
S
?
?
+ 'i sign i; + ? k [h (k; ri) ? h (k; ri?1)] X ? ? + 'i hi ? h (k; ri) sign i; ? ? k ? sign i; + ? k P
+ 'i sign i; ? ? k [h (k; ri) ? h (k; ri?1)] (2.22) As ! 0, clearly the second and fourth terms, (2.20) and (2.22) tend to Z
R' sign( ? k) h(k; r) dx: x
Regarding the rst and the third term, (2.19) and (2.21), it is a simple exercise to verify that solution of the Riemann problem implies that (2.19) and (2.22) are non-negative. Hence the limit satis es the Kruzkov entropy condition. Thus, for smooth r, the front tracking construction will give a limit which is L1 contractive, i.e., (2.15) holds. We now show that also if r is not continuous, there is a weak solution to (2.1) such that (2.15) holds .
12
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
For later use we now write the entropy inequality satis ed by as Z
0
(2.23)
T
Z
R ?
?
?
?
? k 't + sign ? k h ; r ? h k; r 'x dxdt
Z
X
T
0
+
S
X
P
?
?
'i sign i; + ? k [h (k; ri) ? h (k; ri?1)] 'i sign i; ? ? k [h (k; ri) ? h (k; ri?1)] dt
?
Z
(x; T )
R
? k '(x; T ) dx +
Z
R
0 ? k '(x; 0) dx 0:
If r is discontinuous, let be entropy solutions to (2.24) (i" )t + h (i" ; r")x = 0; with initial data i;0 , and r" = r !" for a standard molli er !". Using the functional F we can now show that Theorem 2.2. For i = 1; 2, let (i;0 ; r) be as in Theorem 2.1, and let i" be de ned by (2.24). Then, for any sequence f"g such that " ! 0, there is a subsequence "j such "j ! i in L1loc as " i
j ! 1. Furthermore i are weak solutions to (i )t + h (i ; r)x = 0; i (x; 0) = i;0 (x):
Proof. We rst show the convergence with respect to ", u" = r (e ps1 ; r"). By Lemma 2.2 and (2.11) we have that T:V: (u" (; t)) 4T:V: (u(; 0)) : Hence the total variation of u is bounded independently of ". Also u" satis es the bounds
ku"k1 M ku"(; s) ? u"(; t)k1 M jt ? sj ku"(; t)k1 M; for some constant M independent of ". So we see that u" satis es all the requirements of using
the classical technique of Helly's theorem and a further diagonalization to show that there exists a convergent subsequence. Hence " converges in L1loc to some function . We also have that Z Z
jW ()j = ( ? " ) 't + (h(; r) ? h (r" ; " )) 'x dxdt M (k ? " k1 + kh(; r) ? h (r" ; " )k1 ) :
Since " tends to in L1, the terms on the right hand side can be made arbitrary small, and is a weak solution to (2.1). Letting i" denote the weak solutions of (2.24) with initial data 0i , we have k1 (; t) ? 2 (; t)k1 = "lim k" (; t) ? 2" (; t)k1 k1;0 ? 2;0 k1 ; !0 1 which is (2.15). Note that we do not know whether the weak solution constructed directly by front tracking; f , is equal to . To show this we must show stability with respect to variations in the coecient r. Now we show stability with respect also to the r variable, this is done by a \doubling of the variables" approach similar to Kuznetsov [16] original approach, see also [22]. Before stating the lemma, we remind the reader that for r twice di erentiable, and T:V: (r0 ) bounded, there exists an entropy solution of (2.1) of bounded variation in any bounded time interval. This follows by showing that the approximate solution generated by the Lax-Friedrichs scheme are of bounded variation, and that they satisfy an approximate entropy inequality, see e.g., [19] or [12].
STABILITY OF A RESONANT SYSTEM
13
Lemma 2.5. Let and ^ satisfy Z TZ j ? kj 't+ sign( ? k) [(h(; r) ? h(k; r)) 'x ? (h(k; r)x) '] dxdt
R
0
(2.25)
Z
T
0
(2.26)
Z ^
R
Z
? j(x; T ) ? kj '(x; T ) dx +
R
? k 't+ sign ^ ? k ?
h
Z ^ (x; T )
R
Z
R
j0 ? kj '(x; 0) dx 0; i
h(^; r^) ? h(k; r^) 'x ? (h(k; r^)x ) ' dxdt
?
Z k '(x; T ) dx + ^0 R
? k '(x; 0) dx 0;
for all k and all non-negative test functions '. Assume that either T:V: ((; t)) or constants ^ T:V: (; t) is uniformly bounded for t T , and that either r or r^ are Lipschitz continuous, and that both T:V: (r0 ) ; T:V: (^r0) are bounded. Then
(2.27)
(; T ) ? ^(; T ) M 0 ? ^0 + kr ? r^k1 + T:V: (r ? r^) : 1
1
n
o
for some nite constant M depending on min T:V:x () ; T:V:x ^ and min fkr0 k1 ; kr^0k1 g, for t 2 [0; T0]. Proof. Let !" be the usual approximate function, with j!"j M! =", and set F1(; r; k) = sign ( ? k) (h(; r) ? h(k; r)) ; F2(; r; k) = sign( ? k) h(k; r)x:
Then (2.25) and (2.26) read ZZ
(2.28)
j ? kj 't + F1(; r; k)'x ? F2(; r; k)' dxdt Z
ZZ ^
(2.29)
Z
+ j0 ? kj dx ? j(T ) ? kj ' dx 0
?
k 't + F1(^; r^; k)'x Z + ^0 k dy
? F2 (^; r^; k)' dyds
?
?
Z ^ (T )
? k ' dy 0:
By using the test function '(x; y; s; t) = !" ((x ? y))!" ((t ? s)) and setting k = ^(y; s) in (2.28) and k = (x; t) in (2.29), then integrating (2.28) with respect to y and s and (2.29) with respect to x and t, we nd that Z TZ Z TZ 0 (x; t) ? ^(y; s) 't
R0 R
0
(2.30)
+ (2.31) 0
Z
0
+ F1 (x; t); r(x); ^(y; s) 'x ? F2 (x; t); r(x); ^(y; s) ' dxdt dyds
T
Z Z
T
Z
T
0
Z (x; t)
R0 R
Z
" 0 ; ^(; s) )!" (s) ds ?
T
0
" (; T ); ^(; s) )!" (s) ds
? ^(y; s) 's
+ F1 ^(y; s); r^(y); (x; t) 'y ? F2 ^(y; s); r^(y); (x; t) ' dxdt dyds +
where " is de ned by
Z
0
T
" ^0 ; (; t) )!" (t) dt ?
" (w; z ) =
Z Z
Z
0
T
RR! (x ? y) jw(x) ? z(y)j dxdy: "
" ^(; T ); (; t) )!" (s) dt;
14
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
Using that 'x = ?'y and 't = ?'s , and adding (2.30) and (2.31), we obtain Z
T
0
" (; T ); ^(; s) !"(T ? s) ds +
Z
T
Z Z
T
Z n
Z
T
Z Z
Z n
Z
T
T
0
" ^(; T ); (; t) !" (t ? T ) dt
F1 (x; t); r(x); ^(y; s) ? F1 ^(y; s); r^(y); (x; t)
R0 R ? F2 0 R0 R
0
Z
T
o
'x dxdt dyds
(x; t); r(x); ^(y; s) ? F2 ^(y; s); r^(y); (x; t) Z
T
o
' dxdt dyds
+ " 0 ; ^(; s) ds + " ^0 ; (; t) dt: 0 0 We write this equation as (2.32) L1 (") + L2(") R1 (") + R2(") + R3 (") + R4("): Now standard arguments, see e.g., [15] imply that
lim [ L ( " ) + L ( " )] =
(; T ) ? ^(; T ) 1 2 "!0 1
(2.33) lim [R3(") + R4 (")] =
0 ? ^0
: !0 1 0 Regarding R2, assuming that r is bounded, we have that "
R2(")
Z
T
Z Z
Z
jhr ((x; t); r^(y)) (^r0 (y) ? r0 (x))j '
R0 R
0
M1
T
Z
Z Z
T
+ hr hr (x; t); r^(y) ? ^(y; s); r(x) Z n
T
R R
jr0(x)j ' dxdt dyds
o
jr^0 (y) ? r0 (x)j + ^(y; s) ? (x; t) + jr(x) ? r^(y)j ' dxdt dyds;
0 0 1 where M depends linearly on kr0 k1 but not on " or on kr^k1 . Now Z
Z
0
T
Z Z
T
Z
"!0 jr(x) ? r^(y)j ' dxdt dyds ??? !T
Z
R0 R Rjr(x) ? r^(x)j dx !! 0 T jr0 (x) ? r^0(x)j dx jr0 (x) ? r^0 (y)j ' dxdt dyds ??? 0 R0 R R !! 0 (x; t) ? ^(y; s) ' dxdt dyds ??? (x; t) ? ^(x; t) dxdt: R0 R 0 R
Z
(2.34)
T
0
T
Z Z
Z Z
T
T
Z
Z
"
Z
This provides the bound
"
Z
Z
T
lim jR2(")j M 1 T (kr ? r^k1 + T:V: (r ? r^)) + (2.35) "!0 0 Now we estimate R1. Let
Z
T
(
; t)
; t) ? ^(
1
!
dt :
Q(x; t; y; s) = sign (x; t) ? ^(y; s) h
i
(h((x; t); r(x)) ? h((x; t); r^(y))) ? (h(^(y; s); r(x)) ? h(^(y; s); r^(y))) ;
Then Observe that
R1(") = ZZZZ
ZZZZ
Q(x; t; y; s)'x dxdt dyds:
Q(y; s; y; s)'x dxdt dyds = 0:
Hence we can write R1 as ZZZZ R1(") = fQ(x; t; y; s) ? Q(y; s; y; s)g 'x dxdt dyds:
STABILITY OF A RESONANT SYSTEM
15
Estimating the di erence Q(x; t; y; s) ? Q(y; s; y; s), we nd that
jQ(x; t; y; s) ? Q(y; s; y; s)j M jr(x) ? r^(y)j j(x; t) ? (y; s)j + jr(x) ? r(y)j (y; s) ? ^(y; s) for some constant M depending on h. Consequently
jR1 (")j M
(Z
T
Z Z
T
Z
T
Z Z
T
Z
0
+
Z
jr(x) ? r^(y)j j(x; t) ? (y; s)j j!" 0(x ? y)j !" (t ? s) dxdt dyds
R0 R 0 R0 Rjr(x) ? r(y)j (y; s) ? ^(y; s) j! (x ? y)j ! (t ? s) dxdt dyds
0
"
)
"
=: I1 (") + I2 ("): Now I1 (") M
M
ZZZZ ZZZZ
(jr^(y) ? r(y)j + jr(y) ? r(x)j) j(x; t) ? (y; s)j j!" 0 (x ? y)j !"(t ? s) dxdt dyds
jr(y) ? r^(y)j (j(x; t) ? (y; t)j + j(y; t) ? (y; s)j) j!" 0(x ? y)j !" (t ? s) dxdt dyds
1 2Z
+ M "2M!
+
Z Z
T
t "
R?
0
+
Z
x "
j(x; t) ? (y; s)j dyds dxdt:
?
t "
x "
Since is bounded and measurable, the second term above tends to zero as " becomes small. The integral in the rst term is bounded by
kr ? r^k1
Z
0
T
Z
0
T
!" (t ? s)
Z Z
"
Z
Z
kr ? r^k1 kr ? r^k1
0
RRj(x; t) ? (y; t)j j! (x+? y)j dxdy + j(y; t) ? (y; s)j j! 0 (x ? y)j dx dy ds dt R ? 0 ! (t ? s) Rj! (z)j Rj(x; t) ? (x ? z; t)j dx dz 0 0
Z
T
Z
y "
Z
T
y "
"
Z
"
"
+ 2M" ! k(; t) ? (; s)k1 ds dt
Z
T
?
0
M 2 T:V: ((; t)) + M 3 dt
R
where the constant M 2 depends on z!" 0(z )dz and M 3 depends linearly on T:V: (). Hence lim I1 (") M 4 T kr ? r^k1 ; "!0
for some constant M 4 depending linearly on T:V:x (). We estimate I2 by using the Lipschitz continuity of r. ZZZZ 0 1 lim I2 (") M lim (y; s) ? ^(y; s) jx ? yj j!" (x ? y)j !" (t ? s) dxdt dyds !0
"
!0
"
M 1 "lim !0 M 1M 2 Consequently, (2.36)
ZZ (y; s)
Z
T
0
(
Z
? ^(y; s) jx ? yj j!" 0 (x ? y)j !"(t ? s) dx dy ds
; t) ? ^(; t)
1 dt: Z
lim jR1(")j M 5 T kr ? r^k1 + "!0 0
T
(
; s)
; t) ? ^(
1
!
dt ;
16
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
for some constant M 5 depending linearly on kr0k1 and T:V: (). Collecting the bounds on the terms in (2.32), and using the relation kr ? r^k1 T:V: (r ? r^), we nd that (2.37) ! Z T
6 ^ ^ ^ T (kr ? r^k1 + T:V: (r ? r^)) +
(; t) ? (; s) dt
(; T ) ? (; T ) 0 ? 0 + M 1
1
0
1
for some constant M 6 depending linearly on kr0k1 and T:V: (). The lemma then follows by the Gronwall inequality. Note that for r = r^, this proof will yield Kruzkov's stability result (2.15). Remark. By itself, this lemma is not quite enough to show that the weak solution generated by front tracking; f , is the same weak solution obtained by letting " ! 0 in (2.24). The next section is devoted to obtaining the necessary tools for showing this. 2.5. An estimate of the convergence rate if r is smooth. We can actually extract more information from the entropy inequality for the approximate solutions , (2.18). To this end, we need some technical results. For a function u(x; t), assume that there exist continuous functions x and t R+ 7! R+, with x;t(0) = 0, such that sup
jyj"
(2.38)
sup
jsj
Z
Rju(x + y; t) ? u(x; t)j dx (") Rju(x; t + s) ? u(x; t)j dx ( ) x
Z
t
If we can nd such functions, we say that u has moduli of continuity in x and t. Note that if u is of bounded variation in x, then x can be chosen as C". The next proposition states that functions (; r) that have the property that r (; r) is of bounded variation, have moduli of continuity. Proposition 2.1. Let : R 7! [?; ] and r 7! [a; b] where 0 < a < b. Assume that r (; r)
has bounded variation. Furthermore assume that both r and are constant outside some bounded interval I . Then has a module of continuity x which can be chosen as C"1=2 for some constant C depending on h, a, b, T:V: (r (; r)) and I . Proof. Without loss of generality, we can assume that both and r are right continuous. Let " > 0, and set xi = i" for i 2 Z. Set i = (xi +), and ri = r(xi+). De ne the sets
S" = xi ji =2j p" and ji?1 =2j p" and B" = xi xi 62 S" : p Note that for j =2j > p", j@P r j C1 " for some constant C1 depending on h and a. Since and r are constant outside I , S" 1 = O (1="). Therefore, X ?p 2 p X C2 = " " ji ? i?1 j :
S"
S"
Furthermore since T:V: (r (; r)) is nite X (r ) ? (r ) i i?1 C3 ; B"
for some constant C1 not depending on ". Combining the above we obtain p" X j ? j = p" X j ? j + p" X ji ? i?1 j i i?1 i i?1 i
B"
X C4 (r )
C5 :
B"
i
S"
? (r )i?1 + C2
Since " is arbitrary, this implies the conclusion of the proposition.
STABILITY OF A RESONANT SYSTEM
17
Lemma 2.6 (Kruzkov's interpolation lemma). Let z(x; t) be a bounded measurable function de ned in T = R [0; T ]. For t 2 [0; T ] assume that z possesses a spatial modulus of continuity x that does not depend on t. Suppose that for any 2 C01(R) and any t1,t2 2 [0; T ], Z (z (x; t2) ? z (x; t1)) (x) dx ConstT k 0 k jt2 ? t1 j : (2.39) 1 R Then for t and t + 2 [0; T ] and all " > 0 Z (2.40) jz (x; t + ) ? z (x; t)j dx ConstT j " j + x(") :
R
For a proof of this lemma, see [11]. Now it is straightforward to show that satis es (2.39). Hence we can use the lemma and Proposition 2.1 to conclude that possesses a modulus of continuity t which can be chosen as t( ) = C 1=3: Consequently both the approximate solutions and the limit possess moduli of continuity (2.41) x (") = C"1=2; t( ) = C 1=3; where the constant C does not depend on t for t T . Let !" (x) be a standard molli er as before, and de ne a test function '(x; y; t; s) by '(x; y; t; s) = !" (x ? y)!" (t ? s): For a function u = u(x; t) let " (u; c) be de ned as (2.42) ZZ " (u; k) = ? ju ? kj 't + sign (u ? c) (h(u; r) ? h(k; r))'x ? sign(u ? c) h(k; r)x' dxdt T
Z
+ ju ? kj ' tt==0T dx:
For two functions u and v we de ne the functional " (u; v) as (2.43)
" (u; v) =
ZZ
T
R
" (u; v(y; s)) dyds;
where u = u(x; t) and v = v(y; s). In passing, we note that if u is an entropy solution of (2.1), then (2.44) " (u; v) 0: For two arbitrary functions u and v we have the following result: Lemma 2.7 (Kuznetsov's lemma). Assume that that r is in C (R) and r0 is in L1 (R) and that both r and r0 have moduli of continuity. If u and v are in L1 (T ) and have moduli of continuity in space and time, then ku(; T ) ? v(; T )k1 ku(; 0) ? v(; 0)k1 + " (u; v) + "(v; u)
(2.45)
+ 21 [x (u(; 0); ") + x (v(; 0); ") + x (u(; T ); ") + x (v(; T ); ")] + 21 [t(u(; T ); ") + t(v(; T ); ") + t(u(; 0); ") + t(v(; 0); ")] + kr0k1 khr k1 T sup (x (u(; t); ") + t (u(; t); "))
0tT 0 + CT (kr k1 x(r; ") + x (r0; ")) ;
where x (; ) and t(; ) denote moduli of continuity. The constant C depends on hr and hrr .
18
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
For a proof of this lemma, see Karlsen and Risebro [11]. Applying Kuznetsov's lemma with
u = and v = and using (2.41), we nd
1=2 1=3 :
(; T ) ? (; T ) 0 ? + "( ; ) + C kr0k (2.46) 0 1 1 "+" +" 1 Here we tacitly assume that kr0 k1 > 0, as this will be the case for our primary application of the results following from this. It remains to estimate "( ; ). Let ~ be de ned by the negative of (2.23), then ~ 0, and " ( ; k) = "( ; k) ? ~ + ~ " ( ; k) ? ~ : Now Z Z ? ~ h ( ; r ) ? h( ; r) + h (k; r ) ? h(k; r) j'x j dxdt ( ; k ) ? " T
+
Z
XZ
T
0
i
xi+1
xi
j'(xi ; t) ? '(x; t)j jh(k; r)x j dx dt:
Regarding the rst term on the right, we have that h ( ; r ) = h( ; r ), hence h ( ; r ) ? h( ; r) C; (2.47) since r ? r . Also h (k; r ) ? h(k; r) h (k; r ) ? h (k; r) + h (k; r) ? h(k; r) C (2.48) by the construction of h . To estimate the second term, we must specify how the points xi are chosen. By assumption, r is constant outside a nite interval I = ha; bi. Let x0 = a, if necessary modify such that r(a) = j for some j 2 Z. Then de ne n o (2.49) xi+1 = inf x > xi j jr(x) ? r(xi )j) : If this in mum does not exist, then xi+1 = b. Since r and r0 are of bounded variation, for suciently small , r is monotone in each interval hxi; xi+1i. Then Z
(2.50)
xi+1
xi
j'(xi ; t) ? '(x; t)j jh(k; r)x j dx C
Z
xi+1
Z
xi
x
xi
j'z (z; t)j dz jr0(x)j dx
C (xi+1 ? xi)
Combining (2.47), (2.48) and (2.50) we then nd " ( ; k) C
Z
0
TZ
R+
X i
(xi+1 ? xi )
Z
xi+1
xi
Z
xi+1
xi
j'x (x; t)j dx
j'x (x; t)j dx dt
Recall that we have de ned our approximation such that p (x) ? 0 (x) 0 for all x. Since 0 has compact support, this implies that
0 ? 0 1 C
1=2:
Finally, using our special choice of test function, we nd that (2.51)
; T ) ? (; T ) 1 +C 1=2 + " + kr0 k1 " + "1=2 + "1=3 ;
(
where C depends also on T . Minimizing with respect to " gives
(; T ) ? (; T ) C kr0k3 (2.52) 1 1 To conclude this section, we state our results:
1=4
STABILITY OF A RESONANT SYSTEM
19
Theorem 2.3. Let 0 and r be functions such that the assumptions of Theorem 2.1 hold. Fur-
thermore assume that r is continuous and r0 uniformly bounded, and that 0 and r0 have bounded support. Let denote the entropy solution to t + h(; r)x = 0; (x; 0) = 0 (x); and let denote the front tracking approximation to . Then the bound (2.52) holds, where the constant C only depends on h, T and the support of 0 and r0. 2.6. Entropy solutions with discontinuous r. Let now " denote the solution to (2.24). Since r is bounded we have that k(r" )0 k1 C" ; for some constant C . Furthermore we have established that there is a subsequence of f"g; f"j g
such that
"
j
? 0 1 ! 0;
where 0 is a weak solution of (2.53). If ;" denotes the front tracking approximation to " then the previous theorem; Theorem 2.3 says that
;"
1=4 ? 1 C "3 :
"
Let now denote the front tracking approximation to (2.53) t + h(; r)x = 0; with r bounded but possibly discontinuous. Note that while we must de ne r;" according to (2.49), however we can de ne r to be any function taking values in the set fj gj 2 , such that
lim r ? r 1 = 0: (2.54) !0
Z
Now set = "4 , by choosing subsequences if necessary, we have that
? f ! 0; and " ? 0 ! 0; 1 1 ;" 1 = 4 as (") ! 0. Now de ne r as r , when " = this choice clearly satis es (2.54). Since the initial data coincide, also ;" = , thus
f
? 0 f ? + ? ;" + ;" ? " + " ? 0 1 1 1 1 1
f ? 1 + C "1=4 + " ? 0 1 : Now let " ! 0 and conclude that f = 0 . In other words, also the solution obtained by front tracking converges to the solution obtained by smoothing the coecients and letting the smoothing radius tend to zero. Furthermore by (2.15) this limit is unique. We therefore say that a weak solution to (2.1) is andn entropy solution if it is the L1 limit as " ! 0 of Kruzkov entropy solutions to (2.55) t" + h (" ; r !" )x = 0: Summing up, we have shown: Theorem 2.4. Let (0 ; r) be such that u0 = rc (0 ; r) is in B:V: \ L1 . Then, there exists a unique entropy solution to
t + h(; r)x = 0; (x; 0) = 0 (x): This entropy solution can be constructed by front tracking. Furthermore, if ^ is another entropy solution to the above equation, but with initial data ^0 , then
(
; t) ? ^(; t)
1
0 ? ^0
1 :
20
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
Note that for the speci c equation as (2.1), using the methods in [13], we can actually prove a stronger stability estimate
(2.56)
(; t) ? ^(; t) 0 ? ^0 + CtT:V: (r ? r^) 1
1
for some constant C depending only on r, r^, h and the initial data. This estimate is possible only for those systems on the form (2.1) where the transition curves are not a function of r. In the proof of Theorem 2.4, we have not used this, and our arguments relies only on the fact that the solution of the Riemann problem can transformed into the diagrams in gure 2. While not attempting to describe precisely which scalar conservation laws that have this property, we will display one such equation in the next section. 3. The polymer system In this section we shall show analogous results for the polymer system (1.10), (1.11), as we did in the previous section for the auxiliary system (2.1). To do this we adapt Wagner's general results [23], to the polymer case. For technical reasons, we assume that the relative permeability of the water satis es w (s) = 0; for all s for some constant < 1. This assumption means that a certain amount of the water is bound to the rock. We begin by translating the stability results obtained for smooth coecients in the previous section, to the corresponding results for the polymer system. 3.1. Stability estimates for smooth c. Let now g = f=s. In [22], Tveito and Winther de ned an entropy solution to (1.10), (1.11) as follows: A pair of functions (s; c) is called an entropy solution to (1.10), (1.11) in R [0; T ] if: De nition 3.1. 1. (s; c) take values in [ ; 1] [0; 1], (s; c) is L1 (dx)-Lipschitz continuous in t, s(; t) and cx (; t) are Lipschitz continuous and of bounded variation, and c is Lipschitz continuous.
2. For all non-negative test functions ', all constants q 2 [ ; 1] if Z
(3.1)
0
T
Z
Rjs ? qj ' + sign (s ? q) (f (s; c) ? f (q; c)) ' ? sign (s ? q) f (q; c) ' dxdt ? js(x; T ) ? qj '(x; T ) dx + js(x; 0) ? qj '(x; 0) dx 0: R R t
x
Z
x
Z
3. For almost all (x; t) in R [0; T ], c satis es (3.2) ct + g(s; c)cx = 0: By Theorem 2 in [23], there is a one-to-one correspondence between weak solutions of (1.10), (1.11), satisfying 0 < s 1, and weak solutions of 1 ? f (s; c) = 0 s s (3.3) y c = 0: The new coordinates (y; ) are given by Then
@y = s; @x
@y = ?f (s; c); @t
@ = 0; and @ = 1: @x @t
@ = @ +f @ ; @ =1 @ : @ @t s @x @y s @x
STABILITY OF A RESONANT SYSTEM
21
For smooth c, a Kruzkov entropy solution of (3.3) satis es (3.4) Z TZ 1 f (s; c) 1;c ' ? k ' ? sign 1 ? k ? kf y s s s k 0 + sign 1s ? k kf k1 ; c ' dy d
R
?
Z 1 s(T )
R
y
?
Z k '(T ) dy +
1 s
R0?
k '(0) dy
0
for all constants k 2 [1; 1= ], and all non-negative test functions '. Changing variables to (x; t) we nd that the rst integral above equals Z T Z ( 1 ? k ' + f ' ? sign 1 ? k f (s; c) ? kf 1 ; c 1 '
Rs
0
=
=k
Z
T
0 Z
0
T
Similarly
Z
1 k k
R
Z s
R
?
?
t
s
x
s
1 s 't + sign k
1 ' + sign s k x
Also, if c(y) is smooth, then
?s
? k1
Z 1 s
R
s
k
s
x
)
1 1 + sign s ? k kf k ; c 1s ' s dxdt
x
1 ? k f ? f + kf 1 ; c ' x s s k + sign 1 ? s kf 1 ; c ' dxdt k
f (s; c) ? f
? k ' dy = k
k
x
1 ; c ' ? sign s ? 1 f 1 ; c ' dxdt: x k k k
x
Z s
R
? k1 ' dx:
f (s; c) cx = 0 ct + s
almost everywhere. Consequently, entropy solutions of (1.10), (1.11) in the sense of De nition 3.1, are equivalent to Kruzkov entropy solutions of (3.3). Furthermore, from the stability estimate (2.15), we nd Z Z js(; t) ? s^(; t)j dx js(; t) ? s^(; t)j s(1; t) dx Z 1 1 = s(; ) ? s^(; ) dy Z 1 1 (3.5) s ? s^ dy 0 0 Z 1 = js^ ? s j dx
R
R R R s0 0 0 R 1 js^0 ? s0 j dx R Z
where s and s^ are entropy solutions of (3.3) with initial data s0 and s^0 respectively. It is also now straightforward to translate the stability estimate (2.27) into a corresponding estimate for the polymer system (ks(; t) ? s^(; t)k1 + kc(; t) ? c^(; t)k1 + kcx(; t) ? c^x (; t)k1 ) ? (3.6) M ks0 ? s^0 k1 + kc0 ? c^0k1 + kc0;x ? c^0;x k1 where (s; c) and (^s; c^) are entropy solutions of (1.10), (1.11), with initial data (s0 ; c0) and (^s0 ; c^0). Collecting the bounds obtained in this section, we have:
22
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
Theorem 3.1. There exists a unique entropy solution, in the sense of De nition 3.1 to (1.10),
(1.11). Furthermore two entropy solutions with the same initial c data, satis es (3.5), and two entropy solutions with di erent s and c initial data satis es (3.6). 3.2. Stability of entropy solutions for general c. Let now 1 1 ; and h( ; q) = f 1 ; c(q) : = ; q= s (c) We wish to use front tracking to construct approximate solutions to + h( ; q)y = 0; q = 0 (3.7) (y; 0) = 0 (y); where 0 is in the interval [1; 1= ] and q > 0. Below we have shown the function h as a function of for two di erent r values. Using the solution of the Riemann problem in [4], it is now a matter The function h(θ,r) as a function of θ for two r values 2
1.5
1
h(θ,r)
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
2
2.5
3
ln(θ)
Figure 3. The function h( ; q) as a function of ln( ).
of routine to verify that there exist smooth mappings ( ; q) and r(q), such that the solution of the Riemann problem for (3.7) is given by gure 2. Using this Riemann solution, we can repeat the arguments in section 2 to show that there exists a weak solution for any initial data 0 , and r such that F (( 0 ; r); q(r)) is bounded. In particular, this includes the case where 0 and r are of bounded variation. This means that also the polymer system (1.10), (1.11), has a weak solution for s0 and c0 in BV \ L1 . We can also mimic the de nition of entropy solutions in the case where c0 (x) is discontinuous. Let (s" ; c") be entropy solutions, in the sense of De nition 3.1, to the initial value problem s"t + f (s" ; c" )x = 0; (s" c" )t + (c" f (s" ; c"))x = 0 (3.8) s" (x; 0) = s0 (x); c"(x; 0) = c0 !"(x) where !" is a standard molli er. By using the arguments in the previous section, we nd that f(s" ; c")g converges in L1loc to a unique limit as " ! 0, and that this limit is a weak solution to (1.10), (1.11). Furthermore, this unique limit can be constructed by using front tracking on the corresponding problem (3.7). We de ne this unique limit as an entropy solution to (1.10), (1.11). Hence we have shown Theorem 3.2. Let f (s; c) be as in section 1.1, with f (s; c) = 0 for all s and all c 2 [0; 1]. Assume that s0 and c0 are in BV \ L1 and take values in [ ; 1] and [0; 1] respectively. Then there exists a unique entropy solution to the initial value problem st + f (s; c)x = 0; (sc)t + (cf (s; c))x = 0 s(x; 0) = s0 (x); c(x; 0) = c0(x):
STABILITY OF A RESONANT SYSTEM
23
If s^ is another entropy solution with initial value s^0 , but with the same initial c value, then
ks(; t) ? s^(; t)k1 1 ks0 ? s^0 k1 :
Remark. The above uniqueness results enable us to give an independent proof of the equivalence
between weak (entropy) solutions of (1.10), (1.11) and (3.3). To prove such a result we can do the following: First show that front tracking converge to the unique entropy solutions of (1.10), (1.11) and (3.3) respectively. Then we can use the fact that the front tracking approximations are weak solutions to approximate problems, and the fact that the Rankine-Hugoniot condition for the front tracking approximations to (1.10), (1.11) and (3.3) are equivalent. This means that under the coordinate transformation x 7! y, where ? @y @y = s ; = ? f s ; c ; @x @t
where s , c and f are the front tracking approximations and the approximate ux function respectively, front tracking approximations to (1.10), (1.11) are mapped to front tracking approximations to (3.3). Since front tracking converges to unique entropy solutions, these must also be equivalent. Remark. The front tracking approximation can also yield an existence result without the restriction f (s; c) = 0 for s . This follows by using the functional F and the arguments in [21] or [5]. [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
References P. Baiti and H. K. Jenssen. Well-posedness for a class of conservation laws with l1 data. J. Di . Eqn., 140:161{185, 1997. A. Bressan and R. Colombo. The semigroup generated by 2 2 systems of conservation laws. Arch. Rat. Mech., 133:1{75, 1995. O. Dahl, T. Johansen, A. Tveito, and R. Winther. Multicomponentchromatographyin a two phase enviroment. SIAM J. Appl. Math., 52:65{104, 1992. T. Gimse and N. H. Risebro. Riemann problems with discontinuous ux functions. In Third international conference on hyperbolic problems, theory numrical methods and applications, pages 488{452. Studentlitteratur/Chartwell-Bratt, 1991. T. Gimse and N. H. Risebro. Solution of the Cauchy problem for a conservation law with a discontinuous ux function. SIAM J. Math. Anal, 23:635{648, 1992. H. Holden, L. Holden, and R. H egh-Krohn. A numerical method for rst order nonlinear scalar conservation laws. Comp. Math. Applic., 15:595{602, 1988. E. Isaacson. Global solution of a Riemann problem for a non-strictly hyperbolic system of conservation laws arising in enahanced oil recovery. Preprint, Rockefeller University, New York, 1981. E. Isaacson and B. Temple. Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math., 52:1260{1278, 1992. E. Isaacson and B. Temple. Convergence of the 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math., 55:625{640, 1995. H. K. Jenssen. private communication. K. Karlsen and N. H. Risebro. Viscous and inviscid conservation laws with rough coecients. work in preparation, to appear at http://www.math.ntnu.no/conservation. R. A. Klausen and N. H. Risebro. Well posedness of a 2 2 system of resonant conservation laws. In Proc. of the 8th international conference on hyperbolic problems, Zurich, 1998. R. A. Klausen and N. H. Risebro. Stability of conservation laws with discontinuous coecients. J. Di . Eqn., 157:41{60, 1999. C. Klingenberg and N. H. Risebro. Convex conservation laws with discontinuous coecients, existence, uniqueness and asymtotic behavior. Comm. PDE., 20:1959{1990, 1995. S. N. Kruzkov. First order quasilinear equation in several independent variables. Mat. Sbornik, 10:217{243, 1970. N. N. Kuznetsov. Accuracy of some approximate methods for computing the weak solutions of a rst order quasilinear equation. USSR Comput. Math. and Math. Phys., 16:105{119, 1976. L. Longwai, B. Temple, and W. Jinghua. Suppression of oscillations in Godunov's method for a resonant non-strictly hyperbolic system. SIAM J. Numer. Anal, 32:841{864, 1995. L. Longwai, B. Temple, and J. Wang. Uniqueness and continuous dependence for a resonant non-strictly hyperbolic system. unpublished note.
24
CHRISTIAN KLINGENBERG AND NILS HENRIK RISEBRO
[19] O. A. Oleinik. Discontinuous solutions of nonlinear di erential equations. Amer. Math. Soc. Trans., 26:95{172, 1963. [20] J. Smoller. Shock waves and reaction-di usion equations. Springer, New York, 2 edition, 1995. [21] B. Temple. Global solution of the Caucy problem for a 2 2 non-strictly hyperbolic system of conservation laws. Adv. Appl. Math., 3:335{375, 1982. [22] A. Tveito and R. Winther. Existence, uniqueness and continuous dependence for a system of conservation laws modeling polymer ooding. SIAM J. Math. Anal, 22:905{933, 1991. [23] D. Wagner. Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions. J. Di . Eqn., 68:118{136, 1987. (Klingenberg), University of Wurzburg, Department of Applied Mathematics and Statistics, Am Hubland, D{97074 Wurzburg, Germany
E-mail address :
[email protected]
(Risebro), Department of Mathematics, University of Oslo, P.O. Box 1053,Blindern, N{0316 Oslo, Norway
E-mail address :
[email protected]
