On Some Recent Methods for Nonlinear Partial Differential Equations

On Some Recent Methods for Nonlinear Partial Differential Equations P I E R R E - L O U I S LIONS

CEREMADE, Université Paris-Dauphine, Place de Lattre de Tassigny, F-75775 Paris Cedex 16, France

Dedicated to the memory of Ron DiPerna (1947-1989) 1 Introduction We wish to present here some aspects of a few general methods that have been introduced recently in order to solve nonlinear partial differential equations and related problems in nonlinear analysis. As is well known, nonlinear partial differential equations have become a rather vast subject with a long history of deep and fruitful connections with many other areas of mathematics and various sciences like physics, mechanics, chemistry, engineering sciences, etc. And we shall not pretend to make any attempt at surveying all recent activities in that field. Also, we shall concentrate on rather theoretical issues leaving completely aside more applied issues such as mathematical modelling, numerical questions that go hand in hand in a fundamental way with the theories. For a discussion of the interaction between nonlinear analysis and modern applied mathematics, we refer the reader to the report by Majda [56] in the preceding Congress. We shall mainly discuss here recent methods that have been developed recently for the analysis of the major mathematical models of gas dynamics (and compressible fluid mechanics), namely the Boltzmann equation and compressible Euler and Navier-Stokes equations (essentially in the so-called "isentropic regime"). These methods include velocity averaging, regularization by collisions that we shall apply to the solution of the Boltzmann equation (Section 2 below), and compactness via commutators and in particular compensated compactness, which we illustrate on isentropic compressible Euler and Navier-Stokes equations. This selection of topics (equations and methods) is by no means an exhaustive treatment of all the exciting progresses that have taken place recently in nonlinear partial differential equations: many more important problems have been investigated — see for instance the various reports in this Congress related to Nonlinear Partial Differential Equations — and other methods and theories have been developed. We briefly mention a few in Section 4. And even for the methods that we describe here, much more could be said in particular about applications to other relevant problems. Proceedings of the International Congress of Mathematicians, Zürich, Switzerland 1994 © Birkhäuser Verlag, Basel, Switzerland 1995

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We only hope that our selection will serve as a good illustration of recent activities. It will also emphasize some current trends that go far beyond the material discussed here. The first one is the analysis of the qualitative behavior of solutions (regularity, compactness, classification of possible behaviors, etc.). The second one, related to the preceding one, concerns the structure of specific nonlinearities and its interplay with the behavior (or possible behaviors) of solutions. Finally, this requires theories and methods that are connected with many branches of mathematics and analysis in particular. 2 Boltzmann equation 2.1 Existence and compactness results The Boltzmann equation is given by ^

+ v-Vxf

(x,v)eR2N

= Q(fJ)

,t>0

(1)

where the unknown / is a nonnegative function on R2N x [0,oo), N > 2, \7X denotes the gradient with respect to x, and we denote by x • y or (x, y) the scalar product in RN. The nonlinear operator Q can be written as

Q(f,f) Q+(Lf)

= Q+(f,f)-Q-(fJ)

(2)

= [ dvj

(3)

JRN

Q-(fJ)=[dvJ JRN

JSN~1

JS"-1

du,B(v-v.,u)f'fi

dujB(v-v*,u)fft

= fL(f),

L(f) = f*A V

(4)

where /„ = f(x,v*,t), f = f(x,v',t), ft = f(x,v^,t), A(z) = J^^ B(z,uj)dw, and B = B(z,u) is a given nonnegative function of \z\ and |(2,CJ)|, is called the scattering cross-section or the collision kernel, which depends on the physical interactions of the gas particles (or molecules) and v1 — v — (v — V*,UJ)U) ,

v't = v* 4- (v—V*,UJ)UJ .

(5)

A typical example (the so-called hard spheres case) of B is given by: B = \(Z,UJ)\. We always assume that A e Lloc(RN) a n d ( l + | ^ | 2 ) - 1 • L, 0 as \z\ —> oo, for all R E (0,oo). Of course, we wish to solve (1) given an initial condition that is the values of / at t = 0 /|t=o - h in K2iV • (6) The initial value problem (1),(6) is a deceivingly simple-looking first-order partial differential equation with nonlinear (quadratic) nonlocal terms. It is a relevant model for the study of a rarefied gas and is currently used for flights in the upper layers of the atmosphere (Mach 20-24, altitude of 70-120 km). The statistical description of a gas in terms of the evolution of the density / of molecules was

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Pierre-Louis Lions

originally obtained by Boltzmann [6] (see also Maxwell [57], [58]). There is a long history of important mathematical contributions to the study of (1) by Hilbert [31], Carleman [8], [9] etc. Further details on the derivation of (1) and references to earlier mathematical contributions can be found in Grad [28], Cercignani [10], and DiPerna and Lions [18]. The major mathematical difficulty of (1), (6) is the lack of a priori estimates on solutions: only bounds on / in L1 (with weights) and on / log / in L1 are known! Nevertheless, the following result, taken from [18],[20], holds: THEOREM 2.1. Let / 0 > 0 satisfy:

/o(l + \x\2 + \v\2 + I log /o|) dx dv < oo. Then there exists a global weak solution of (1),(6) f G C([0,oo);L 1 (M^)) satisfying sup

/

£G[0,oo) o) JR2N

f(t)(l + \x-vt\2

+ \v\2 + |log f(t)|) dx dv < oo

and the following entropy inequality for [

/(*) log /(*) dxdv+\

JR2JV

where D[f] = j[R2N dvdv*

[ dsf 4 JQ

JSN-I

allt>0

JRN

B du(fft-fft)

dxD[f]

< f

/ 0 log / 0 dx dv

(7)

JR2N

log ££-.

REMARKS 2.1. (i) We do not want to give here the precise definition of global weak solutions as it is a bit too technical. Let us mention that the notion introduced in [18], [20] is modified in Lions [48] (additional properties are imposed on / in [48]).

(ii) Further regularity properties of solutions are an outstanding open problem. It is only known that the regularity of solutions is not "created by the evolution" and has to come from the initial condition /o- It is tempting to think, in view of the results shown in [48] (see sections 2.2, 2.3 below), that, at least in the model case when B = tp(\z\, ^^-) with ip E C£°((0,oo) x (0,1)), / is smooth if / 0 is smooth. Related to the regularity issue is the uniqueness question: uniqueness of weak solutions is not known (it is shown in [48] that any weak solution is equal to a solution with improved bounds assuming that the latter exists!). (iii) The assumption made upon B corresponds to the so-called angular cut-off. (iv) Boundary conditions for Boltzmann's equation can be treated: see Hamdache [29] for an analogue of the above result in that case. Realistic boundary conditions require some new a priori estimates and are treated in Lions [46]. (v) Other kinetic models of physical and mathematical interest can be studied by the methods of proof of Theorem 2.1: see for instance DiPerna and Lions [19], Arkeryd and Cercignani [2], Esteban and Perthame [22], and Lions [48].

On Some Recent Methods for Nonlinear Partial Differential Equations

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The strategy of proof for Theorem 2.1 is a classical one, which is almost always the one followed for the proofs of global existence results: one approximates the problem by a sequence of simpler problems having the same structure (and the same a priori bounds) for which one shows easily the existence of global solutions, and then one tries to pass to the limit. This strategy is also useful for the mathematical analysis of numerical methods because one can view numerical solutions as approximated solutions or solutions of approximated problems. This is wiry the main mathematical problem behind the proof of Theorem 2.2 is the analysis of the behavior of sequences of solutions (we could as well consider approximated solutions... ) and in particular of passages to the limit in the equation. This is a delicate question because the available a priori bounds only yield weak convergences that are not enough to pass to the limit in nonlinear terms, This theme will be recurrent in this[report_(as_it was_already_in Majda's report [56]). We thus consider a sequence of (weak or even smooth) nonnegative solutions fn of (1) corresponding to initial conditions (6) with /b replaced by ffî and we assume sup / ft(l + \x\2 + \v\2 + \]Qgff\)dxdv < oo (8) n>lJR™

sup sup /

/ n ( t ) ( l + | z - ^ | 2 + H 2 + |log fn\)dxdv

< oo

(9)

71>1 t>0 JR™

sup [

dv D[fn]

dt [

n>lJO

< oo .

(10)

JRH

Without loss of generality — extracting subsequences if necessary — we may assume that / £ , / " converge weakly in L ^ R 2 ^ ) , L ^ R 2 ^ x (0,T))(V T E (0,oo)) respectively to fo,f. THEOREM

2.2. We have for all T\J fni/jdv->

/ JRK

n

j

E

Cg°(R^)}

fyjdv

in

L

1

(0, oo) ^

x (0,T)) ,

f Q+(fJ)iPdv,^ n

JRN

I Q-(r,n*pdv^ [ Q-(fj)rpdv \ JRN

(11)

JRK

I Q+(fnJn)^dv^ JRN

T,Re

n

( 12 )

J^N

in measure for \x\ < R , t E (0,T) ; and f is a global weak solution of (1), (6). THEOREM

Q+(f\D

2.3. (1) We have for all R,T E (0,oo) -> Q + ( / , / )

in measure for \x\ 0. As indicated in [47] (see also the recent result of Desvillettes [16]) this fact might be related to the angular cutoff assumption because grazing collisions seem to generate some compactification ("nonlinear hypoelliptic features" in the model studied in [47]). 2.2 Velocity averaging A typical example of the so-called velocity averaging results is the following 2.4. Let m > 0 ; let 9 E [0,1); and let f,g E LP(R% x R f x R t ) with 1 < p < 2. We assume

THEOREM

d

l+v-Vxf ut

= (-Ax,t

+ l)9/2(-Av

+ l)m'2g

in

V'(R2N+1).

(14)

Then, for all VJ E CQ°(RN), JRN f(x,v,t)yj(v) dv belongs to the (Besov) space B*'P(RN x R) — and thus to HS'>P(RN) for all 0 < s' < s — where s = ( 1 -

2.3. (i) If m = 0, JRN fipdv E Hs>p with s = 2=^. The above exponent s is optimal in general (this is shown in a work to appear by the author). Similar results are available if 2 < p < oo or in more general settings: we refer the reader to DiPerna, Lions, and Meyer [21]. REMARKS

(ii) Such velocity averages are known in statistical physics (or mechanics) as macroscopic quantities. The above result shows that transport equations induce some improved partial regularity on velocity averages (by some kind of dispersive effect). (iii) The first results in this direction were obtained in Golse, Perthame, and Sentis [27], Golse, Lions, Perthame, and Sentis [26] (where the case m = 0 is considered). The case m > 0, p = 2, was treated in DiPerna and Lions [19] while the general case is due to DiPerna, Lions, and Meyer [21] — a slight improvement of the Besov space can be found in Bézard [5]. Two related strategies of proof are proposed in [21] that both rely on Fourier analysis, one uses some harmonic analysis, namely product Hardy spaces and interpolation theory, while the second one uses classical multipliers theory and careful Littlewood-Paley dyadic decompositions. However, the main idea is rather elementary and described below in extremely rough terms. As indicated in the preceding remark, we give a caricatural (but accurate!) explanation of the phenomena illustrated by Theorem 2.4. If we Fourier transform

On Some Recent Methods for Nonlinear Partial Differential Equations

145

(14) in (x,t), we see that we gain decay (^regularity) in (f, r) — dual variables of (x,t) — provided \r + v • £| > 6\v • £| for some 6 > 0. On the other hand, the set of v on which we do not gain that regularity, namely {v E Supp (VJ) / \r + v ' f| < ^IC^OI}» n a s a Pleasure of order 6, and hence contributes little to the integral (JRn f(^,v,r)yj(v)dv). Balancing the two contributions, we obtain some (fractional) regularity. Of course, such improved regularity yields local compactness (in (x, t)) of the velocity averages and leads (after some work) to (11). 2.3 Gain terms and Radon transforms

We set for

J , ^ ^ ) Q^/.S)" =

l~ dvj

JRN

JS"-1

durB(v-v*,uj)f'gl

(15)~

and we assume (to simplify the presentation) that B satisfies: — y{\z\, u| ) (this is always the case in the context of (1)) and tp E N N CQ°((0, oo) x (0,1)). We denote by A1(R ) the space of bounded measures on R .

B(Z,UJ)

2.5. The operator Q+ from M(RN) into H^^ÇR1*) is bounded for all s E R.

THEOREM

x HS(RN)

and HS(RN)

x

M(RN)

REMARK 2.4. This result is taken from Lions [48] using generalized Radon transforms; a variant of this proof making direct connection with the classical Radon transform has been recently given by Wennberg [72] (this proof, contrarily to the one in [48], does not extend to more general situations such as collision models for mixtures or relativistic models — this case is treated in Andréasson [1]). The above gain of regularity ( ^ - ^ derivatives) can be shown by writing Q + or its adjoint as a "linear combination" of translates of some Radon-like transforms given by

f

Ryj(v)

B(v,uj)yj((v,u)uj)duj

\tvj£C^(RN)

,

(16)

or Rijj(v) =

f

B(v,uj)yj(v-(v,uj)u))duj

,

Viß e C%°(RN) .

(17)

In both cases, one integrates (/? over the set {(V,UJ)UJ \ UJ E SN~1} = {V — (V,UJ)UJ \ UJ E SN~1}, which is the sphere centered at | and of radius -^. These operators are rather special Fourier integral operators often called generalized Radon transforms (see for instance Phong and Stein [62], Stein [66]). The crucial fact is that the set over which VJ is integrated "moves" with v — except that all these spheres go through 0, but this does not create difficulties because B vanishes if (V,UJ)UJ = 0 or if v~(v, U))UJ — 0. This is the main reason why one can prove that R is bounded from ^ ( R ^ ) into H**1**1^) for all s E R (^f1 comes from the stationary phase principle... ).

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Pierre-Louis Lions

3 Compressible Euler and Navier-Stokes equations The compressible Euler and Navier-Stokes equations are the basic models for the evolution of a compressible gas. In the case of aeronautical applications, the main difference between the domains of validity of the Boltzmann equation and the Euler-Navier-Stokes systems is the altitude of the aircraft. This indicates that there should be a transition from the Boltzmann model to those mentioned here. Mathematically, this corresponds to replacing B by ^ B in (1) and letting e go to 0 (at least formally): as is well known, one recovers, taking velocity averages of the limit / (i.e. p = JRN f dv, pu = JRN fvdv, pE = JRN f\v\2 dv), the compressible Euler equations (with 7 = ^j^) — see Cercignani [10] for more details. This heuristic limit (and related limits) remains completely open from a mathematical viewpoint: partial results can be found in Nishida [61], and Ukai" and Asano [71], and recent progress based upon the material described in Section 2 above is due to Bardos, Golse, and Levermore [4]. Related problems are described in Varadhan's report in this Congress. The compressible Euler and Navier-Stokes equations take the following form: - ^ + div (pu) = 0

x E RN , t > 0

(18)

—- (pu) + div (pu (g> u) - XAu - (A+/i) V div u + Vp = 0 xeRN , t>0 (19) at and an equation for the pressure p (or equivalently for the total energy or the temperature) that we do not wish to write for reasons explained below. The unknowns p, u correspond respectively to the density of the gas (p > 0) and its velocity u (where u(x, t) E R ^ ) . The constants À, a axe the viscosity coefficients of the fluid: if À = fi — 0, the above system is called the compressible Euler equations, whereas if A > 0, 2A + p > 0, it is called the compressible Navier-Stokes equations. Despite the long history of these problems, the global existence of solutions "in the large" is still open for the full (i.e. with the temperature equation) systems except in the case of compressible Navier-Stokes equations when N = 1: in that case, general existence and uniqueness results can be found in Kazhikov and Shelukhin [37], Kazhikov [36], Serre [64], [63], and Hoff [34]. This is why we shall restrict ourselves here to the so-called "isentropic" (or barotropic) case where one postulates that p is a function of p only, and in order to fix ideas we take p = ap1

,

a>0

,

7 > 1.

(20)

This condition is a severe restriction from the mechanical viewpoint (in the NavierStokes case, it essentially means considering the adiabatic case and neglecting the viscous heating). Mathematically, it leads to an interesting model problem that is supposed to capture some of the difficulties of the exact systems. Of course we complement (18)-(19) with initial conditions p\t=o = Po

,

pu\t=o = rn0

in

R^

(21)

where po > 0, mo are given functions on R ^ . We study the case of compressible isentropic Euler equations in Section 3.1. The analogous problem for Navier-Stokes equations is considered in Section 3.3.

On Some Recent Methods for Nonlinear Partial Differential Equations

147

3.1 ID isentropic gas dynamics We thus consider the following system

g + 2 H _ „ , £M + ^ „ W ) . „

ieK|i>0

(22)

where p > 0, and a > 0, 7 > 1 are given constants. Without loss of generality (by a simple scaling) we can take a = ^ ' (to simplify some of the constants below). As is well known for such systems of nonlinear hyperbolic (first-order) equations, singularities develop in finite time: even if po = p + p\ > 0 on R with p E R, p > 0, pijtio E CQ°(R), then ux and px become infinite in finite time (see Lax [38], [39], [41], and Majda [54], [55] for more details). In addition, bounded solutions of (22), (21) are not unique and additional requirements known as (Lax) entropy conditions on the solutions are needed (Lax [41],[40], see also the report by Dafermos in this Congress). In the case of (22), these requirements take the following form (see DiPerna [17], Chen [11], and Lions, Perthame, and Tadmor [53]): ^ [ip(p, pu)] + ^

[yj(p, pu)} < 0

in

X>'(R x (0, oo))

(23)

and tp, VJ are given by

JR

(24) r

VJ = JR

1

2 x

jdv[6v+(l-6)u]uj(v)(p y- -(v-u) )+

where UJ is an arbitrary convex function on R such that UJ" is bounded on R, A\ —

3-^y

A

— 2=1

— 2(7-1)' ° — 2 •

3.1. Letpo,mQ E L°°(R) satisfy: p 0 > 0, |mo| < Cp 0 a-e. inR for some C > 0. Then there exists (p,u) E L°°(R x (0,oo)) (p > 0) solution of (21)-(22) satisfying (23). THEOREM

As explained in Section 2.2, the proof of the existence results depends very much upon the stability and compactness results shown below (in fact one approximates (22) by the vanishing viscosity method; i.e., adding — e ^ 2 , —e —^2 in the equations respectively satisfied by p, pu where E > 0, and one lets e go to 0). We thus consider a sequence (pn,un) of solutions of (22) satisfying (23) and we assume that (pn,un) is bounded uniformly in n in L°°(R x (0,oo)) (pn > 0 a.e.). Without loss of generality, we may assume that (pn,un) converges weakly in L°°(R x (0,oo))-weak* to some (p,u) E L°°(R x (0,oo)) (p > 0 a.e.). The main mathematical difficulty is the lack of any a priori estimate (except for 7 = 3, the so-called monoatomic case) that would ensure the pointwise compactness needed to pass to the limit in pn(un)2 or (p71)1. 3.2. pn,pnun converge in measure on (-R,R) x (0,T) (for all 0 < R,T < 00J to p,pu respectively. And (p,u) is a solution of (22) satisfying (23). THEOREM

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Pierre-Louis Lions

REMARKS 3.1. (i) This result shows that the hyperbolic system (22) has compact ifying properties because initially at t = 0 we did not require that pn or pnun converge in measure.

(ii) Theorem 3.2 is essentially due to DiPerna [17] if 7 = §f±f (k > 1), Chen [11] if 1 < 7 < | . It is shown in Lions, Perthame, and Tadmor [53] if 7 > 3 and in Lions, Perthame, and Sóuganidis [51] if 1 < 7 < 3. The existence result (Theorem 3.1) for 1 < 7 < 00 is taken from [51]. (iii) The proofs in [53],[51] use two main tools: the method introduced by Tartar [69] (and developed by DiPerna [17]) which combines the compensated-compactness theory of Tartar [68], [69], Murat [60] and the entropy inequalities (23), and the kinetic formulation of (22) introduced in [52], [53] where one adds a new "velocity" variable, and writes the unknowns (p, pu) in terms of macroscopic quantities (velocity averages) associated with a density f(x,v,t) that has a fixed "profile" in v (a "pseudo-maxwellian"). This formulation connects the Boltzmann theory as described in Section 2 and the study of compressible hydrodynamic (or gas dynamics) macroscopic models. More details on this new approach are to be found in Perthame's report in this Congress. In the next section, we present some aspects of the compensated-compactness theory. 3.2 Compensated compactness and Hardy spaces One important point in the compensated-compactness theory developed by Tartar [68], [69] and F. Murat [60] is the systematic detection of nonlinear quantities that enjoy "weak compactness" properties. A typical example known as the div-curl example — it is precisely the one used in the proof of Theorem 3.2 — is given by the following result taken from [60]. 3.3. Let (En,Bn) converge weakly to (E,B) in LP(RN)N x Lq(RN)N 71 n with 1

2. We assume that cuilE , div B are relatively compact in W~liP(RN), W~1,q(RN) respectively. Then, En • Bn converges weakly (in the sense of measures or in distributions sense) to E • B.

THEOREM

3.2. Let US sketch a proof. We write: En = V-Kn + Ën where àìvE71 = 0, É is compact in U>(RN) (Hodge-De Rham decompositions), ir71 E Lpoc(RN), V?rn E LP(RN). Then, we only have to pass to the limit in Bn- W 1 = div (7rnBn)7rndivBn. The first term passes to the limit because irn is compact in Lfoc(RJV) (Rellich-Kondrakov theorem) while the second term also does because âivBn is relatively compact in ^ - ^ ( R ^ ) and Virn is bounded in LP(RN). As shown in Coifman, Lions, Meyer, and Semmes [12], the above nonlinear phenomenon is intimately connected with some general results in harmonic analysis associated with the (multi-dimensional) Hardy spaces denoted here by ^ ( R ^ ) (0 < p < 1): see Stein and Weiss [67], Fefferman and Stein [23], and Coifman and Weiss [14] for more details on Hardy spaces. In particular, the following result holds. REMARK n

On Some Recent Methods for Nonlinear Partial Differential Equations

149

3.4. Let E E LP(RN) satisfy curl E = 0 in V'(RN), let B E Lg(RN) satisfy divB = 0 in V'(RN) with 1 < p,q < oo, £ = ± + ± < 1 + ^ . T/ie?z E-BGJÏ^R"). THEOREM

REMARKS 3.3. (i) This result is taken from [12] (and was inspired by a surprising observation due to Müller [59]).

(ii) The relations between the weak compactness result (Theorem 3.3) and the regularity result (Theorem 3.4) are made clear in [12] and follow from some general considerations on dilation and translation invariant multilinear forms that enjoy a crucial cancellation property (JRN E • B dx = 0 in Theorem 3.4 above). (iii) Theorem 3.4 is one of the tools used in the proof by Hélein [30] of the regularity of two-dimensional harmonic maps. (iv) It is shown in [12] that any element of H\(RN) can be decomposed in a series ]C n >i KEn • Bn where ||£;n||L2 = ||B n ||L a = li d i v S n = cui\En = 0, E n > l l A n | < OO. If we denote by Rk the Riesz transform (= öfc(—A) -1 / 2 ), then, under the conditions of Theorem 3.4, there exists % E LP(RN) such that E = Rn. And E- B = B-Rïï = B-Rïï + (R- B)TT because R • B = ( - A ) " 1 / 2 div B = 0. Then we can recover the case r = 1 in Theorem 3.4 using the iJi-BMO duality and the result on commutators due to Coifman, Rochberg, and Weiss [13]: indeed, we then obtain f(Rkg) + (Rkf)g £ fi^R*) for each k > 1, / E LP(RN), g E L«(RN), K j x o o , | + | = 1. 3.3 Isentropic Navier-Stokes equations We now consider the system ( dp 1

-^ + div (pu) = 0 , ^P

+ div (pu u) - XAu - (X+p)V div u + aS/p1 = 0 ,

( 25 )

xeRN,t>0, where a > 0, 1 < 7 < 00, A > 0, 2A + p > 0, p(x,t) > 0 on R ^ x (0,00), with the initial conditions (21) that are required to satisfy

\ mo = yfpövn THEOREM 3.5.

a.e. with

^o E L2(RN)

.

We assume (26) and 7 > § if N = 2, 7 > \%f N = 3, 7 > ^ */

AT > 4. Then tfiere eztós a solution (p,u) E L°°(0,oo;L^(R i V ))nL 2 (0,r;iJ 1 (5 / î )) ( V Ä , r E (0,oo)) o/(25),(21) satisfying in addition: p E C([0,oo);L p (R N )) z/

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Pierre-Louis Lions

1 < p < 7 , p\u\2 e L ^ O . o o j L 1 ^ ) ) , p£L*(J8LN x (0,T)) for 1 < q < 7 + | f - 1 i/iV>2. / lp(t)Ht)\2 Jn 2

+ -^-p(ty 7 —1

1

< Z2

dx + [dsf X\Vu\2 + 7o Jn pò

(X+ß){äivu)2dx C2Y'\

+ — - r Po dx 7-I

/or almost allt > 0. REMARKS 3.4. (i) This result is taken from Lions [45] (see also [50]). If TV = 1, more general results are available and we refer to Serre [63], Hoff [32],[33].

(ii) If AT > 2, the uniqueness and further regularity of solutions are completely open as is the case of a general 7 > 1. The case 7 = 1 is also an interesting mathematical problem (see [49]). (iii) Of course, the equations contained in (25) hold in the sense of distributions. (iv) The preceding result is rather similar to the results obtained by Leray [42], [43], [44] on the global existence of weak solutions of three-dimensional incompressible Navier-Stokes equations satisfying an energy inequality like (27). Despite many important contributions (like the partial regularity results obtained by Caffarelli, Kohn, and Nirenberg [7]), the uniqueness and regularity of solutions are still open questions. As explained in the previous sections, the above existence result is based upon a convergence result for sequences of solutions pn,un satisfying uniformly in n the properties mentioned in the above result. Hence, without loss of generality, we may assume that (pn,un) converge weakly to (p,u) in L'y(RN x (0,T)) x L2(0,T;H1(BR)) (V R,T E (0,oo)). Then it is shown in [45], [49] that if p% (= pn\t=Q) converges in L ^ R ^ ) , then pn converges in C([0, T\\ D>(RN))nL