Conservation laws with vanishing nonlinear diffusion and dispersion1

Nonlinear Analysis 36 (1999) 213 – 230

Conservation laws with vanishing nonlinear diusion and dispersion1 Philippe G. LeFloch a;∗ , Roberto Natalini b a

Centre de Mathematiques Appliquees and CNRS URA 756, Ecole Polytechnique, 91128 Palaiseau, France b Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, 00161 Roma, Italy Received 21 February 1997; accepted 12 March 1997

Keywords: Nonlinear dispersive equations; Korteweg–de Vries equation; Pseudo-viscosity; Burgers equation; Shock waves; Measure-valued solutions

1. Introduction This paper investigates the convergence of the smooth solutions u ;  to the initial value problem for a nonlinear diusive–dispersive equation: @t u + @x f(u) = @x (@x u) − @3x u u(x; 0) = u0;  (x);

(x; t) ∈ R × (0; ∞);

x ∈ R;

(1.1) (1.2)

where the parameters ¿0 and ¿0 tend to zero and u0;  is an approximation of a given initial condition u0 : R → R. The ux f(u) and the (possibly degenerate) viscosity () are given smooth functions satisfying certain assumptions to be listed shortly. When  = 0, Eq. (1.1) reduces to the generalized Korteweg–de Vries (KdV) equation, the original one corresponding to the special ux f(u) = u 2 =2. On the other hand, when  = 0, Eq. (1.1) reduces to a nonlinear (possibly degenerate) parabolic equation. The rst case was extensively studied from mathematical and numerical standpoints; see, for instance [6, 14, 15] and Section 2 below for additional background. For  = 0 and under suitable assumptions, Eq. (1.1) can be viewed as a simpli ed model of the ∗

Corresponding author. E-mail: le [email protected]. This work was partially carried out during a visit of the rst author to the Istituto per le Applicazioni del Calcolo. 1

0362-546X/98/$19.00 ? 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 0 1 2 - 1

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pseudo viscosity approximation proposed by von Neumann and Richtmyer [27], and Lax and Wendro [22–24]. See also Whitham [36]. We refer to [25] for a mathematical study of this equation and to [29, 28] for the numerical analysis. A large literature is available on this type of equation. See later parts of this introduction for additional references. In this paper, under an assumption on the relative size of the parameters  and , we prove that the sequence {u ;  } converges as  and  → 0 to an entropy solution u of the rst order, hyperbolic conservation law @t u + @x f(u) = 0:

(1.3)

The following two conditions on f and are assumed throughout this paper: (A1 ) There exist constants C1 ¿0 and m¿1 such that |f0 (u)| ≤ C1 (1 + |u|m−1 )

for all u ∈ R:

(A2 ) The function () is nondecreasing and satis es (0) = 0. Note that the ux need not be a convex function. It follows from (A2 ) that () ≥ 0 for all  ∈ R. The following assumptions will also be in use: (B1 ) There exist constants C2 ; C3 ; N ¿0 and r ≥ 1 such that C2 ||3r ≤ () ≤ C3 ||3r

for all || ≥ N:

The function therefore is at least quadratic at in nity and may vanish on a bounded interval. Finally, two other assumptions on the function will be required for certain results below. (B2 ) There exist C4 ¿0 and N ¿0 such that () ≥ C4 ||3

for all || ≥ N:

(B3 ) There exist C5 ¿0 and r ≥ 1 such that () ≥ C5 ||3r

for all  ∈ R:

Both (B2 ) and (B3 ) are dierent assumptions on the lower bound estimate in (B1 ); condition (B2 ) is actually a consequence of (B1 ). On the other hand (B3 ) is just a re nement of (B1 ) for small values of . Note that the uniform bounds to be derived below will be obtained under condition (B2 ) or condition (B3 ), while, for the convergence proofs, we shall use condition (B1 ). So for our main results we need only assume either (B1 ), or (B1 ) and (B3 ). One typical example of a function satisfying (A2 ) and (B1 )–(B3 ) is given by () = ||3r−2 

(r ≥ 1):

(1.4)

Recall that, for the Cauchy problem for Eq. (1.3), the existence and uniqueness of an entropy solution were rst proved by Kruzkov [17]. The vanishing viscosity limit, corresponding to  = 0 and  → 0, was studied by several authors in the special case of a linear (and therefore nondegenerate) diusion () = . This activity started with Oleinik’s [26] and Kruzkov’s [17] works. For more general viscosity coecients and under the assumptions (A2 )–(B1 ), the convergence was proved in [25].

P.G. LeFloch, R. Natalini / Nonlinear Analysis 36 (1999) 213 – 230

215

The vanishing dispersion limit ( = 0 and  → 0) has also been widely investigated; see Lax and Levermore [21], [7, 33, 34], and the survey paper by Lax [20]. The main source of diculty lies in the highly oscillating behavior of the solutions u 0;  , which do not converge in a strong topology. The full equation (1.1) arises, at least for the linear diusion () =  (the so called Korteweg–de Vries–Burgers equation), as a model of the nonlinear propagation of dispersive and dissipative waves [36]. For related studies, refer to [1–3, 11, 12]. Traveling waves are studied in [2, 10] and (partial) analytical results are found in [16, 6]. For a study of numerical approximations with similar features as Eq. (1.1), see [4]. The vanishing dispersive and diusion limit, both  and  tending to zero, was rst studied by Schonbek [30]. She assumed that () =  and used the compensated compactness method introduced by Tartar [32]. In particular, she proved that the sequence {u ;  } converges to a weak solution to Eq. (1.3) (but not necessarily the unique entropy one) under the assumption that either  = 0(2 ) for f(u) = u 2 =2, or  = 0(3 ) for arbitrary subquadratic ux-functions f (i.e. take m = 2 in (A1 )). Indeed certain constraints on  and  are required: the viscosity eects should dominate the dispersion eects. When the dispersion eects are dominant, it is known that {u ;  } do not converge to a weak solution of Eq. (1.3) (a fortiori to the entropy one). For recent development on the intermediate case that dispersion and diusion are kept in balance, we refer to Hayes and LeFloch [9]. Here we consider the general nonlinear equation (1.1) and, under assumption (B1 ) and the condition  ≤ C(5−m)=(3−m) (m¡3), we show that the solutions u ;  converge to the entropy weak solution of Eq. (1.3) (Theorem 4.1). Under assumptions (B1 ) and (B3 ) and when  ≤ C (5−m)=(r(5−m)−1) (m¡5 − 1=r; r ≥ 1), the same result holds true (Theorem 4.2). Furthermore, our Theorem 4.3 below improves upon Schonbek’s result in [30] in the case that () =  and f is subquadratic. The main tool we use here to deal with these singular limits is the concept of entropy measure-valued solution to Eq. (1.3). Measure-valued solutions were introduced by DiPerna [7] (see Section 2) to represent the weak-? limits of solutions to Eq. (1.1). According to DiPerna’s theory, the strong convergence of an approximate sequence follows once one obtains (i) uniform bounds in L∞ (0; T ; Lq (R)) for all T ¿0 with q¿m (where m is given by (A1 ), (ii) weak consistency of the sequence with the entropy inequalities and the strong consistency with the initial data (cf. conditions (2.5) and (2.6) of Section 2). Such a framework can be applied to establish the convergence of dierence schemes; see [5]. Because of the nature of the operator on the right-hand side of Eq. (1.1), no maximum principle is available for the solutions u ;  and the Lq (uniform) estimates are more natural. We actually use here the Lq version of DiPerna’s result derived by Szepessy in [31]. One basic estimate, requiring no restriction on the parameters  and , is the standard energy estimate based on the quadratic function, u 2 (Lemma 3.1). The derivation of the L q estimates with larger exponents (q¿2) is based on the nonlinearity of the viscosity coecient and requires suitable restriction on  and .

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The problem studied in this paper is motivated by a numerical method proposed by von Neumann and Richtmyer [27, 29] and based on the pseudo-viscosity approximation () = ||. An important advantage of the nonlinear (quadratic) diusion term () is that it adds sharply localized viscosity near discontinuities. The original Lax–Wendro scheme uses [22–24] that technique: a minimal amount of numerical viscosity is added in order to prevent both the formation of unphysical (entropy violating) shocks and the generation of highly oscillatory approximate solutions. 2. Preliminaries This section contains short background material on Young measures, entropy measure-valued (m.-v.) solutions (Section 2.1) and dispersive equations (Section 2.2). 2.1. Young measures and measure-valued solutions Following Schonbek [30], we state a representation theorem for the Young measures associated with a given sequence of uniformly bounded functions on L q . In the whole of this subsection, q ∈ (1; ∞) is xed. The corresponding setting in L∞ was rst established by Tartar [32]. Lemma 2.1. Let {uj } be a uniformly bounded sequence in L∞ (R+ ; L q (R)). Then there exists a subsequence {uj0 } and a weakly-? measurable mapping  : R × R+ → Prob (R) taking its values in the space of nonnegative measures with unit total mass ( probability measures) such that; for all functions g ∈ C(R) satisfying g(u) = 0(|u|r )

as |u| → ∞

(2.1)

for some r ∈ [0; q); the following limit representation holds ZZ Z ZZ 0 g(uj (x; t))(x; t) dx dt = g() d(x; t) ()(x; t) dx dt lim 0 j →∞

R×R+

R×R+

R

(2.2) for all  ∈ C0∞ (R ×R+ ). The measure-valued function (x; t) is a Young measure associated with the sequence {uj }. The following result reveals the connection between the structure of  and the strong convergence of the initial sequence. Lemma 2.2. Suppose that  is a Young measure associated with a sequence {uj }; uniformly bounded in L∞ (R+ ; L q (R)). For u ∈ L∞ (R+ ; L q (R)); the following statements are equivalent: (i) limj→∞ uj = u in L∞ (R+ ; Lrloc (R)) for some r ∈ [1; q); (ii) (x; t) () = u(x; t) () for almost every (x; t) ∈ R × R+ .

P.G. LeFloch, R. Natalini / Nonlinear Analysis 36 (1999) 213 – 230

217

In (ii) above, the notation u(x; t) () = ( − u(x; t)) is used for the Dirac mass de ned by ZZ ZZ hu(x; t) ; g(·)i(x; t) dx dt = g(u(x; t))(x; t) dx dt R×R+

R×R+

for all g ∈ C(R) satisfying Eq. (2.1) and all  ∈ C0∞ (R ×R+ ). Following DiPerna [7] and Szepessy [31], we now de ne the measure-valued (m.-v.) solutions to the rst-order Cauchy problem @t u + @x f(u) = 0; u(x; 0) = u0 (x);

(2.3) x ∈ R:

(2.4)

De nition 2.1. Assume that f satis es the growth condition (2.1) and u0 ∈ L1 (R) ∩ L q (R). A Young measure  associated with a sequence {uj }, which is assumed to be uniformly bounded in L∞ (R+ ; L q (R)), is called an entropy m.-v. solution to the problem (2.3)–(2.4) if @t h(·) ; | − k|i + @x h(·) ; sgn( − k) (f() − f(k))i ≤ 0 in the distributional sense for all k ∈ R and, for all intervals I ⊆ R, Z Z 1 T h(x; t) ; | − u0 (x)|i dx dt = 0: lim+ T →0 T 0 I

(2.5)

(2.6)

A function u ∈ L∞ (R+ ; L1 (R) ∩ L q (R)) is an entropy weak solution to Eqs. (2.3) and (2.4) in the sense of Kruzkov [17] and Volpert [35] if and only if the Dirac measure u(·) is an entropy m.-v. solution. In the case p = +∞, existence and uniqueness of such solutions were proved in [17]. The following results on entropy m.-v. solutions were proved in [31]: Theorem 2.3 states that entropy m.-v. solutions are actually Kruzkov’s solutions. Theorem 2.4 states that the problem has a unique L q solution. Theorem 2.3. Assume that f satis es Eq. (2:1) and u0 ∈ L1 (R) ∩ L q (R). Suppose that  is an entropy m.-v. solution to Eqs. (2:3) and (2:4). Then there exists a function w ∈ L∞ (R+ ; L1 (R) ∩ L q (R)) such that (x; t) = w(x; t)

(2.7)

for almost every (x; t) ∈ R×R+ . Theorem 2.4. Assume that f satis es Eq. (2:1) and u0 ∈ L1 (R) ∩ L q (R). Then there exists a unique entropy solution u ∈ L∞ (R+ ; L1 (R) ∩ L q (R)) to Eqs. (2:3) and (2:4) which; moreover; satis es ku(· ; t)kLr (R) ≤ ku0 kLr (R)

(2.8)

for almost every t ∈ R+ and all r ∈ [1; q]. The measure-valued mapping (x; t) = u(x; t) is the unique entropy m.-v. solution of the same problem.

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Combining Theorems 2.3 and 2.4 and Lemma 2.2, we obtain our main convergence tool. Corollary 2.5. Assume that f satis es Eq. (2:1) and u0 ∈ L1 (R) ∩ L q (R). Let {uj } be a sequence; uniformly bounded in L∞ (R+ ; L q (R)) for q ≥ 1; and let  be a Young measure associated with this sequence. If  is an entropy m.-v. solution to Eqs. (2:3) and (2:4); then lim uj = u

j→∞

in L∞ (R+ ; Lrloc (R))

for all r ∈ [1; q); where u ∈ L∞ (R+ ; L q (R)) is the unique entropy solution to Eqs. (2:3) and (2:4). Proof. We only need to show that, if  is a Dirac mass, the sequence converges strongly. Consider a strictly convex entropy U satisfying the growth condition (2.1) and such that U (uj ) − U (u) − U 0 (u) (uj − u) ≥ C |uj − u|r

(2.9)

holds. As j → ∞ the left-hand side of Eq. (2.9) converges to h; U i−U (u)−U 0 (u) (u− u) = 0, since  is a Dirac. The right-hand side of Eq. (2.9) converges to zero almost everywhere (after extraction of a subsequence of {uj }, if necessary). 2.2. Nonlinear dispersive equations Consider the fully nonlinear, KdV-type equation in one space variable @t u + F(u; @x u; @x2 u; @x3 u) = 0

(2.10)

for (x; t) ∈ R ×(0; T ) together with the Cauchy data u(x; 0) = u0 (x)

for x ∈ R:

(2.11)

In recent years this class of problems has been extensively studied; see for instance [14–16] and references cited therein. In particular, under suitable assumptions on F, these equations enjoy a gain of regularity of the solutions with respect to their initial data (cf. [6, 13, 18] and references above). A complete theory of global existence still remains to be developed however, and presenting a complete review is far beyond the aim of this subsection. Here we only recall a result – that is sucient to deal with Eq. (1.1) – of local existence and uniqueness of the smooth solutions to Eq. (2.10). More details and proofs can be found in the paper by Craig et al. [6] (see also [16]). Using the notation U = (u0 ; u1 ; u2 ; u3 ), the assumptions on the smooth function F = F(U ) are as follows: (H1 ) There exists ¿0 such that @u3 F(U ) ≥ ¿0

for all U ∈ R 4 ;

(H2 ) @u2 F(U ) ≤ 0 for all U ∈ R 4 . Eq. (1.1) does satisfy these hypotheses provided ¿0 and (A2 ) holds.

P.G. LeFloch, R. Natalini / Nonlinear Analysis 36 (1999) 213 – 230

219

Theorem 2.6 (Uniqueness). Let T ¿0 be xed and assume that F satis es (H1 ) and (H2 ). For any u0 ∈ H 7 (R) there exists at most one solution u ∈ L∞ ((0; T ); H 7 (R)) to the problem given by Eqs. (2:10) and (2:11). Theorem 2.7 (Existence). Assume that F satis es (H1 ) and (H2 ). Let N ≥ 7 be an integer and C0 ¿0 be a given constant. There exists a time T ¿0; depending only on C0 ; such that for all u0 ∈ H N (R) with ku0 kH 7 ≤ C0 ; there exists at least one solution u ∈ L∞ ((0; T ); H N (R)) to the problem given by Eqs. (2:10) and (2:11). The setting above based on a Sobolev norm of relatively high order (H 7 (R)) is the optimal result provided by the current techniques of analysis of dispersive equations. The global existence of smooth solutions to Eqs. (1.1) and (1.2) appears to be an open problem. In the following we restrict attention to a time T∗ ∈ (0; ∞] chosen such that the problem given by Eqs. (1.1) and (1.2) is well-posed in the strip R × (0; T∗ ). 3. A priori estimates In this section we consider a sequence {u;  } of smooth solutions of Eqs. (1.1) and (1.2) vanishing at in nity. We assume that the initial data {u0;  } are smooth functions with compact support, uniformly bounded in L1 (R) ∩ Lq (R) for a suitable q¿1. In what follows, whenever it does not lead to confusion, we omit the indices  and . All constants are denoted by C; C1 ; : : : . We begin with the natural energy estimate based on the quadratic function u 2 =2. Observe that arbitrary convex functions cannot be easily used here because of the dispersive term. Lemma 3.1. For T ∈ (0; T∗ ), one has Z TZ Z Z u 2 (x; T ) dx + 2 (ux (x; t))ux (x; t) dx dt = u02 (x) dx: R

0

R

R

(3.1)

Proof. We multiply Eq. (1.1) by u and then integrate it in x ∈ R. After integrating by parts we obtain Z Z Z Z d 1 u 2 dx + uf0 (u)ux dx = − (ux )ux dx −  u uxxx dx: (3.2) dt 2 R R R R The second terms on both sides of Eq. (3.2) vanish identically, since we can write them in a conservative form: uf0 (u)ux = G(u)x with G 0 = uf0 (u) and u uxxx = (u uxx − 12 ux2 )x : The estimate Eq. (3.1) follows from Eq. (3.2).

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At this stage, in view of assumption (A2 ) and Lemma 3.1, we have obtained a uniform control of u in L∞ (0; T∗ ; L 2 (R)) and  (ux )ux in L1 ((0; T∗ ) × R). Our next aim is to derive an estimate of u in the L∞ norm, which will be uniform in  but not in . We rst derive a second energy-type estimate. Lemma 3.2. Let F be de ned by F 0 (u) = f(u). For every T ∈ (0; T∗ ), we have  2

Z R

ux2 (x; T ) dx Z

 = 2

R

Z −

u0;2 x (x) dx

R

Z F(u(x; T )) dx +  Z

−

R

T

R

0

Z F(u0 (x)) dx + 

Z

T

2 0 uxx (ux ) dx dt

Z

0

R

f0 (u) (ux )ux dx dt:

(3.3)

Proof. Multiplying Eq. (1.1) by f(u) + uxx we obtain Z 0 =

R

{F(u)t + H (u)x − f(u) (ux )x + f(u)uxxx } dx

Z

+

R

2 0 {uxx ut + uxx f(u)x − uxx (ux ) + uxxx uxx } dx

where H 0 = ff0 . After integration by parts in space, we have d dt

Z 

 Z Z  2 0 F(u) − ux2 dx + f0 (u) (ux )ux dx −  uxx (ux ) dx = 0; 2 R R

which gives the desired conclusion. √ Using the bound for ux in L∞ (0; T∗ ; L 2 (R)) that follows from Lemma 3.2, we now estimate u in the L∞ norm. Lemma 3.3. If m¡5 in assumption (A1 ), there exists a constant C¿0 such that |u(x; t)| ≤ C−1=(5−m) for all (x; t) ∈ R × (0; T∗ ). Proof. In view of (A2 ) and for all u ∈ R, we have |F(u)| ≤ C(1 + |u| m+1 ):

(3.4)

P.G. LeFloch, R. Natalini / Nonlinear Analysis 36 (1999) 213 – 230

221

The main idea is to use Eq. (3.3) to control ux2 in terms of F(u), the latter being estimated using the growth condition. From Eq. (3.3) we deduce that for T ∈ (0; T∗ ) Z Z TZ  2 0 ux2 (x; T ) dx +  uxx (ux ) dx dt 2 R R 0 Z Z Z TZ  2 = f0 (u) (ux )ux dx dt u (x) dx + (F(u(x; T )) − F(u0 (x))) dx +  2 R 0; x R 0   Z TZ m−1 2 ≤ C + Cku(·; T )kL∞ (R) ku(·; T )kL 2 (R) +  (ux )ux dx dt ; 0

R

where we used (A2 ) as well. (We do not explicitly write the terms involving the initial data as they are uniformly bounded.) Therefore in view of Eq. (3.1), we arrive at Z Z TZ  2 2 0 u (x; T ) dx +  uxx (ux ) dx dt ≤ C + Cku(·; T )kLm−1 ∞ (R) : 2 R x R 0 Using the Cauchy–Schwarz inequality, we obtain Z x |u(x; T )| 2 ≤ 2 |u(y; T )ux (y; T )| dy −∞

√ 2 ≤ √ ku(·; t)kL 2 (R) kux (·; T )kL 2 (R)  C 1=2 ≤ √ ku0 kL 2 (R) (1 + ku(·; T )kLm−1 : ∞ (R) )  Hence, for all t ∈ (0; T∗ ), C ku(·; t)k4L∞ (R) ≤ √ (1 + ku(·; t)kLm−1 ∞ (R) ): 

(3.5)

Since m¡5, the growth of the left-hand side of Eq. (3.5) exceeds the growth of the right-hand side. Setting y = ku(·; t)kL∞ and studying the inequality y4 ≤ C−1 (1 + y m−1 ), it is easily checked that Eq. (3.5) implies  1=(5−m) ! 2C ku(·; t)kL∞ (R) ≤ max 1; (3.6) ≤ C−1=(5−m) :  Using the same arguments as in the proof of Lemma 3.3 we also get: Lemma 3.4. For any T ∈ (0; T∗ ) we have Z Z TZ 1 2 0 ux2 (x; T ) dx +  uxx (ux )dx dt ≤ C−4=(5−m) : 2 R R 0

(3.7)

Finally we derive uniform bounds in L∞ (0; T∗ ; Lq (R)) with q ≤ 5. The following result provides a uniform estimate in L5 , improving upon the L 2 bound in Lemma 3.1.

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Here we take advantage of the nonlinearity property of stated in (B2 ) (for large values of  = ux ). Proposition 3.5. Assume that (B2 ) holds and m¡3 in (A1 ). There exists a constant C¿0, depending only on the initial data, such that, for all small enough  and , sup ku(·; t)k5L5 (R) ≤ C(1 + T∗ + −1 (3−m)=(5−m) ):

t∈(0; T∗ )

(3.8)

Proof. Set (u) = |u|5 , multiply Eq. (1.1) by 0 (u), and integrate on R × (0; T ). It follows that Z TZ Z (u(x; T )) dx +  00 (u) (ux )ux dx dt R

R

0

Z

 = (u0 (x)) dx − 2 R

Z

T

Z

0

R

000 (u)ux3 dx dt:

(3.9)

On the other hand, according to (B2 ), there exists a constant C such that ||3 ≤ C(1 + ()) for all  ∈ R. Using the latter condition in Eq. (3.1), we estimate the second term in the right-hand side of Eq. (3.9): Z T Z Z TZ 000 3  (u)u dx dt ≤ C |u| 2 |ux |3 dx dt 1 x 0

R

0

Z ≤ C2

T

0

Z

R

|u| 2 (1 + (ux )ux ) dx dt

≤ C3 T sup ku(t)kL 2 (R) + t∈(0; T )

C kukL2∞ (R×(0; T )) 

≤ CT + C(1 + −1 −2=(5−m) ); where the latter inequality follows from the L∞ bound in Lemma 3.3. Returning to Eq. (3.9), we obtain Eq. (3.8). Under the stronger assumption (B3 ), we have a sharper estimate. Proposition 3.6. Assume that (B3 ) holds for a given r ≥ 1 and that m¡q := 5 − 1=r in (A1 ). There exists a constant C¿0, depending only on the initial data, such that, for all small enough  and  and every T ∈ (0; T∗ ), sup ku(·; t)kqLq (R) ≤ C(1 + T 1−1=r −1=r (q−m)=(5−m) ):

t∈(0; T )

(3.10)

P.G. LeFloch, R. Natalini / Nonlinear Analysis 36 (1999) 213 – 230

223

Proof. As in the proof of Proposition 3.5, we consider formula (3.9) but now with (u) = |u|q with q = 5 − 1=r. Using Eqs. (3.1) and (3.4), assumption (B3 ) yields Z T Z Z TZ 000 3  (u)u dx dt ≤ C |u|q−3 |ux |3 dx dt x 0

R

R

0

Z ≤C

T

0

Z R

r 0 (q−3)

|u|

1=r ≤ C−1=r kuk1=r L∞ (R) T

1=r 0 Z dx dt

0

0

T

1=r

Z R

(ux )ux dx dt

sup ku(·; t)k2=r L2 (R)

t∈(0; T )

0

≤ CT 1=r −1=r −1=r(5−m) ; where 1=r +1=r 0 = 1 and since r 0 (q−3) = 2+r 0 =r. Now Eq. (3.10) follows from formula (3.9). Estimates (3.8) and (3.10) hold only thanks to the nonlinear form of the viscosity term. When  = 0, these estimates reduce to bounds that are uniform in , which is expected for conservation laws with viscosity but no dispersion. On the other hand, most of our estimates blow-up when  → 0 and provide no control of Lq norms. The L2 bound in Lemma 3.1 is uniform for  arbitrary and  → 0. Loosely speaking, this is also the case of the estimate in Proposition 3.6 if r could be chosen to be ∞, in which case Eq. (3.10) becomes a uniform L5 estimate. Such a choice of r is not allowed however, cf. (B3 ). For the sake of completeness, we nally state an analogous estimate for linear diffusions which was proved by Schonbek [30]. Proposition 3.7. Let () =  and take m = 2 in (A1 ). For any T ∈ (0; T∗ ); there exists a constant CT ¿0; depending only on the initial data; such that for  ≤ C3 sup ku(·; t)kL4 (R) ≤ CT :

t∈(0; T )

(3.11)

4. Convergence results In Section 3, under certain assumptions on the functions f; , and the parameters  and ; we have established several uniform bounds for the sequence {u;  } of solutions to the Cauchy problem given by Eqs. (1.1) and (1.2). Assume again that the initial data u0;  are smooth with compact support and that there exists a limiting function u0 ∈ L1 (R) ∩ Lq (R) and a suitable q¿1 (speci ed below) such that, if  = O(), lim u0;  = u0

→0

in L1 (R) ∩ Lq (R):

(4.1)

Returning to the proofs of Section 3, it is not hard to see that the following conditions on the initial data are sucient for the estimates therein to hold uniformly with respect

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P.G. LeFloch, R. Natalini / Nonlinear Analysis 36 (1999) 213 – 230

to a class of initial data: ;  ku0;  kL2 (R) + ku0;  kLq (R) + 1=2 ku0; x kL2 (R) ≤ C:

Indeed Eq. (3.1) is uniform in (; ) provided ku0;  kL2 (R) remains uniformly bounded. ;  Deriving estimate (3.3) requires that ku0;  kLq (R) and 1=2 ku0; x kL2 (R) remain uniformly bounded. In this section we prove the strong convergence of the sequence u;  . Theorem 4.1. Assume that (B1 ) holds and m¡3 in (A1 ). Let u ;  be a sequence of smooth solutions to Eqs. (1:1) and (1:2); de ned on R × (0; T∗ ); vanishing at in nity and associated with initial data satisfying Eq. (4:1) with q = 5. If there is a constant C¿0 such that  ≤ C(5−m)=(3−m) ; then the (whole) sequence u;  converges in Ls (0; T∗ ; Lp (R)) (s¡∞ and p¡5) to a function u ∈ L∞ (0; T∗ ; L5 (R)); which is the unique entropy solution to Eqs. (2:3) and (2:4). Theorem 4.2. Assume that (B1 ) and (B3 ) hold and m¡5 − 1=r in (A1 ). Let u;  be a sequence of smooth solutions to Eqs. (1:1) and (1:2) de ned on R × (0; T∗ ); vanishing at in nity; and associated with initial data satisfying Eq. (4:1) with q = 5 − 1=r. If there is a constant C¿0 such that  ≤ C(5−m)=(r(5−m)−1) ; then the (whole) sequence converges in Ls (0; T∗ ; Lp (R)) (s¡∞ and p¡q) to a function u ∈ L∞ (0; T∗ ; Lq (R)); q = 5 − 1=r; which is the unique entropy solution to Eqs. (2:3) and (2:4). In the case () = ; we improve upon a result in [30] (cf. Theorem 5.1 therein). Theorem 4.3. Assume m = 2 in (A1 ) and () = . Let u;  be a sequence of smooth solutions to Eqs. (1:1) and (1:2) de ned on R × (0; T∗ ); vanishing at in nity; and associated with initial data satisfying Eq. (4:1) with q = 4: If  = o(3 ); the (whole) sequence u;  of solutions of Eqs. (1:1) and (1:2) converges to a function u ∈ L∞ (0; T∗ ; L4 (R)); which is the unique entropy solution to Eqs. (2:3) and (2:4). Similar convergence results can be proven for the case m = 3 or for some special

ux-functions along the lines of what was done in [30] (cf. Sections 4 and 5 therein). Theorems 4.1 and 4.2 follow easily by simply using the following general result and the Lq bounds derived in Section 3, Proposition 3.5 and Proposition 3.6, respectively. Theorem 4.4. Assume that (B1 ) holds. Let u;  be a sequence of smooth solutions to Eqs. (1:1) and (1:2) on R × (0; T∗ ); vanishing at in nity and associated with initial data satisfying Eq. (4:1). If the sequence is uniformly bounded in L∞ (0; T∗ ; Lq (R)) for q¿m and  = o(1=r ); then the (whole) sequence converges in u ∈ Ls (0; T∗ ; Lq (R)) (s¡∞ and p¡q) to a function u ∈ L∞ (0; T∗ ; Lq (R)); which is the unique entropy solution to Eqs. (2:3) and (2:4). Note that the technique of proof used here (compensated compactness method) does not allow one to derive an error estimate for the dierence u;  − u. We give a proof of, rst, Theorem 4.4, then Theorem 4.3.

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225

Proof of Theorem 4.4. First of all we establish that, for any convex function (u) such that 0 ; 00 ; 000 are uniformly bounded on R; ;  := @t (u;  )+@x Q(u;  ) converges to a nonpositive measure in D0 (R × R+ ); (4.2) where Q0 = f0 0 . To begin with, observe that ;  = (0 (u) (ux ))x − 00 (u) (ux )ux − (0 (u)uxx )x + 00 (u)ux uxx = T1 + T2 + T3 + T4 : To estimate T1 , we use (B1 ) which implies | ()| ≤ C(1 + ||3r−1 ) for all . For any given  ∈ C0∞ (R × (0; T∗ ));  ≥ 0; and for p = 3r=(3r − 1)¿1 (p0 , below, being the conjugate exponent of p), we get Z Z hT1 ; i = 0 (u) (ux )x dx dt Z Z 1=p p(3r−1) ≤ Ckx kL1 (R×(0;T∗ )) + Ckx kLp0 (R×(0;T∗ )) |ux | : So using the derivative estimate in Eq. (3.1) and (B1 ) again hT1 ; i ≤ C( + 1=3r ) ≤ C1=3r :

(4.3)

The second term T2 is nonpositive: Z Z 00 (u) (ux )ux  dx dt ≤ 0: hT2 ; i = −

(4.4)

To estimate T3 ; we use the energy estimate (3.1) and the fact that is at least quadratic: Z Z 0 (u)uxx x dx dt hT3 ; i =  R×(0; T∗ )

Z Z =

R×(0; T∗ )

((0 (u)ux )x − 00 (u)ux2 )x dx dt

Z Z ≤ −

R×(0; T∗ )

0 (u)ux xx dx dt + 

Z Z ≤ C + C

supp 

3r

|ux | dx dt

1=3r

Z Z R×(0; T∗ )

ux2 |x | dx dt

+ C−2=3r ;

where supp  denotes the support of  in R × (0; T ). Thus hT3 ; i ≤ C(1 + C−2=3r + −1=3r ) ≤ C(1 + C−2=3r ):

(4.5)

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P.G. LeFloch, R. Natalini / Nonlinear Analysis 36 (1999) 213 – 230

Finally we deal with T4 as follows: Z Z 00 |hT4 ; i| =   (u)ux uxx  dx dt

Z Z Z Z 1 000 (u) ux3  dx dt =  00 (u)ux2 x dx dt +  2 Z Z (|ux |2 + |ux |3 ) dx dt; ≤ C supp 

so |hT4 ; i| ≤ C(−2=3r + −1=r ) ≤ C−1=r :

(4.6)

If  = o(1=r ), Eq. (4.2) follows immediately from the estimates (4.3) – (4.6). To apply Corollary 2.5 we show that Eqs. (2.5) and (2.6) are satis ed for a Young measure  associated with the sequence u;  . It is a standard matter to deduce, for all convex entropy pairs, @t h(·) ; ()i + @x h(·) ; Q()i ≤ 0 from the convergence property (4.2). Namely it follows from the de nition of the Young measure that the terms in Eq. (4.2) converge (in the sense of distributions) to their “natural” limits, (u;  ) * h; i;

Q(u;  ) * h; Qi:

Inequality (2.5) (for all k ∈ R) then follows by a standard regularization of the function |u−k|. To show that the initial condition Eq. (2.6), is satis ed, we combine the entropy inequalities and the weak consistency property, as was suggested by DiPerna [7]. We follow the detailed argument in Szepessy [31]. Consider the function g() = ||r for 1¡r¡min(2; q), and set G(; 0 ) := g() − g(0 ) − g0 (0 )( − 0 ) ≥

r(r − 1) ( − 0 )2 : 2 (1 + || + |0 |)2−r

(4.7)

Let I ⊆ R be a closed and bounded interval. Using the Jensen inequality, the Cauchy– Schwarz inequality, and the convexity inequality (4.7), it is easily checked that Z Z 1 T h(x; t) ; | − u0 (x)|i dx dt T 0 I 1=2  Z TZ 1 ≤ CI h(x; t) ; G(; u0 (x))i dx dt : (4.8) T 0 I Let { n } ∈ C0∞ (R) be a sequence of test functions such that lim

n→∞

n

= g0 (u0 )

0

in Lr (R)

P.G. LeFloch, R. Natalini / Nonlinear Analysis 36 (1999) 213 – 230

227

with 1=r + 1=r 0 = 1. Using the uniform bound in Lq available for {u0;  }, we get Z

Z

T

h(x; t) ; G(; u0 (x))i dx dt

I

0

Z ≤

Z

T

Z

R

0

h(x; t) ; u0 − i

n dx dt + T

R\I

|u0 |2 dx + 2T ku0 kLr (R) kg0 (u0 )

− n kLr0 (R) :

(4.9)

Taking an increasing sequence of compact sets Ki covering R, i.e. such that I ⊂ K1 ⊂ K2 S∞ ⊂ · · · and i=1 Ki = R, we have Z

Z

T

Z h(x; t) ; G(; u0 (x))i dx dt ≤

I

0

T

Z Ki

0

h(x; t) ; G(; u0 (x))i dx dt:

This, together with Eq. (4.9) with I replaced by Ki , yields 1 T

Z

T

Z h(x; t) ; Q(; u0 (x))i dx dt

I

0

1 ≤ T

Z

T

Z R

0

h(x; t) ; u0 (x) − i

n

dx dt + 2ku0 kLr (R) kg0 (u0 ) −

n kLr 0 (R) ;

(4.10)

since Z lim

i→∞

R\Ki

|u0 |2 dx = 0:

Therefore, in view of Eqs. (4.8) and (4.10), the strong consistency property (2.6) will be established if we show Z

1 lim T →0+ T

0

T

Z R

h(x; t) ; u0 (x) − i

n

dx dt ≤ 0

(4.11)

for all n. By de nition of the Young measure (Eq. (2.2)), we have 1 T

Z 0

T

Z R

h(x; t) ; u0 (x) − i

n

Z

1 ; →0 T

dx dt = lim

T

0

Z

(u0 (x) − u;  (x; t))

n

dx dt:

On the other hand, we can write Z R

(u0 (x) − u0;  (x; t)) n (x) dx −

1 T

Z T Z Z 0

R

0

t

@s u;  (x; s) ds

 n (x) dx dt

=: A + B:

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P.G. LeFloch, R. Natalini / Nonlinear Analysis 36 (1999) 213 – 230

In view of the weak consistency property (4.1), the term A tends to zero as  → 0. By arguing as in the derivation of Eq. (4.3), we nd  Z Z Z t 1 T (−@x f(u;  ) + @x ( (ux;  )) − @3x u;  n (x) dx dt B=− T 0 R 0 Z TZ Z t 1 =− (f(u;  )@x n −  (ux;  )@x n + u;  @3x n ) ds dx dt T 0 R 0 ≤ Cn T: This shows inequality (4.11). The proof of Theorem 4.4 is completed. Proof of Theorem 4.3. It is based on modi ed estimates in inequalities (4.5) and (4.6), arguing as in Schonbek [30]. Namely, to estimate T3 , we use the energy estimate (3.1) and the fact that () = , as follows: Z Z Z Z |0 (u)||ux ||xx | dx dt +  ux2 |x | dx dt |hT3 ; i| ≤  R×(0; T )

Z Z ≤ C

supp 

R×(0; T )

|ux |2 dx dt

1=2

Z Z + C

supp 

|ux |2 dx dt

≤ C(−1=2 + −1 ): Finally, we estimate T4 in the following way, using Eq. (3.7) in Lemma 3.4, Z Z 00  (u)ux uxx  dx dt |hT4 ; i| =  Z Z ≤ C

supp 

|ux |2 dx dt

1=2 Z Z supp 

|uxx |2 dx dt

1=2

≤ C−1=2 −2=3 −1=2 ≤ C1=3 −1 : Then these two terms tends to zero, if  = o(3 ). Acknowledgements The authors would like to thank Luis Vega for helpful conversations about nonlinear dispersive equations, and Brian T. Hayes for fruitful discussions on the content of this paper. References [1] J.L. Bona, W.G. Pritchard, L.R. Scott, An evaluation of a model equation for water waves, Philos. Trans. Roy. Soc. London Ser. A 302 (1981) 457– 510.

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