Conservation laws with vanishing nonlinear diffusion and dispersion
aa r X i v : . [ m a t h . A P ] N ov CONSERVATION LAWS WITH VANISHINGNONLINEAR DIFFUSION AND DISPERSION
Philippe G. LeFloch Current address : Laboratoire Jacques-Louis LionsCentre National de la Recherche ScientifiqueUniversit´e de Paris 64, Place Jussieu, 75252 Paris, France.and
Roberto Natalini Istituto per le Applicazioni del Calcolo “M. Picone”Consiglio Nazionale delle RicercheViale del Policlinico 13700161 Roma, Italy
ABSTRACT . We study the limiting behavior of the solutions to a class of conservationlaws with vanishing nonlinear diffusion and dispersion terms. We prove the convergenceto the entropy solution of the first order problem under a condition on the relative sizeof the diffusion and the dispersion terms. This work is motivated by the pseudo-viscosityapproximation introduced by Von Neumann in the 50’s.
Key words and phrases:
Nonlinear dispersive waves, Korteweg-de Vries equation, pseudo-viscosity, Burgers equation, shock waves, measure-valued solutions.Published in : Nonlinear Analysis 36 (1999), 213–230.This work was partially carried out in the Summer 1992 during a visit of the first author at theIstituto per le Applicazioni del Calcolo E-mail address: [email protected]. E-mail address: [email protected] Conservation Laws with Diffusion and Dispersion
1. Introduction
This paper is concerned with the convergence of smooth solutions u = u ε,δ tothe initial value problem for the nonlinear dispersive equation ∂ t u + ∂ x f ( u ) = ε ∂ x β ( ∂ x u ) − δ ∂ x u, ( x, t ) ∈ IR × (0 , ∞ ) , (1.1)with the initial condition u ( x,
0) = u ε,δ ( x ) , x ∈ IR , (1.2)where the parameters ε > δ > u ε,δ is an ap-proximation of a given initial condition u : IR → IR. The flux f = f ( u ) andthe (degenerate) viscosity β = β ( λ ) are given smooth functions satisfying certainassumptions to be listed shortly. When ε = 0, the equation (1.1) reduces to thegeneralized Korteweg-de Vries (KdV) equation, the original one corresponding tothe special flux f ( u ) = u /
2. On the other hand, when δ = 0, we recover a non-linear degenerate parabolic equation. The first case was extensively studied frommathematical and numerical standpoints; see for instance [13, 14, 6] and the ref-erences therein (see also Section 2 for some background). For δ = 0 and undersuitable assumptions, this equation was treated in [22] as a simplified model of thepseudo viscosity approximation proposed by von Neumann and Richtmyer [24] andthen studied in [26, 25] for numerical purposes.In this paper we prove, under an assumption on the relative size of the parameters ε and δ , that the sequence { u ε,δ } converges to an entropy solution u of the firstorder hyperbolic conservation law ∂ t u + ∂ x f ( u ) = 0 (1.3)as ε and δ →
0. Throughtout the paper, we assume the following conditions on f and β :(A ) There are two constants C > m > | f ′ ( u ) | ≤ C (1 + | u | m − ) for all u ∈ IR . (A ) The function β = β ( λ ) is non-decreasing and satisfies β (0) = 0.Observe that, from (A ), we can deduce β ( λ ) λ ≥ λ ∈ IR. The followingassumption will also be in use:(B ) There exist constants C , C , N >
0, and r ≥ C | λ | r ≤ β ( λ ) λ ≤ C | λ | r for all | λ | ≥ N. The function β , therefore, is at least quadratic and could vanish on a boundedinterval. Two other assumptions on the function β will be required for certainresults below: .G. LeFloch and R. Natalini 3 (B ) There are constants C > N > β ( λ ) λ ≥ C | λ | for all | λ | ≥ N. (B ) There are constants C > r ≥ β ( λ ) λ ≥ C | λ | r for all λ ∈ IR . Observe that both (B ) and (B ) are refinements to the lower bound estimatein (B ), and that (B ) is actually a consequence of (B ). For our main theorems,we shall assume either (B ), or (B ) and (B ). The condition (B ) will be relevantonly in the course of the discussion of the proofs. One typical example of a function β satisfying (A ) and (B ) (and also (B ), (B )) is given by β ( λ ) = | λ | r − λ ( r ≥ . (1.4)Recall that, for the Cauchy problem for (1.3), existence and uniqueness of anentropy solution were first proved by Kruzkov [16] (Section 2 below). The vanishingviscosity limit , which corresponds to δ = 0 and ε →
0, was studied by severalauthors in the special case of a linear (and therefore non-degenerate) diffusionterm β ( λ ) = λ ; this activity started with Oleinik’s [23] and Kruzkov’s [16] works.For more general viscosity coefficients and under the assumptions (A )–(B ), theconvergence was proved in [22].The vanishing dispersion limit ( ε = 0 and δ →
0) has also been widely investi-gated, see Lax-Levermore [20], [7, 30, 31], and the survey paper by Lax [19]. Themain source of difficulty lies in the highly oscillating behavior of the solutions u ,δ ,which do not converge in a strong topology.The complete equation (1.1) arises, at least for the linear diffusion β ( λ ) = λ (the so called Korteweg-de Vries-Burgers equation), as a model of the nonlinearpropagation of dispersive and dissipative waves in many different physical systems.For related studies, let us refer to [1, 2, 3, 10, 11]. Travelling waves were studied in[2] and [9] and (partial) analytical results can be found in [15] and [6]. For a studyof numerical approximations with similar features as (1.1), see [4].The vanishing dispersive and diffusion limit (both ε and δ tend to zero) wasfirst studied by Schonbek [27]. She assumed β ( λ ) = λ and used the compensatedcompactness method introduced by Tartar [29]. In particular, she proved thatthe sequence { u ε,δ } converges to a weak solution to (1.3) (but not necessarily theunique entropy one) under the assumption that either δ = 0( ε ) for f ( u ) = u / δ = 0( ε ) for arbitrary subquadratic flux-functions f (i.e. take m = 2 in (A )).Our inequalities on ε and δ requires that viscosity dominates dispersion, whichis expected since { u ε,δ } are known not to converge to a weak solution of (1.3)(a fortiori to the entropy one) when dispersion effects are dominant.Here we consider the general nonlinear equation (1.1) and we show that, underthe assumption (B ) and if δ ≤ C ε − m − m ( m < u ε,δ converge to Conservation Laws with Diffusion and Dispersion the entropy weak solution of (1.3) (Theorem 4.1). Under the assumptions (B ) and(B ) and when δ ≤ C ε − mr (5 − m ) − ( m < − /r, r ≥ β ( λ ) = λ and f is subquadratic.The main tool necessary to deal with these singular limits is the concept ofentropy measure-valued solution to the equation (1.3). Measure-valued solutionswere introduced by DiPerna [7] (see Section 2) to represent the weak- ⋆ limits ofthe solutions of (1.1), According to DiPerna’s theory, the strong convergence of anapproximate sequence follows once one obtains(i) uniform bounds in L ∞ (0 , T ; L q (IR)) for all T > q > m (where m is givenby (A );(ii) weak consistency of the sequence with the entropy inequalities and the strongconsistency with the initial data (Cf. the conditions (2.5) and (2.6) of Section2).Such a framework can be used to establish as well the convergence of differenceschemes as shown in [5]. Observe however that, because of the oscillatory behav-ior of u ε,δ , no maximum principle is available for the equation (1.1) and the L q (uniform) estimates are more natural. Therefore we actually use the L q version ofDiPerna’s result derived by Szepessy in [28].One basic estimate that does not require any restriction on the parameters ε and δ is the so-called energy estimate based on the simplest quadratic function u (Lemma 3.1). The derivation of the necessary L q estimates with larger exponents( q >
2) is based on the nonlinearity of the viscosity coefficient β and requiressuitable restriction on ε and δ .The problem studied in this paper is motivated by the numerical method pro-posed by Von Neumann and Richtmyer [24] (see also [26]) and based on the pseudo-viscosity approximation β ( λ ) = | λ | λ . An important advantage of the nonlinear(quadratic) diffusion term β ( λ ) is that it adds sharply localized viscosity near shocksand a small quantity (possibly zero if β vanishes for λ <
0) elsewhere. The pseudo-viscosity idea was proposed by Von Neumann to stabilize an unstable scheme suchas the Lax-Wendroff scheme [21, 18, 8]: a minimal amount of numerical viscosityis 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, entropymeasure valued (mv) solutions (Section 2.1), and dispersive equations (Section 2.2).
Following Schonbek [27], we describe a representation theorem for Young mea-sures associated with a sequence of uniformly bounded functions of L q where q ∈ (1 , ∞ ) is fixed in the whole of this subsection. The corresponding settingin L ∞ was first established by Tartar [29]. .G. LeFloch and R. Natalini 5 Lemma 2.1.
Let { u j } be a uniformly bounded sequence in L ∞ (IR + ; L q (IR)) . Thenthere exists a subsequence { u j ′ } and a weak- ⋆ measurable, mapping ν : IR × IR + → Prob (IR) taking its values in the spaces of non-negative measures with total massone (probability measures) such that, for all functions g ∈ C (IR) satisfying g ( u ) = 0( | u | r ) as | u | → ∞ (2.1) for some r ∈ [0 , q ) , the following limit representation holds lim j ′ →∞ Z Z IR × IR + g ( u j ′ ( x, t )) φ ( x, t ) d x d t = Z Z IR × IR + Z IR g ( λ ) d ν ( x,t ) ( λ ) φ ( x, t ) d x d t (2.2) for all φ ∈ C ∞ (IR × IR + ) . The measure-valued function ν ( x,t ) is a Young measure associated with the se-quence { u j } . The following result reveals the connection between the structure of ν and the strong convergence. Lemma 2.2.
Suppose that ν is a Young measure associated with a sequence { u j } that is uniformly bounded in L ∞ (IR + ; L q (IR)) . For u ∈ L ∞ (IR + ; L q (IR)) , thefollowing statements are equivalent: (i) lim j →∞ u j = u in L ∞ (IR + ; L r loc (IR)) for some r ∈ [1 , q ) ; (ii) ν ( x,t ) = δ u ( x,t ) for almost every ( x, t ) ∈ IR × IR + . In (ii) above, the notation δ u ( x,t ) is used for the Dirac mass defined by Z Z IR × IR + < δ u ( x,t ) , g ( · ) > φ ( x, t ) d x d t = Z Z IR × IR + g ( u ( x, t )) φ ( x, t ) d x d t for all g ∈ C (IR) satisfying (2.1) and all φ ∈ C ∞ (IR × IR + ). Following DiPerna[7] and [28], we now define the measure-valued solutions to the first order Cauchyproblem ∂ t u + ∂ x f ( u ) = 0 , (2.3) u ( x,
0) = u ( x ) , x ∈ IR . (2.4) Definition 2.1.
Assume that f satisfies the growth condition (2.1) and u ∈ L (IR) ∩ L q (IR) . A Young measure ν associated with a sequence { u j } , which isassumed to be uniformly bounded in L ∞ (IR + ; L q (IR)) , is called an entropy mv so-lution to the problem (2.3)-(2.4) if ∂ t h ν ( · ) , | λ − k |i + ∂ x h ν ( · ) , sgn ( λ − k )( f ( λ ) − f ( k )) i ≤ in the distributional sense for all k ∈ IR and, for all interval I ⊆ IR , lim T → + T Z T Z I h ν ( x,t ) , | λ − u ( x ) |i d x d t = 0 . (2.6) Conservation Laws with Diffusion and Dispersion
Remark.
A function u ∈ L ∞ (IR + ; L (IR) ∩ L q (IR)) is an entropy weak solutionto (2.3)-(2.4) in the sense of Kruˇzkov [16] if and only if the Dirac measure δ u ( · ) isan entropy mv solution. In the case p = + ∞ , existence and uniqueness of suchsolutions were proved in [16]. The following results of entropy mv solutions wereproved in [28]: Theorem 2.3 states that entropy mv solutions are actually Kruzkov’ssolutions. Theorem 2.4 states that the problem has a unique L q solution. Theorem 2.3.
Assume that f satisfies (2.1) and u ∈ L (IR) ∩ L q (IR) . Supposethat ν is an entropy mv solution to (2.3)-(2.4) . Then there exists a function w ∈ L ∞ (IR + ; L (IR) ∩ L q (IR)) such that ν ( x,t ) = δ w ( x,t ) (2.7) for almost every ( x, t ) ∈ IR × IR + . Theorem 2.4.
Assume that f satisfies (2.1) and u ∈ L (IR) ∩ L q (IR) . Then thereexists a unique entropy solution u ∈ L ∞ (IR + ; L (IR) ∩ L q (IR)) to (2.3)-(2.4) which,moreover, satisfies k u ( · , t ) k L r (IR) ≤ k u k L r (IR) (2.8) for almost every t ∈ IR + and for all r ∈ [1 , q ] . Moreover the measure-valued mapping ν ( x,t ) = δ u ( x,t ) is the unique entropy mv solution of the same problem. Combining Theorems 2.3 and 2.4 and Lemma 2.2 we obtain our main convergencetool.
Corollary 2.5.
Assume that f satisfies (2.1) and u ∈ L (IR) ∩ L q (IR) . Let { u j } be a sequence of functions that are uniformly bounded in L ∞ (IR + ; L q (IR)) for q ≥ ,and let ν be a Young measure associated with this sequence. If ν is an entropy mvsolution to (2.3)-(2.4) , then lim j →∞ u j = u in L ∞ (IR + ; L rloc (IR)) (2.9) for all r ∈ [1 , q ) , where u ∈ L ∞ (IR + ; L q (IR)) is the unique entropy solution to (2.3)-(2.4) . Consider the fully nonlinear, KdV-type equation in one space variable ∂ t u + F ( u, ∂ x u, ∂ x u, ∂ x u ) = 0 (2.10)for ( x, t ) ∈ IR × (0 , T ) together with the Cauchy data u ( x,
0) = u ( x ) for x ∈ IR . (2.11)In recent years this class of problems has been extensively studied; see for in-stance [13, 14, 15] and the references cited therein. In particular, under suitable .G. LeFloch and R. Natalini 7 assumptions, these equations enjoy a gain of regularity of the solutions with respectto their initial data (Cf. [6, 12, 17] and the references above). We observe howeverthat a complete theory of global existence remains to be developped. Presenting acomplete review is far beyond the aim of this presentation. Here we need only recalla result –that is sufficient to deal with (1.1)– of local existence and uniqueness otthe smooth solutions to the equation (2.10). More details and proofs can be foundin the paper by Craig-Kappeler-Strauss [6] (see also [15]).Using the notation U = ( u , u , u , u ), the assumptions on the smooth function F = F ( U ) are as follows:(H ) There exists δ > ∂ u F ( U ) ≥ δ > U ∈ IR ;(H ) ∂ u F ( U ) ≤ U ∈ IR .The equation (1.1) does satisfy these hypotheses provided δ > ) holds. Theorem 2.6 (Uniqueness) . Let
T > be fixed and assume F satisfies (H )-(H ) .For any u ∈ H (IR) there is at most one solution u ∈ L ∞ ((0 , T ); H (IR)) to theproblem (2.10)-(2.11) . Theorem 2.7 (Existence) . Assume F satisfies (H )-(H ) . Let N be an integer ≥ and let C > be a given constant. There exists a time T > , depending onlyon C , such that for all u ∈ H N (IR) with k u k H ≤ C , there exists at least onesolution u ∈ L ∞ ((0 , T ); H N (IR)) to the problem (2.10)-(2.11) . Observe that the setting above based on a Sobolev norm of relatively high order( H (IR)) is the optimal result provided by the current techniques of analysis ofdispersive equations. The global existence of smooth solutions to (1.1)-(1.2) appearsto be an open problem. In the following we tacitly restrict attention to a time T ∗ ∈ (0 , ∞ ] chosen such that the problem (1.1)-(1.2) is well-posed in the stripIR × (0 , T ∗ ).
3. A Priori Estimates
In this section we consider a sequence { u ε,δ } of smooth solutions of (1.1)-(1.2)that vanish at infinity. We assume also that the initial data { u ε,δ } are smoothfunctions with compact support, and are uniformly bounded in L (IR) ∩ L q (IR) fora suitable q >
1. In what follows, whenever it does not lead to confusion, we omitthe indices ε and δ . Similarly all constants are denoted by C, C , · · · . We beginwith the natural energy estimate based the quadratic function u /
2. (Arbitraryconvex functions could not be easily used here because of the dispersive term.)
Lemma 3.1.
For any
T > , it holds that Z IR u ( x, T ) d x + 2 ε Z T Z IR β ( u x ( x, t )) u x ( x, t ) d x d t = Z IR u ( x ) d x. (3.1) Conservation Laws with Diffusion and Dispersion
Proof.
We multiply (1.1) by u and integrate in space. Integrating by parts weobtaindd t Z IR u d x + Z IR uf ′ ( u ) u x d x = − ε Z IR β ( u x ) u x d x − δ Z IR u u xxx d x. (3.2)The second terms on both sides of (3.2) vanish identically, since we can write theseterms in a conservative form: u f ′ ( u ) u x = ( G ( u )) x with G ′ = u f ′ ( u ) and u u xxx = ( u u xx − u x ) x . The estimate (3.1) follows therefore from (3.2). (cid:3)
At this stage, in view of the assumption (A ) and Lemma 3.1, we have a uniformcontrol of u in L ∞ (IR + ; L (IR)) and ε β ( u x ) u x in L ((0 , T ) × IR) for every
T > u in L ∞ norm, which will beuniform in ε but not in δ . We first provide a second energy-type estimate. Lemma 3.2.
Let F be defined by F ′ ( u ) = f ( u ) . For every T > , we have δ Z IR u x ( x, T ) d x − Z IR F ( u ( x, T )) d x + ε δ Z T Z IR u xx β ′ ( u x ) d x d t = δ Z IR u ,x ( x ) d x − Z IR F ( u ( x )) d x + ε Z T Z IR f ′ ( u ) β ( u x ) u x d x d t. (3.3) Proof.
Multiplying (1.1) by f ( u ) + δu xx we obtain the equality0 = Z IR { F ( u ) t + H ( u ) x − εf ( u ) β ( u x ) x + δf ( u ) u xxx } d x + Z IR { δ u xx u t + δu xx f ( u ) x − δu xx β ′ ( u x ) + δu xxx u xx } d x where we have set H ′ = f f ′ . After integration by parts in space, we havedd t Z F ( u ) d x − δ u x d x + Z IR εf ′ ( u ) β ( u x ) u x d x − ε δ Z IR u xx β ′ ( u x ) d x = 0 , which gives the desired conclusion. (cid:3) Using the bound for √ δ u x in L ∞ (IR + ; L (IR)) that follows from Lemma 3.2, weare now able to estimate u in the L ∞ norm. .G. LeFloch and R. Natalini 9 Lemma 3.3. If m < in the assumption (A ) , there exists a constant C > suchthat | u ( x, t ) | ≤ C δ − − m (3.4) for all ( x, t ) ∈ IR × IR + .Proof. In view of (A ) and for all u ∈ IR, we have | F ( u ) | ≤ C (1 + | u | m +1 ) . The main idea is to use (3.3) to control δ u x in terms of F ( u ), the latter beingestimated from the above growth condition. We deduce from (3.3) that δ Z IR u x ( x, T ) d x + ε δ Z T Z IR u xx β ′ ( u x ) d x d t = δ Z IR u ,x ( x ) d x + Z IR (cid:0) F ( u ( x, T )) − F ( u ( x )) (cid:1) d x + ε Z T Z f ′ ( u ) β ( u x ) u x d x d t ≤ C + C k u ( · , T ) k m − L ∞ (IR) k u ( · , T ) k L (IR) + ε Z T Z IR β ( u x ) u x d x d t ! , where we used (A ) as well. (We do not explicitly write the terms involving theinitial data as it is assumed to be a smooth function.) Therefore in view of (3.1),we arrive at δ Z IR u x ( x, T ) d x + ε δ Z T Z IR u xx β ′ ( u x ) d x d t ≤ C + C k u ( · , T ) k m − L ∞ (IR) . Using the Cauchy-Schwartz inequality, we obtain | u ( x, t ) | ≤ Z x −∞ | u ( y, t ) u x ( y, t ) | d y ≤ √ δ k u ( · , t ) k L (IR) √ δ k u x ( · , t ) k L (IR) ≤ C √ δ k u k L (IR) (cid:16) k u ( · , t ) k m − L (IR) (cid:17) / . Hence, for all t > k u ( · , t ) k L ∞ (IR) ≤ Cδ (cid:16) k u ( · , t ) k m − L ∞ (IR) (cid:17) . (3.5)Now, since m <
5, the growth of the left hand side of (3.5) exceeds the growthof the right hand side. So we can check that (3.5) implies k u ( · , t ) k L ∞ (IR) ≤ max , (cid:18) Cδ (cid:19) − m ! ≤ C δ − − m . Namely, setting y = k u ( · , t ) k L ∞ , we have for y > y ≤ Cδ (cid:0) y m − (cid:1) . (3.6)If y ≤ y > (cid:0) Cδ (cid:1) − m , thenwe would deduce that y > Cδ y m − > Cδ (cid:0) y m − (cid:1) , which would contradict the inequality (3.6). (cid:3) In view of the proof of Lemma 3.3 we also state the following result:
Lemma 3.4.
For any
T > we have Z IR u x ( x, T ) d x + ε Z T Z IR u xx β ′ ( u x ) d x d t ≤ C δ − − m . (3.7)Finally we derive uniform bounds in L ∞ ((0 , T ); L q (IR)) with q ≤
5. The follow-ing result provides a uniform estimate in the Lebesgue space L (IR) (so improvingupon the L bound in Lemma 3.1) by taking advantage of the nonlinearity propertyof β (for large values of λ = u x ) as stated in (B ). Proposition 3.5.
Assume that the assumption (B ) holds and m < in the as-sumption (A ) . There exists a constant C > , which depends only on the initialdata, such that, for all δ and ε small enough and for every T > , we have sup t ∈ (0 ,T ) k u ( · , t ) k L (IR) ≤ C (cid:16) T + ε − δ − m − m (cid:17) . (3.8) Proof.
Set η ( u ) = | u | . Multiply (1.1) by η ′ ( u ) and integrate on IR × (0 , T ). Itfollows easily that Z IR η ( u ( x, T )) d x + ε Z T Z IR η ′′ ( u ) β ( u x ) u x d x d t = Z IR η ( u ( x )) d x − δ Z T Z IR η ′′′ ( u ) u x d x d t. (3.9)On the other hand, according to (B ), there exists a constant C such that for all λ ∈ IR | λ | ≤ C (1 + β ( λ ) λ ) . .G. LeFloch and R. Natalini 11 Using the latter in the energy estimate (3.1), we are able to control the second termin the right hand side of (3.9): (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z η ′′′ ( u ) u x d x d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z T Z IR | u | | u x | d x d t ≤ C Z T Z IR | u | (cid:0) β ( u x ) u x (cid:1) d x d t ≤ C T sup t ∈ (0 ,T ) k u ( t ) k L (IR) + Cε k u k L ∞ (IR × (0 ,T )) ≤ C T + C (cid:16) ε − δ − m − m (cid:17) . where the latter inequality follows from the L ∞ bound in Lemma 3.3. Returningto (3.9), we obtain (3.8). (cid:3) Under the stronger assumption (B ), we have a sharper estimate. Proposition 3.6.
Assume that (B ) holds for a given r ≥ and that m < q ≡ − /r in (A ) . There exists a constant C > , which depends only on the initialdata, such that, for all δ and ε small enough and every T > , sup t ∈ (0 ,T ) k u ( · , t ) k qL q (IR) ≤ C (cid:16) T − /r + ε − /r δ − q − m − m (cid:17) . (3.10) Proof.
As in the proof of Proposition 3.5, we consider the formula (3.9) but nowwith η ( u ) = | u | q with q = 5 − /r . The assumption (B ) yields, using (3.1) and(3.4), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z IR η ′′′ ( u ) u x d x d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z T Z IR | u | q − | u x | d x d t ≤ C Z T Z IR | u | r ′ ( q − d x d t ! /r ′ Z T Z IR β ( u x ) u x d x d t ! /r ≤ C ε − /r k u k /rL ∞ (IR) T /r ′ sup t ∈ (0 ,T ) k u ( · , t ) k /rL (IR) ≤ C T /r ′ ε − /r δ − r (5 − m ) , where 1 /r + 1 /r ′ = 1 and since r ′ ( q −
3) = 2 + r ′ /r . Now (3.10) follows from theformula (3.9). (cid:3) Observe that the estimates (3.8) and (3.10) hold only thanks to the nonlinearform of the viscosity term. Note that when δ = 0, these estimates reduce to boundthat are uniform in ε , which is expected for conservation laws with viscosity but no dispersion. On the other hand, most of our estimates blow-up when taking ε = 0and provide no control of L q norms. Observe however that the L bound in Lemma3.1 is uniform for δ arbitrary and ε →
0. Loosely speaking, this is also the caseof the estimate in Proposition 3.6 if r could be chosen to be r = ∞ , in which case(3.10) becomes a uniform L estimate. Such a choice of r is not allowed however,cf. (B ).For the sake of completeness, we finally state an analogous estimate for lineardiffusions which was proved by Schonbek [27]. Proposition 3.7.
Let β ( λ ) = λ and take m = 2 in (A ) . For any T > , thereexists a constant C T > , which depends only on the initial data, such that for δ ≤ ν ε sup t ∈ (0 ,T ) k u ( · , t ) k L (IR) ≤ C T . (3.11)
4. Convergence Results
In Section 3, we have established several uniform bounds for the sequence { u ε,δ } of solutions to the Cauchy problem (1.1)-(1.2) under certain assumptions on thefunctions f , β , and the parameters ε and δ . Assume again that the initial data u ε,δ are smooth with compact support and that there exists a limiting function u ∈ L (IR) ∩ L q (IR) and a suitable q > δ = O ( ε ),lim ε → u ε,δ = u in L (IR) ∩ L q (IR) . (4.1)Returning to the proofs of Section 3, it is not hard to see that the following con-ditions on the initial data are sufficient for the estimates therein to hold uniformlywith respect to a class of initial data: k u ε,δ k L (IR) + k u ε,δ k L q (IR) δ / k u ε,δ ,x k L (IR) ≤ C. In this section we prove the strong convergence of the sequence u ε,δ . Theorem 4.1.
Assume that (B ) holds and m < in (A ) . Let u ε,δ be a sequenceof smooth solutions to (1.1)-(1.2) on IR × (0 , T ) (for a given T > ), which vanish atinfinity and are associated with initial data satisfying (4.1) with q = 5 . If there is aconstant C > such that δ ≤ C ε − m − m , then the (whole) sequence u ε,δ converges to afunction u ∈ L ∞ ((0 , T ); L (IR)) , which is the unique entropy solution to (2.3)-(2.4) . Theorem 4.2.
Assume that (B ) and (B ) hold and m < − /r in (A ) . Let u ε,δ be a sequence of smooth solutions to (1.1)-(1.2) on IR × (0 , T ) (for a given T > ), which vanish at infinity and are associated with initial data satisfying (4.1) with q = 5 − /r . If there is a constant C > such that δ ≤ C ε − mr (5 − m ) − , thenthe (whole) sequence converges to a function u ∈ L ∞ ((0 , T ); L q (IR)) , q = 5 − /r ,which is the unique entropy solution to (2.3)-(2.4) . Let us give an analogous statement in the case β ( λ ) = λ , which improve upon[27] (Cf. Theorem 5.1 therein). .G. LeFloch and R. Natalini 13 Theorem 4.3.
Assume m = 2 in (A ) and let β ( λ ) = λ . If δ ≤ C ε , the whole se-quence u ε,δ of solutions of (1.1)-(1.2) converges to a function u ∈ L ∞ ((0 , T ); L (IR)) ,which is the unique entropy solution to (2.3)-(2.4) . Similar convergence results can be proven for the case m = 3 or for some specialflux-functions along the lines of what was done in [27] (Cf. Sections 4 and 5 therein).Theorems 4.1-4.2 follow easily by simply using the following general result and the L q bounds derived in Section 3, Proposition 3.5 and Proposition 3.6 respectively. Theorem 4.4.
Assume that (B ) holds. Let u ε,δ be a sequence of smooth solutionsto (1.1)-(1.2) on IR × (0 , T ) (for a given T > ) associated with initial data satisfying (4.1) . If the sequence is uniformly bounded in L ∞ ((0 , T ); L q (IR)) for q > m and δ = o ( ε /r ) , then the (whole) sequence converges to a function u ∈ L ∞ ((0 , T ); L q (IR)) ,which is the unique entropy solution to (2.3)-(2.4) .Proof of Theorem 4.4. First of all let us establish that, for any convex function η = η ( u ) such that η ′ , η ′′ , η ′′′ are uniformly bounded on IR, we haveΛ ε,δ = ∂ t η ( u ε,δ ) + ∂ x Q ( u ε,δ ) ⇀ D ′ (IR × IR + ) , (4.2)where Q ′ = f ′ η ′ . To begin with, observe thatΛ ε,δ = ε ( η ′ ( u ) β ( u x )) x − εη ′′ ( u ) β ( u x ) u x − δ ( η ′ ( u ) u xx ) x + δη ′′ ( u ) u x u xx = T + T + T + T . To estimate T , we use the assumption (B ), which implies | β ( λ ) | ≤ C (cid:0) | λ | r − (cid:1) for all λ . For any given θ ∈ C ∞ (IR × (0 , T )), θ ≥
0, and for p = r r − > p ′ being the conjugate exponent of p ), we get h T , θ i = (cid:12)(cid:12)(cid:12)(cid:12)Z Z ε θ x η ′ ( u ) β ( u x ) d x d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ε k θ x k L (IR × (0 ,T )) + C ε k θ x k L p ′ (IR × (0 ,T )) (cid:18)Z Z | u x | p (3 r − (cid:19) /p , so using the derivative estimate in (3.1) and (B ) again: h T , θ i ≤ C (cid:0) ε + ε / r (cid:1) ≤ C ε / r . (4.3)The second term T is nonpositive, namely h T , θ i = − Z Z εη ′′ ( u ) β ( u x ) u x θ d x d t ≤ . (4.4) To estimate T , we use the energy estimate (3.1) and the fact that β is at leastquadratic. We write h T , θ i = δ Z Z IR × (0 ,T ) θ x η ′ ( u ) u xx d x d t = δ Z Z IR × (0 ,T ) θ x (cid:0) ( η ′ ( u ) u x ) x − η ′′ ( u ) u x (cid:1) d x d t ≤ − δ Z Z IR × (0 ,T ) θ xx η ′ ( u ) u x + δ Z Z IR × (0 ,T ) | θ x | u x d x d t ≤ C δ + C δ (cid:18)Z Z supp θ | u x | r d x d t (cid:19) / r + C ε − / r , where supp θ denotes the support of the function in IR × (0 , T ), thus h T , θ i ≤ C δ (cid:16)
C ε − / r + ε − / r (cid:17) ≤ C (cid:16) C ε − / r (cid:17) . (4.5)Finally we deal with T as follows: |h T , θ i| = (cid:12)(cid:12)(cid:12)(cid:12) δ Z Z η ′′ ( u ) u x u xx θ d x d t (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) δ Z Z θ x η ′′ ( u ) u x + δ Z Z θη ′′′ ( u ) 12 u x (cid:12)(cid:12)(cid:12)(cid:12) ≤ C δ (cid:18)Z Z supp θ | u x | + | u x | d x d t (cid:19) , so |h T , θ i| ≤ C δ (cid:16) ε − / r + ε − /r (cid:17) ≤ C ε − /r . (4.6)Therefore, if δ = o ( ε /r ), (4.2) follows immediately from the estimate (4.3)-(4.6).To apply Corollary 2.5 we have to show that (2.5) and (2.6) are satisfied for a Youngmeasure ν associated with the sequence u ε,δ . It is a standard matter to deduce, forall convex entropy pairs, ∂ t h ν ( · ) , η ( λ ) i + ∂ x h ν ( · ) , Q ( λ ) i ≤ k ∈ IR then followsby using a standard regularization of the function | u − k | . Concerning the initialdata and in orderto establish (2.6), we now combine the entropy inequalities and the weak consis-tency property as was suggested by DiPerna [7]. We follow the detailled argumentsgiven in [28]. .G. LeFloch and R. Natalini 15 Consider the function g ( λ ) = | λ | r for 1 < r < min (2,q), and set G ( λ, λ ) ≡ g ( λ ) − g ( λ ) − g ′ ( λ )( λ − λ ) ≥ r ( r − λ − λ ) (1 + | λ | + | λ | ) − r . (4.7)Let I ⊆ IR be be a closed and bounded interval. Using the Jensen inequality, theCauchy-Schwartz inequality, and the above convexity inequality (4.7), it is easilychecked that 1 T Z T Z I h ν ( x,t ) , | λ − u ( x ) |i d x d t ≤ C I T Z T Z I h ν ( x,t ) , G ( λ, u ( x )) i d x d t ! / . (4.8)Let { ψ n } ∈ C ∞ (IR) be a sequence of test-functions such thatlim u →∞ ψ n = g ′ ( u ) = in L r ′ (IR) , where 1 = 1 /r + 1 /r ′ . Using the uniform bound in L q available for the sequence { u ε,δ } , we get Z T Z I h ν ( x,t ) , G ( λ, u ( x )) i d x d t ≤ Z T Z IR h ν ( x,t ) , u − λ i ψ n d x d t + T Z IR r I | u | d x + 2 T k u k L r (IR) k g ′ ( u ) − ψ n k L r ′ (IR) . (4.9)Taking an increasing sequence of compact sets K i covering IR, i.e. such that I ⊂ K ⊂ K ⊂ . . . and S ∞ i =1 K i = IR, we have Z T Z I h ν ( x,t ) , G ( λ, u ( x )) i d x d t ≤ Z T Z K i h ν ( x,t ) , G ( λ, u ( x )) i d x d t, which, together with (4.9) where I is replaced by K i , yields1 T Z T Z I h ν ( x,t ) , Q ( λ, u ( x )) i d x d t ≤ T Z T Z IR h ν ( x,t ) , u ( x ) − λ i ψ n d x d t + 2 k u k L r (IR) k g ′ ( u ) − ψ n k L r ′ (IR) , (4.10) since lim i →∞ Z IR r K i | u | d x = 0 . Therefore, in view of (4.8) and (4.10), the strong consistency property (2.6) willbe established if we show thatlim T → + T Z T Z IR h ν ( x,t ) , u ( x ) − λ i ψ n d x d t ≤ n ∈ N . By definition of the Young measure (Cf. (2.2)), we have1 T Z T Z IR h ν ( x,t ) , u ( x ) − λ i ψ n d x d t = lim ε,δ → T Z T Z (cid:0) u ( x ) − u ε,δ ( x, t ) (cid:1) ψ n d x d t. On the other hand, we can write= Z IR (cid:0) u ( x ) − u ε,δ ( x, t ) (cid:1) ψ n ( x ) d x − T Z T Z IR (cid:18)Z t ∂ s u ε,δ ( x, s ) d s (cid:19) ψ n ( x ) d x d t ≡ A + B. The term A tends to zero as ε → B = − T Z T Z IR (cid:18)Z t ( − ∂ x f ( u ε,δ ) + ε∂ x ( β ( u ε,δx )) − δ∂ x u ε,δ (cid:19) ψ n ( x ) d x d t = − T Z T Z IR Z t (cid:0) f ( u ε,δ ) ∂ x ψ n − εβ ( u ε,δx ) ∂ x ψ n + δu ε,δ ∂ x ψ n (cid:1) d s d x d t ≤ C n T. This leads to the inequality (4.11). The proof of Theorem 4.4 is completed. (cid:3)
The proof of Theorem 4.3 is based on slightly modified estimates in the inequal-ities (4.5) and (4.6), which can be derived by arguing as in [27]. The details of theproof are omitted.
Acknowledgments.
The authors would like to thank Luis Vega for helpful con-versations about nonlinear dispersive equations, and Brian T. Hayes for generaldiscussions on the content of this paper.
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