ZERO DIFFUSION-DISPERSION-SMOOTHING LIMITS FOR SCALAR CONSERVATION LAW WITH DISCONTINUOUS FLUX FUNCTION

We consider multidimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach of 𝐻-measures to investigate the zero diffusion-dispersion-smoothing limit.


Introduction
We consider the convergence of smooth solutions u u ε t, x with t, x ∈ R × R d of the nonlinear partial differential equation International Journal of Differential Equations for p > 2 and every l > 0. The aim is to show convergence to a weak solution of the corresponding hyperbolic conservation law: We refer to this problem as the zero diffusion-dispersion-smoothing limit.
In the case when the flux f is at least Lipschitz continuous, it is well known that the Cauchy problem corresponding to 1.3 has a unique admissible entropy solution in the sense of Kružhkov 1 or measure valued solution in the sense of DiPerna 2 .The situation is more complicated when the flux is discontinuous and it has been the subject of intensive investigations in the recent years see, e.g., 3 and references therein .The one-dimensional case of the problem is widely investigated using several approaches numerical techniques 3, 4 , compensated compactness 5, 6 , and kinetic approach 7, 8 .In the multidimensional case there are only a few results concerning existence of a weak solution.In 9 existence is obtained by a two-dimensional variant of compensated compactness, while in 10 the approach of H-measures 11, 12 is used for the case of arbitrary space dimensions.Still, many open questions remain such as the uniqueness and stability of solutions.
A problem that has not yet been studied in the context of conservation laws with discontinuous flux, and which is the topic of the present paper, is that of zero diffusiondispersion limits.When the flux is independent of the spatial and temporal positions, the study of zero diffusion-dispersion limits was initiated in 13 and further addressed in numerous works by LeFloch et al. e.g., 14-17 .The compensated compactness method is the basic tool used in the one-dimensional situation for the so-called limiting case in which the diffusion and dispersion parameters are in an appropriate balance.On the other hand, when diffusion dominates dispersion, the notion of measure valued solutions 2, 18 is used.More recently, in 19 the limiting case has also been analyzed using the kinetic approach and velocity averaging 20 .
The remaining part of this paper is organized as follows.In Section 2 we collect some basic a priori estimates for smooth solutions of 1.1 .In Section 3 we look into the diffusiondispersion-smoothing limit for multidimensional conservation laws with a flux vector which is discontinuous with respect to spatial variable.In doing so we rely on the a priori estimates from the previous section in combination with Panov's H-measures approach 10 .Finally, in Section 4 we restrict ourselves to the one-dimensional case for which we obtain slightly stronger results using the compensated compactness method.

A priori Inequalities
Assume that the flux f in 1.1 is smooth in all variables.Consider a sequence u ε,δ ε,δ of solutions of ∂ t u div x f t, x, u εdiv x b ∇u δ d j 1 ∂ 3 x j x j x j u, u x, 0 u 0 x , x ∈ R d .

2.1
International Journal of Differential Equations We assume that u ε,δ ε,δ has enough regularity so that all formal computations below are correct.So, following Schonbek 13 , we assume that for every ε, δ > 0 we have u ε,δ ∈ L ∞ 0, T ; H 4 R d .
Later on, we will assume that the initial data u 0 depends on ε.In this section, we will determine a priori inequalities for the solutions of problem 2.1 .
To simplify the notation we will write u ε instead of u ε,δ .We will need the following assumptions on the diffusion term b λ b 1 λ , . . ., b n λ .H1 For some positive constants C 1 , C 2 we have 2.2 H2 The gradient matrix Db λ is a positive definite matrix, uniformly in λ ∈ R d , that is, for every λ, ∈ R d , there exists a positive constant C 3 such that we have

2.3
We use the following notation:

2.4
In the sequel, for a vector valued function g g 1 , . . ., g d defined on R × R d × R, we denote 2.5 The partial derivative ∂ x i in the point t, x, u , where u possibly depends on t, x , is defined by the formula In particular, the total derivative D x i and the partial derivative ∂ x i are connected by the identity Finally we use

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With the previous conventions, we introduce the following assumption on the flux vector f.H3 The growth of the velocity variable u and the spatial derivative of the flux f are such that for some C, α > 0, p ≥ 1, and every l > 0, we have max where μ ∈ M R × R d is a bounded measure and, accordingly, the above inequality is understood in the sense of measures .Now, we can prove the following theorem.
Theorem 2.1.Suppose that the flux function f f t, x, u satisfies (H3) and that it is Lipschitz continuous on R ×R d ×R.Assume also that initial data u 0 belongs to L 2 R d .Under conditions (H1)-(H2) the sequence of solutions u ε ε>0 of 2.1 for every t ∈ 0, T satisfies the following inequalities: for some constants C 4 and C 5 .
Proof.We follow the procedure from 19 .Given a smooth function η η u , u ∈ R, we define

2.12
If we multiply 2.1 by η u , it becomes

2.13
Choosing here η u u 2 /2 and integrating over 0, t × R d , we get

2.14
where the second equality sign is justified by the following partial integration:

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As for inequality 2.11 , we start by using 2.14 , namely, where C R dv/ 1 |v| 1 α .From here, using H3 , we conclude in particular that for some constant C 11 independent of ε.
Next, we differentiate 2.1 with respect to x k and multiply the expression by ∂ x k u.Integrating over R d , using integration by parts and then summing over k 1, . . ., d, we get:

2.18
Integrating this over 0, t and using the Cauchy-Schwarz inequality and condition H2 , we find

2.19
International Journal of Differential Equations 7 where C 3 is independent of ε.Then, using Young's inequality the constant C 3 is the same as previously mentioned we obtain

The Multidimensional Case
Consider the following initial-value problem.Find u u t, x such that For the flux f f 1 , . . ., f d we need the following assumption, denoted H4 .H4a For the flux f f t, x, u , t, x, u ∈ R ×R d ×R, we assume that f ∈ C R; BV R × R d and that for every l ∈ R we have max u∈ −l,l |f t, x, u

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H4b There exists a sequence f f 1 , . . ., f d , ∈ 0, 1 , such that f f t, x, u ∈ C 1 R × R d × R , satisfying for some p > 2 and every l ∈ R : where C i , i 1, 2, 3, and C are constants, while the function β : R → R is such that lim ρ → 0 β ρ 0. In the case when we have only vanishing diffusion, it is usually possible to obtain uniform L ∞ bound for the corresponding sequence of solutions under relatively mild assumptions on the flux and initial data see, e.g., 9, 10 .In the case when we have both vanishing diffusion and vanishing dispersion, we must assume more on the flux in order to obtain even much weaker bounds see Theorem 3.2 .We remark that demand on controlling the flux at infinity is rather usual in the case of conservation laws with vanishing diffusion and dispersion see, e.g., 16, 17, 19 .Remark 3.1.For an arbitrary compactly supported, nonnegative where β is a positive function tending to zero as → 0. In the case when the flux We also need to assume that the flux f is genuinely nonlinear, that is, for every is nonconstant on every nondegenerate interval of the real line.
International Journal of Differential Equations 9 We will analyze the vanishing diffusion-dispersion-smoothing limit of the problem where the flux f satisfies the conditions H4b .We denote the solution of 3.5 -3.6 by u ε u ε t, x .We assume that We also assume that ε → 0 and δ δ ε → 0 as ε → 0. We want to prove that under certain conditions, a sequence of solutions u ε ε>0 of 3.5 -3.6 converges to a weak solution of problem 3.1 as ε → 0. To do this in the multidimensional case we use the approach of H-measures, introduced in 11 and further developed in 10, 21 .In the one-dimensional case, we use the compensated compactness method, following 13 .
In order to accomplish the plan we need the following a priori estimates.
Theorem 3.2 a priori inequalities .Suppose that the flux f t, x, u satisfies (H4).Also assume that the initial data u 0 satisfies 3.7 .Under these conditions the sequence of smooth solutions u ε ε>0 of 3.5 -3.6 satisfies the following inequalities for every t ∈ 0, T : 3.9 for some constants C 10 , C 11 , C 12 (the constants C 4 , C 5 are introduced in Theorem 2.1).
Proof.For every fixed , the function f f 1 , . . ., f d is smooth, and, due to H4 , we see that f satisfies H3 .This means that we can apply Theorem 2.1.

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Replacing the flux f by f from 3.5 and u 0 by u 0,ε from 3.6 in 2.10 and 2.11 , we get

3.11
To proceed, we use assumption H4 .We have 3.12 which together with 3.10 immediately gives 3.8 .Similarly, combining H4 and 3.11 , and arguing as in 3.12 , we get 3.9 .
In this section, we will inspect the convergence of a family u ε ε>0 of solutions to 3.5 -3.6 in the case when for the function b appearing in the right-hand side of 3.5 .This is not an essential restriction, but we will use it in order to simplify the presentation.Thus, we use the following theorem which can be proved using the H-measures approach see, e.g., 10, Corollary 2 and Remark 3 .We let θ denote the Heaviside function.
Theorem 3.3 see 10 .Assume that the vector f t, x, u is genuinely nonlinear in the sense of We can now prove the following theorem.
Theorem 3.4.Assume that the flux vector f is genuinely nonlinear in the sense of 3.4 and that it satisfies (H4).Furthermore, assume that and that u 0,ε satisfies 3.7 .Then, there exists a subsequence of the family u ε ε>0 of solutions to 3.5 -3.6 that converges to a weak solution of problem 3.1 .
Proof.We will use Theorem 3.3.Since it is well known that the family u ε ε>0 of solutions of problem 3.5 -3.6 is not uniformly bounded, we cannot directly apply the conditions of Theorem 3.3.Take an arbitrary C 2 function S S u , u ∈ R, and multiply the regularized equation 3.5 by S u ε .As usual, put q t, x, u u 0 S v ∂ u f dv, q q 1 , . . ., q d .

3.16
We easily find that 3.17 We will apply this formula repeatedly with different choices for S u .
In order to apply Theorem 3.3, we will consider a truncated sequence T l u ε ε>0 , where the truncation function T l is defined for every fixed l ∈ N as

3.18
We will prove that the sequence T l u ε ε>0 is precompact for every fixed l.Denote by u l a subsequential limit in L 1 loc of the family T l u ε ε>0 , which gives raise to a new sequence u l l>1 that we prove converges to a weak solution of 3.1 .
To carry out this plan, we must replace T l by a C 2 regularization T l,σ : R → R. We define T l,σ : R → R by T l,σ 0 0 and

3.19
Next, we want to estimate T l,σ u ε ∇u ε L 2 R ×R d .To accomplish this, we insert the functions T ± l,σ for S in 3.17 where T ± l,σ are defined by T ± l,σ 0 0 and

3.21
Notice that

3.22
By inserting S u −T l,σ u , q q t, x, u

3.23
International Journal of Differential Equations 13 Similarly, for S u T − l,σ u , q q − t, x, u u 0 T − l,σ v ∂ u f dv, we have from 3.17

3.24
Adding 3.23 to 3.24 , we get x j x j u ε dxdt.

3.25
From 3.22 and the definition of q − and q , it follows x j x j u ε dxdt.

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Without loss of generality, we can assume that l > 1.Having this in mind, we get from H4 and 3.26

3.28
These estimates follow from H4 and the a priori estimates 3.8 , 3.9 .If in addition we use the assumption ε from 3.15 , we conclude

3.29
Thus, in view of 3.27 , 3.30 which is the sought for estimate for Next, take a function U ρ z satisfying U ρ 0 0 and

3.31
Clearly, U ρ is convex, and U ρ z → θ z in L p loc R as ρ → 0, for any p < ∞; as before, θ denotes the Heaviside function.
Inserting S u ε U ρ T l,σ u ε − c in 3.17 , we get

3.32
We rewrite the previous expression in the following manner: where

3.34
To continue, we assume that σ depends on ε in the following way:

3.35
From here, we will prove that the sequence T l u ε ε>0 satisfies the assumptions of Theorem 3.3.Accordingly, we need to prove that the left-hand side of 3.33 is precompact in To accomplish this, we use Murat's lemma 22, Chapter 1, Corollary 1 .More precisely, we have to prove the following.
i When the left-hand side of 3.33 is written in the form div Q ε , we have ii The right-hand side of 3.33 is of the form M loc,B H −1 loc,c , where M loc,B denotes a set of families which are locally bounded in the space of measures, and H −1 loc,c is a set of families precompact in H −1 loc .First, since T l u ε is uniformly bounded by l, we see that i is satisfied.To prove ii , we consider each term on the right-hand side of 3.33 .First we prove that

3.36
We have

3.37
Since the function θ z − c z − c is Lipschitz continuous in z with the Lipschitz constant one, and, according to definition of U ρ , it holds |U ρ z − θ z z| ≤ 1/2ρ, we conclude from the last expression

3.38
From this and assumptions 3.15 and 3.35 on σ σ ε and ρ ρ ε , it follows that as ε → 0 in L p loc for all p < ∞.Thus, since we can take p 2 as well we see that Γ 1,ε ∈ H −1 loc,c .Next, we will prove that International Journal of Differential Equations from which we conclude

International Journal of Differential Equations 19
Consider now each term on the right-hand side of 3.43 .Since T l is a continuous function and T l u ∈ −l, l , the function f t, x, T l u is uniformly continuous in u ∈ R. Therefore, we have pointwise on R × R d : −→ 0 as σ −→ 0.

3.46
We pass to Γ 2 2,ε .We have to distinguish between different cases depending on the relative size of c and l.Consider first the case when |c| ≤ l, in which case we have T l c c and T l,σ c c. Thus, For c > l we have c ≥ l σ for a σ small enough, and therefore θ T l,σ u ε − c ≡ 0. On the other hand, for c < −l we have c ≤ −l − σ, and so θ T l,σ u ε − c ≡ 1.Thus, the problematic case is when c < −l.In this case, we have instead of 3.47

3.50
where C is the constant given by 3.2d .Similarly, from 3.2d and since |T l v | ≤ 1, we have

3.51
from which we conclude that Γ 3 2,ε is bounded in L 2 loc .From assumptions 3.15 and 3.35 , as well as for the estimates 3.46 -3.51 , it follows that the expression from 3.43 is bounded in L 2 loc from which it follows that Γ 2,ε ∈ H −1 loc,c .The next term is

International Journal of Differential Equations 21
According to H4 , it is clear that Γ 3,ε ∈ M loc,B .Indeed, since |U ρ |, |T l,σ | ≤ 1 we have from 3.2b and 3.2e for a constant K 6 , implying the claim.
Next, we claim that

3.54
Due to a priori estimates 3.8 and 3.9 and, again, the fact that |T l,σ |, |U ρ | ≤ 1, we see that for every i 1, . . ., d

3.56
Since |U ρ | ≤ 1 and |T l,σ | ≤ 1σ we have from 3.30 recall 3.28 for the definition of the constants K j for j 3, 4, 5

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In view of a priori estimates 3.8 and 3.9 , and assumptions 3.15 and 3.35 , it holds

3.59
for some constants K 7 and K 8 .The second estimate holds since Finally, we will prove that

3.60
First, notice that supp U ρ 0, , and therefore

3.61
Assume first that |c| > l.Since we can choose ρ σ see 3.35 arbitrarily small, we can assume that |c| > l σ.In that case U ρ T l,σ u ε − c / 0 only if l σ ≤ T l,σ u ε ≤ l σ ρ which is never fulfilled according to the definition of T l,σ see 3.22 .So, in this case, 3.62 Next we assume that |c| < l.As before, we can assume that |c| < l − ρ since we can choose ρ σ arbitrarily small.From 3.61 , we see that according to 3.30 we put there l c .Finally, assume that |c| l.From 3.61 and 3.19 , we conclude

3.64
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Collecting the previous items, due to the properties of Γ i,ε , i 1, . . ., 7, it follows from 3.33 that

3.66
Therefore, we see that ii is satisfied and we can use Murat's lemma to conclude that

3.67
Thus we conclude that the conditions of Theorem 3.3 are satisfied, and we find that for every l > 0 the sequence

The One-Dimensional Case
We will analyze the convergence of the sequence u ε ε>0 of solutions to 3.5 -3.6 in the one dimensional case.Unlike the situation we had in the previous section, we will assume that the flux is continuously differentiable with respect to u.This will enable us to optimize the ratio δ/ε 2 .We will work under the following assumptions on the flux f f t, x, u denoted H4 .H4a For the flux f f t, x, u we assume that H4b There exists a sequence f >0 defined on R × R × R, smooth in t, x ∈ R × R, and continuously differentiable in u ∈ R, satisfying for some p > 2 and every l > 0: where C i , i 1, 2, 3, and C are constants independent on t, x, u ∈ R × R d × R.

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Under these assumptions we will prove the following.
i Without assuming nondegeneracy of the flux, the sequence u ε ε>0 converges along a subsequence to a solution of 3.1 -11 in the distributional sense when δ O ε 2 and O ε less stringent assumptions than in the multidimensional case .
ii If, in addition, we assume and that f is genuinely nonlinear in the sense of 4.12 , the sequence u ε ε>0 of solutions of problem 3.5 -3.6 is strongly precompact in Remark 4.1.The proof relies on a priori inequalities 3.8 and 3.9 .Notice that thanks to H4a , we can take β 1 in inequality 3.9 .
We will need the fundamental theorem of Young measures.
Theorem 4.2 see 24 .Assume that the sequence Then, there exists a subsequence (not relabeled) u ε k and a sequence of probability measures such that the limit exists in the distributional sense for all g measurable with respect to t, x ∈ R × R d , continuous in u ∈ R and satisfying uniformly in t, x : for a constant C independent of u, and q such that 0 ≤ q < p.The limit is represented by the expectation value for almost all points t, x ∈ R × R d .
One refers to such a sequence of measures ν ν t,x as the Young measures associated to the sequence u ε k k∈N . Furthermore, if and only if ν y δ u y a.e.

4.7
Before we continue, we need to recall the celebrated Div-Curl lemma.
Lemma 4.3 Div-Curl .Let Q ⊂ R 2 be a bounded domain, and suppose that By η n one denotes the truncation of the function η: and q n t, x, λ the corresponding entropy flux.If for every n ∈ N one has div t,x η n t, x, u ε , q n t, x, u ε ∈ H −1 loc,c R × R , 4.11 then the limit function u is a weak solution of 1.3 .Furthermore, if the flux function f f t, x, λ is twice differentiable with respect to λ, and is genuinely nonlinear, that is, for every t, x ∈ R × R d the mapping on nondegenerate intervals, then u ε ε>0 converges strongly along a subsequence to u in L 1 loc R × R .
Proof.We will apply the method of compensated compactness as in 13 .
First, notice that according to Theorem 4.2 there exist a subsequence u ε k ⊂ u ε and a sequence of probability measures I n v ∂ λ f t, x, u ε , and −ψ n t, x, u ε , φ n u ε , where φ ∈ C 1 R is an arbitrary entropy, and ψ n is the entropy flux corresponding to φ n .Here I n and φ n denote the smooth truncation functions of I and φ, respectively cf. 4.10 .
According to 4.11 , we can apply the Div-Curl lemma on the given vector fields.Hence, we get after letting ε → 0 along a subsequence: and that it is genuinely nonlinear in the sense of 4.12 .
Then, take arbitrary entropies and η 2 ∈ C 1 R , and denote by q 1 t, x, u and q 2 u , respectively, their corresponding entropy fluxes.Assume that η i , q i , η i q i , i 1, 2, satisfy 4.4 for q < 2. Notice that ∂ u η 1 depends explicitly on t, x , International Journal of Differential Equations while D u η 2 does not.Denote by η 1,n and η 2,n the appropriate smooth truncations cf.4.10 and by q 1,n and q 2,n the corresponding entropy fluxes, that is,

4.23
Due to 4.11 and the Div-Curl lemma the following commutation relation holds:

4.24
Letting n → ∞ as in 4.20 , we get

4.25
Next, recall that the function f satisfies 4.4 for q 1.Therefore, the following entropy-entropy fluxes are admissible: that is, that ν t,x δ u t,x a.e. on R × R implying strong L 1 loc convergence of u ε ε>0 along a subsequence see Theorem 4.2 .Now we are ready to prove the main theorem of the section.Proof.Assume that η t, x, λ , t, x, λ ∈ R × R 2 is a function such that η ∈ C 2 R; L ∞ ∩ BV R t × R x .As usual, denote by η n the truncation given by 4.10 , and let the entropy flux corresponding to η n and f be q n t, x, u u ∂ v η n t, x, v ∂ v f t, x, v dv.

4.31
According to Lemma 4.4, it is enough to prove that for every fixed n ∈ N the expression div η n t, x, u ε t, x , q n t, x, u ε t, x is precompact in H −1 loc R × R .

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In order to prove the latter, take the following mollifier η n,ε t, x, u η n •, •, u 1/ε 1/2 ω t/ε 1/4 ω x/ε 1/4 , where ω is a nonnegative real function with unit mass.Denote the entropy flux corresponding to η n and f by q n,ε t, x, u u ∂ v η n,ε t, x, v ∂ v f t, x, v dv.

4.32
Recall that here and in the sequel we assume that O ε .Actually, we can take ε without loss of generality.
Notice that according to the assumptions on η and the choice of the mollifier η n,ε we have

20 International
to 3.2a , and O 1 comes from 3.35 .Journal of Differential Equations

Theorem 4 . 5 .
Assume that δ δ ε O ε 2 , O ε , ε → 0, 4.30 and u 0 ∈ H 1 R .Assume that the flux function f from 3.1 with d 1 satisfies (H4 ).Assume also that the function b from 3.5 satisfies (H1) and (H2).Then a subsequence of solutions u ε k ⊂ u ε of problem 3.5 -3.6 converges in the sense of distributions to a weak solution of problem 3.1 .If the flux functionf ∈ C 2 R; BV R × R x ∩ L ∞ R × R × R x ,and if it is genuinely nonlinear in the sense of 4.12 , then a subsequence of solutions u ε k ⊂ u ε of problem 3.5 -3.6 converges strongly in L 1 R × R to a weak solution of 3.1 .
distributional sense for all g measurable with respect to t, x ∈ R ×R, continuous in u ∈ R, and satisfying 4.4 for some q ∈ R such that 0 ≤ q < p, and is represented by the expectation value Next, notice that the function f satisfies 4.4 for q 1.Indeed, from H4a , it follows ∂ u |f| sgn f ∂ u f ≤ C, and from here |f| ≤ C 1 |u| , for a constant C which depends on the constant C and the L ∞ bound of the function f.From this, we conclude that for the flux function f t, x, v we have lim Notice that for |λ| < n it holds ψ n t, x, λ sgn λ − u t, x f t, x, λ − f t, x, u t, x .Therefore, we have from 4.18 Then, following 6 , we insert the last quantities in 4.25 which yields the following relation: only if f t, x, is constant for all between u t, x and λ.Still, this is not possible according to the genuine nonlinearity condition 4.12 .Thus, from this and 4.27 , we conclude that 2 d − f t, x, λ − f t, x, u