International Journal of Differential Equations

1. Introduction

We consider the convergence of smooth solutions with of the nonlinear partial differential equation

as and . Here is the Caratheodory flux vector such that

for and every . 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 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 -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 diffusion-dispersion 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 diffusion-dispersion-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.

2. A priori Inequalities

Assume that the flux in (1.1) is smooth in all variables. Consider a sequence of solutions of

We assume that has enough regularity so that all formal computations below are correct. So, following Schonbek [13], we assume that for every we have .

Later on, we will assume that the initial data depends on . In this section, we will determine a priori inequalities for the solutions of problem (2.1).

To simplify the notation we will write instead of .

We will need the following assumptions on the diffusion term .

(H1) For some positive constants we have

(H2) The gradient matrix is a positive definite matrix, uniformly in , that is, for every , there exists a positive constant such that we have

We use the following notation:

In the sequel, for a vector valued function defined on , we denote

The partial derivative in the point , where possibly depends on , is defined by the formula

In particular, the total derivative and the partial derivative are connected by the identity

Finally we use

With the previous conventions, we introduce the following assumption on the flux vector .

(H3) The growth of the velocity variable and the spatial derivative of the flux are such that for some , , and every , we have

where 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 satisfies (H3) and that it is Lipschitz continuous on . Assume also that initial data belongs to . Under conditions (H1)-(H2) the sequence of solutions of (2.1) for every satisfies the following inequalities: for some constants and .

Proof. We follow the procedure from [19]. Given a smooth function , , we define If we multiply (2.1) by , it becomes Choosing here and integrating over , we get where the second equality sign is justified by the following partial integration:
Now inequality (2.10) follows from (2.14), using (H1).
As for inequality (2.11), we start by using (2.14), namely, where .
From here, using (H3), we conclude in particular that for some constant independent of .
Next, we differentiate (2.1) with respect to and multiply the expression by . Integrating over , using integration by parts and then summing over we get: Integrating this over and using the Cauchy-Schwarz inequality and condition (H2), we find where is independent of . Then, using Young's inequality (the constant is the same as previously mentioned) we obtain Multiplying this by , using , and applying (2.17), we conclude This inequality is actually inequality (2.11) when we take ,.

3. The Multidimensional Case

Consider the following initial-value problem. Find such that

where is a given initial data.

For the flux we need the following assumption, denoted (H4).

(H4a) For the flux , , we assume that and that for every we have , .

(H4b) There exists a sequence , , such that , satisfying for some and every :

where , , and are constants, while the function is such that .

In the case when we have only vanishing diffusion, it is usually possible to obtain uniform 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 and with total mass one denote and , where is a positive function tending to zero as . In the case when the flux is bounded, straightforward computation shows that the sequence satisfies (H4b) with .

We also need to assume that the flux is genuinely nonlinear, that is, for every and every , the mapping

is nonconstant on every nondegenerate interval of the real line.

We will analyze the vanishing diffusion-dispersion-smoothing limit of the problem

where the flux satisfies the conditions (H4b). We denote the solution of (3.5)-(3.6) by . We assume that

We also assume that and as . We want to prove that under certain conditions, a sequence of solutions of (3.5)-(3.6) converges to a weak solution of problem (3.1) as . To do this in the multidimensional case we use the approach of -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 satisfies (H4). Also assume that the initial data satisfies (3.7). Under these conditions the sequence of smooth solutions of (3.5)-(3.6) satisfies the following inequalities for every : for some constants (the constants are introduced in Theorem 2.1).

Proof. For every fixed , the function is smooth, and, due to (H4), we see that satisfies (H3). This means that we can apply Theorem 2.1.
Replacing the flux by from (3.5) and by from (3.6) in (2.10) and (2.11), we get To proceed, we use assumption (H4). We have 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 of solutions to (3.5)-(3.6) in the case when

for the function 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 -measures approach (see, e.g., [10, Corollary and Remark ]). We let denote the Heaviside function.

Theorem 3.3 (see [10]). Assume that the vector is genuinely nonlinear in the sense of (3.4). Then each family such that for every the distribution is precompact in contains a subsequence convergent in .

We can now prove the following theorem.

Theorem 3.4. Assume that the flux vector is genuinely nonlinear in the sense of (3.4) and that it satisfies (H4). Furthermore, assume that and that satisfies (3.7). Then, there exists a subsequence of the family 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 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 function , , and multiply the regularized equation (3.5) by . As usual, put
We easily find that We will apply this formula repeatedly with different choices for .
In order to apply Theorem 3.3, we will consider a truncated sequence , where the truncation function is defined for every fixed as We will prove that the sequence is precompact for every fixed . Denote by a subsequential limit (in ) of the family , which gives raise to a new sequence that we prove converges to a weak solution of (3.1).
To carry out this plan, we must replace by a regularization . We define by and
Next, we want to estimate . To accomplish this, we insert the functions for in (3.17) where are defined by and Notice that By inserting , in (3.17) and integrating over , we get Similarly, for , , we have from (3.17) Adding (3.23) to (3.24), we get From (3.22) and the definition of and , it follows Without loss of generality, we can assume that . Having this in mind, we get from (H4) and (3.26) where , , are constants such that (cf. (3.8) and (3.9)) These estimates follow from (H4) and the a priori estimates (3.8), (3.9). If in addition we use the assumption from (3.15), we conclude Thus, in view of (3.27), which is the sought for estimate for .
Next, take a function satisfying and Clearly, is convex, and in as , for any ; as before, denotes the Heaviside function.
Inserting in (3.17), we get We rewrite the previous expression in the following manner: where
To continue, we assume that depends on in the following way:
From here, we will prove that the sequence 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 ]). More precisely, we have to prove the following.
(i) When the left-hand side of (3.33) is written in the form , we have for .
(ii) The right-hand side of (3.33) is of the form , where denotes a set of families which are locally bounded in the space of measures, and is a set of families precompact in .
First, since is uniformly bounded by , 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 We have Since the function is Lipschitz continuous in with the Lipschitz constant one, and, according to definition of , it holds , we conclude from the last expression
From this and assumptions (3.15) and (3.35) on and , it follows that as in for all . Thus, (since we can take as well) we see that .
Next, we will prove that Indeed, Since if and if , from which we conclude with
Consider now each term on the right-hand side of (3.43). Since is a continuous function and , the function is uniformly continuous in . Therefore, we have pointwise on : Since , , Lebesgue's dominated convergence theorem yields , where as . Thus, we conclude
We pass to . We have to distinguish between different cases depending on the relative size of and . Consider first the case when , in which case we have and . Thus, where appears due to (3.2a), and comes from (3.35).
For we have for a small enough, and therefore . On the other hand, for we have , and so . Thus, the problematic case is when . In this case, we have instead of (3.47) implying since in for , and .
It remains to estimate . Noticing that , we get