Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function
H. Holden,1,2K. H. Karlsen,3and D. Mitrovic1
Academic Editor: Philippe G. LeFloch
Received02 Apr 2009
Revised24 Aug 2009
Accepted24 Sept 2009
Published03 Dec 2009
Abstract
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 -measures to investigate the zero diffusion-dispersion-smoothing limit.
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 KruΕΎhkov [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., [14β17]). 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)β(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)β(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
where is the constant given by (3.2d). Similarly, from (3.2d) and since , we have
from which we conclude that is bounded in . From assumptions (3.15) and (3.35), as well as for the estimates (3.46)β(3.51), it follows that the expression from (3.43) is bounded in from which it follows that . The next term is
According to (H4), it is clear that . Indeed, since we have from (3.2b) and (3.2e)
for a constant , implying the claim. Next, we claim that
Due to a priori estimates (3.8) and (3.9) and, again, the fact that , we see that for every
in . Therefore, . Further, we claim that
Since and we have from (3.30) (recall (3.28) for the definition of the constants for )
for some constant , according to assumptions (3.15) and (3.35) on , , , and . Thus, we see that . Next, we need to show
In view of a priori estimates (3.8) and (3.9), and assumptions (3.15) and (3.35), it holds
for some constants and . The second estimate holds since and implying . Therefore, . Finally, we will prove that
First, notice that , and therefore
Assume first that . Since we can choose (see (3.35)) arbitrarily small, we can assume that . In that case only if which is never fulfilled according to the definition of (see (3.22)). So, in this case,
Next we assume that . As before, we can assume that since we can choose arbitrarily small. From (3.61), we see that if implying that . Thus,
according to (3.30) (we put there ). Finally, assume that . From (3.61) and (3.19), we conclude
From here and since , it follows (recall that we assume )
according to (3.30). From here, (3.62) and (3.63), we conclude (3.60). Collecting the previous items, due to the properties of , , it follows from (3.33) that
Therefore, we see that (ii) is satisfied and we can use Murat's lemma to conclude that
Thus we conclude that the conditions of Theorem 3.3 are satisfied, and we find that for every the sequence is precompact in . Since the sequence is uniformly bounded in , from [23, Lemma ], we conclude that is precompact in .
4. The One-Dimensional Case
We will analyze the convergence of the sequence of solutions to (3.5)β(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 . This will enable us to optimize the ratio . We will work under the following assumptions on the flux denoted (H4).
(H4a) For the flux we assume that and .
(H4b) There exists a sequence defined on , smooth in , and continuously differentiable in , satisfying for some and every :
where , , and are constants independent on .
Under these assumptions we will prove the following.
(i)Without assuming nondegeneracy of the flux, the sequence converges along a subsequence to a solution of (3.1)β(11) in the distributional sense when and (less stringent assumptions than in the multidimensional case).(ii)If, in addition, we assume , and that is genuinely nonlinear in the sense of (4.12), the sequence of solutions of problem (3.5)β(3.6) is strongly precompact in when .
Remark 4.1. The proof relies on a priori inequalities (3.8) and (3.9). Notice that thanks to (H4a), we can take in inequality (3.9).
We will need the fundamental theorem of Young measures.
Theorem 4.2 (see [24]). Assume that the sequence is uniformly bounded in , . Then, there exists a subsequence (not relabeled) and a sequence of probability measures
such that the limit
exists in the distributional sense for all measurable with respect to , continuous in and satisfying uniformly in :
for a constant independent of , and such that . The limit is represented by the expectation value
for almost all points . One refers to such a sequence of measures as the Young measures associated to the sequence . Furthermore,
if and only if
Before we continue, we need to recall the celebrated Div-Curl lemma.
Lemma 4.3 (Div-Curl). Let be a bounded domain, and suppose that
in as . Assume also that the two sequences and lie in a (common) compact subset of , where and . Then along a subsequence
Lemma 4.4. Assume that converges weakly in to a function . Assume that , , is a function satisfying (4.4) with such that . By one denotes the truncation of the function :
and the corresponding entropy flux. If for every one has
then the limit function is a weak solution of (1.3). Furthermore, if the flux function is twice differentiable with respect to , and is genuinely nonlinear, that is, for every the mapping
on nondegenerate intervals, then converges strongly along a subsequence to in .
Proof. We will apply the method of compensated compactness as in [13]. First, notice that according to Theorem 4.2 there exist a subsequence and a sequence of probability measures
such that the limit
exists in the distributional sense for all measurable with respect to , continuous in , and satisfying (4.4) for some such that , and is represented by the expectation value
for almost all points . Next, notice that the function satisfies (4.4) for . Indeed, from (H4a), it follows , and from here , for a constant which depends on the constant and the bound of the function . From this, we conclude that for the flux function we have
To continue, notice that
Take in (4.10), and consider the vector fields where , and , where is an arbitrary entropy, and is the entropy flux corresponding to . Here and denote the smooth truncation functions of and , 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 along a subsequence:
where
Then, put . Notice that for it holds . Therefore, we have from (4.18)
It is clear that for every fixed the right-hand side of (4.20) tends to zero as implying (due to the Lebesgue dominated convergence theorem)
Now, a standard procedure gives (see, e.g., [6, Remark ])
where . From here, it follows that is a weak solution to (3.1). This concludes the first part of the lemma. For the details of the procedure one should consult, for example, [13]. Now, assume that , and that it is genuinely nonlinear in the sense of (4.12). Then, take arbitrary entropies and , and denote by and , respectively, their corresponding entropy fluxes. Assume that , , , , satisfy (4.4) for . Notice that depends explicitly on , while does not. Denote by and the appropriate smooth truncations (cf. (4.10)) and by and the corresponding entropy fluxes, that is,
Due to (4.11) and the Div-Curl lemma the following commutation relation holds:
Letting as in (4.20), we get
Next, recall that the function satisfies (4.4) for . Therefore, the following entropy-entropy fluxes are admissible:
Then, following [6], we insert the last quantities in (4.25) which yields the following relation:
By the CauchyβSchwarz inequality
with the equality only if is constant for all between and . Still, this is not possible according to the genuine nonlinearity condition (4.12). Thus, from this and (4.27), we conclude that
that is, that a.e. on implying strong convergence of along a subsequence (see Theorem 4.2).
Now we are ready to prove the main theorem of the section.
Theorem 4.5. Assume that
and . Assume that the flux function from (3.1) with satisfies (H4). Assume also that the function from (3.5) satisfies (H1) and (H2). Then a subsequence of solutions of problem (3.5)β(3.6) converges in the sense of distributions to a weak solution of problem (3.1). If the flux function , and if it is genuinely nonlinear in the sense of (4.12), then a subsequence of solutions of problem (3.5)β(3.6) converges strongly in to a weak solution of (3.1).
Proof. Assume that , is a function such that . As usual, denote by the truncation given by (4.10), and let the entropy flux corresponding to and be
According to Lemma 4.4, it is enough to prove that for every fixed the expression is precompact in . In order to prove the latter, take the following mollifier , where is a nonnegative real function with unit mass. Denote the entropy flux corresponding to and by
Recall that here (and in the sequel) we assume that . Actually, we can take without loss of generality. Notice that according to the assumptions on and the choice of the mollifier we have
for a constant . Then, applying (3.17) with replaced by , we find
Now, we apply a similar procedure as in the multidimensional case. Combining (H4b') and (4.33), we get
for a constant , implying boundedness of the subintegral expression in the sense of measures. Similarly, for a constant
implying boundedness of the subintegral expression in the sense of measures. Then, combining (4.33) with (3.8) and (3.9) we infer (see estimation of ) that
is bounded in . Next,
is precompact in since , , , and from (3.8) and (3.9) (see also Remark 4.1) we have
Similarly, by (3.8) and (3.9) (see the estimation of again)
Next, due to (H4b) and the well-known properties of the convolution, it holds for a constant independent on :
for arbitrary implying
Similarly, it is easy to see that
and thus
From (4.35)β(4.44) and the fact that , we conclude using Murat's lemma that
Finally, relying on Lemma 4.4 we conclude the proof of the theorem.
Acknowledgments
The work is supported in part by the Research Council of Norway. The work was supported by the Research Council of Norway through the Projects Nonlinear Problems in Mathematical Analysis, Waves In Fluids and Solids, and an Outstanding Young Investigators Award (KHK). This paper was written as part of the the International Research Program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008β2009.
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