Global Solutions for a Simplified Shallow Elastic Fluids Model

Abstract

The Cauchy problem for a simplified shallow elastic fluids model, one system of Temple’s type, is studied and a global weak solution is obtained by using the compensated compactness theorem coupled with the total variation estimates on the first and third Riemann invariants, where the second Riemann invariant is singular near the zero layer depth . This work extends in some sense the previous works, (Serre, 1987) and (Leveque and Temple, 1985), which provided the global existence of weak solutions for strictly hyperbolic system and (Heibig, 1994) for strictly hyperbolic system with smooth Riemann invariants.

1. Introduction

In [1], the authors proposed a new reduced model for gravity-driven free-surface flows of shallow elastic fluids. It is obtained by an asymptotic expansion of the upper-convected Maxwell model for elastic fluids. The viscosity is assumed small, but the relaxation time is kept finite. The simplified system of (5.6) in [1] is as follows: Since the variable in the last equation is independent of the first four equations in (1), we remove it and let , to obtain the following conservation system:

By simple calculations, three eigenvalues of system (2) are with corresponding Riemann invariants and all characteristic fields are of the Temple type, namely, the shock waves and rarefaction waves coincide.

In this paper, we will study the Cauchy problem of system (2) with bounded initial data and prove the existence of global weak solutions by using the vanishing viscosity method coupled with the compensated compactness argument.

The hyperbolic systems of two conservation laws whose shock and rarefaction curves coincide were first described by Temple in [2], and later the global existence of weak solutions for such systems, with any bounded variation initial condition, were proven by Serre in [3] (also [4] for a special system), where the key technique is that the total variation of the Riemann invariants is decreasing in time. Serre's idea was extended by Heibig [5] to study any strictly hyperbolic system of Temple type, with smooth flux functions. For system (2), both the flux functions and the second Riemann invariant are singular near the zero layer depth . So, the technique proposed in [3, 5] cannot be used to study system (2) directly. Fortunately, we may accept the method in [3] to obtain that the total variation, of the first and third Riemann invariants, is decreasing in time, which implies the compactness of and for the viscosity solutions given by the Cauchy problem. Then we may apply for the div-curl lemma to the pairs of functions and , where is a constant, to give a very short proof of the existence of global weak solutions for the Cauchy problem (2) and (5).

Theorem 1. Suppose (I) where is a small positive constant and , are suitable constants satisfying (II) the total variations of and are bounded, then the Cauchy problem (2) and (5) has a bounded measurable solution , satisfying system (2) in the sense of distributions.

The existing result given in [5] is based on that for any system of Temple type; there exists a strictly convex entropy (Lemma  1, [5]). However, in our case, the existence of a strictly convex entropy is not obvious since all pairs of entropy-entropy flux of (2) are constructed in the following theorem.

Theorem 2. All the entropies of (2) are in the form where , , and are arbitrary functions, with corresponding entropy flux Furthermore when , , and are all convex, the entropy is convex.

We will prove Theorems 1 and 2 in Sections 2 and 3, respectively.

2. Proof of Theorem 1

We will prove Theorem 1 by several lemmas.

Consider the Cauchy problem for the related parabolic system with initial data where is a mollifier.

Lemma 3. If the initial data satisfy (6) and (7), then for any fixed , the viscosity solution of the Cauchy problem (10) and (11) exists.

Proof of Lemma 3. We substitute the first equation into the second and third equations in (10), respectively, to obtain Then by applying the maximum principle to (12), we have due to the initial conditions given in (6), where are Riemann invariants given in (4).
Using these estimates, we have the following bounded estimates on : due to the condition in (7). Since the boundedness of , we have the positive, lower estimate by using Theorem  1.0.2 in [6] and the first equation in (10), where could tend to zero as the time tends to infinity or tends to zero.
The standard theory of semilinear parabolic systems, namely, the local existence and the a priori bounded estimates in (14) and (15), gives us the existence of global smooth solution for the Cauchy problem (10) and (11). Lemma 3 is proven.

Lemma 4. If the total variations of and are bounded, then the viscosity solutions satisfy where is a positive constant, which is independent of .

Proof of Lemma 4. Let . Choose in (12). Differentiating (12) with respect to and then multiplying the sequence of smooth functions to the result, we have If we choose a convex , and , as , then we let in (17) and we have in the sense of distributions. Integrating (18) in , we have since is bounded. So, we proved the first estimate in (16). Similarly we can prove the second estimate in (16). Lemma 4 is proven.

Lemma 5. where , are any smooth functions.

Proof of Lemma 5. Using (16) and (14), we know that and are uniformly bounded both in and in and so compact in due to Murat's lemma [7]. Lemma 5 is proven.

Lemma 6. are compact in .

Proof of Lemma 6. Multiplying    to the first equation in (10) (for simplicity, we omit the superscript ), we have Since is bounded in , multiplying a suitable nonnegative test function to (22), we have that So the right-hand side of the first equation in (10) is compact in , which implies the compactness of .
We substitute the first equation into the second in (10) to obtain Then we multiply (24) by , the first equation in (10) by , then add the result to obtain Choosing a strictly convex function and multiplying a suitable nonnegative test function to (25), we have that Similarly we can prove that Using (23), (26), (27), and (20), we may prove the compactness of the last two terms in (21). Lemma 6 is proven.

Now we give the proof of Theorem 1. Let be the weak star limits of .

Using the div-curl lemma [6, 8] to the following pairs of functions, respectively: where is a constant, we have Therefore we may prove that the limit satisfies (2) in the sense of distributions by letting go to zero in (10). Theorem 1 is proven.

3. Proof of Theorem 2

In this section, we will prove Theorem 2. It is easy to prove that any pair of entropy-entropy flux , , of system (2), must satisfy the following system: Then we have which is Using the first and third equations in (32), we obtain Resolving the second equation in (32), we have where is an arbitrary function of .

We rewrite the first equation in (32) as Then where is an arbitrary function of .

Differentiating (36) with respect to and using the first equation in (32), we have Differentiating (34) with respect to and using (33) and (37), we have Moreover, differentiating (34) with respect to and differentiating (36) with respect to and using the second equation in (32), we have We resolve (38) and (39) to obtain where , are two arbitrary functions.

Integrating (34) with respect to , we have where , , and is a suitable function of such that given in (41) must satisfy (36). Thus satisfies and so or . Thus (8) is proven.

To prove (9), we remove all the terms on the right-hand side of (12) to obtain Multiplying the first equation in (43) by and the first equation in (2) by , then adding the result, we have Thus the entropy flux corresponding to the entropy is .

Similarly, we may prove that the entropy flux corresponding to the entropy is , the flux corresponding to the entropy is . Thus (9) is proven.

Let , and . By simple calculations and for any vector Thus is convex when is convex. Similarly we can prove both and are convex when , are convex. Theorem 2 is proven.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the Natural Science Foundation of Zhejiang Province of China (Grant no. LY12A01030 and no. LZ13A010002) and the National Natural Science Foundation of China (Grant no. 11271105).