Global Existence of Solutions for a Nonstrictly Hyperbolic System

Abstract

We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly bounded estimates and when is increasing (similarly, and when is decreasing) for the -viscosity and -flux approximation solutions of nonhomogeneous, resonant system without the restriction or as given in Klingenberg and Lu (1997), where and are Riemann invariants of nonhomogeneous, resonant system; is a uniformly bounded function of depending only on the function given in nonhomogeneous, resonant system, and is the bound of . Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions.

1. Introduction

The following system describes the evolution of an isothermal fluid in a nozzle with discontinuous cross-sectional area , where and stand for the density and the particle velocity of the fluid under consideration, respectively, and denotes the pressure function (See [1]). The existence of global weak solutions for the Cauchy problem or the initial boundary value problem of system (1) has been studied in [1–3]. In [4–6], the authors showed the global existence of BV entropy solutions to a more general class of nonhomogeneous, resonant system by the generalized Glimm scheme.

The Riemann problem for a more general resonant system of equations, was resolved in [7], where and is a smooth function.

To study the existence of entropy solutions of the Cauchy problem (1), the main difficulty is to establish boundedness of solutions because the equations are not in conservative form and the Conley-Chuey-Smoller principle of invariant regions does not apply (See [1] for the details about the physical background of system (1) and its difficulty in analysis). For the polytropic gas and the adiabatic exponent , the definition of a finite energy solution (unbounded) is given and its existence is obtained by using the compensated compactness method in [1].

For smooth solution, system (1) is equivalent to the following conservation laws of three equations: or the system of two equations where . When is a constant, system (3) or system (4) itself has many different physical backgrounds. For instance, it is a scaling limit system of Newtonian dynamics with long-range interaction for a continuous distribution of mass in (cf. [8, 9]) and also a hydrodynamic limit for the Vlasov equation (cf. [10]). Its global weak solution was obtained by using the random choice method [2] in [11] and by using the compensated compactness theory in [12, 13].

By simple calculations, two eigenvalues of system (4) are with corresponding Riemann invariants where is a constant.

The existence of weak solutions for the Cauchy problem (4) with bounded initial data was first studied in [12], where is a smooth function and the technical condition or on the initial data is imposed for obtaining the a-priori, uniform estimate of or .

Without the condition or , the reasonable estimate, depending on the variable , was first obtained in [2] for system (1) when , by using a modified Godunov scheme, and in [14] for general pressure function and smooth function by using the compensated compactness.

In this paper, using the vanishing viscosity method and the maximum principle coupled with the flux approximation proposed in [15] for the homogeneous system of isentropic gas dynamics, we extend the results in [2, 12] to the Cauchy problem (4)–(7) for any bounded initial data and for the function satisfying the conditions , .

We first construct the sequence of hyperbolic systems to approximate system (4), where denotes the flux approximation constant and the approximation pressure , and is the smooth approximation of , being a mollifier. If is a monotonic function, as required in Theorem 2, and and converge to zero much faster than , then it is easy to prove that and satisfy Second, we add the viscosity terms to the right-hand side of (8) to obtain the following parabolic system: with initial data where are given in (7).

Lemma 1. Let or equivalently for two constants and . If is increasing or equivalently , then we can choose a function satisfying , , and where the positive constants , and depend on , and , but are independent of .

The proof of Lemma 1 is trivial.

By applying the maximum principle to the Cauchy problem (11)-(12), we first obtain the estimates and when is increasing (similarly and when is decreasing) for a suitable positive, bounded function given in Lemma 1; then by using the compensated compactness theory and the already existed compact frameworks given in [12, 13], we give the following global existence theorem of weak solutions.

Theorem 2. Let and for two positive constants and .
(A) Let be increasing and we choose satisfying all conditions in Lemma 1 and, moreover, or , where and for a sufficiently small constant . Then the Riemann invariants and of system (4) with respect to the approximated solutions of the Cauchy problem (11)-(12) satisfy the estimate if and if .
(B) For such function and the initial data satisfying the conditions in Part , if either , , or , where is a positive constant, then there exists a subsequence of , which converges pointwise to a pair of bounded functions as , and tend to a zero, and the limit is a weak entropy solution of the Cauchy problem (4)–(7).

Definition 3. For integrable function , a pair of bounded measurable functions is called a weak entropy solution of the Cauchy problem (4)–(7), if hold for all test function and holds for any nonnegative test function , where is a pair of convex entropy-entropy flux of system (4).

We can easily construct many functions , and satisfying the conditions in Theorem 2.

Example 4. Let and choose Then (13) is satisfied since and satisfies all the conditions in Theorem 2. In fact, we may choose and then and for a positive constant .

We are going to prove Theorem 2 in the next section.

2. Proof of Theorem 2

By simple calculations, two eigenvalues of system (8) are with corresponding right eigenvectors and Riemann invariants which are similar to the Riemann invariants of system (4) given by (6).

We multiply (11) by and , respectively, to obtain Let . Then or or where and .

Using the first equation of (11), we have the a-priori estimate . Since the conditions on in Theorem 2, the following two terms on the left-hand side of (28): Now, we consider the other terms on the left-hand side of (28).

First, we have from that Second, we have from , that and so since .

Thus, (28) is reduced to the following inequality about : and we can prove that or if applying for the maximum principle to (34).

To prove the estimate of , we have from (25) that where .

Let where is the upper bound of and , and are the bounds of and obtained from the local solution. Then

We have from (35)–(37) that We argue by assuming that (38) is violated for at a point in . Let be the least upper bound of values of at which . Then, by the continuity we see that at some points . So , , and at ; that is, But from (35) and (36), Since on ; then Thus, at from the relation of , and given by (23). So the right-hand side of (40) is negative, which yields a conclusion contradicting (39). So (38) is proved. Therefore for any point in , which yields the desired estimate if we let in (42), and, hence, complete the proof of Part (A) in Theorem 2.

For the homogeneous case (), the convergence of as , and tend to zero in the Part (B) was given in [13] when , , and given in [12] when by using the compensated compactness theory [16, 17] coupled with some basic ideas of the kinetic formulation [18, 19].

Now, we are going to prove the convergence of as , and tend to zero for the inhomogeneous system (11).

Any entropy-entropy flux pair of the original hyperbolic system (4) satisfies the additional system. Consider Eliminating the from (44), we have Similarly, any entropy-entropy flux pair of the approximated hyperbolic system (8) satisfies By eliminating the from (46), we have also the same entropy equation (45). Therefore, system (4) and system (8) have the same entropies.

For any entropy-entropy flux pair of system (4), by multiplying to system (11), we have where is the entropy flux of the approximated system (8) corresponding to entropy . Since is uniformly integrable, then the last term on the right-hand side of system (47) is compact in , for some , by the Sobolev embedding theorems. It is obvious that the term on the left-hand side is compact in . Therefore, using the same techniques given in [12, 13] for the homogeneous system, we may prove that is compact in and so the convergence of as , and tend to zero. Furthermore, the limit satisfies (16).

If precisely using (10), we can prove that the limit satisfies the following conservation form: In fact, we multiply the first equation in (11) by to obtain which yields (48) when goes to zero.

Since both systems (4) and (8) have the same entropies, we can easily prove that the limit satisfies the entropy condition (17). So we complete the proof of Theorem 2.

Conflict of Interests

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

Acknowledgments

Yun-guang Lu and De-yin Zheng’s 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) and Guo-qiang Song’s work was partially supported by the Natural Science Foundation of Anhui Education Committee (Grant no. KJ2012A171).