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On hyperbolic systems of conservation laws. (English. Russian original) Zbl 1125.35065

Differ. Equ. 39, No. 5, 701-711 (2003); translation from Differ. Uravn. 39, No. 5, 663-6673 (2003).
From the introduction: The present paper deals with a priori estimates and the absence of global smooth solutions of the Cauchy problem for hyperbolic systems of conservation laws. The main goal is to illustrate the method of integral relations for the proof of the blow-up of smooth solutions of such systems.
It is well known that, for general quasilinear hyperbolic systems, the Cauchy problem with general initial data has no global smooth solutions. This is caused by the blow-up of the solutions in finite time, also known as the gradient catastrophe. The proof of this assertion is usually based on Riemann invariants or quasi-invariants. This approach was developed for one-dimensional systems \((N =1)\). For \(M=2\), i.e., in the case of \(2\times 2\) systems, one can obtain nonimprovable closed-form conditions for the appearance of the gradient catastrophe.
In the case of systems with \(M>2\), the statement of conditions for the appearance of the gradient catastrophe either is implicit or pertains to small perturbations of the initial data.
In the present paper, we suggest an approach based on integral inequalities for specially defined nonlinear functions, which are associated with the original quasilinear system. This approach is a development of the nonlinear capacity method [E. Mitidieri and, Trudy Mat. Inst. RAN, 234, 1–383 (2001; Zbl 0987.35002)], used for nonlinear equations and systems with strictly positive nonlinear right-hand side. The present paper is close to [S. I. Pokhozhaev, Sb. Math. 194, No. 1, 151–164 (2003); translation from Mat. Sb., 194, 147–160 (2003; Zbl 1049.35130)], where the above-mentioned approach was suggested for scalar multidimensional conservation laws, i.e., for the case in which \(M=1\) and \(N\geq 1\).
In Section 1, we present the general scheme of our approach. In Section 2, we consider the application to a nonisoentropic one-dimensional gas motion and derive the main integral relations. The main a priori estimate is obtained in Section 3. In Section 4, we derive an integral criterion of the gradient catastrophe for a nonisoentropic gas motion. The time of appearance of the gradient catastrophe is estimated above in Section 5. In Section 6, we consider the equations of isoentropic gas motion, for which we derive an integral condition for the gradient catastrophe and estimate the blow-up time.

MSC:

35L65 Hyperbolic conservation laws
35B45 A priori estimates in context of PDEs
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