Abstract
Let u(t, x) be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations. It is impossible that the solutions belong to, for example, H1 because by Sobolev embedding theorem H1 functions are Hölder continuous. However, the author notes that from any point (t, x), he can draw a generalized characteristic downward which meets the initial axis at y = α(t, x). If he regards u as a function of (t, y), it indeed belongs to H1 as a function of y if the initial data belongs to H1. He may call this generalized persistence (of high regularity) of the entropy weak solutions. The main purpose of this paper is to prove some kinds of generalized persistence (of high regularity) for the scalar and 2 × 2 Temple system of hyperbolic conservation laws in one space dimension.
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This work was supported by the National Natural Science Foundation of China (No. 12171097), Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education of China, Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University and Shanghai Science and Technology Program (No. 21JC1400600).
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Zhou, Y. Generalized Persistence of Entropy Weak Solutions for System of Hyperbolic Conservation Laws. Chin. Ann. Math. Ser. B 43, 499–508 (2022). https://doi.org/10.1007/s11401-022-0342-5
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DOI: https://doi.org/10.1007/s11401-022-0342-5