Nonuniqueness of Unbounded Solutions of the Cauchy Problem for Scalar Conservation Laws
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This article is aimed at studying the Cauchy problem for a first-order quasi-linear equation with a flow function of power type and unbounded initial data of power or exponential type. It is known that the Cauchy problem in the class of locally bounded functions may have several solutions. We describe all entropy solutions of this problem, which can be represented in a special form. It is shown that after the first discontinuity line (shock wave), these solutions eventually exhibit the same behavior, and their nonuniqueness actually amounts to the choice of the first shock wave.
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- 3.A. Yu. Goritsky, S. N. Kruzhkov, and G. A. Chechkin, First-Order Partial Differential Equations: A Textbook [in Russian], The Center of Appl. Stud. at the Faculty of Mech. and Math. of Moscow State Univ. (1999).Google Scholar
- 5.E. Yu. Panov, “Well-posedness classes of locally bounded generalized entropy solutions of the Cauchy problem for quasilinear first-order equations,” Fundam. Prikl. Mat., 12, No. 5, 175–188 (2006).Google Scholar
- 6.A. Yu. Goritsky, “Construction of an unbounded entropy solution of the Cauchy problem with countably many shock waves,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 2, 3–6 (1999).Google Scholar