Optimal results on TV bounds for scalar conservation laws with discontinuous flux

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:258

Article

Ghoshal, Shyam Sundar^1,2,3 

[1] Univ Paris 06, Lab Jacques Louis Lions, Paris, France

[2] Univ Toulouse 3, Inst Math Toulouse, F-31062 Toulouse, France

[3] GSSI, Laquila, Italy

关键词: Hamilton-Jacobi equation;    Scalar conservation laws;    Discontinuous flux;    Explicit formula;    Characteristic lines;    BV function;   

DOI  :  10.1016/j.jde.2014.10.014

来源: Elsevier

【 摘 要 】

This paper is concerned with the total variation of the solution of scalar conservation law with discontinuous flux in one space dimension. One of the main unsettled questions concerning conservation law with discontinuous flux was the boundedness of the total variation of the solution near interface. In [1], it has been shown by a counter-example at T = 1, that the total variation of the solution blows up near interface, but in that example the solution become of bounded variation after time T > 1. So the natural question is what happens to the BY-ness of the solution for large time. Here we give a complete picture of the bounded variation of the solution for all time. For a uniform convex flux with only L-infinity data, we obtain a natural smoothing effect in BV for all time t > T-0. Also we give a counter-example (even for a BV data) to show that the assumptions which have been made are optimal. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free