Structure of entropy solutions to general scalar conservation laws in one space dimension

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:428

Article

Bianchini, Stefano¹  Yu, Lei² 

[1] SISSA, I-34136 Trieste, Italy

[2] Fudan Univ, Shanghai Ctr Math Sci, Shanghai 200433, Peoples R China

关键词: Scalar conservation law;    Entropy solution;    Front tracking approximations;    Coarea formula;    Global structure of solution;   

DOI  :  10.1016/j.jmaa.2015.03.006

来源: Elsevier

【 摘 要 】

In this paper, we show that the entropy solution u of a scalar conservation law enjoys the following properties: u is continuous outside a 1-rectifiable set Xi subset of R+ x R, up to a countable set, for each point ((t) over bar,(x) over bar) E E there exist two cone shaped regions arbitrarily close to half planes where u is left/right continuous at ((t) over bar,(x) over bar) We provide examples showing that these estimates are nearly optimal. In order to achieve these regularity results, we extend the wave representation of the front tracking approximate solutions to the entropy solution. This representation can be interpreted as some sort of Lagrangian representation of the solution to the nonlinear scalar PDE, and implies a fine structure on the level sets of u. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free