Global solutions to one-dimensional shallow water magnetohydrodynamic equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:401

Article

Gu, Feng¹  Lu, Yun-guang¹  Zhang, Qiong² 

[1] Hangzhou Normal Univ, Dept Math, Hangzhou, Zhejiang, Peoples R China

[2] Beijing Inst Technol, Dept Math, Beijing 100081, Peoples R China

关键词: Magnetohydrodynamic equations;    Global weak solutions;    Compensated compactness theory;   

DOI  :  10.1016/j.jmaa.2012.12.042

来源: Elsevier

【 摘 要 】

In this paper, we study the Cauchy problem for the one-dimensional shallow water magnetohydrodynamic equations. The main difficulty is the case of zero depth (h 0) since the nonlinear flux function P(h) is singular and the definition of solution is not clear near h = 0. First, assuming that h has a positive and lower bound, we establish the pointwise convergence of the viscosity solutions by using the div-curl lemma from the compensated compactness theory to special pairs of functions (c, f(epsilon)), and obtain a global weak entropy solution. Second, under some technical conditions on the initial data such that the Riemann invariants (w, z) are monotonic and increasing, we introduce a variant of the vanishing artificial viscosity to select a weak solution. Finally, we extend the results to two special cases, where P(h) is for the polytropic gas or for the Chaplygin gas. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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