FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS VIA

doi:10.1007/s10473-019-0107-8

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (1): 83-93.doi: 10.1007/s10473-019-0107-8

FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS VIA

王泽军, 张琦

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

收稿日期:2017-11-30 修回日期:2018-08-15 出版日期:2019-02-25 发布日期:2019-03-13
通讯作者: Zejun WANG E-mail:wangzejun@gmail.com
作者简介:Qi ZHANG,zhangqinju@gmail.com
基金资助:
Zejun Wang's research was supported in part by NSFC (11671193) and the Fundamental Research Funds for the Central Universities (NE2015005). Qi Zhang's research was supported in part by NSFC (11271182 and 11501290).

FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS VIA

Zejun WANG, Qi ZHANG

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received:2017-11-30 Revised:2018-08-15 Online:2019-02-25 Published:2019-03-13
Contact: Zejun WANG E-mail:wangzejun@gmail.com
Supported by:
Zejun Wang's research was supported in part by NSFC (11671193) and the Fundamental Research Funds for the Central Universities (NE2015005). Qi Zhang's research was supported in part by NSFC (11271182 and 11501290).

摘要/Abstract

摘要： In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x=0. First, we prove a simple but useful property of Lax-Oleinik formula (Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)-qF'(q) + L(F'(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T^*. Meanwhile, we can give the equation of the shock and an explicit value of T^*.

关键词: scalar conservation law, Lax-Oleinik formula, Riemann problem, asymptotic behavior

Abstract: In this paper, we use Lax-Oleinik formula to study the asymptotic behavior for the initial problem of scalar conservation law u_t + F(u)_x=0. First, we prove a simple but useful property of Lax-Oleinik formula (Lemma 2.7). In fact, denote the Legendre transform of F(u) as L(σ), then we can prove that the quantity F(q)-qF'(q) + L(F'(q)) is a constant independent of q. As a simple application, we first give the solution of Riemann problem without using of Rankine-Hugoniot condition and entropy condition. Then we study the asymptotic behavior of the problem with some special initial data and prove that the solution contains only a single shock for t > T^*. Meanwhile, we can give the equation of the shock and an explicit value of T^*.

Key words: scalar conservation law, Lax-Oleinik formula, Riemann problem, asymptotic behavior

王泽军, 张琦. FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS VIA[J]. 数学物理学报(英文版), 2019, 39(1): 83-93.

Zejun WANG, Qi ZHANG. FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS VIA[J]. Acta mathematica scientia,Series B, 2019, 39(1): 83-93.

参考文献

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FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS VIA

FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS VIA

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