THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA∗

THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗

THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗

THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗

THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗

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doi:10.1007/s10473-023-0414-y

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1668-1674.doi: 10.1007/s10473-023-0414-y

Fei WU, Zejun WANG, Fangqi CHEN^†

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China; Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing 211106, China

收稿日期:2022-04-25 发布日期:2023-08-08
通讯作者: †Fangqi CHEN, E-mail: fangqichen1963@126.com
作者简介:Fei WU, E-mail: wufei003@nuaa.edu.cn; Zejun WANG, E-mail: wangzejun@gmail.com
基金资助:
*Zejun Wang was partially supported by the NSFC (11671193); Fangqi Chen was partially supported by the NSFC (12172166, 11872201).

Fei WU, Zejun WANG, Fangqi CHEN^†

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China; Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT, Nanjing 211106, China

Received:2022-04-25 Published:2023-08-08
Contact: †Fangqi CHEN, E-mail: fangqichen1963@126.com
About author:Fei WU, E-mail: wufei003@nuaa.edu.cn; Zejun WANG, E-mail: wangzejun@gmail.com
Supported by:
*Zejun Wang was partially supported by the NSFC (11671193); Fangqi Chen was partially supported by the NSFC (12172166, 11872201).

摘要： In this paper, we study the global existence of BV solutions of the initial value problem for the isentropic p-system, where the state equation of the gas is given by $P=Av^{-\gamma}$ . For $\gamma>1$ , the general existence result for large initial data has not been obtained. By using the Glimm scheme, Nishida, Smoller and Diperna successively obtained the global existence results for $(\gamma-1)\mbox{TV}(v_0(x),u_0(x))$ being small. In the present paper, by adopting a rescaling technique, we improve these results and obtain the global existence result under the condition that $(\gamma-1)^{\gamma+1}({\rm TV}(v_{0}(x)))^{\gamma-1}(\mbox{TV}(u_{0}(x)))^{2}$ is small, which implies that, for fixed $\gamma>1$ , either $\mbox{TV}(v_{0}(x))$ or $\mbox{TV}(u_{0}(x))$ can be arbitrarily large.

关键词: conservation laws, p-system, vanishing viscosity method, BV solutions

Abstract: In this paper, we study the global existence of BV solutions of the initial value problem for the isentropic p-system, where the state equation of the gas is given by $P=Av^{-\gamma}$ . For $\gamma>1$ , the general existence result for large initial data has not been obtained. By using the Glimm scheme, Nishida, Smoller and Diperna successively obtained the global existence results for $(\gamma-1)\mbox{TV}(v_0(x),u_0(x))$ being small. In the present paper, by adopting a rescaling technique, we improve these results and obtain the global existence result under the condition that $(\gamma-1)^{\gamma+1}({\rm TV}(v_{0}(x)))^{\gamma-1}(\mbox{TV}(u_{0}(x)))^{2}$ is small, which implies that, for fixed $\gamma>1$ , either $\mbox{TV}(v_{0}(x))$ or $\mbox{TV}(u_{0}(x))$ can be arbitrarily large.

Key words: conservation laws, p-system, vanishing viscosity method, BV solutions

Fei WU, Zejun WANG, Fangqi CHEN. THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1668-1674.

Fei WU, Zejun WANG, Fangqi CHEN. THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1668-1674.

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