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

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].Acta mathematica scientia,Series B, 2023, 43(4): 1668-1674.

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