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.
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
.
DOI: 10.1007/s10473-023-0414-y
[1] Bressan A.Hyperbolic Systems of Conservation Laws: The 1-Dimensional Cauchy Problem. Oxford: Oxford Univ Press, 2000
[2] Bressan A, Liu T P, Yang T. $L^{1}$ stability estimates for $n\times n$ conservation laws. Arch Ration Mech Anal, 1999, 149(1): 1-22
[3] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math1965, 18: 697-715
[4] Lax P D. Hyperbolic systems of conservation laws II. Comm Pure Appl Math1957, 10: 537-566
[5] Lions P L, Perthame B, Tadmor E. Kinetic formulation of the isentropic gas dynamics and p-system. Comm Math Phys1994, 163: 415-431
[6] Lions P L, Perthame B, Souganidis P E. Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm Pure Appl Math, 1996, 49: 599-638
[7] Nishida T. Global solution for an initial boundary value problem of a quasilinear hyperbolic system. Proc Japan Acad, 1968, 44(7): 642-646
[8] Bakhvalov N S. The existence in the large of a regular solution of a quasilinear hyperbolic system. Ussr Comp Math and Math Phys, 1970, 10: 969-980
[9] Nishida T, Smoller J. Solutions in the large for some nonlinear hyperbolic conservation laws. Comm Pure Appl Math, 1973, 26(2): 183-200
[10] Diperna R J. Existence in the large for quasilinear hyperbolic conservation laws. Arch Ration Mech Anal1973, 52(3): 244-257
[11] Yang W, Liu S. Large BV solution for isentropic gas dynamics system with initial data far from vacuum. Applied Mathematics Letter, 2021, 114: 106914
[12] Dafermos C M.Hyperbolic Conservation Laws in Continuum Physics. Berlin: Springer, 2005: 436-460
