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Global solution for an initial boundary value problem of a quasilinear hyperbolic system. (English) Zbl 0167.10301
From the introduction: We consider the following system of equations \[ \partial v/\partial t-\partial u/\partial x=0,\quad \partial u/\partial t+\partial (a^2/v)/\partial x=0,\tag{1} \] which is the simplest equation in gas dynamics (Lagrangian form in the isothermal case: \(p-a^2/v\), \(a\) is a constant \(>0\)), where \(v\) is the specific volume, \(u\) is the speed of the gas.
Here we consider the Cauchy problem in \(t\geq 0\), \(-\infty<x<+\infty\) for (1) with the initial values \[ v(0,x))v_0(x),\quad u(0,x)=u_0(x)\quad\text{for}\;-\infty<x<+\infty\tag{2} \] and also the piston problem (an initial boundary value problem) in \(t\geq 0\), \(x\geq 0\) for (1) with the boundary values
\[ v(0,x)=v_0(x),\quad u(0,x)=u_0(x)\;\text{for}\; x\geq 0,\quad u(t,0)=u_1(t)\;\text{for}\;t\geq 0,\tag{3} \] where \(v_0(x)\), \(u_0(x)\), \(u_1(t)\) are bounded functions with locally bounded variation and \(v_0(x)\geq\delta=\text{const.}>0\).
We see that the Cauchy problem (1), (2) and the piston problem (1), (3) have generalized solutions in the large. We use the Glimm’s (or slightly modified) difference scheme [J. Glimm, Commun. Pure Appl. Math. 18, 697–715 (1965; Zbl 0141.28902)] for the proof of the existence theorems.
There are many articles [listed in the references] which treat the existence theorem of the solution in the large for the initial value problem of the quasilinear hyperbolic system of equations, where the system is more general than in this paper, but the initial value is more restricted.

35L50 Initial-boundary value problems for first-order hyperbolic systems
35D99 Generalized solutions to partial differential equations
Full Text: DOI
[1] S. K. Godunov: J. of Num. Math, and Math. Phys., Vol. 1, 622-637 (1961). · Zbl 0129.19701 · doi:10.1016/0041-5553(63)90172-1
[2] J. Glimm: Comm. Pure Appl. Math., Vol. 18, 697-715 (1965). · Zbl 0141.28902 · doi:10.1002/cpa.3160180408
[3] J. Glimm and P. D. Lax: Bull. Amer. Math. Soc, Vol. 73, 105 (1967). · Zbl 0146.33803 · doi:10.1090/S0002-9904-1967-11666-5
[4] Zhang Tong and Guo Yu-Fa: Chinese Math., Vol. 7, 90-101 (1965).
[5] J. L. Johnson and J. A. Smoller: J. of Math, and Mech., Vol. 17, 561-576 (1967). · Zbl 0165.11402
[6] J. L. Johnson and J. A. Smoller: Global solutions of hyperbolic systems of conservation laws in two dependent variables (preprint). · Zbl 0164.12102 · doi:10.1090/S0002-9904-1968-12082-8
[7] J. L. Johnson: Ph. D. thesis, University of Michigan (1967).
[8] M. Yamaguti and T. Nishida: Funkcialaj Ekvacioj, Vol. 11 (1968). · Zbl 0183.38601
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