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Uniqueness and stability of Riemann solutions with large oscillation in gas dynamics. (English) Zbl 1029.76045

The authors study a system of conservation laws for Euler system of gas dynamics in Lagrangian coordinates \(\partial_t u+\partial_xp=0\), \(\partial_t u+\partial_xu=0\), \(\partial_t (e+u^2/2)+\partial_x(pu)=0\), where \(u,p,v\) and \(e\) are velocity, pressure, specific volume (\(v=1/\rho \), \(\rho \) density) and internal energy of the fluid, respectively. The Riemann problem for this system is a special Cauchy problem with initial data, taking two different constant states for \(x<0\) and for \(x>0\). The solutions are known as Riemann solutions or simple waves (states). The large-time behavior of the solutions in \(L^{\infty }\cap BV_{\text{loc}}(\mathbb{R}_+^2)\) is of interest here. The authors study the asymptotic stability of classical Riemann solution \(R(x/t)\) for the considered Riemann problem. The uniqueness of Riemann solution under entropy condition (entropy solution) in the class \(L^{\infty }\cap BV\) with arbitrarily large oscillation is proved. For this purpose the authors use the global behavior of shock curves in phase space, and the singularity of centered rarefaction waves near the center in physical plane. The uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large \(L^{1}\cap L^{\infty }\cap BV\) perturbation of Riemann initial data, as long as the corresponding solutions belongs to \(L^{\infty }\) and have local bounded total variation satisfying a natural condition for its growth with time.

MSC:

76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q35 PDEs in connection with fluid mechanics
35L65 Hyperbolic conservation laws
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