Abstract
In the current paper, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock waves have been studied in several papers. The main concern in the current paper is a $2 × 2$ system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock waves. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves.
Highlights
We consider large time behavior of solutions to a scalar conservation laws with an artificial heat flux ut + f (u)x + qx = 0 (1)over a one-dimensional full space R := (−∞, ∞)
The first aim of the current paper is to show existence and asymptotic stability of the viscous shock wave for the equation (1) with Cattaneo’s law, which is a 2 × 2 system of hyperbolic equations and called a system of Cattaneo-type through the current paper
Since this case is deduced to a scalar viscous conservation laws (4), existence and asymptotic stability of the viscous shock wave are already proved
Summary
Scalar viscous conservation laws, weighted energy method, relaxation limit, initial layer. The first aim of the current paper is to show existence and asymptotic stability of the viscous shock wave for the equation (1) with Cattaneo’s law, which is a 2 × 2 system of hyperbolic equations and called a system of Cattaneo-type through the current paper. Since this case is deduced to a scalar viscous conservation laws (4), existence and asymptotic stability of the viscous shock wave are already proved. C and C denote several generic positive constants
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