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Relative entropy in diffusive relaxation. (English) Zbl 1277.35246

The authors consider the system of isentropic gas dynamics with friction and the diffusive limit of it, being a single porous media equation for the density. Weak entropy solutions of this system are compared directly to a smooth solutions of the limit equation using a relative entropy analysis of the Lyapunov type. This provides a convergence to solutions of the equation that stay away from vacuum. The rate of convergence is estimated. The method is applied also to other types of systems, including the \(p\)-system with damping and viscoelasticity of the memory type.

MSC:

35L60 First-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
35Q31 Euler equations
35B25 Singular perturbations in context of PDEs
76N15 Gas dynamics (general theory)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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