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Pointwise convergence to shock waves for viscous conservation laws. (English) Zbl 0902.35069

The goal of the present paper is to study the behaviour of the perturbations of shock waves for nonlinear viscous conservation laws. The author uses a new pointwise approach, which consists of a time-asymptotic expansion, construction of approximate Green functions, and analysis of nonlinear wave interactions corresponding to different characteristic families. As an immediate consequence, the pointwise estimates yield optimal \(L^p\)-convergence of the perturbation to the shock and diffusion waves, \(1 \leq p \leq \infty\). This new approach of obtaining pointwise estimates based on Green functions for the linearized systems and the analysis of nonlinear wave interactions can be also applied to stability analysis of nonclassical shocks and various types of waves.

MSC:

35L65 Hyperbolic conservation laws
76L05 Shock waves and blast waves in fluid mechanics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35L67 Shocks and singularities for hyperbolic equations
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