Vanishing Viscosity with Short Wave Long Wave Interactions for Systems of Conservation Laws
نویسنده
چکیده
Motivated by Benney’s general theory, we propose new models for short wave long wave interactions when the long waves are described by nonlinear systems of conservation laws. We prove the strong convergence of the solutions of the vanishing viscosity and short wave long wave interactions systems by using compactness results from the compensated compactness theory and new energy estimates obtained for the coupled systems. We analyse several of the representative examples such as scalar conservation laws, general symmetric systems, nonlinear elasticity, nonlinear electromagnetism.
منابع مشابه
The Riemann problem admitting δ′-shock wave type solution (the vanishing viscosity approach)
In this paper, using the vanishing viscosity method, a solution of the Riemann problem for the system of conservation laws ut + ( u ) x = 0, vt + 2 ( uv ) x = 0, wt + 2 ( v + uw ) x = 0 with the initial data (u(x, 0), v(x, 0), w(x, 0)) = { (u−, v−, w−), x < 0, (u+, v+, w+), x > 0, is constructed. This problem admits a δ′-shock wave type solution, which is a new type of singular solutions to sys...
متن کاملHyperbolic Conservation Laws An Illustrated Tutorial
These notes provide an introduction to the theory of hyperbolic systems of conservation laws in one space dimension. The various chapters cover the following topics: 1. Meaning of a conservation equation and definition of weak solutions. 2. Hyperbolic systems. Explicit solutions in the linear, constant coefficients case. Nonlinear effects: loss of regularity and wave interactions. 3. Shock wave...
متن کاملSharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems
Consider a general strictly hyperbolic, quasilinear system, in one space dimension ut +A(u)ux = 0, (1) where u 7→ A(u), u ∈ Ω ⊂ R , is a smooth matrix-valued map. Given an initial datum u(0, ·) with small total variation, let u(t, ·) be the corresponding (unique) vanishing viscosity solution of (1) obtained as limit of solutions to the viscous parabolic approximation ut +A(u)ux = μuxx, as μ→ 0....
متن کاملStable Viscosity Matrices for Systems of Conservation Laws
A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, II, + f(u), =0, u E R”, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u, + f(u), = v(Du,), as v -t 0 if D is in ...
متن کاملVanishing Viscosity Limit for Initial-Boundary Value Problems for Conservation Laws
The convergence of the vanishing viscosity method for initialboundary value problems is analyzed for nonlinear hyperbolic conservation laws through several representative systems. Some techniques are developed to construct the global viscous solutions and establish the H−1 compactness of entropy dissipation measures for the convergence of the viscous solutions with general initial-boundary cond...
متن کامل