Vanishing Viscosity with Short Wave–Long Wave Interactions for Systems of Conservation Laws

• Published in: Archive for rational mechanics and analysis Vol. 196; no. 3; pp. 981 - 1010
• Main Authors: Dias, João Paulo, Figueira, Mário, Frid, Hermano
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-06-2010; Springer; Springer Nature B.V
• Subjects: Applied classical electromagnetism; Classical Mechanics; Classical transport; Complex Systems; Electromagnetic wave propagation, radiowave propagation; Electromagnetism; electron and ion optics; Exact sciences and technology; Fluid- and Aerodynamics; Fundamental areas of phenomenology (including applications); Mathematical and Computational Physics; Physics; Physics and Astronomy; Solid mechanics; Static elasticity (thermoelasticity...); Statistical physics, thermodynamics, and nonlinear dynamical systems; Structural and continuum mechanics; Theoretical; Transport processes; Cauchy Problem; Wave Interaction; Entropy Solution; Nonlinear Elasticity; Weak Solution; Conservation laws; Long wave radiation; Viscosity; Strong convergence; Electromagnetism; Non linear elasticity; Wave interaction; Modelling; Non linear effect; Strong solution
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-009-0273-2

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 compensated compactness theory and new energy estimates obtained for the coupled systems. We analyze several of the representative examples, such as scalar conservation laws, general symmetric systems, nonlinear elasticity and nonlinear electromagnetism.