Nonlinear Conservation Laws and Applications pp 433-445 | Cite as
Viscous System of Conservation Laws: Singular Limits
Abstract
We continue our analysis of the Cauchy problem for viscous system of conservation, under natural assumptions. We examine in which way does the existence time depend upon the viscous tensor B(u). In particular, we consider singular limits, where the rank of the symbol \(B(u; \xi)\) drops at the limit. This covers a lot of situations, for instance that of the limit of the Navier-Stokes-Fourier system towards the Euler- Fourier system, or that of the vanishing viscosity. We emphasize the symmetry of the dissipation tensor, an hypothesis which is reminiscent to the Onsager’s reciprocity relations. We find it useful in this asymptotic context, when establishing uniform estimates.
Key words
Systems of conservation laws dissipative structure entropy singular limit Onsager’s reciprocity relationsPreview
Unable to display preview. Download preview PDF.
Reference
- 1.S. Benzoni-Gavage and D. Serre. Multi-dimensional hyperbolic partial differential equations. First order systems and applications. Oxford Mathematical Monographs, Oxford University Press (2007).Google Scholar
- 2.G. Billet, V. Giovangigli, and G. de Gassowski. Impact of volume viscosity on a shock-hydrogen-bubble interaction. Combustion Theory and Modelling, 12 (2008), pp. 221–248.CrossRefMATHMathSciNetGoogle Scholar
- 3.C. Dafermos and L. Hsiao. Development of singularities in solutions of the equations of nonlinear thermoelasticity. Quarterly of applied mathematics, 44 (1986), pp. 463–474.MATHMathSciNetGoogle Scholar
- 4.S. Kawashima. Systems of a hyperbolic parabolic type with applications to the equations of magnetohydrodynamics. PhD Thesis, Kyoto University (1983).Google Scholar
- 5.L. Onsager. Reciprocal relations in irreversible processes. I. Phys. Rev., 37 (1931), pp. 405–426.MATHGoogle Scholar
- 6.D. Serre. Sur l´équation monodimensionnelle d’un fluide visqueux, compressible et conducteur de chaleur. C. R. Acad. Sci. Paris Sér. I Math., 303 (1986), pp. 703–706.MATHMathSciNetGoogle Scholar
- 7.D. Serre. The structure of dissipative viscous system of conservation laws. Physica D 239 (2010), pp. 1381–1386.MathSciNetGoogle Scholar
- 8.D. Serre. Local existence for viscous system of conservation laws: Hs-data with s > 1 + d/2. Proceedings of C.A.S. program on PDEs, Oslo 2009. H. Holden & K. Kalrsen eds. Contemporary Mathematics, Vol. 526. American Mathematical Society, Providence, RI., pp. 339–358.Google Scholar