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Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. (English) Zbl 0896.35078

The authors study viscous perturbations of quasilinear hyperbolic systems in several dimensions as the viscosity goes to zero. The boundary is noncharacteristic for the hyperbolic system. In particular, they describe the boundary layer which arises near the boundary and give a sufficient condition for the convergence of the solution to the solution of some mixed hyperbolic problem with some nonlinear maximal dissipative boundary conditions. A counterexample is given when this condition is not satisfied, and the solution blows up as the viscosity goes to zero.

MSC:

35L50 Initial-boundary value problems for first-order hyperbolic systems
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35L60 First-order nonlinear hyperbolic equations
35B25 Singular perturbations in context of PDEs
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[2] Bardos, C.; Brezis, D.; Brezis, H., Perturbations singulières et prolongements maximaux d’opérateurs positifs, Arch. Rational Mech. Anal., 53, 69-100 (1973) · Zbl 0281.47028
[3] Bardos, C.; Rauch, J., Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Amer. Math. Soc., 270, 377-408 (1982) · Zbl 0485.35010
[6] Egorov, Y. V.; Shubin, M. A., Partial Differential Equations I. Partial Differential Equations I, Encyclopaedia of Mathematical Sciences, 30 (1988), Springer-Verlag: Springer-Verlag Berlin/New York
[7] Gisclon, M., Etude des conditions aux limites pour un système strictement hyperbolique, via l’approximation parabolique, J. Math. Pures Appl., 75, 485-508 (1996) · Zbl 0869.35061
[8] Gisclon, M.; Serre, D., Etude des conditions aux limites pour un système strictement hyperbolique, via l’approximation parabolique, C.R. Acad. Sci. Paris, Série I, 319, 377-382 (1994) · Zbl 0808.35075
[9] Grenier, E., Couches limites de systèmes paraboliques nonlinéaires caractéristiques, C.R. Acad. Sci. Paris, 323, 1013-1017 (1996) · Zbl 0863.35010
[11] Guès, O., Couches limites pour des problèmes mixtes hyperboliques, Séminaire E.D.P. de l’Ecole Polytechnique (1994) · Zbl 0886.35091
[12] Guès, O., Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites, Ann. Inst. Fourier, 973-1006 (1995) · Zbl 0831.34023
[13] Guès, O., Viscous boundary layers and high frequency oscillations, (Rauch, J.; Taylor, M., Singularities and Oscillations (1997), Springer-Verlag: Springer-Verlag Berlin/New York), 61-77 · Zbl 0877.35072
[14] Guès, O., Problèmes mixtes hyperboliques quasilinéaires caractéristiques, Comm. Partial Differential Equations, 15, 595-645 (1990)
[16] Koppel, N.; Howard, L. N., Bifurcations and trajectories joining critical points, Adv. in Math., 18, 306-358 (1975) · Zbl 0361.34026
[17] Kreiss, H.-O.; Lorenz, J., Initial boundary value problems and the Navier-Stokes equations, Pure and Applied Mathematics (1989), Academic Press: Academic Press London
[18] Lions, J.-L., Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lectures Notes in Math. (1973), Springer-Verlag: Springer-Verlag Boston/New York · Zbl 0268.49001
[19] Lax, P. D.; Philips, R. S., Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math., 13, 427-455 (1960) · Zbl 0094.07502
[20] Ladyzhenskaya, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and quasilinear equations of parabolic type, Moscou 1967, Trans. Math. Monographs (1968), Amer. Math. Soc: Amer. Math. Soc Providence
[21] Majda, A., Compressible fluid flow and systems of conservation laws in several space variables, Appl. Math. Sci. (1984), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0537.76001
[22] Métivier, G., Problèmes mixtes non linéaires et stabilité des chocs multidimensionnels, Sém. Bourbaki, 671 (1987) · Zbl 0659.35063
[23] Rauch, J., Boundary value problems as limits of problems in all space, Sém. Goulaouico Schwartz, Ecole Polytechnique (1978) · Zbl 0435.35052
[24] Rauch, J.; Massey, F., Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189, 303-318 (1974) · Zbl 0282.35014
[25] Serre, D., Systèmes de lois de conservation II (1996), Diderot editeur, Arts et Sciences
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