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

Grenier, Emmanuel; Guès, Olivier

J. Differ. Equations 143, No. 1, 110-146 (1998).

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.

Reviewer: Wang Jinghua (Beijing)

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

Keywords:

nonlinear maximal dissipative boundary conditions

References:

[1]	K. Asano, Zero viscosity limit of the incompressible Navier-Stokes equation, I and II; K. Asano, Zero viscosity limit of the incompressible Navier-Stokes equation, I and II · Zbl 0711.76023
[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
[4]	R. E. Caflisch, M. Sanmartino, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space I. Existence for Euler and Prandtl equations; R. E. Caflisch, M. Sanmartino, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space I. Existence for Euler and Prandtl equations · Zbl 0913.35102
[5]	R. E. Caflisch, M. Sanmartino, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space II. Construction of the Navier-Stokes solution; R. E. Caflisch, M. Sanmartino, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space II. Construction of the Navier-Stokes solution · Zbl 0913.35103
[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
[10]	E. Grenier, Boundary layers of characteristic nonlinear parabolic equations, J. Math. Pures Appl.; E. Grenier, Boundary layers of characteristic nonlinear parabolic equations, J. Math. Pures Appl. · Zbl 0914.35032
[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) · Zbl 0712.35061
[15]	J. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, 25, Am. Math. Soc. Providence, Rhode Island; J. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, 25, Am. Math. Soc. Providence, Rhode Island · Zbl 0642.58013
[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 · Zbl 0689.35001
[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 · Zbl 0164.12302
[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 · Zbl 0930.35002