Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems


Author: F. Rousset
Journal: Trans. Amer. Math. Soc. 355 (2003), 2991-3008
MSC (2000): Primary 35K50, 35L50, 35L65, 76H20
DOI: https://doi.org/10.1090/S0002-9947-03-03279-3
Published electronically: March 17, 2003
MathSciNet review: 1975409
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity $\varepsilon$, $u^\varepsilon_t+F(u^\varepsilon)_x =\varepsilon(B(u^\varepsilon) u^\varepsilon_x )_x .$ When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that $u^\varepsilon$ converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small. This generalizes previous results obtained for $B$ invertible and the linear study of Serre and Zumbrun obtained for a pure Dirichlet's boundary condition.


References [Enhancements On Off] (What's this?)

  • 1. M. Gisclon, Étude des conditions aux limites pour un système strictement hyperbolique, via l'approximation parabolique, J. Math. Pures Appl. (9) 75 (1996), no. 5, 485-508. MR 97f:35129
  • 2. M. Gisclon and D. Serre, Étude des conditions aux limites pour un système strictement hyberbolique via l'approximation parabolique, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 4, 377-382. MR 95e:35119
  • 3. J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational Mech. Anal. 95 (1986), no. 4, 325-344. MR 88b:35127
  • 4. E. Grenier and O. Guès, Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems, J. Differential Equations 143 (1998), no. 1, 110-146. MR 98j:35026
  • 5. E. Grenier and F. Rousset, Stability of one-dimensional boundary layers by using Green's functions, Comm. Pure Appl. Math. 54 (2001), no. 11, 1343-1385. MR 2003a:35126
  • 6. S. Kawashima, Systems of a hyperbolic parabolic type with applications to the equations of magnetohydrodynamics, Ph.D. thesis, Kyoto University (1983).
  • 7. G. Kreiss and H.-O. Kreiss, Stability of systems of viscous conservation laws, Comm. Pure Appl. Math. 51 (1998), no. 11-12, 1397-1424. MR 2000c:35156
  • 8. T. T. Li and W. C. Yu, Boundary value problems for quasilinear hyperbolic systems, Duke University Mathematics Department, Durham, N.C., 1985. MR 88g:35115
  • 9. C. Mascia and K. Zumbrun, Stability of viscous shock profiles for dissipative symmetric hyperbolic-parabolic systems, Preprint (2001).
  • 10. A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system compressible viscous gas, Commun. Math. Physics 222 (2001), 449-474. MR 2002m:76083
  • 11. F. Rousset, The boundary conditions coming from the real vanishing viscosity method, Discrete Continuous Dynamical Systems (to appear).
  • 12. D. Serre and K. Zumbrun, Boundary layer stability in real vanishing viscosity limit, Commun. Math. Phys. 221 (2001), 267-292.
  • 13. Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249-275. MR 86k:35107
  • 14. K. Zumbrun, Multidimensional stability of planar viscous shock waves., Advances in the theory of shock waves, Prog. Nonlinear Differential Equations Appl. 47, Birkhaüser, Boston, MA, 307-516 (2001). (English). MR 2002k:35200
  • 15. K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (1998), no. 3, 741-871. MR 99m:35157

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35K50, 35L50, 35L65, 76H20

Retrieve articles in all journals with MSC (2000): 35K50, 35L50, 35L65, 76H20


Additional Information

F. Rousset
Affiliation: ENS Lyon, UMPA (UMR 5669 CNRS), 46, allée d’Italie, 69364 Lyon Cedex 07, France
Address at time of publication: Laboratoire Dieudonné, Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
Email: frousset@umpa.ens-lyon.fr, frousset@math.unice.fr

DOI: https://doi.org/10.1090/S0002-9947-03-03279-3
Received by editor(s): January 30, 2002
Received by editor(s) in revised form: December 13, 2002
Published electronically: March 17, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society