Long-time stability of noncharacteristic viscous boundary layers
Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé no. 6, 15 p.

We report our results on long-time stability of multi–dimensional noncharacteristic boundary layers of a class of hyperbolic–parabolic systems including the compressible Navier–Stokes equations with inflow [outflow] boundary conditions, under the assumption of strong spectral, or uniform Evans, stability. Evans stability has been verified for small-amplitude layers by Guès, Métivier, Williams, and Zumbrun. For large–amplitudes, it may be checked numerically, as done in one–dimensional case for isentropic gas by Costanzino, Humpherys, Nguyen, and Zumbrun.

Nguyen, Toan 1 ; Zumbrun, Kevin 2

1 Institut de Mathématiques de Jussieu Université Pierre et Marie Curie (Paris 6)
2 Department of Mathematics Indiana University Bloomington IN 47402
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Nguyen, Toan; Zumbrun, Kevin. Long-time stability of noncharacteristic viscous boundary layers. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé no. 6, 15 p. http://archive.numdam.org/item/SEDP_2009-2010____A6_0/

[BHRZ] B. Barker, J. Humpherys, K. Rudd, and K. Zumbrun. Stability of viscous shocks in isentropic gas dynamics, to appear, Comm. Math. Phys. | MR | Zbl

[Bra] Braslow, A.L., A history of suction-type laminar-flow control with emphasis on flight research, NSA History Division, Monographs in aerospace history, number 13 (1999).

[BDG] T. J. Bridges, G. Derks, and G. Gottwald, Stability and instability of solitary waves of the fifth- order KdV equation: a numerical framework, Phys. D, 172(1-4):190–216, 2002. | MR | Zbl

[Br1] L. Q. Brin. Numerical testing of the stability of viscous shock waves, PhD thesis, Indiana University, Bloomington, 1998. | MR

[Br2] L. Q. Brin. Numerical testing of the stability of viscous shock waves, Math. Comp., 70(235):1071–1088, 2001. | MR | Zbl

[BrZ] L. Q. Brin and K. Zumbrun. Analytically varying eigenvectors and the stability of viscous shock waves, Mat. Contemp., 22:19–32, 2002, Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). | MR | Zbl

[CHNZ] N. Costanzino, J. Humpherys, T. Nguyen, and K. Zumbrun, Spectral stability of noncharacteristic boundary layers of isentropic Navier–Stokes equations, to appear, Arch. Ration. Mech. Anal. | MR | Zbl

[GR] Grenier, E. and Rousset, F., Stability of one dimensional boundary layers by using Green’s functions, Comm. Pure Appl. Math. 54 (2001), 1343-1385. | MR | Zbl

[GMWZ1] O. Guès, G. Métivier, M. Williams, and K. Zumbrun. Multidimensional viscous shocks I: degenerate symmetrizers and long time stability, J. Amer. Math. Soc. 18 (2005), no. 1, 61–120. | MR | Zbl

[GMWZ5] O. Guès, G. Métivier, M. Williams, and K. Zumbrun. Existence and stability of noncharacteristic hyperbolic-parabolic boundary-layers. Preprint, 2008.

[GMWZ6] O. Guès, G. Métivier, M. Williams, and K. Zumbrun. Viscous boundary value problems for symmetric systems with variable multiplicities J. Differential Equations 244 (2008) 309–387. | MR | Zbl

[HZ] P. Howard and K. Zumbrun, Stability of undercompressive viscous shock waves, in press, J. Differential Equations 225 (2006), no. 1, 308–360. | MR | Zbl

[HLZ] J. Humpherys, O. Lafitte, and K. Zumbrun. Stability of viscous shock profiles in the high Mach number limit, (Preprint, 2007).

[HLyZ1] Humpherys, J., Lyng, G., and Zumbrun, K., Spectral stability of ideal-gas shock layers, Preprint (2007). | MR

[HLyZ2] Humpherys, J., Lyng, G., and Zumbrun, K., Multidimensional spectral stability of large-amplitude Navier-Stokes shocks, in preparation.

[HoZ1] D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J. 44 (1995), no. 2, 603–676. | MR | Zbl

[HoZ2] D. Hoff and K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys. 48 (1997), no. 4, 597–614. | MR | Zbl

[HuZ] J. Humpherys and K. Zumbrun. An efficient shooting algorithm for evans function calculations in large systems, Physica D, 220(2):116–126, 2006. | MR | Zbl

[KK] Y. Kagei and S. Kawashima Stability of planar stationary solutions to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 266 (2006), 401-430. | MR | Zbl

[KNZ] S. Kawashima, S. Nishibata, and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003), no. 3, 483–500. | MR | Zbl

[MaZ3] C. Mascia and K. Zumbrun. Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Ration. Mech. Anal., 169(3):177–263, 2003. | MR | Zbl

[MaZ4] C. Mascia and K. Zumbrun. Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Ration. Mech. Anal., 172(1):93–131, 2004. | MR | Zbl

[MN] Matsumura, A. and Nishihara, K., Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys., 222 (2001), no. 3, 449–474. | MR | Zbl

[MZ] Métivier, G. and Zumbrun, K., Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems, Memoirs AMS, 826 (2005). | Zbl

[N2] T. Nguyen, On asymptotic stability of noncharacteristic viscous boundary layers, SIAM J. Math. Analysis, to appear. | MR

[NZ1] T. Nguyen and K. Zumbrun, Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic-parabolic systems, J. Maths. Pures et Appliquées, to appear. | MR

[NZ2] T. Nguyen and K. Zumbrun, Long-time stability of multi-dimensional noncharacteristic viscous boundary layers, Preprint, 2008 | MR

[RZ] M. Raoofi and K. Zumbrun, Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems Preprint, 2007. | MR

[S] H. Schlichting, Boundary layer theory, Translated by J. Kestin. 4th ed. McGraw-Hill Series in Mechanical Engineering. McGraw-Hill Book Co., Inc., New York, 1960. | MR | Zbl

[SZ] Serre, D. and Zumbrun, K., Boundary layer stability in real vanishing-viscosity limit, Comm. Math. Phys. 221 (2001), no. 2, 267–292. | MR | Zbl

[YZ] S. Yarahmadian and K. Zumbrun, Pointwise Green function bounds and long-time stability of large-amplitude noncharacteristic boundary layers, Preprint (2008). | MR

[Z2] K. Zumbrun. Multidimensional stability of planar viscous shock waves. In Advances in the theory of shock waves, volume 47 of Progr. Nonlinear Differential Equations Appl., pages 307–516. Birkhäuser Boston, Boston, MA, 2001. | MR | Zbl

[Z3] K. Zumbrun. Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In Handbook of mathematical fluid dynamics. Vol. III, pages 311–533. North-Holland, Amsterdam, 2004. With an appendix by Helge Kristian Jenssen and Gregory Lyng. | MR

[Z4] K. Zumbrun. Planar stability criteria for viscous shock waves of systems with real viscosity. In Hyperbolic systems of balance laws, volume 1911 of Lecture Notes in Math., pages 229–326. Springer, Berlin, 2007. | MR | Zbl