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.

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author = {Nguyen, Toan and Zumbrun, Kevin},
title = {Long-time stability of noncharacteristic viscous boundary layers},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
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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://www.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 2403609 | Zbl 1171.35071

[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 1946769 | Zbl 1047.37053

[Br1] L. Q. Brin. Numerical testing of the stability of viscous shock waves, PhD thesis, Indiana University, Bloomington, 1998. | MR 2613047

[Br2] L. Q. Brin. Numerical testing of the stability of viscous shock waves, Math. Comp., 70(235):1071–1088, 2001. | MR 1710652 | Zbl 0980.65092

[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 1965784 | Zbl 1044.35057

[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 2505363 | Zbl 1169.76051

[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 1846801 | Zbl 1026.35015

[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 2114817 | Zbl 1058.35163

[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 2376200 | Zbl 1138.35052

[HZ] P. Howard and K. Zumbrun, Stability of undercompressive viscous shock waves, in press, J. Differential Equations 225 (2006), no. 1, 308–360. | MR 2228699 | Zbl 1102.35069

[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 2563632

[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 1355414 | Zbl 0842.35076

[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 1471469 | Zbl 0882.76074

[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 2253406 | Zbl 1101.65082

[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 2238883 | Zbl 1117.35062

[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 2005853 | Zbl 1038.35057

[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 2004135 | Zbl 1035.35074

[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 2048568 | Zbl 1058.35160

[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 1888084 | Zbl 1018.76038

[MZ] Métivier, G. and Zumbrun, K., Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems, Memoirs AMS, 826 (2005). | Zbl 1074.35066

[N2] T. Nguyen, On asymptotic stability of noncharacteristic viscous boundary layers, SIAM J. Math. Analysis, to appear. | MR 2644918

[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 2565843

[NZ2] T. Nguyen and K. Zumbrun, Long-time stability of multi-dimensional noncharacteristic viscous boundary layers, Preprint, 2008 | MR 2672797

[RZ] M. Raoofi and K. Zumbrun, Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems Preprint, 2007. | MR 2488696

[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 122222 | Zbl 0096.20105

[SZ] Serre, D. and Zumbrun, K., Boundary layer stability in real vanishing-viscosity limit, Comm. Math. Phys. 221 (2001), no. 2, 267–292. | MR 1845324 | Zbl 0988.35028

[YZ] S. Yarahmadian and K. Zumbrun, Pointwise Green function bounds and long-time stability of large-amplitude noncharacteristic boundary layers, Preprint (2008). | MR 2628016

[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 1842778 | Zbl 0989.35089

[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 2099037

[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 2348937 | Zbl 1138.35061