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.
@article{SEDP_2009-2010____A6_0,
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"},
note = {talk:6},
pages = {1--15},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2009-2010},
language = {en},
url = {http://www.numdam.org/item/SEDP_2009-2010____A6_0/}
}
TY - JOUR AU - Nguyen, Toan AU - Zumbrun, Kevin TI - Long-time stability of noncharacteristic viscous boundary layers JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:6 PY - 2009-2010 SP - 1 EP - 15 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_2009-2010____A6_0/ LA - en ID - SEDP_2009-2010____A6_0 ER -
%0 Journal Article %A Nguyen, Toan %A Zumbrun, Kevin %T Long-time stability of noncharacteristic viscous boundary layers %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:6 %D 2009-2010 %P 1-15 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://www.numdam.org/item/SEDP_2009-2010____A6_0/ %G en %F SEDP_2009-2010____A6_0
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 | 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
