Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

Guès, Olivier¹ ; Métivier, Guy² ; Williams, Mark³ ; Zumbrun, Kevin⁴

¹ LATP, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille, France
² MAB, Université de Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France
³ University of North Carolina, Department of Mathematics, CB 3250, Phillips Hall, Chapel Hill, NC 27599
⁴ Indiana University, Department of Mathematics, Rawles Hall, Bloomington, IN 47405

Journal of the American Mathematical Society, Tome 18 (2005) no. 1, pp. 61-120

Résumé

We use energy estimates to study the long time stability of multidimensional planar viscous shocks $\psi (x_1)$ for systems of conservation laws. Stability is proved for both zero mass and nonzero mass perturbations, and some of the results include rates of decay in time. Shocks of any strength are allowed, subject to an appropriate Evans function condition. The main tools are a conjugation argument that allows us to replace the eigenvalue equation by a problem in which the $x_1$ dependence of the coefficients is removed and degenerate Kreiss-type symmetrizers designed to cope with the vanishing of the Evans function for zero frequency.

Détail

DOI : 10.1090/S0894-0347-04-00470-9

Bibliographie

[1] Coppel, W. A. Stability and asymptotic behavior of differential equations 1965

[2] Chazarain, Jacques, Piriou, Alain Introduction to the theory of linear partial differential equations 1982

[3] Francheteau, Jacques, Mã©Tivier, Guy Existence de chocs faibles pour des systèmes quasi-linéaires hyperboliques multidimensionnels C. R. Acad. Sci. Paris Sér. I Math. 1998 725 728

[4] Freistã¼Hler, Heinrich, Liu, Tai-Ping Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws Comm. Math. Phys. 1993 147 158

[5] Freistã¼Hler, H., Szmolyan, P. Spectral stability of small shock waves Arch. Ration. Mech. Anal. 2002 287 309

[6] Gardner, Robert A., Zumbrun, Kevin The gap lemma and geometric criteria for instability of viscous shock profiles Comm. Pure Appl. Math. 1998 797 855

[7] Goodman, Jonathan Nonlinear asymptotic stability of viscous shock profiles for conservation laws Arch. Rational Mech. Anal. 1986 325 344

[8] Goodman, Jonathan Remarks on the stability of viscous shock waves 1991 66 72

[9] Goodman, Jonathan Stability of viscous scalar shock fronts in several dimensions Trans. Amer. Math. Soc. 1989 683 695

[10] Goodman, Jonathan, Miller, Judith R. Long-time behavior of scalar viscous shock fronts in two dimensions J. Dynam. Differential Equations 1999 255 277

[11] Guã¨S, Olivier, Mã©Tivier, Guy, Williams, Mark, Zumbrun, Kevin Multidimensional viscous shocks. II. The small viscosity limit Comm. Pure Appl. Math. 2004 141 218

[12] Guã¨S, Olivier, Williams, Mark Curved shocks as viscous limits: a boundary problem approach Indiana Univ. Math. J. 2002 421 450

[13] Hoff, David, Zumbrun, Kevin Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves Z. Angew. Math. Phys. 1997 597 614

[14] Hoff, David, Zumbrun, Kevin Pointwise Green’s function bounds for multidimensional scalar viscous shock fronts J. Differential Equations 2002 368 408

[15] Hoff, David, Zumbrun, Kevin Asymptotic behavior of multidimensional scalar viscous shock fronts Indiana Univ. Math. J. 2000 427 474

[16] Howard, Peter Pointwise estimates on the Green’s function for a scalar linear convection-diffusion equation J. Differential Equations 1999 327 367

[17] Howard, Peter Pointwise Green’s function approach to stability for scalar conservation laws Comm. Pure Appl. Math. 1999 1295 1313

[18] Kapitula, Todd M. Stability of weak shocks in 𝜆–𝜔 systems Indiana Univ. Math. J. 1991 1193 1219

[19] Kapitula, Todd On the stability of travelling waves in weighted 𝐿^{∞} spaces J. Differential Equations 1994 179 215

[20] Kapitula, Todd, Sandstede, Bjã¶Rn Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations Phys. D 1998 58 103

[21] Kawashima, Shuichi, Matsumura, Akitaka, Nishihara, Kenji Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas Proc. Japan Acad. Ser. A Math. Sci. 1986 249 252

[22] Kreiss, Heinz-Otto Initial boundary value problems for hyperbolic systems Comm. Pure Appl. Math. 1970 277 298

[23] Kreiss, Gunilla, Kreiss, Heinz-Otto Stability of systems of viscous conservation laws Comm. Pure Appl. Math. 1998 1397 1424

[24] Kreiss, Gunilla, Kreiss, Heinz-Otto, Petersson, N. Anders On the convergence to steady state of solutions of nonlinear hyperbolic-parabolic systems SIAM J. Numer. Anal. 1994 1577 1604

[25] Lax, Peter D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves 1973

[26] Liu, Tai-Ping Nonlinear stability of shock waves for viscous conservation laws Mem. Amer. Math. Soc. 1985

[27] Liu, Tai-Ping Pointwise convergence to shock waves for viscous conservation laws Comm. Pure Appl. Math. 1997 1113 1182

[28] Liu, Tai-Ping, Xin, Zhou Ping Stability of viscous shock waves associated with a system of non-strictly hyperbolic conservation laws Comm. Pure Appl. Math. 1992 361 388

[29] Liu, Tai-Ping, Zeng, Yanni Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws Mem. Amer. Math. Soc. 1997

[30] Liu, Tai-Ping, Zumbrun, Kevin Nonlinear stability of an undercompressive shock for complex Burgers equation Comm. Math. Phys. 1995 163 186

[31] Liu, Tai-Ping, Zumbrun, Kevin On nonlinear stability of general undercompressive viscous shock waves Comm. Math. Phys. 1995 319 345

[32] Majda, Andrew The stability of multidimensional shock fronts Mem. Amer. Math. Soc. 1983

[33] Majda, Andrew The existence of multidimensional shock fronts Mem. Amer. Math. Soc. 1983

[34] Mascia, Corrado, Zumbrun, Kevin Pointwise Green’s function bounds and stability of relaxation shocks Indiana Univ. Math. J. 2002 773 904

[35] Mascia, C., Zumbrun, K. Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems Comm. Pure Appl. Math. 2004 841 876

[36] Mascia, Corrado, Zumbrun, Kevin Pointwise Green function bounds for shock profiles of systems with real viscosity Arch. Ration. Mech. Anal. 2003 177 263

[37] Mascia, Corrado, Zumbrun, Kevin Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems Arch. Ration. Mech. Anal. 2004 93 131

[38] Matsumura, Akitaka, Nishihara, Kenji On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas Japan J. Appl. Math. 1985 17 25

[39] Mã©Tivier, Guy Problèmes mixtes non linéaires et stabilité des chocs multidimensionnels Astérisque 1987

[40] Mã©Tivier, Guy Stability of multidimensional shocks 2001 25 103

[41] Plaza, Ramon, Zumbrun, Kevin An Evans function approach to spectral stability of small-amplitude shock profiles Discrete Contin. Dyn. Syst. 2004 885 924

[42] Rousset, Frederic Viscous approximation of strong shocks of systems of conservation laws SIAM J. Math. Anal. 2003 492 519

[43] Szepessy, Anders, Xin, Zhou Ping Nonlinear stability of viscous shock waves Arch. Rational Mech. Anal. 1993 53 103

[44] Zumbrun, Kevin Multidimensional stability of planar viscous shock waves 2001 307 516

[45] Zumbrun, Kevin, Howard, Peter Pointwise semigroup methods and stability of viscous shock waves Indiana Univ. Math. J. 1998 741 871

[46] Zumbrun, K., Serre, D. Viscous and inviscid stability of multidimensional planar shock fronts Indiana Univ. Math. J. 1999 937 992