##
**Stability of multidimensional viscous shocks for symmetric systems with variable multiplicities.**
*(English)*
Zbl 1197.35045

The author establishes long-time stability of multidimensional viscous shocks of a general class of symmetric hyperbolic-parabolic systems with variable multiplicities, notably including the equations of compressible magnetohydrodynamics (MHD) in dimensions more than one. The obtained result extends the existing result established by Zumbrun for systems with characteristics of constant multiplicity to the ones with variable multiplicity, yielding the first such stability result for (fast) MHD shocks. At the same time, the author drops the technical assumption on the structure of the so-called glancing set that was used in the previous investigations. The key idea for the improvements is to use a new simple argument for obtaining a bound for the \(L^1\to L^p\) resolvent in the low-frequency regimes by employing the recent construction of the degenerate Kreiss symmetrizers by Gues, Metivier, Williams, and Zumbrun. Thus, at the low-frequency resolvent bound level, our analysis gives an alternative to the earlier pointwise Green-function approach of Zumbrun. High-frequency solution operator bounds have previously been established entirely by nonlinear energy estimates.

Reviewer: Michael I. Gil’ (Beer-Sheva)

### MSC:

35B35 | Stability in context of PDEs |

35L60 | First-order nonlinear hyperbolic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35L67 | Shocks and singularities for hyperbolic equations |

76W05 | Magnetohydrodynamics and electrohydrodynamics |

### References:

[1] | 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), 61–120. · Zbl 1058.35163 |

[2] | -, Viscous boundary value problems for symmetric systems with variable multiplicities , J. Differential Equations 244 (2008), 309–387. · Zbl 1138.35052 |

[3] | -, Existence and stability of noncharacteristic hyperbolic-parabolic boundary-layers , to appear in Arch. Ration. Mech. Anal. |

[4] | D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J. 44 (1995), 603–676. · Zbl 0842.35076 |

[5] | -, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys. 48 (1997), 597–614. · Zbl 0882.76074 |

[6] | J. Humpherys, G. Lyng, and K. Zumbrun, Spectral stability of ideal-gas shock layers , to appear in Arch. Ration. Mech. Anal. · Zbl 1422.76122 |

[7] | -, Multidimensional spectral stability of large-amplitude Navier-Stokes shocks , in preparation. |

[8] | G. Kreiss and H.-G. Kreiss, Stability of systems of viscous conservation laws, Comm. Pure Appl. Math. 51 (1998), 1397–1424. · Zbl 0935.35013 |

[9] | B. Kwon and K. Zumbrun, Asymptotic behavior of multidimensional scalar relaxation shocks , to appear in J. Hyperbolic Differ. Equ. · Zbl 1188.35115 |

[10] | C. Mascia and K. Zumbrun, Pointwise Green function bounds for shock profiles of systems with real viscosity , Arch. Ration. Mech. Anal. 169 (2003), 177–263. · Zbl 1035.35074 |

[11] | -, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems , Arch. Ration. Mech. Anal. 172 (2004), 93–131. · Zbl 1058.35160 |

[12] | G. MéTivier and K. Zumbrun, Hyperbolic boundary value problems for symmetric systems with variable multiplicities , J. Differential Equations 211 (2005), 61–134. · Zbl 1073.35155 |

[13] | -, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems , Mem. Amer. Math. Soc. 175 (2005), no. 826. · Zbl 1074.35066 |

[14] | T. Nguyen, On asymptotic stability of noncharacteristic viscous boundary layers, · Zbl 1211.35049 |

[15] | T. Nguyen and K. Zumbrun, Long-time stability of multi-dimensional noncharacteristic viscous boundary layers , · Zbl 1402.35228 |

[16] | K. Zumbrun, “Multidimensional stability of planar viscous shock waves” in Advances in the Theory of Shock Waves , Progr. Nonlinear Differential Equations Appl. 47 , Birkhäuser, Boston, 2001, 307–516. · Zbl 0989.35089 |

[17] | -, “Stability of large-amplitude shock waves of compressible Navier-Stokes equations” in Handbook of Mathematical Fluid Dynamics, Vol. III , with an appendix by H. K. Jenssen and G. Lyng, North-Holland, Amsterdam, 2004, 311–533. · Zbl 1222.35156 |

[18] | K. Zumbrun, “Planar stability criteria for viscous shock waves of systems with real viscosity” in Hyperbolic Systems of Balance Laws , Lecture Notes in Math. 1911 , Springer, Berlin, 2007, 229–326. · Zbl 1138.35061 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.