Stability of Noncharacteristic Viscous Boundary <br />Layers
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | METIVIER, Guy | |
| dc.date.accessioned | 2024-04-04T03:18:37Z | |
| dc.date.available | 2024-04-04T03:18:37Z | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/194427 | |
| dc.description | 1 Introduction 3 <br />2 Hyperbolic-parabolic boundary value problems 8 <br />2.1 Structure of equations . . . . . . . . . . . . . . . . . . . . . . 8 <br />2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 11 <br />3 Layers profiles 12 <br />3.1 The profile equation . . . . . . . . . . . . . . . . . . . . . . . 12 <br />3.2 Existence of profiles . . . . . . . . . . . . . . . . . . . . . . . 13 <br />3.3 The inviscid boundary conditions . . . . . . . . . . . . . . . . 14 <br />3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 <br />4 Spectral stability 19 <br />4.1 The linearized equations . . . . . . . . . . . . . . . . . . . . . 19 <br />4.2 Structure of the linearized equations . . . . . . . . . . . . . . 21 <br />4.3 Evans functions and Lopatinski determinant; weak stability . 25 <br />4.4 Maximal estimates and uniform spectral stability conditions 28 <br />4.4.1 Low and medium frequencies . . . . . . . . . . . . . . 28 <br />4.4.2 High frequencies . . . . . . . . . . . . . . . . . . . . . 29 <br />4.4.3 The inviscid case . . . . . . . . . . . . . . . . . . . . . 31 <br />5 The Zumbrun-Serre-Rousset Theorem and the reduced low <br />frequency problem 32 <br />5.1 Transversality is necessary . . . . . . . . . . . . . . . . . . . . 32 <br />5.2 The reduced problem . . . . . . . . . . . . . . . . . . . . . . . 33 <br />5.3 The ρ → 0 limit for Evans functions . . . . . . . . . . . . . . 34 <br />5.4 The ρ → 0 limit for maximal estimates . . . . . . . . . . . . . 36 <br />5.5 Viscous instabilities . . . . . . . . . . . . . . . . . . . . . . . 38 <br />6 LF and MF symmetrizers 42 <br />6.1 The method of symmetrizers . . . . . . . . . . . . . . . . . . 42 <br />6.2 Elliptic points and MF symmetrizers . . . . . . . . . . . . . . 45 <br />6.3 LF symmetrizers . . . . . . . . . . . . . . . . . . . . . . . . . 47 <br />7 Symmetrizers for nonelliptic blocks; Examples 50 <br />7.1 Simple hyperbolic points . . . . . . . . . . . . . . . . . . . . . 50 <br />7.2 Simple glancing modes . . . . . . . . . . . . . . . . . . . . . . 51 <br />7.3 Hyperbolic modes with constant multiplicity . . . . . . . . . 53 <br />7.4 Smoothly diagonalizable hyperbolic modes . . . . . . . . . . . 54 <br />7.4.1 The inviscid case . . . . . . . . . . . . . . . . . . . . . 55 <br />7.4.2 The viscous case . . . . . . . . . . . . . . . . . . . . . 56 <br />7.5 Totally nonglancing modes and symmetrizable systems . . . . 56 <br />8 Main results from [M ́eZu2] and [GMWZ6] 58 <br />8.1 Hyperbolic multiple roots . . . . . . . . . . . . . . . . . . . . 58 <br />8.2 The decoupling condition . . . . . . . . . . . . . . . . . . . . 60 <br />8.3 The hyperbolic block structure condition . . . . . . . . . . . . 62 <br />8.4 The hyperbolic-parabolic case . . . . . . . . . . . . . . . . . . 69 <br />8.5 Existence of K-families of symmetrizers . . . . . . . . . . . . 72 <br />9 The high frequency analysis 73 <br />9.1 The main high frequency estimate . . . . . . . . . . . . . . . 73 <br />9.2 Spectral analysis of the symbol . . . . . . . . . . . . . . . . . 77 <br />9.3 Analysis of the hyperbolic block. . . . . . . . . . . . . . . . . 83 <br />9.3.1 The genuine coupling condition . . . . . . . . . . . . . 83 <br />9.3.2 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 86 <br />9.3.3 About Assumption (H9) . . . . . . . . . . . . . . . . . 88 <br />9.4 Proof of Theorem 9.2 . . . . . . . . . . . . . . . . . . . . . . 88 <br />9.4.1 In the cone Cδ . . . . . . . . . . . . . . . . . . . . . . 88 <br />9.4.2 Analysis in the central zone . . . . . . . . . . . . . . . 90<br />10 Linear stability 91 <br />10.1 Linearized equations, spectral stability conditions . . . . . . . 91 <br />10.2 Maximal stability estimates . . . . . . . . . . . . . . . . . . . 93 <br />10.3 Hints for the proof . . . . . . . . . . . . . . . . . . . . . . . . 94 <br />10.4 Further steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 <br />11 Nonlinear stability 100 <br />11.1 Statement of the problem and main result . . . . . . . . . . . 100 <br />11.2 High order approximate solutions . . . . . . . . . . . . . . . . 102 <br />11.3 Parabolic methods . . . . . . . . . . . . . . . . . . . . . . . . 103 <br />11.4 Hyperbolic-like methods . . . . . . . . . . . . . . . . . . . . . 106 | |
| dc.description.abstractEn | The material included in these lectures is taken from a series of joint pa- <br />pers with O.Gu`es, M.Williams and K.Zumbrun. They concern the linear <br />and nonlinear stability of viscous boundary layers which arise when one <br />considers small viscosity parabolic perturbations of noncharacteristic multi- <br />dimensional hyperbolic boundary value problems. | |
| dc.language.iso | en | |
| dc.title.en | Stability of Noncharacteristic Viscous Boundary <br />Layers | |
| dc.subject.hal | Mathématiques [math] | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| hal.identifier | cel-00113903 | |
| hal.version | 1 | |
| hal.audience | Non spécifiée | |
| hal.origin.link | https://hal.archives-ouvertes.fr//cel-00113903v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=METIVIER,%20Guy& |
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