Stability of Noncharacteristic Viscous Boundary <br />Layers

1 Introduction 3 2 Hyperbolic-parabolic boundary value problems 8 2.1 Structure of equations . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 11 3 Layers profiles 12 3.1 The profile equation . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Existence of profiles . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 The inviscid boundary conditions . . . . . . . . . . . . . . . . 14 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Spectral stability 19 4.1 The linearized equations . . . . . . . . . . . . . . . . . . . . . 19 4.2 Structure of the linearized equations . . . . . . . . . . . . . . 21 4.3 Evans functions and Lopatinski determinant; weak stability . 25 4.4 Maximal estimates and uniform spectral stability conditions 28 4.4.1 Low and medium frequencies . . . . . . . . . . . . . . 28 4.4.2 High frequencies . . . . . . . . . . . . . . . . . . . . . 29 4.4.3 The inviscid case . . . . . . . . . . . . . . . . . . . . . 31 5 The Zumbrun-Serre-Rousset Theorem and the reduced low frequency problem 32 5.1 Transversality is necessary . . . . . . . . . . . . . . . . . . . . 32 5.2 The reduced problem . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 The ρ → 0 limit for Evans functions . . . . . . . . . . . . . . 34 5.4 The ρ → 0 limit for maximal estimates . . . . . . . . . . . . . 36 5.5 Viscous instabilities . . . . . . . . . . . . . . . . . . . . . . . 38 6 LF and MF symmetrizers 42 6.1 The method of symmetrizers . . . . . . . . . . . . . . . . . . 42 6.2 Elliptic points and MF symmetrizers . . . . . . . . . . . . . . 45 6.3 LF symmetrizers . . . . . . . . . . . . . . . . . . . . . . . . . 47 7 Symmetrizers for nonelliptic blocks; Examples 50 7.1 Simple hyperbolic points . . . . . . . . . . . . . . . . . . . . . 50 7.2 Simple glancing modes . . . . . . . . . . . . . . . . . . . . . . 51 7.3 Hyperbolic modes with constant multiplicity . . . . . . . . . 53 7.4 Smoothly diagonalizable hyperbolic modes . . . . . . . . . . . 54 7.4.1 The inviscid case . . . . . . . . . . . . . . . . . . . . . 55 7.4.2 The viscous case . . . . . . . . . . . . . . . . . . . . . 56 7.5 Totally nonglancing modes and symmetrizable systems . . . . 56 8 Main results from [M ́eZu2] and [GMWZ6] 58 8.1 Hyperbolic multiple roots . . . . . . . . . . . . . . . . . . . . 58 8.2 The decoupling condition . . . . . . . . . . . . . . . . . . . . 60 8.3 The hyperbolic block structure condition . . . . . . . . . . . . 62 8.4 The hyperbolic-parabolic case . . . . . . . . . . . . . . . . . . 69 8.5 Existence of K-families of symmetrizers . . . . . . . . . . . . 72 9 The high frequency analysis 73 9.1 The main high frequency estimate . . . . . . . . . . . . . . . 73 9.2 Spectral analysis of the symbol . . . . . . . . . . . . . . . . . 77 9.3 Analysis of the hyperbolic block. . . . . . . . . . . . . . . . . 83 9.3.1 The genuine coupling condition . . . . . . . . . . . . . 83 9.3.2 Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 86 9.3.3 About Assumption (H9) . . . . . . . . . . . . . . . . . 88 9.4 Proof of Theorem 9.2 . . . . . . . . . . . . . . . . . . . . . . 88 9.4.1 In the cone Cδ . . . . . . . . . . . . . . . . . . . . . . 88 9.4.2 Analysis in the central zone . . . . . . . . . . . . . . . 9010 Linear stability 91 10.1 Linearized equations, spectral stability conditions . . . . . . . 91 10.2 Maximal stability estimates . . . . . . . . . . . . . . . . . . . 93 10.3 Hints for the proof . . . . . . . . . . . . . . . . . . . . . . . . 94 10.4 Further steps . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 11 Nonlinear stability 100 11.1 Statement of the problem and main result . . . . . . . . . . . 100 11.2 High order approximate solutions . . . . . . . . . . . . . . . . 102 11.3 Parabolic methods . . . . . . . . . . . . . . . . . . . . . . . . 103 11.4 Hyperbolic-like methods . . . . . . . . . . . . . . . . . . . . . 106The material included in these lectures is taken from a series of joint pa- pers with O.Gu`es, M.Williams and K.Zumbrun. They concern the linear and nonlinear stability of viscous boundary layers which arise when one considers small viscosity parabolic perturbations of noncharacteristic multi- dimensional hyperbolic boundary value problems