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Pointwise semigroup methods and stability of viscous shock waves (1998)
| Content Provider | CiteSeerX |
|---|---|
| Author | Zumbrun, Kevin Howard, Peter |
| Abstract | Abstract. Considered as rest points of ODE on L p, stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact which precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or totally compressive case ([Sat], [K.2], resp.), each of which can be reduced to the standard semigroup setting by Sattinger's method of weighted norms. We overcome this di culty in the general case by the introduction of new, pointwise semigroup techniques, generalizing ear-lier work of Howard [H.1], Kapitula [K.1-2], and Zeng [Ze,LZe]. These techniques allow us to do \hard " analysis in PDE within the dynamical systems/semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, su cient for the analysis of linear and nonlinear stability. The method is general, and should nd applications |
| File Format | |
| Volume Number | 47 |
| Journal | Indiana Univ. Math. J |
| Language | English |
| Publisher Date | 1998-01-01 |
| Access Restriction | Open |
| Subject Keyword | Pointwise Semigroup Method Viscous Shock Wave Stationary Viscous Shock Wave Linearized Operator Compressive Case Global Pointwise Bound Dynamical System Semigroup Framework Stationary Mode Zeng Ze Hard Quot Invariant Subspace General Case Spectral Gap Rest Point Nonlinear Stability Su Cient Usual Analysis Semigroup Perspective Standard Semigroup Setting Di Culty Critical Case Ear-lier Work Shock Stability Standard Semigroup Method Essential Spectrum Weighted Norm Pointwise Semigroup Technique |
| Content Type | Text |
| Resource Type | Article |
