Cauchy problem and quasi-stationary limit for the Maxwell-Landau-Lifschitz and Maxwell-Bloch equations Eric Dumas?, Franck Sueur† June 18, 2010 Abstract In this paper we continue the investigation of the Maxwell-Landau-Lifschitz and Maxwell-Bloch equations. In particular we extend some previous results about the Cauchy problem and the quasi-stationary limit to the case where the magnetic permeability and the electric permittivity are variable. Keywords: Maxwell equations, Bloch equation, Landau-Lifschitz equation, quasi-stationary limit, energy esti- mates, compensated compactness, Strichartz estimates. MSC: 35L45, 35Q60. Contents 1 Introduction 2 2 Main results 3 2.1 An abstract setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Cauchy problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Quasi-stationary limits . . . . . . . . . . . . . . . . . .
- into account
- strichartz estimates
- space dimension
- estimates performed
- global finite energy
- hilbert space
- finite energy
- maxwell-landau- lifschitz system