Title: Moving interfaces and quasilinear parabolic evolution equations /
Authors: Prüss, Jan, 1951-; Simonett, Gieri, 1959-
Year: 2016
Subjects: Boundary value problems.; Interfaces (Physical sciences) Mathematics.; Boundary value problems; Interfaces (Physical sciences) Mathematics
Abstract: In this monograph, the authors develop a comprehensive approach for the mathematical analysis of a wide array of problems involving moving interfaces. It includes an in-depth study of abstract quasilinear parabolic evolution equations, elliptic and parabolic boundary value problems, transmission problems, one- and two-phase Stokes problems, and the equations of incompressible viscous one- and two-phase fluid flows. The theory of maximal regularity, an essential element, is also fully developed. The authors present a modern approach based on powerful tools in classical analysis, functional analysis, and vector-valued harmonic analysis. The theory is applied to problems in two-phase fluid dynamics and phase transitions, one-phase generalized Newtonian fluids, nematic liquid crystal flows, Maxwell-Stefan diffusion, and a variety of geometric evolution equations. The book also includes a discussion of the underlying physical and thermodynamic principles governing the equations of fluid flows and phase transitions, and an exposition of the geometry of moving hypersurfaces.
Contents: Preface -- Basic Notations -- General References -- Part I Background -- 1.Problems and Strategies -- 2.Tools from Differential Geometry -- Part II Abstract Theory -- 3.Operator Theory and Semigroups -- 4.Vector-Valued Harmonic Analysis -- 5.Quasilinear Parabolic Evolution Equations -- Part III Linear Theory -- 6.Elliptic and Parabolic Problems -- 7.Generalized Stokes Problems -- 8.Two-Phase Stokes Problems -- Part IV Nonlinear Problems -- 9.Local Well-Posedness and Regularity -- 10.Linear Stability of Equilibria -- 11.Qualitative Behaviour of the Semiflows -- 12.Further Parabolic Evolution Problems -- Biographical Comments -- Outlook and Future Challenges -- References -- List of Figures -- List of Symbols -- Subject Index.