Moving Interfaces and Quasilinear Parabolic Evolution Equations Preface Basic Notations General References Part I Background 1Problems and Strategies 2.Tools from Differential Geometry Part II Abstract Theory 3Operator 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 Semiows 12.Further Parabolic Evolution Problems Biographical Comments Outlook and Future Challenges References List of Figures List of Symbols Subject Index. 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.