Moving Interfaces and Quasilinear Parabolic Evolution Equations

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Birkhäuser, Jul 25, 2016 - Mathematics - 609 pages

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.

 

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Contents

1 Problems and Strategies
3
2 Tools from Differential Geometry
43
Part II Abstract Theory
87
3 Operator Theory and Semigroups
88
4 VectorValued Harmonic Analysis
149
5 Quasilinear Parabolic Evolution Equations
195
Part III Linear Theory
231
6 Elliptic and Parabolic Problems
232
Part IV Nonlinear Problems
417
9 Local WellPosedness and Regularity
418
10 Linear Stability of Equilibria
451
11 Qualitative Behaviour of the Semiflows
491
12 Further Parabolic Evolution Problems
515
Bibliographical Comments
571
Outlook and Future Challenges
585
List of Figures
587

7 Generalized Stokes Problems
311
8 TwoPhase Stokes Problems
363

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