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AuthorCaps, Oliver. author
TitleEvolution Equations in Scales of Banach Spaces [electronic resource] / by Oliver Caps
ImprintWiesbaden : Vieweg+Teubner Verlag, 2002
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Descript 309 p. 2 illus. online resource


The book provides a new functional-analytic approach to evolution equations by considering the abstract Cauchy problem in a scale of Banach spaces. Conditions are proved characterizing well-posedness of the linear, time-dependent Cauchy problem in scales of Banach spaces and implying local existence, uniqueness, and regularity of solutions of the quasilinear Cauchy problem. Many applications illustrate the generality of the approach. In particular, using the Fefferman-Phong inequality unifying results on parabolic and hyperbolic equations generalizing classical ones and a unified treatment of Navier-Stokes and Euler equations is described. Assuming only basic knowledge in analysis and functional analysis the book provides all mathematical tools and is aimed for students, graduates, researchers, and lecturers


1 Tools from functional analysis -- 1.1 A brief introduction into the theory of semigroups -- 1.2 Selfadjoint operators -- 1.3 Generators of analytic semigroups and their powers -- 1.4 Fractional Powers of operators of positive type -- 1.5 Complex interpolation spaces -- 1.6 Time-dependent, linear evolution equations -- 2 Well-posedness of the time-dependent linear Cauchy problem -- 2.1 Properties of well-posed linear Cauchy problems in scales of Banach spaces -- 2.2 Scales of Banach spaces generated by families of closed operators -- 2.3 Commutator estimates and scales of Banach spaces -- 2.4 Characterization of well-posedness of the Cauchy problem.. -- 2.5 Sufficient conditions for well-posedness of the Cauchy problem -- 3 Quasilinear Evolution Equations -- 3.1 Semilinear Evolution Equations -- 3.2 Commutator estimates and quasilinear evolution equations -- 3.3 A local existence and uniqueness result for quasilinear evolution equations -- 3.4 Regularity for quasilinear evolution equations in scales of Banach spaces -- 4 Applications to linear, time-dependent evolution equations -- 4.1 Pseudodifferential operators and weighted Sobolev spaces -- 4.2 Pseudodifferential evolution equations in scales of weighted Sobolev spaces -- 4.3 Essential selfadjointness of pseudodifferential operators -- 4.4 Evolution equations in C0(IRn) and Feller semigroups -- 4.5 Evolution equations in scales of Lq-Sobolev spaces -- 4.6 An application to a degenerate-elliptic boundary value problem -- 4.7 Evolution equations on networks -- 5 Applications to quasilinear evolution equations -- 5.1 Estimates of Nash-Moser type for differential operators -- 5.2 Quasilinear evolution equations in Sobolev spaces -- 5.3 Degenerate Navier-Stokes equations -- 5.4 The generalized Kadomtsev-Petviashvili equation -- 5.5 Quasilinear evolution equations in scales of Lq-Sobolev spaces -- 5.6 First order hyperbolic evolution equations in the C0k-scale

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