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.
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.
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.
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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