General parabolic mixed order systems in Lp and applications Robert Denk, Mario Kaip

Format:

Book

Author/Creator:

Denk, Robert, 1964-

Contributor:

Kaip, Mario

Series:

Operator theory, advances and applications ; 0255-0156 239,

Operator Theory: Advances and Applications 239 0255-0156

Language:

English

Subjects (All):

Operator theory.

Mathematics.

Differential equations, Partial.

Mathematical physics.

mathematics.

applied mathematics.

Physical Description:

1 online resource

Place of Publication:

Cham Birkhäuser 2013

Cham Birkha�user 2013

System Details:

PDF

text file

Summary:

In this text, a theory for general linear parabolic partial differential equations is established, which covers equations with inhomogeneous symbol structure as well as mixed order systems. Typical applications include several variants of the Stokes system and free boundary value problems. We show well-posedness in Lp-Lq-Sobolev spaces in time and space for the linear problems (i.e., maximal regularity), which is the key step for the treatment of nonlinear problems. The theory is based on the concept of the Newton polygon and can cover equations that are not accessible by standard methods as, e.g., semigroup theory. Results are obtained in different types of non-integer Lp-Sobolev spaces as Besov spaces, Bessel potential spaces, and Triebel-Lizorkin spaces. The latter class appears in a natural way as traces of Lp-Lq-Sobolev spaces. We also present a selection of applications in the whole space and on half-spaces. Among others, we prove well-posedness of the linearizations of the generalized thermoelastic plate equation, the two-phase Navier-Stokes equations with Boussinesq-Scriven surface, and the Lp-Lq two-phase Stefan problem with Gibbs-Thomson correction

Contents:

Introduction and Outline

1 The joint time-space H(infinity)-calculus

2 The Newton polygon approach for mixed-order systems.-3 Triebel-Lizorkin spaces and the Lp-Lq setting.- 4 Application to parabolic differential equations

Machine generated contents note: 1. joint time-space H[∞]-calculus

1.1. joint H[∞]-calculus for tuples of operators

a). Sectorial and bisectorial operators, R-boundedness

b). Joint H[∞]-calculus

1.2. Vector-valued Sobolev spaces

a). Interpolation of Banach spaces

b). Retractions and coretractions

c). Definition of Sobolev spaces

1.3. time-space derivative

a). Fourier multipliers

b). Vector-valued space and time derivatives

c). Joint space-time H[∞]-calculus

2. Newton polygon approach for mixed-order systems

2.1. Inhomogeneous symbols and the Newton polygon

a). Inhomogeneous symbols and principal parts

b). Newton polygons and order functions

2.2. N-parameter-ellipticity and N-parabolicity

a). N-parameter-elliptic symbols and SN(Lt x Lx)

b). Partition of the co-variable space

c). Equivalent characterization of SN(Lt x Lx)

2.3. H[∞]-calculus of N-parabolic mixed-order systems

a). H[∞]-calculus of N-parabolic symbols

b). Mixed-order systems on spaces of mixed scales

c). Remarks on the compatibility condition

3. Triebel-Lizorkin spaces and the Lp-Lq-setting

3.1. Vector-valued Triebel-Lizorkin spaces and interpolation

3.2. Anisotropic Triebel-Lizorkin spaces and representation by intersections

3.3. Auxiliary results on Bessel-valued Triebel-Lizorkin spaces

a). joint time-space H[∞]-calculus on Bessel-valued Triebel-Lizorkin spaces

b). H[∞]-calculus of N-parabolic symbols on Bessel-valued Triebel-Lizorkin spaces

3.4. Mixed-order systems on Triebel-Lizorkin spaces

3.5. Singular integral operators on Lp-Lq

a). Singular integral operators

b). Extension symbols

4. Application to parabolic differential equations

4.1. generalized Lp-Lq Stokes problem on Ω = Rn

a). Remarks on homogeneous Sobolev spaces

b). generalized Stokes problem

4.2. generalized Lp-Lq thermo-elastic plate equations on Ω = Rn

4.3. linear Lp-Lq Cahn-Hilliard-Gurtin problem in Ω = Rn

4.4. compressible fluid model of Korteweg type on Ω = Rn

4.5. linear three-phase problem on Ω = Rn

4.6. spin-coating process

4.7. Two-phase Navier-Stokes equations with Boussinesq-Scriven surface and gravity

4.8. Lp-Lq two-phase Stefan problem with Gibbs-Thomson correction

Notes:

Includes bibliographical references and index

Other Format:

Printed edition:

ISBN:

9783319020006

3319020005

3319019996

9783319019994

OCLC:

864999735

Publisher Number:

10.1007/978-3-319-02000-6

Access Restriction:

Restricted for use by site license