The Stokes equations
Author(s)
Bibliographic Information
The Stokes equations
（Mathematical research = Mathematische Forschung, Bd. 76）
Akademie Verlag , VCH Publishers, c1994
1st ed
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Note
Includes bibliographical references and index
Description and Table of Contents
Description
In the first part of this text a theory of solvability for the stationary Stokes equations in exterior domains is developed. Varnhorn proves the existence of strong solutions in Sobolev spaces and uses a localization principle and the divergence equation to deduce further properties of the solution (uniqueness, asymptotics). The second part considers the resolvent equations with methods of potential theory. The author presents the explicit fundamental tensor for general space dimension and solves the boundary value problems via systems of integral equations. New a-priori norm estimates for the solution are developed independently of small resolvent parameters. This leads to the uniform boundedness of the semigroup associated with the Stokes operator, also for the case of a two-dimensional exterior domain. In the third part Varnhorn approximates the nonstationary equations with help of the resolvent and proves convergence of optimal order in a scale of Sobolev spaces. This includes the nonlinear Navier-Stokes equations, which can be regularized with methods of time delay.
Table of Contents
- Part 1 The stationary equations: on hydrodynamical potential theory
- boundary integral operators
- existence and uniqueness in Sobolev spaces
- the divergence operator
- estimates in exterior domains. Part 2 The resolvent equations: the fundamental tensors
- properties of the boundary layer potentials
- interior and exterior boundary value problems
- asymptotics of the solution
- norm estimates independent of the resolvent parameter. Part 3 The nonstationary equations: on the boundedness of the semigroup
- a semidiscrete approximation procedure
- the method of Crank-Nicholson
- time delay for the nonlinear equations. (Part contents).
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