The Navier-Stokes Equations with Low-Regularity Data in Weighted Function Spaces.
Technische Universität, Darmstadt
[Ph.D. Thesis], (2007)
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|Item Type:||Ph.D. Thesis|
|Title:||The Navier-Stokes Equations with Low-Regularity Data in Weighted Function Spaces|
We consider the Navier-Stokes equations in a bounded domain. It is our aim to develop a solution theory which requires a regularity of the data that is as low as possible. This means at the same time that one obtains a class of solutions that is so large that the solutions possess a priori no weak derivatives. This in turn makes it necessary to introduce a notion of solutions that is more general than the one of weak solutions, the so-called very weak solutions. We study this problem in weighted Lebesgue-, Sobolev- and Bessel Potential spaces, where the weight function is taken from the class of Muckenhoupt weights. As a preparation we study the Laplace equation as well as the divergence equation in weighted function spaces. Moreover we construct a linear operator that extends functions defined on the boundary to functions defined on the domain. Next we investigate the linearized Stokes equations. In the stationary as well as in the instationary case one obtains the solvability with respect to the most general data that are considered in the work, by dualization of strong solutions. However, these solutions in general do not possess enough regularity to make their restriction to the boundary well-defined. Boundary values are meaningful only after a restriction to data that can be decomposed to a distribution on the domain and a distribution on the boundary. With the help of complex interpolation between the very weak and the strong solutions the solution theory of stationary and instationary Stokes equations can be extended to weighted Bessel potential spaces. This in turn requires a characterization of the interpolation spaces of the corresponding spaces of data and solutions. Finally we examine the nonlinear Navier-Stokes equations. In the stationary as in the instationary case one obtains existence and uniqueness of solutions for small data. In the instationary case this smallness can be realized by restricting the problem to a short time interval.
|Place of Publication:||Darmstadt|
|Uncontrolled Keywords:||Sehr Schwache Lösung|
|Classification DDC:||000 Allgemeines, Informatik, Informationswissenschaft > 000 Allgemeines, Wissenschaft|
|Divisions:||04 Department of Mathematics|
|Date Deposited:||17 Oct 2008 09:22|
|Last Modified:||07 Dec 2012 11:52|
|Referees:||Farwig, Prof.Dr. Reinhard and Simader, Prof.Dr. Christian|
|Advisors:||Farwig, Prof.Dr. Reinhard|
|Refereed:||1 February 2007|