Very Weak Solutions to the Navier-Stokes Equations in General Unbounded Domains

Beschreibung

We study the instationary incompressible Navier-Stokes system for domains having smooth but noncompact boundary. More precisely we consider general uniform C2-domains. We do not impose any other geometric condition on the domain. Therefore, domains are included, where the usual Helmholtz decomposition fails to hold. Adopting an abstract functional analytic approach which is based on duality, interpolation and maximal regularity of the Stokes operator, we construct so called very weak solutions u, which are contained in spaces. We prove local existence and global uniqueness of such solutions, as well as certain regularity properties. We apply this theory to find regularity criteria for weak solutions to the Navier-Stokes system in the sense of Leray and Hopf. Moreover, we construct so called mild solutions using Katö's method.
We study the instationary incompressible Navier-Stokes system for domains having smooth but noncompact boundary. More precisely we consider general uniform C2-domains. We do not impose any other geometric condition on the domain. Therefore, domains are included, where the usual Helmholtz decomposition fails to hold. Adopting an abstract functional analytic approach which is based on duality, interpolation and maximal regularity of the Stokes operator, we construct so called very weak solutions u, which are contained in spaces. We prove local existence and global uniqueness of such solutions, as well as certain regularity properties. We apply this theory to find regularity criteria for weak solutions to the Navier-Stokes system in the sense of Leray and Hopf. Moreover, we construct so called mild solutions using Katö's method.

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Paul Felix Riechwald

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