On Kato's method for Navier--Stokes Equations

Details On Kato's method for Navier--Stokes Equations On Kato's method for Navier--Stokes Equations

2007-09-13

arxiv.org/abs/0709.2067v3

Mathematics

Analysis of PDEs

Revised version, to appear in Journal of Mathematical Fluid Mechanics. New section on solutions in Morrey-spaces added. 35 pag

Scientific paper

10.1007/s00021-008-0270-5

We investigate Kato's method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier-Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in ${\mathbb R}^3$ and irregular domains in ${\mathbb R}^n$.

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