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Heat and Navier-Stokes equations in supercritical function spaces. (English) Zbl 1326.35232

Let \[ \partial_t \mathbf{u} - \Delta \mathbf{u} + \mathbb{P} \operatorname{div} (\mathbf{u} \otimes \mathbf{u}) =0 \] be the usual Navier-Stokes equations, where \(\mathbb{P}\) stands for the Leray projector, subject to the initial data \(u(\cdot,0) = u_0\). The paper deals with unique strong solutions of these equations and, as a forerunner, non-linear heat equations, in \(\mathbb{R}^n \times (0,T)\) in terms of distinguished inhomogeneous spaces \(A^s_{p,q} (\mathbb{R}^n)\), \(A\in \{B,F \}\), of Besov-Triebel-Lizorkin type.

MSC:

35Q30 Navier-Stokes equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35K05 Heat equation
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
35D35 Strong solutions to PDEs
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