Forces for the Navier–Stokes Equations and the Koch and Tataru Theorem

We consider the Cauchy problem for the incompressible Navier–Stokes equations on the whole space R 3 , with initial value u → 0 ∈ BMO - 1 (as in Koch and Tataru’s theorem) and with force f → = div F where smallness of F ensures existence of a mild solution in absence of initial value. We study the i...

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Bibliographic Details
Published in:Journal of mathematical fluid mechanics 2023-08, Vol.25 (3), Article 51
Main Author: Gilles, Lemarié-Rieusset Pierre
Format: Article
Language:English
Subjects:
Cauchy problems
Classical and Continuum Physics
Existence theorems
Fluid flow
Fluid mechanics
Fluid- and Aerodynamics
Ladyzhenskaya Centennial Anniversary
Mathematical analysis
Mathematical Methods in Physics
Navier-Stokes equations
Physics
Physics and Astronomy
Theoretical mathematics
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Summary:We consider the Cauchy problem for the incompressible Navier–Stokes equations on the whole space R 3 , with initial value u → 0 ∈ BMO - 1 (as in Koch and Tataru’s theorem) and with force f → = div F where smallness of F ensures existence of a mild solution in absence of initial value. We study the interaction of the two solutions and discuss the existence of global solution for the complete problem (i.e. in presence of initial value and forcing term) under smallness assumptions. In particular, we discuss the interaction between Koch and Tataru solutions and Lei-Lin’s solutions (in L 2 F - 1 L 1 ) or solutions in the multiplier space M ( H ˙ t , x 1 / 2 , 1 ↦ L t , x 2 ) .
ISSN:1422-6928
1422-6952
DOI:10.1007/s00021-023-00788-6