Finite time blow up for a Navier-Stokes like equation

Details Finite time blow up for a Navier-Stokes like equation Finite time blow up for a Navier-Stokes like equation

1999-11-29

arxiv.org/abs/math/9911223v2

Proc. A.M.S., 129, (2001), 3017-3023

Mathematics

Analysis of PDEs

Also available at http://www.math.missouri.edu/~stephen/preprints/

Scientific paper

We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to show the limitations of the so called semigroup method for the Navier-Stokes equation. We also consider the possibility of existence of solutions with initial data in the Besov space $\dot B^{-1,\infty}_\infty$. We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation.

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