Finite time blow up for a Navier-Stokes like equation

Montgomery-Smith, Stephen

doi:10.1090/S0002-9939-01-06062-2

Finite time blow up for a Navier-Stokes like equation
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by Stephen Montgomery-Smith PDF

Proc. Amer. Math. Soc. 129 (2001), 3025-3029 Request permission

Abstract:

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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