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Finite time blow up for a Navier-Stokes like equation

Author: Stephen Montgomery-Smith
Journal: Proc. Amer. Math. Soc. 129 (2001), 3025-3029
MSC (2000): Primary 35Q30, 46E35; Secondary 34G20, 37L05, 47D06, 47H10
Published electronically: April 17, 2001
MathSciNet review: 1840108
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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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Additional Information

Stephen Montgomery-Smith
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Keywords: Navier-Stokes equation, semigroup, fixed point method, Triebel-Lizorkin space, Besov space
Received by editor(s): March 1, 2000
Published electronically: April 17, 2001
Additional Notes: The author was partially supported by NSF grant DMS 9870026.
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society

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