Finite Time Blow up for a Navier-stokes like Equation

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 Ḃ ∞ . We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation. In this paper, we consider a simplified model for the Navier-Stokes equation — what we call the cheap Navier-Stokes equation. For this equation, we show that for sufficiently large initial data, that we get blow up in finite time. The purpose of this is not to indicate the possibility that the Navier-Stokes equation might blow up in finite time — indeed the author strongly believes the opposite. Rather, the purpose of this paper is to show limitations in some of the methods used in studying the Navier-Stokes equation. Let us consider the following version of the Navier-Stokes equation: ∂u ∂t = ∆u− P (div(u⊗ u)), where t 7→ ut is an R valued function, or tempered distribution, on R. Here P denotes the Leray projection that takes a vector field to its divergence free part. A tremendous amount of work has been done on the very hard problem of determining if the solutions exist, if they are unique, and to which spaces they belong. One approach, the one we consider in this paper, is to consider mild solutions using what is often called the semigroup approach. This is described in [Ca1], and is used in many papers, for example, [FK], [K], [GM], [KT]. An example of this 1991 Mathematics Subject Classification. Primary 35Q30, 46E35; Secondary 34G20, 37L05, 47D06, 47H10.