Navier-Stokes like equation

**Stephen Montgomery-Smith ^{1}
Department of Mathematics
University of Missouri
Columbia, MO 65211
stephen@math.missouri.edu
http://faculty.missouri.edu/~stephen
**

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
.
We give initial data in this space for which there is no reasonable
solution for the Navier-Stokes like equation.

Keywords: Navier-Stokes equation, semigroup, fixed point method, Triebel-Lizorkin space, Besov space.

A.M.S. Classification (2000): Primary 35Q30, 46E35; Secondary 34G20, 37L05, 47D06, 47H10.

Keywords: Navier-Stokes equation, semigroup, fixed point method, Triebel-Lizorkin space, Besov space.

A.M.S. Classification (2000): Primary 35Q30, 46E35; Secondary 34G20, 37L05, 47D06, 47H10.

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:

However, if one studies all these papers, one sees that they
do not use all of the properties of the Navier-Stokes equation.
Indeed, the methods seem to apply equally well to the the following
equation, the *cheap Navier-Stokes equation*:

Here is a scalar valued function or tempered distribution on . The semigroup approach is to consider a mild solution of the cheap Navier-Stokes equation, that is, a solution to the equation , where , with being a space of tempered distributions on :

The method used to find a fixed point of is to show that is a contraction mapping on or on some subset of . (Oftentimes one cannot obtain the result by working directly with . For example, in the case one has to work with some space embedded into , for as is shown by Oru [O], the map is not even continuous on .)

The semigroup method by itself never seems to be able to obtain global results, that is, results valid for both and the size of arbitrarily large. To obtain global results, one has to appeal to other kinds of estimates for the Navier-Stokes equation, for example, energy estimates (see, for example, [KF] or [Ca2]).

The purpose of this paper is to show that the semigroup method really is limited in this way. For if we could obtain global results for the Navier-Stokes equation using only the semigroup method, then the same methods would also apply to the cheap Navier-Stokes equation. This would then contradict the main result of this paper and its corollary.

Thus, if one is going to obtain global results for the Navier-Stokes equation, one has to consider properties of the bilinear form that are not shared by the bilinear form . Perhaps there is some mysterious cancellation property in the first bilinear form that causes global regularity for the Navier-Stokes equation. In any case, the semigroup technique in of itself, at least in the manner in which it has been applied to date, is not going to solve the problem.

Let denote the first unit vector in , and for , , let denote . Let denote the Fourier transform of with respect to .

We also present another result about the
cheap Navier-Stokes
equation. For the Navier-Stokes equation (and hence also for the
cheap Navier-Stokes equation), it turns out that the natural spaces
in which to consider solutions are of the form
, where
is *scale invariant*, that is,
for all
.
Authors have considered which is the
largest scale invariant space for which one gets existence
results. For example, recently Koch and Tataru [KT] showed such
results when is the space of derivatives of functions with
bounded mean oscillation.

In fact, it can be shown (arguing similarly as in Frazier, Jawerth and Weiss in [FJW] for the minimality of ) that all scale invariant spaces of distributions, that also contain all Schwartz functions, are contained in the Besov space . Cannone [Ca1] was able to obtain results for the Navier-Stokes equation in the space , , but left open the case corresponding to the space .

We will respond to this last case negatively for the cheap Navier-Stokes equation. Although this does not answer the question for the Navier-Stokes equation, the author believes that it is very possible that there is a similar non-existence result for the Navier-Stokes equation. But this would say more about the nature of the space than about the Navier-Stokes equation itself.

The crucial observation for all these results is that the cheap Navier-Stokes equation is that

(here denotes convolution). Thus we see that if , and that , then .

**Proof of Theorem 1:**
Using the usual embedding theorems, it may be seen that all of the
Triebel-Lizorkin and Besov spaces
embed into
for some
.
Let us recall the definition of the Besov space
.
Let be some
tempered Schwartz function
whose Fourier transform is non-negative,
whose support is ``mostly'' contained
in a band about
, and such that
is uniformly bounded above and below.
Here
.
Then
is the space of
distributions on
for
which the norm
is finite.

Write for the function , and set . Since and are real valued, we see that must be an even function. Hence we quickly see that has norm equal to , and is supported in .

We will show by induction
that
for ,
where
,
, and
.
The case follows since
.
Suppose that our desired inequality holds for . Then

whenever , that is, .

Let
.
Since
,

It is clear that this is infinite if .

**Proof of Corollary 2:**
Suppose for a contradiction that there is a solution
. By the methods of [K],
we know that there is a
number
, depending only upon
, such that
for every
that there is a mild solution
to the cheap Navier-Stokes equation,
with , that is obtained by starting with
,
and iterating a function similar to that defined above.
By the uniqueness result of Furioli, Lemarié-Rieusset
and Terraneo [FLT],
we have that for
.
From this it is clear that if
, then
for
. Applying this argument
several times, we see that
for
.
Then by Theorem 1, is not in any
Triebel-Lizorkin space, and hence in particular is not in
.

**Proof of Theorem 3:**
Suppose we have a non-negative solution
to
the cheap Navier-Stokes equation with . It is clear that

which is infinite on the support of for , and hence cannot be in .

The author would like to thank Marco Cannone for many useful discussions and helpful remarks.