# Trapping regions for the Navier-Stokes equations.

1 Introduction

In this paper, we examine the three-dimensional Navier-Stokes equations, which model the flow of incompressible fluids:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where v > 0 is viscosity, p is pressure, u is velocity, and t > 0 is time. We shall assume that both u and p are periodic in x. For simplicity, we take the period to be one. The first equation is Newton's Second Law, force equals mass times acceleration, and the second equation is the assumption that the fluid is incompressible.

Mattingly and Sinai [5] attempted to show that smooth solutions to 3D Navier Stokes equations exist for all initial conditions u(x, 0) = [u.sup.0](x) [member of] [C.sup.[infinity]] by dealing with an equivalent form of the Navier-Stokes equations for periodic boundary conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where the vorticity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Their strategy was as follows: Represent the equations (2) as a Galerkin system in Fourier space with a basis {[e.sup.2[pi]ikx]}k[member of][Z.sup.3]. A finite dimensional approximation of this Galerkin system can be associated to any finite subset Z of [Z.sup.3] by setting [u.sup.(k)](t) = [[omega].sup.(k)](t) = 0 for all k outside of Z. For each finite dimensional approximation of this Galerkin system, consider the system of coupled ODEs for the Fourier coefficients. Then construct a subset [OMEGA](K) of the phase space (the set of possible configurations of the Fourier modes) so that all points in [OMEGA](K) possess the desired decay properties. In addition, construct [OMEGA](K) so that it contains the initial data. Then show that the dynamics never cause the sequence of Fourier modes to leave the subset [OMEGA](K) by showing that the vector field on the boundary of [OMEGA](K) points into the interior of [OMEGA](K).

Unfortunately, their strategy only worked for the 3D Navier-Stokes equations when the Laplacian operator A in (2) was replaced by another similar linear operator. (Their strategy was in fact successful for the 2D Navier-Stokes equations.) In this paper, we attempt to apply their strategy to the original equations (1).

2 Navier-Stokes equations in Fourier space

Moving to Fourier space where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

let us consider the system of coupled ODEs for a finite-dimensional approximation to the Galerkin-system corresponding to (1),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

[summation over (i=1,2,3)] [k.sub.i][u.sup.(k).sub.i] = 0, (5)

where Z is a finite subset of [Z.sup.3] in which [u.sup.(k)](t) = [p.sup.(k)](t) = = 0 for each k [member of] [Z.sup.3] outside of Z. Like the Mattingly and Sinai paper, in this paper, we consider a generalization of this Galerkin-system:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

[summation over (i=1,2,3)] [k.sub.i][u.sup.(k).sub.i] = 0, (7)

where [alpha] [greater than or equal to] 2. Multiplying each of the first three equations by [k.sub.i] for i = 1,2,3 and adding the resulting equations together, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

since [[summation].sub.i=1,2,3] [k.sub.i][[du.sup.(k).sub.i]/dt] = 0 (by equation (7)). Then substituting the above calculated expression for [p.sup.(k)] in terms of u into (6) we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

And since [[summation].sub.j=1,2,3] [r.sub.j][u.sup.(r).sub.j] = 0 and [q.sub.j] + [r.sub.j] = [k.sub.j], we can substitute [k.sub.j] for [q.sub.j]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

3 A new theorem

Now, we state and prove the following theorem:

Theorem: Let {[u.sup.(k)](t)} satisfy (10), where [alpha] > 2.5. And let 1.5 < s < [alpha] - 1. Suppose there exists a constant [C.sub.0] > 0 such that [absolute value of [u.sup.(k)(0)] [less than or equal to] [C.sub.0][[absolute value of k].sup.-s], for all k [member of] [Z.sup.3]. Then there exists a constant C > [C.sub.0] such that [absolute value of [u.sup.(k)](t)] [less than or equal to] C[[absolute value of k].sup.-s], for all k [member of] [Z.sup.3] and all t > 0. (The constants, [C.sub.0] and C, are independent of the set Z defining the Galerkin approximation.)

Proof: By the basic energy estimate (see [1,2,7]), there exists a constant E [greater than or equal to] 0 such that for each t [greater than or equal to] 0 and for any finite-dimensional Galerkin approximation defined by Z [subset] [Z.sup.3], we have [[summation].sub.k[member of]Z] [[summation].sub.i=1,2,3] [[absolute value of [u.sup.(k).sub.i](t)].sup.2] [less than or equal to] E. Hence, for any K > 0, we can find a C > [C.sub.0] such that [absolute value of R([u.sup.(k)])] [less than or equal to] C[[absolute value of k].sup.-s] and [absolute value of J([u.sup.(k)])] [less than or equal to] C[[absolute value of k].sup.-s], for all t [greater than or equal to] 0 and k [member of] [Z.sup.3] with [absolute value of k] [less than or equal to] K. Now let us consider the set,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

We will show that if K is chosen large enough, any point starting in [OMEGA](K) cannot leave [OMEGA] (K), because the vector field along the boundary [partial derivative][OMEGA](K) is pointing inward, i.e., [OMEGA](K) is a trapping region. Since the initial data begins in [OMEGA](K), proving this would prove the theorem.

We pick a point on [partial derivative][OMEGA](K) where R([u.sup.([bar.k]).sub.i]) or J([u.sup.([bar.k]).sub.i]) = [+ or -] C[[absolute value of [bar.k]].sup.-s] for some [bar.k] [member of] Z such that [[absolute value of [bar.k]] > K and some i [member of] {1,2,3}. (For definiteness, we shall assume that R([u.sup.[bar.k].sub.i]) = C[[absolute value of [bar.k]].sup.-s], but the same line of argument which follows also applies to the other possibilities.) Then the following inequalities hold when K is chosen large enough:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

This establishes that the vector field points inward along the boundary of [OMEGA](K) for all t > 0. So the trajectory never at any time leaves [OMEGA](K). Then we have the desired estimate that [absolute value of [u.sup.(k)](t)] [less than or equal to] C[[absolute value of k].sup.-s] for all t > 0.

4 Discussion

Just as in the 1999 paper by Mattingly and Sinai [5], an existence and uniqueness theorem for solutions follows from our theorem by standard considerations (see [1,2,7]). The line of argument is as follows: By the Sobolev embedding theorem, the Galerkin approximations are trapped in a compact subset of [L.sup.2] of the 3-torus. This guarantees the existence of a limit point which can be shown to satisfy (10), where Z = [Z.sup.3]. Using the regularity inherited from the Galerkin approximations, one then shows that there exists a unique solution to the generalized 3D Navier-Stokes equations where [alpha] > 2.5.

The inequality (12) in the proof of our Theorem is not necessarily true when [alpha] = 2. Because of this, there is nothing preventing the solutions to (10) from escaping the region [OMEGA](K) when [alpha] = 2. Hence, there is no logical reason why the standard 3D Navier-Stokes equations must always have solutions, even when the initial velocity vector field is smooth; if they do always have solutions, it is due to probability (see [6]) and not logic, just like the Collatz 3n + 1 Conjecture and the Riemann Hypothesis (see [3,4]). Of course, it is also possible that there is a counterexample to the famous unresolved conjecture that the Navier-Stokes equations always have solutions when the initial velocity vector field is smooth. But as far as the author knows, nobody has ever found such a counterexample.

Submitted on October 15, 2014 / Accepted on October 22, 2014

References

[1.] Constantin P., Foias C. Navier-Stokes Equations. University of Chicago Press, Chicago, 1988.

[2.] Doering C., Gibbon J. Applied analysis of the Navier-Stokes equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 1995.

[3.] Feinstein C. Complexity Science for Simpletons. Progress in Physics, 2006, issue 3, 35-42.

[4.] Feinstein C. The Collatz 3n +1 Conjecture is Unprovable. Global Journal of Science Frontier Research, Mathematics & Decision Sciences, 2012, v. 12, issue 8, 13-15.

[5.] Mattingly J., Sinai Y. An elementary proof of the existence and uniqueness theorem for the Navier-Stokes equations. Commun. Contemp. Math. 1, 1999, no. 4, 497-516.

[6.] Montgomery-Smith S., Pokorny M. A counterexample to the smoothness of the solution to an equation arising in fluid mechanics. Commentationes Mathematicae Universitatis Carolinae, 2002, v. 43, issue 1, 61-75.

[7.] Temam R. Navier-Stokes equations: Theory and numerical analysis. Volume 2 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, revised edition, 1979.

Craig Alan Feinstein

2712 Willow Glen Drive, Baltimore, Maryland 21209. E-mail: cafeinst@msn.com
COPYRIGHT 2015 Progress in Physics
No portion of this article can be reproduced without the express written permission from the copyright holder.