Navier and stokes meet the wavelet

Abstract

We work in the space ℱ = ℱ_ε of divergence-free measurable vector fields onR ³ complete in the norm ‖ ‖′, where

for some fixed ε>0.B(x, R) is the ball of radiusR centered atx. Given an initial velocity distribution$\vec v$(0) in ℱ, we find$\vec v$(x,t) for 0≦t≦T=T(‖v(0)‖'),T>0, such that$\vec v$(x,t) is the unique strong solution of the Navier-Stokes equations, in a suitable sense.

We expand$\vec v$′(x,t) =$\vec v$(x,t) −$\vec v$(x, 0) in terms of divergence-free vector wavelets

$$\vec v'(x,t) = \sum\limits_\alpha {c_\alpha (t)\vec u_\alpha (x)}$$

. The Navier-Stokes equations become an infinite set of integral equations for thec _α (t). In an appropriate space one realizes thec _α (t) satisfying the equations as the fixed point of a contraction mapping. The thus unique solution is the strong solution mentioned above.