Changing viscosity by spatial rescaling

Statement

Suppose $(u,p,f)$ solves the Navier–Stokes equations with viscosity $\nu_0>0$ . For a desired $\nu>0$ , set $a=\sqrt{\nu/\nu_0}$ and

u^{(\nu)}(x,t)=a u(x/a,t),\quad p^{(\nu)}(x,t)=a^2p(x/a,t),\quad f^{(\nu)}(x,t)=a f(x/a,t).

These fields solve the equations with viscosity $\nu$ . Time and its endpoints are unchanged.

Verification and norms

The time derivative, advection, pressure gradient, and force each acquire a factor $a$ . The diffusion term is $\nu a^{-1}\Delta u=a\nu_0\Delta u$ . The divergence is the original divergence at $x/a$ . In dimension $n$ ,

\|u^{(\nu)}(t)\|_2^2=a^{n+2}\|u(t)\|_2^2,\qquad \nu\int\!\|\nabla u^{(\nu)}\|_2^2dt =a^{n+2}\nu_0\int\!\|\nabla u\|_2^2dt.

A spatial support $K$ becomes $aK$ , and $\partial_x^\alpha\partial_t^m f^{(\nu)}=a^{1-|\alpha|}(\partial_x^\alpha\partial_t^m f)(x/a,t)$ . These are bounds for each fixed positive viscosity, not assertions uniform as $\nu\to0$ .