Rescaling unit viscosity to positive viscosity

Statement

Let $(u_1,p_1,f_1)$ solve the Navier–Stokes equations with viscosity $1$ . For $\nu>0$ , set

u_\nu(t,x)=\sqrt\nu\,u_1(t,x/\sqrt\nu),\quad p_\nu(t,x)=\nu p_1(t,x/\sqrt\nu),\quad f_\nu(t,x)=\sqrt\nu\,f_1(t,x/\sqrt\nu).

Then $(u_\nu,p_\nu,f_\nu)$ solves the equations with viscosity $\nu$ .

Verification and energy

The time, advection, pressure, and viscous terms all become $\sqrt\nu$ times their unit-viscosity versions. Divergence remains zero. By change of variables, the kinetic energy in dimension $d$ is $E_\nu(t)=\nu^{1+d/2}E_1(t)$ , whenever finite. The spatial domain is also dilated by $\sqrt\nu$ ; on a torus this changes its periods.

References

Charles Fefferman, Existence and Smoothness of the Navier–Stokes Equation.