Delayed parabolic rescaling at a fixed terminal time

Statement

Let smooth Navier–Stokes fields $(u,p,f)$ be defined for $0\le s<T$ , vanish for $s$ in an initial neighborhood of zero, and have a fixed compact spatial support. For $\lambda\ge1$ , put $t_0=T(1-\lambda^{-2})$ . On $t_0\le t<T$ , define

\widetilde u(x,t)=\lambda u(\lambda x,\lambda^2(t-t_0)),\quad \widetilde p(x,t)=\lambda^2p(\lambda x,\lambda^2(t-t_0)),\quad \widetilde f(x,t)=\lambda^3f(\lambda x,\lambda^2(t-t_0)).

Extending all three by zero for $t<t_0$ gives smooth fields satisfying the same equation and viscosity. This uses the parabolic scaling symmetry.

Support and endpoint

The initial vanishing makes the time gluing smooth. A spatial support $K$ shrinks to $\lambda^{-1}K$ , while $\lambda^2(T-t_0)=T$ , so the old terminal time is reached at the same new terminal time. If the original force is defined smoothly for later $s$ , its displayed rescaling is used there as well. This rescaling can place all supports inside a fundamental cube before periodization.