Uniqueness against a smooth compactly supported reference flow

Statement

Let $(u,p)$ be a smooth Navier–Stokes solution on $\mathbb R^3\times[0,T]$ , $\nu>0$ , with the velocity supported in one fixed compact spatial set. If $(v,P)$ is another smooth solution with the same viscosity, force and initial velocity, and $v\in L^\infty_tL^2_x$ , then $v=u$ . The pressure $P$ may have unrestricted spatial growth.

This is a comparison theorem using localized difference energy, not an existence assertion.

Pressure first

Set $w=v-u$ and $g=v\otimes v-u\otimes u$ . Uniform $L^2$ bounds give $g\in L^\infty_tL^1_x$ . Pressure-gradient identification replaces $\nabla(P-p)$ by the canonical pressure gradient. Pairing this distributional identity with compactly supported time-space tests replaces the pressure flux in the energy identity by its canonical counterpart for almost every time.

Absorb the boundary fluxes

Choose $\phi=1$ near the unit ball, $0\le\phi\le1$ , and set

E_R=\int\phi_R^8|w|^2,\quad A_R=\|\phi_R^4\nabla w\|_2,\quad B_R=\|\phi_R^4w\|_6.

The cutoff Sobolev bound gives $B_R\le C(A_R+R^{-1}\|w\|_2)$ . For large $R$ , the reference velocity vanishes on the cutoff's derivative support, so the transport flux is at most $C R^{-1}B_R^{3/2}$ ; the Laplacian flux is at most $C R^{-2}$ . The pressure estimate contributes only powers $3/2,1/2,3/4$ of $B_R$ . Absorb each subquadratic power into $\nu A_R^2/2$ . Constants may depend on $T,\nu,u$ and the competitor's energy bound, but not on $R$ . The result is

\frac12 E_R'+\frac\nu2 A_R^2\le\|\nabla u\|_\infty E_R+C/R.

Since $E_R(0)=0$ , Gronwall gives $E_R(t)\le C_T/R$ . On every fixed ball the cutoff equals one for sufficiently large $R$ ; taking $R\to\infty$ proves $w=0$ . Smoothness extends the equality to every time. No global gradient bound for the competitor was assumed.