Cutoff pressure-flux estimate for a velocity difference

Statement

Let $u\in L^2\cap L^6(\mathbb R^3)$ , $w\in L^2\cap H^1_{\mathrm{loc}}$ , and $g=w\otimes w+w\otimes u+u\otimes w$ . Let $\pi_*$ be its canonical pressure. For $0\le\phi\le1$ smooth and compactly supported, put $\phi_R(x)=\phi(x/R)$ , $\chi_R=\phi_R^8$ , and $B_R=\|\phi_R^4w\|_6$ . If $R\ge1$ and $\|w\|_2+\|u\|_2+\|u\|_6\le M$ , then

\left|\int\pi_* w\cdot\nabla\chi_R\right| \le C_{M,\phi}R^{-1}\left((B_R+1)B_R^{1/2}+R^{-3/4}B_R^{3/4}\right).

The pressure is locally integrable for these inputs, and the pairing is well-defined.

Local and commutator parts

For $T_{ij}=R_iR_j$ , write

\phi_R^4\pi_* =\sum_{i,j}T_{ij}(\phi_R^4g_{ij}) +\sum_{i,j}[M_{\phi_R^4},T_{ij}]g_{ij}.

The first sum has $L^{3/2}$ norm at most $C_M(B_R+1)$ : use $\|\phi_R^4w_iw_j\|_{3/2}\le B_R\|w\|_2$ and $\|\phi_R^4w_i u_j\|_{3/2}\le\|w\|_2\|u\|_6$ . The second has $L^{4/3}$ norm at most $C_MR^{-3/4}$ . These identities extend from compact smooth approximations using $L^1$ convergence of $g$ , the negative Sobolev pressure bound, and the commutator bound.

Since $|\nabla\chi_R|\le C R^{-1}\phi_R^7$ , it remains to pair these terms with $\phi_R^3w$ . Hölder interpolation gives $\|\phi_R^3w\|_3\le B_R^{1/2}\|w\|_2^{1/2}$ and $\|\phi_R^3w\|_4\le B_R^{3/4}\|w\|_2^{1/4}$ , proving the estimate. Local integrability follows by choosing cutoffs equal to one on each compact set.