Finite initial higher Sobolev norms give uniform Galerkin bounds on the common base interval

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Fix $N\ge 1$, $\nu \ge 0$, a real $s>5\mathrm{/}2$, and $u_0\in H_{\sigma }^s$ with $Y_0=\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^s}$, and let $T$ and $u_N:[0,T]\to X_N$ be as in The higher-order energy inequality and a common Galerkin lifespan with $\mathrm{\parallel }{u}_N(t){\mathrm{\parallel }}_{H^s}\le 2Y_0$. For every real $m\ge s$ with $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^m}<+\mathrm{\infty }$ define $B_m:=\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^m}{e}^{4C_m{Y}_{0}T},B_m^{\mathrm{\prime }}:=\nu B_m+3C_{m-2}{B}_{m-1}^2,$ where $C_m$ is the commutator constant of The transport term is orthogonal to itself and the tame commutator estimate $\mathrm{\parallel }[{\mathrm{\Lambda }}^m,u\cdot \mathrm{\nabla }]u{\mathrm{\parallel }}_{L^2}\le C_m\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}$ and $3C_{m-2}$ is the product constant of Discrete Young inequality, the Sobolev product bound, and the transport product bound on the torus at order $m-2$ (so that $B_m^{\mathrm{\prime }}$ is defined for $m>7\mathrm{/}2$). For lower orders $0\le r<s$ set $B_r:=B_s$; then $\mathrm{\parallel }{u}_N(t){\mathrm{\parallel }}_{H^r}\le B_r$ follows from monotonicity of the weights and the base bound. At any order where an initial norm is infinite, take $B_r=+\mathrm{\infty }$ and the corresponding $B_r\mathrm{\prime }=+\mathrm{\infty }$ by convention, reading the estimate as an extended upper bound. Then for $0\le t\le T$:

1. $\mathrm{\parallel }{u}_N(t){\mathrm{\parallel }}_{H^m}\le B_m$ for every real $m\ge s$; in particular all bounds hold on the single interval $[0,T]$ whose length depends only on $Y_0$ and $s$, uniformly in $N$.
2. $\mathrm{\parallel }{u}_N^{\mathrm{\prime }}(t){\mathrm{\parallel }}_{H^{m-2}}\le B_m^{\mathrm{\prime }}$ for every real $m>7\mathrm{/}2$; in particular, for the integer orders of the main theorem ($s\ge 4$, $m\ge s$) these are the uniform-in-$N$ bounds on $L^{\mathrm{\infty }}(0,T;H^m)$ and $L^{\mathrm{\infty }}(0,T;H^{m-2})$ of the Galerkin approximations.

• The point of the statement is the word common: the interval does not shrink when the order $m$ is raised, because all the higher norms are controlled on the same base interval $[0,T]$.
• For smooth $u_0$ every $B_m$ is finite, which is the situation of the main theorem; for $u_0\in H^s$ only the bounds with $m=s$ are asserted, and they are what the compactness argument uses in the base norm.