Local smooth solutions of the periodic incompressible Euler and constant-viscosity equations on ${\mathbb{T}}^3$

By [F1] each Galerkin approximation $u_N$ is defined on $[0,T]$ and bounded by $B_m$ in $H^m$ and by $B_m^{\mathrm{\prime }}$ in $H^{m-2}$ for every $m\ge s$; by [F2] there is a subsequence $(N_j)$ and $u\in {\bigcap }_{m\ge s}C([0,T],H^m)$ with $u_{N_j}\to u$ in $C([0,T],H^{m-1})$ for every $m\ge s+1$, ${\sup }_{[0,T]}\mathrm{\parallel }u(t){\mathrm{\parallel }}_{H^m}\le B_m$, and $u(0)=u_0$.

Fix $m\ge s+1$. Since $(u_{N_j}\cdot \mathrm{\nabla })u_{N_j}-(u\cdot \mathrm{\nabla })u=(u_{N_j}-u)\cdot \mathrm{\nabla }{u}_{N_j}+u\cdot \mathrm{\nabla }(u_{N_j}-u)$, [F3] gives $\mathrm{\parallel }(u_{N_j}\cdot \mathrm{\nabla })u_{N_j}-(u\cdot \mathrm{\nabla })u{\mathrm{\parallel }}_{H^{m-2}}\le 3C_{m-2}(\mathrm{\parallel }{u}_{N_j}-u{\mathrm{\parallel }}_{H^{m-2}}\mathrm{\parallel }{u}_{N_j}{\mathrm{\parallel }}_{H^{m-1}}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{m-2}}\mathrm{\parallel }{u}_{N_j}-u{\mathrm{\parallel }}_{H^{m-1}})\to 0$ uniformly in $t$. Moreover $P_{N_j}P((u_{N_j}\cdot \mathrm{\nabla })u_{N_j})\to P((u\cdot \mathrm{\nabla })u)$ in $C([0,T],H^{m-2})$: the difference is $(P_{N_j}-P)P((u_{N_j}\cdot \mathrm{\nabla })u_{N_j})+P(P((u_{N_j}\cdot \mathrm{\nabla })u_{N_j})-P((u\cdot \mathrm{\nabla })u))$, whose first term is bounded by $(2\pi (N_j+1){)}^{-1}\mathrm{\parallel }P((u_{N_j}\cdot \mathrm{\nabla })u_{N_j}){\mathrm{\parallel }}_{H^{m-1}}\le (2\pi (N_j+1){)}^{-1}3C_{m-1}{B}_{m-1}{B}_m$ by [F4] and the second tends to $0$ by the previous estimate. Step 1.1 at order $m+1$ also gives $u_{N_j}\to u$ in $C([0,T],H^m)$, so $\mathrm{\Delta }{u}_{N_j}\to \mathrm{\Delta }u$ in $C([0,T],H^{m-2})$ by its bounded multiplier. Hence $F_{N_j}(u_{N_j})\to \nu \mathrm{\Delta }u-P((u\cdot \mathrm{\nabla })u)=:G$ in $C([0,T],H^{m-2})$.

Each Galerkin solution satisfies the integral identity $u_{N_j}(t)=P_{N_j}{u}_0+{\int }_0^t{F}_{N_j}(u_{N_j})(s)ds$ in $H^{m-2}$, because it is a finite-dimensional smooth coefficient curve: apply If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz, claim 1, to its real coefficient coordinates, whose continuous integrals agree with tagged sums by uniform continuity, then apply the bounded coordinate map. At $t=0$ the identity is the zero-integral convention of [F5]. Passing to the limit coefficientwise (justified by the uniform convergence of step 2.1 and $P_{N_j}{u}_0\to u_0$ of [F2], together with the continuity of coefficient functionals and [F5]) gives $u(t)=u_0+{\int }_0^{t}G(s)ds$ in $H^{m-2}$; by [F5] again, $t\mapsto u(t)$ is differentiable in $H^{m-2}$ with $u^{\mathrm{\prime }}(t)=G(t)$, i.e. $u^{\mathrm{\prime }}=\nu \mathrm{\Delta }u-P((u\cdot \mathrm{\nabla })u)$ holds in $C([0,T],H^{m-2})$; since $m\ge s+1$ was arbitrary, it holds for every real $m\ge s$.

The curve satisfies the hypotheses of [F6]: for every real $p$ it is continuous and differentiable into $H^p$ (take an integer $m\ge \max (s+1,p+2)$; continuity in $H^p$ follows from continuity in $H^m$ and $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^p}\le \mathrm{\parallel }v{\mathrm{\parallel }}_{H^m}$, and the identity $u^{\mathrm{\prime }}=\nu \mathrm{\Delta }u-P((u\cdot \mathrm{\nabla })u)$ holds in $H^{m-2}\subseteq H^p$), it is divergence-free and real by [F2]. Hence $u\in C^{\mathrm{\infty }}([0,T]\times {\mathbb{T}}^3;{\mathbb{R}}^3)$ and the differentiated identities hold.

Collecting: $u(0)=u_0$ by [F2]; the equation holds in every $H^{m-2}$, $m\ge s$, by step 3.1, hence pointwise after step 4.1; the divergence-free condition and the bound ${\sup }_{[0,T]}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^m}\le B_m$ hold by [F2]. The mean is preserved by passing the Galerkin identity $\int u_N(t)=\int P_N{u}_0=\int u_0$ to the limit. If $Y_0=0$ then $u_0=P_N{u}_0=0$ for every $N$, so $u_N\equiv 0$ and $u\equiv 0$. This proves claims 1-3.