Global periodic weak viscous existence

For $N\ge 1$, let $X_N=\mathrm{ran}{\mathrm{\Pi }}_N\cap H$ and solve in this finite-dimensional real space ${\mathrm{\partial }}_t{u}_N+{\mathrm{\Pi }}_{N}P\mathrm{div}(u_N\otimes u_N)=\nu \mathrm{\Delta }{u}_N,u_N(0)={\mathrm{\Pi }}_N{u}_0\mathrm{.}$ Choose real orthonormal coordinates on $X_N$, whose finite Fourier coefficients satisfy the real conjugation and solenoidal conditions. The right side preserves this real space and is a polynomial vector field $F_N:{\mathbb{R}}^m\to {\mathbb{R}}^m$. It is continuous on the open time-state domain $\mathbb{R}\times {\mathbb{R}}^m$; its derivative is bounded on each closed coefficient ball, so integration along line segments gives a Lipschitz bound there. Thus [F6] gives a unique local solution through the projected initial value. Pairing with $u_N$ removes $P,{\mathrm{\Pi }}_N$ by orthogonality, and periodic integration by parts gives $\int (u_N\cdot \mathrm{\nabla })u_N\cdot u_N=\int \mathrm{div}(u_N\mathrm{\mid }{u}_N{\mathrm{\mid }}^2\mathrm{/}2)=0$. Thus $\mathrm{\parallel }{u}_N(t){\mathrm{\parallel }}_2^2+2\nu {\int }_s^t\mathrm{\parallel }\mathrm{\nabla }{u}_N{\mathrm{\parallel }}_2^{2}d\tau =\mathrm{\parallel }{u}_N(s){\mathrm{\parallel }}_2^2$ on every local existence interval.

By the overlap uniqueness in [F6], all local solutions through the initial data glue on their interval union to one maximal solution. The energy identity bounds its coordinates in the fixed finite-dimensional real space $X_N$ for all nonnegative times of existence. The polynomial vector field is bounded on that closed coefficient ball, so the coordinate curve is uniformly Lipschitz in time. If its positive endpoint $\beta$ were finite, completeness of ${\mathbb{R}}^m$ would give a limit $a_{\ast }$ as $t\uparrow \beta$. Apply [F6] at $(\beta ,a_{\ast })$ and join the resulting local solution to the old curve. The derivatives match because both approach $F_N(a_{\ast })$, so this is a $C^1$ solution extending the maximal interval, a contradiction. Therefore every $u_N$ is forward global and the energy identity holds for all $0\le s\le t$.

For fixed $k$, the nonlinear coefficient is bounded by $C\mathrm{\mid }{\kappa }_k\mathrm{\mid }\mathrm{\parallel }{u}_N{\mathrm{\parallel }}_2^2$ and the viscous coefficient by $\nu \mathrm{\mid }{\kappa }_k{\mathrm{\mid }}^2\mathrm{\parallel }{u}_N{\mathrm{\parallel }}_2$. The energy identity supplies uniform $L^{\mathrm{\infty }}H$ and $L^{2}V$ bounds. The fixed-mode derivative bound is therefore uniform in $N$; [F3] gives subsequential compactness on each finite horizon, including strong convergence in $L^2(0,m;H)$. Applying [F5] diagonally over integer horizons yields one subsequence that retains uniform coefficient convergence and converges strongly in $L^2(0,m;H)$ for every integer $m$. The limits on overlapping horizons agree because their Fourier coefficient curves are the same; hence they define one global limit $u\in C_w([0,\mathrm{\infty });H)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{2}V$ with the stated energy bounds.

For a finite Fourier divergence-free test, uniform coefficient convergence passes the time-linear terms, strong $L_t^{2}H$ convergence passes the quadratic term by [F3], and the viscous term passes after integration by parts as $+\nu (u_N,\mathrm{\Delta }\varphi )$, using strong $L_t^{2}H$ convergence. A smooth periodic test and its time derivative have uniformly convergent Fourier truncations on their compact time support, by repeated integration by parts in space; the uniform energy bound controls the truncation errors. Thus the weak identity in [F2] holds for every admissible test, including its initial term because ${\mathrm{\Pi }}_N{u}_0\to u_0$ in $H$.

For each fixed $t$, finite coefficient convergence gives $\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2\le {\mathrm{lim\hspace{0.17em}inf}}_N\mathrm{\parallel }{u}_N(t){\mathrm{\parallel }}_2^2$. For the gradient integral on $[s,t]$, each finite Fourier sum is the limit of the corresponding Galerkin sum and is bounded above by the full Galerkin integral; increasing the finite sets gives the lower-semicontinuous bound for ${\int }_s^t\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_2^2$. At $s=0$, use ${\mathrm{\Pi }}_N{u}_0\to u_0$ in the Galerkin equality to obtain the initial energy inequality for every $t$. By step 2.2, the selected sequence converges strongly in $L^2(0,m;H)$ to $u$ for every positive integer $m$. Countable Choice therefore selects one increasing subsubsequence $(N_j)$ such that $\mathrm{\parallel }{u}_{N_j}-u{\mathrm{\parallel }}_{L^2(0,j;H)}^2\le 2^{-j}$ for each $j$. For each integer $m$, Tonelli [F3] gives ${\sum }_{j\ge m}\mathrm{\parallel }{u}_{N_j}(s)-u(s){\mathrm{\parallel }}_H^2<\mathrm{\infty }$ for almost every $s\in (0,m)$; intersecting these full-measure sets over $m$ gives one full-measure $S\subset (0,\mathrm{\infty })$ with strong $H$ convergence at every $s\in S$. For each such $s$, passing the Galerkin equality at $s$ and any fixed $t\ge s$ gives the restart inequality; since the lower-semicontinuity bounds hold for every fixed $t$, this establishes the inequality for all $t\ge s$.

The coefficient convergence implies $u(t)\rightharpoonup u_0$ in $H$ as $t\downarrow 0$. The initial energy inequality gives ${\mathrm{lim\hspace{0.17em}sup}}_{t\downarrow 0}\mathrm{\parallel }u(t){\mathrm{\parallel }}_2\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2$; weak convergence and the Hilbert identity for $\mathrm{\parallel }u(t)-u_0{\mathrm{\parallel }}_2^2$ then imply strong convergence to $u_0$. Together the compactness, weak-limit, and energy arguments verify every clause of the Leray–Hopf definition; the finite-dimensional local existence and continuation used above are supplied by [F6].