Global bounded no-slip Leray–Hopf existence

If $\mathrm{\Omega }=\mathrm{\varnothing }$, [F10] and the closure formulas in the Statement give $H_0^1=H=V=\{0\}$. These are complete separable real Hilbert spaces with their zero pairings, so [F5] applies. Necessarily $u_0=0$; define the actual map $u(t)=0$ for every $t\ge 0$. Every fixed $H$ pairing is the constant zero function, so $u\in C_w([0,\mathrm{\infty });H)$. The constant zero simple map proves strong measurability in $H$ and $V$, and every required $L^{\mathrm{\infty }}$ or local $L^2$ norm is zero. The initial value and strong trace are immediate. Every test curve into $V$ is zero, so the tested identity is $0=0$; both energy inequalities likewise reduce to $0\le 0$. Taking $S=(0,\mathrm{\infty })$ gives the required full-measure restart set. This proves every assertion in the empty case directly, without an orthonormal sequence or a Galerkin ODE on a zero-dimensional target. Henceforth suppose $\mathrm{\Omega }\mathrm{\ne }\mathrm{\varnothing }$, so the nonempty-domain closure, basis and compactness suppliers [F1]–[F4] apply.

Let $(e_j)$ be the basis from [F2], and set $u_N={\sum }_{j=1}^N{a}_j(t)e_j$ with initial value the $H$-projection of $u_0$. Solve $(u_N^{\mathrm{\prime }},e_i{)}_H+{\int }_{\mathrm{\Omega }}(u_N\cdot \mathrm{\nabla })u_N\cdot e_{i}dx+\nu (\mathrm{\nabla }{u}_N,\mathrm{\nabla }{e}_i)=0,i\le N\mathrm{.}$ Because the $e_j$ are real and $H$-orthonormal, the displayed system has identity mass matrix and is an autonomous polynomial ODE $a^{\mathrm{\prime }}=F_N(a)$ on ${\mathbb{R}}^N$. Its open time-state domain is $\mathbb{R}\times {\mathbb{R}}^N$; it is continuous and its derivative is bounded on every closed coefficient ball, giving a Lipschitz bound there by integration along line segments. Hence [F9] gives its unique local solution. Pairing with $u_N$ and summing the equations cancels the transport term because $\mathrm{div}{u}_N=0$ and $\int (u_N\cdot \mathrm{\nabla })u_N\cdot u_N=\int \mathrm{div}(u_N\mathrm{\mid }{u}_N{\mathrm{\mid }}^2\mathrm{/}2)=0$ for its compact support. Thus $\mathrm{\parallel }{u}_N(t){\mathrm{\parallel }}_2^2+2\nu {\int }_s^t\mathrm{\parallel }\mathrm{\nabla }{u}_N{\mathrm{\parallel }}_2^{2}d\tau =\mathrm{\parallel }{u}_N(s){\mathrm{\parallel }}_2^2\mathrm{.}$ The overlap uniqueness in [F9] glues all local solutions through the initial value on their interval union. The energy identity bounds the coefficient vector of this maximal solution for nonnegative times; $F_N$ is bounded on that closed ball, so the curve is uniformly Lipschitz in time. A finite positive endpoint would therefore have a limit $a_{\ast }\in {\mathbb{R}}^N$. The local solution supplied by [F9] at that endpoint and state joins to the old one with matching derivative $F_N(a_{\ast })$, producing a $C^1$ extension and contradicting maximality. Hence $u_N$ is forward global.

For fixed $i$ and $N\ge i$, integration by parts and the energy bound give $\mathrm{\mid }(u_N^{\mathrm{\prime }}(t),e_i)\mathrm{\mid }\le \mathrm{\parallel }\mathrm{\nabla }{e}_i{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{u}_N(t){\mathrm{\parallel }}_2^2+\nu \mathrm{\parallel }\mathrm{\nabla }{e}_i{\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\nabla }{u}_N(t){\mathrm{\parallel }}_2\mathrm{.}$ Integrating from $s$ to $t$ and applying Cauchy–Schwarz bounds coefficient increments by $C_i\mathrm{\mid }t-s\mathrm{\mid }+D_i\mathrm{\mid }t-s{\mathrm{\mid }}^{1\mathrm{/}2}$, uniformly in $N$ on each finite horizon. Diagonal extraction over the countable basis and integer horizons gives uniformly convergent coefficient curves. Their finite sums define for every $t$ a bounded $H$ element; approximation of an arbitrary $H$ test by finite basis sums proves weak continuity as in [F5].

For each integer $m\ge 1$, the energy equality bounds $u_N$ in the separable Hilbert space $X_m=L^2(0,m;V)$. Completeness follows by selecting from a Cauchy sequence a subsequence with summable $X_m$-norm differences, applying [F6] to obtain pointwise summability in complete $V$, and then using the $L^2$ triangle inequality for convergence. Separability follows by approximating strongly measurable maps by finite-valued simple maps, their scalar indicators by the countable Fourier span from [F7], and their values by a countable dense subset of $V$ from [F2]. In any separable Hilbert space, a bounded sequence has a weakly convergent subsequence: diagonalize coordinates in a countable orthonormal basis; finite squared sums bound the limiting coordinates, their square-summable series defines a vector, and finite-basis approximation plus the uniform norm bound gives convergence against every vector. Apply this successively on $m=1,2,\dots$ to nested subsequences of the coefficient-convergent sequence from step 2.1, and take the diagonal subsequence. It converges weakly in every $X_m$. The limits agree on overlapping horizons: for each basis vector $e_j$ from [F2], the map $w\mapsto (w,e_j{)}_H$ is continuous from $V$ to $\mathbb{R}$, so the induced map $X_m\to L^2(0,m)$ preserves weak convergence, while step 2.1 gives strong $L^2$ convergence of that coefficient to its fixed uniform limit. Uniqueness of weak limits identifies the coefficient of each $X_m$ limit with that same curve; the countable basis then identifies the $H$-valued limits almost everywhere. Thus one $u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(0,\mathrm{\infty };V)$ is obtained on all finite horizons.

On each integer horizon $m$, [F3] gives ${\int }_0^m\mathrm{\parallel }(I-{\mathrm{\Pi }}_K)u_N(t){\mathrm{\parallel }}_2^{2}dt\le {\delta }_K^2{\int }_0^m\mathrm{\parallel }{u}_N(t){\mathrm{\parallel }}_V^{2}dt,$ uniformly in $N$. The weak limit in step 3.1 has the same tail bound: restriction and $I-{\mathrm{\Pi }}_K:L^2(0,m;V)\to L^2(0,m;H)$ are bounded linear maps, and the Hilbert norm is weakly lower semicontinuous. For each fixed $K$, the first $K$ coefficients converge uniformly by step 2.1. Choose $K$ to make both $u_N$ and $u$ tails small in $L^2(0,m;H)$, then pass to the limit in the finitely many coefficients; this proves strong convergence in $L^2(0,m;H)$ for every $m$. A further diagonal subsequence can be chosen with $\mathrm{\parallel }{u}_{N_j}-u{\mathrm{\parallel }}_{L^2(0,j;H)}^2\le 2^{-j}$. For each fixed $m$, Tonelli [F6] gives ${\sum }_{j\ge m}\mathrm{\parallel }{u}_{N_j}(t)-u(t){\mathrm{\parallel }}_H^2<\mathrm{\infty }$ for almost every $t\in (0,m)$. Intersecting these full-measure sets over integer $m$ gives a single full-measure set $S\subset (0,\mathrm{\infty })$ on which $u_{N_j}(t)\to u(t)$ strongly in $H$. The identity $\mathrm{\parallel }{u}_{N_j}\otimes u_{N_j}-u\otimes u{\mathrm{\parallel }}_{L^1((0,m)\times \mathrm{\Omega })}\le (\mathrm{\parallel }{u}_{N_j}{\mathrm{\parallel }}_{L^2((0,m)\times \mathrm{\Omega })}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2((0,m)\times \mathrm{\Omega })})\mathrm{\parallel }{u}_{N_j}-u{\mathrm{\parallel }}_{L^2((0,m)\times \mathrm{\Omega })}$ then gives strong tensor convergence on every finite horizon.

For a test field taking values in a fixed finite span of the $e_j$, the Galerkin identity is exact for all sufficiently large $N$. Uniform coefficient convergence passes the time and initial terms, strong tensor convergence from step 4.1 passes the transport term, and weak convergence in $L^{2}V$ passes viscosity. To pass to any compactly supported $C^1$ curve $\varphi :[0,\mathrm{\infty })\to V$, let $Q_K$ be the $V$-orthogonal projection onto $\mathrm{span}(e_1,\dots ,e_K)$. It is a contraction and converges strongly to the identity by [F2]; finite nets of the compact images of $\varphi$ and ${\varphi }_t$ make this convergence uniform in time. The nonlinear error is bounded by the uniform $L_t^{4\mathrm{/}3}{V}^{\ast }$ norm from [F4] times $\mathrm{\parallel }\varphi -Q_K\varphi {\mathrm{\parallel }}_{L_t^{4}V}$; the time and viscosity errors tend to zero by the $L^{\mathrm{\infty }}H$ and $L^{2}V$ bounds. Thus the full weak identity holds.

For each fixed $t$, convergence of the finite coefficients gives $\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2\le {\mathrm{lim\hspace{0.17em}inf}}_N\mathrm{\parallel }{u}_N(t){\mathrm{\parallel }}_2^2$. Weak convergence in $L^{2}V$ restricted to any $[s,t]$ gives lower semicontinuity of the gradient integral there. At $s=0$, $u_N(0)\to u_0$ strongly, so passing the Galerkin equality proves the initial energy inequality for every $t$. At every time $s$ in the full-measure set from step 4.1, $u_N(s)\to u(s)$ strongly; passing the same equality on $[s,t]$ gives the restart inequality for every $t\ge s$. Coefficient convergence gives $u(t)\rightharpoonup u_0$ as $t\downarrow 0$, and the initial energy inequality gives the norm limsup; the Hilbert identity then gives strong convergence to $u_0$. Together the Galerkin, compactness, weak-limit, and energy arguments prove the stated existence, with the finite-dimensional local existence and continuation supplied by [F9].