MINI backward Euler converges subsequentially to a solenoidal energy weak velocity

Since $u_0\in H_{\sigma }={\overline{V_{\sigma }}}^{L^2}$, for any $ϵ>0$ choose $v\in V_{\sigma }$ with $\mathrm{\parallel }{u}_0-v{\mathrm{\parallel }}_2<ϵ$. By [F3], ${\mathrm{\Pi }}_{j}v\in Z_j$ and $\mathrm{\parallel }{\mathrm{\Pi }}_{j}v-v{\mathrm{\parallel }}_2\to 0$. The best-approximation property [F4] gives $\mathrm{\parallel }{P}_{Z_j}{u}_0-u_0{\mathrm{\parallel }}_2\le \mathrm{\parallel }{\mathrm{\Pi }}_{j}v-u_0{\mathrm{\parallel }}_2\le \mathrm{\parallel }{\mathrm{\Pi }}_{j}v-v{\mathrm{\parallel }}_2+ϵ$. Letting $j\to \mathrm{\infty }$ and then $ϵ\downarrow 0$ proves strong convergence of the discrete initial data.

Apply [F1] to every mesh and partition. The energy inequality gives uniform bounds for ${\max }_n\mathrm{\parallel }{u}_j^n{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }{\bar{u}}_j{\mathrm{\parallel }}_{L^{2}V}$; it also gives $\mathrm{\parallel }{\bar{u}}_j{\mathrm{\parallel }}_{L^{\mathrm{\infty }}H}\le M$.

Fix a finite orthonormal family $e_1,\dots ,e_m\in V_{\sigma }$. For each mesh, ${\mathrm{\Pi }}_j{e}_i\in Z_j$ by [F3], so testing the discrete equation [F13] with ${\mathrm{\Pi }}_j{e}_i$ removes pressure. The backward difference of $a_{j,i}^n=(u_j^n,{\mathrm{\Pi }}_j{e}_i{)}_2$ is therefore bounded by $\mathrm{\mid }(D_{\tau }{a}_{j,i}^n)\mathrm{\mid }\le C_{e_i}(\mathrm{\parallel }{f}_j^n{\mathrm{\parallel }}_2+\nu \mathrm{\parallel }{u}_j^n{\mathrm{\parallel }}_V+\mathrm{\parallel }{u}_j^n{\mathrm{\parallel }}_2^{1\mathrm{/}2}\mathrm{\parallel }{u}_j^n{\mathrm{\parallel }}_V^{3\mathrm{/}2})\mathrm{.}$ The first two terms are bounded in $L^2(0,T)$ by [F1], and the last in $L^{4\mathrm{/}3}(0,T)$ by [F11], because its $4\mathrm{/}3$ power is at most $M^{2\mathrm{/}3}C\mathrm{\parallel }{u}_j^n{\mathrm{\parallel }}_V^2$. Thus the coefficient condition of [F2] holds for every finite family.

By [F2], after a subsequence ${\bar{u}}_j\to u$ strongly in $L^2(0,T;L^2)$. The energy bounds [F1] and [F9] give, after a further subsequence, ${\bar{u}}_j\rightharpoonup u$ in $L^2(0,T;V)$. Apply [F10] to the scalar functions $t\mapsto \mathrm{\parallel }{\bar{u}}_j(t)-u(t){\mathrm{\parallel }}_2$ to obtain almost-everywhere-in-time strong convergence along a further subsequence. The uniform $L^{\mathrm{\infty }}H$ bound then implies $u\in L^{\mathrm{\infty }}(0,T;H)$.

For any $q\in C_c^{\mathrm{\infty }}(D)$, apply the face-average interpolant to $q-\mathrm{\mid }D{\mathrm{\mid }}^{-1}{\int }_{D}q$ and subtract the mean of its interpolant to obtain $q_j\in Q_j$ converging in $L^2$, using the approximation estimate in Boundary-preserving face-average P1 interpolation. Since $b(u_j^n,q_j)=0$, [F6] gives ${\int }_{D}q\mathrm{div}{u}_j^n=0$ for the $q_j$ part and ${\int }_D\mathrm{div}{u}_j^n=0$ for the mean part. The $L^2$ approximation error times the uniformly bounded $L^2(0,T;V)$ divergence norm tends to zero by Holder. Passing to the weak gradient limit proves $\mathrm{div}u=0$ a.e.; hence $u\in L^2(0,T;V_{\sigma })$.

Let $v\in V_{\sigma }$ and $\eta \in C^1([0,T])$ vanish near $T$. Test each time step [F13] by ${\mathrm{\Pi }}_{j}v\in Z_j$, multiply by the left-endpoint value $\eta (t_{n-1}^j)$, and sum. Discrete summation by parts gives $-(u_j^0,{\mathrm{\Pi }}_{j}v{)}_2\eta (0)-{\sum }_{n=1}^{N_j-1}(u_j^n,{\mathrm{\Pi }}_{j}v{)}_2(\eta (t_n^j)-\eta (t_{n-1}^j))$; the terminal term is zero for all fine partitions because $\eta$ vanishes near $T$. The sum is exactly the integral of $-{\bar{u}}_j\cdot {\mathrm{\Pi }}_{j}v{\eta }^{\mathrm{\prime }}$ on the nonterminal cells, so the time contribution converges to $-(u_0,v{)}_2\eta (0)-{\int }_0^T(u,v{)}_2{\eta }^{\mathrm{\prime }}$. Moving its initial term to the right side gives the required positive initial trace. Uniform convergence of the left-endpoint step approximation to $\eta$, [F3], and step 1.1 justify these limits. The viscosity term converges by weak $L^{2}V$ convergence and ${\mathrm{\Pi }}_{j}v\to v$ in $V$; the force averages converge by [F7] and the same uniform time-test approximation.

For almost every $t$, choose the first time node $t_j(t)\ge t$ and sum the work-form energy inequality of [F1] through that node. Then ${\bar{u}}_j(t)=u_j^{n_j(t)}$, so step 1.4 gives the required strong convergence of the kinetic-energy term for almost every $t$. By [F7], $f_j\to f$ in $L^{2}H$ and ${\bar{u}}_j\to u$ in $L^{2}H$, so their work integrals converge on $[0,t]$. The overshoot interval $[t,t_j(t)]$ contributes at most $\mathrm{\parallel }{f}_j{\mathrm{\parallel }}_{L^{2}H}M\sqrt{{\tau }_{\max }^{}}$ by Cauchy--Schwarz and the uniform $L^{\mathrm{\infty }}H$ bound, hence tends to zero. The gradient sum through $t_j(t)$ is bounded below by its integral through $t$; weak $L^{2}V$ convergence and [F9] give lower semicontinuity on $[0,t]$. The initial projections converge by step 1.1. This proves the energy inequality for almost every $t$.

The test interpolation error in convection vanishes: continuity from [F11] bounds its integral by $C\mathrm{\parallel }{\mathrm{\Pi }}_{j}v-v{\mathrm{\parallel }}_V\mathrm{\parallel }{\bar{u}}_j{\mathrm{\parallel }}_{L^{2}V}^2\mathrm{\parallel }\eta {\mathrm{\parallel }}_{\mathrm{\infty }}$. The uniform left-endpoint test approximation from step 2.2 permits replacing the discrete weights by $\eta$. For fixed $v\in V_{\sigma }$, strong $L^2{L}^2$ convergence and the uniform $L^2{L}^6$ bound imply strong $L^2{L}^3$ and strong $L^2{L}^4$ convergence of velocities by Holder interpolation. In the skew formula, $u_j\otimes v$ then converges strongly in $L^2{L}^2$ while $\mathrm{\nabla }{u}_j$ converges weakly in $L^2{L}^2$; $u_j\otimes u_j$ converges strongly in $L^1{L}^2$. Thus the convective term passes to $c(u;u,v)$. This proves the limiting momentum equation for all solenoidal tests.

By [F14], $L^2(D{)}^d$ is separable, so balls with centers in a countable dense set and positive rational radii form a countable base. The space $H_{\sigma }={\overline{V_{\sigma }}}^{L^2}$ from [F12] is a closed subspace and hence Hilbert by [F14]; intersecting that countable base with $H_{\sigma }$ gives a countable base there, and [A1] selects one point from each nonempty member to make a dense set. For every $v\in V_{\sigma }$, steps 2.2 and 3.1 give the scalar weak equation with $h_v(t)=(f(t),v{)}_{L^2}-a_{\nu }(u(t),v)-c(u(t);u(t),v)$. This function is integrable: its three terms lie in $L^2(0,T)$, $L^2(0,T)$, and $L^{4\mathrm{/}3}(0,T)$ respectively by the data, energy bound [F1], and convection bound [F11]; finite $T$ embeds these spaces into $L^1$. The uniform $L^{\mathrm{\infty }}H$ bound follows from step 1.4. Apply [F15] with $H=H_{\sigma }$ and $X=V_{\sigma }$ to obtain a unique weakly continuous representative with initial value $u_0$. Its energy inequality is already proved in step 2.3.

Steps 1.4, 2.1–2.3, 3.1, and 4.1 give the strong and weak convergence, divergence constraint, solenoidal momentum equation, weakly continuous initial trace, and energy inequality required by Solenoidal energy weak velocity for incompressible Navier–Stokes. The stronger pressure-bearing PE06 claim is not inferred here and remains escalated as stated.