Weak velocity limits do not supply the smooth pressure-observation estimate

Assume both Countable Choice and the Axiom of Choice. Let $D\subset {\mathbb{R}}^d$, $d\in \{2,3\}$, be a nonempty bounded connected Lipschitz polyhedral domain, let $0<T<\mathrm{\infty }$, let $\nu >0$, let $V=H_0^1(D;{\mathbb{R}}^d)$, $Q=L_0^2(D)$, $V_{\sigma }=\{v\in V:\mathrm{div}v=0\}$, and $H_{\sigma }={\overline{V_{\sigma }}}^{L^2(D{)}^d}$. Define $\epsilon (v)=(\mathrm{\nabla }v+\mathrm{\nabla }{v}^{\mathsf{T}})\mathrm{/}2$, $a_{\nu }(v,w)=2\nu {\int }_D\epsilon (v):\epsilon (w)dx$, $b(v,q)=-{\int }_{D}q\mathrm{div}vdx$, and $c(a;v,w)=\frac{1}{2}{\int }_D((a\cdot \mathrm{\nabla })v\cdot w-(a\cdot \mathrm{\nabla })w\cdot v)dx$. On each matching shape-regular affine simplicial mesh, let $V_{h,j}$ be the MINI continuous piecewise-linear velocity space enriched by one interior bubble per simplex and let $Q_{h,j}$ be the continuous piecewise-linear pressure space with zero mean. Set $Z_j=\{v_{h,j}\in V_{h,j}:b(v_{h,j},q_{h,j})=0\text{ for every }{q}_{h,j}\in Q_{h,j}\}$. Take $f\in L^2(0,T;L^2(D{)}^d)$ and $u_0\in H_{\sigma }$. Use a mesh family with maximum cell diameters $h_j\to 0$ and time partitions $0=t_0^j<\cdots <t_{N_j}^j=T$ whose maximum steps ${\tau }_{\max ,j}$ tend to zero, without a relation between these limits. On each mesh, initialize by the $L^2$ projection of $u_0$ onto $Z_j$ and choose any actual mixed backward Euler MINI solution; write ${\bar{u}}_j(t)=u_j^n$ on $(t_{n-1}^j,t_n^j]$. A subsequence satisfies ${\bar{u}}_j\to u$ strongly in $L^2(0,T;L^2(D{)}^d)$ and weakly in $L^2(0,T;V)$, where $u\in L^{\mathrm{\infty }}(0,T;H_{\sigma })\cap L^2(0,T;V_{\sigma })$ is weakly continuous into $H_{\sigma }$, has $u(0)=u_0$, and obeys, for every fixed $v\in V_{\sigma }$ and $\eta \in C^1([0,T])$ vanishing near $T$, $-{\int }_0^T(u(t),v{)}_{L^2}{\eta }^{\mathrm{\prime }}(t)dt+{\int }_0^T(a_{\nu }(u(t),v)+c(u(t);u(t),v))\eta (t)dt={\int }_0^T(f(t),v{)}_{L^2}\eta (t)dt+(u_0,v{)}_{L^2}\eta (0),$ and, for almost every $t\in (0,T)$, $\frac{1}{2}\mathrm{\parallel }u(t){\mathrm{\parallel }}_{L^2}^2+\nu {\int }_0^t\mathrm{\parallel }\mathrm{\nabla }u(s){\mathrm{\parallel }}_{L^2}^{2}ds\le \frac{1}{2}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{L^2}^2+{\int }_0^t(f(s),u(s){)}_{L^2}ds\mathrm{.}$ The subsequential-limit theorem does not assert pressure regularity or a pressure-observation error estimate. The accumulated-pressure observation proposition on this page additionally requires the strong no-slip regularity $u\in W^{1,\mathrm{\infty }}(0,T;H^2(D{)}^d\cap V_{\sigma })$ and $p\in W^{1,\mathrm{\infty }}(0,T;H^1(D)\cap Q)$. Thus the weak-limit theorem alone does not supply that pressure estimate. This scope remark does not assert that a pressure is absent or rule out a separately proved pressure-primitive result.

solenoidal energy weak velocity

smooth MINI observation interface

MINI subsequential velocity-limit theorem

smooth accumulated-pressure observation bound