PE06 instantaneous pressure refutations and retained primitive estimates

The original PE06 plan required two instantaneous pressure conclusions: a pressure-bearing subsequential weak solution with $p\in L^{4\mathrm{/}3}(0,T;L_0^2(D))$, and the pressure part of the smooth-branch velocity/pressure error estimate. The unchanged MINI backward Euler converges subsequentially to a solenoidal energy weak velocity proves the solenoidal velocity equation and its energy inequality; it does not state a pressure-bearing solution in the regular-pressure class Energy weak solutions of incompressible Navier–Stokes. The unchanged Smooth-branch velocity error for backward Euler MINI proves the velocity estimate only. The generic regular-pressure conclusion is now refuted under its exact original hypotheses by the square construction described in Remarks below; the independent smooth instantaneous rate is also refuted under its unchanged hypotheses by the graded mass-initialization construction in Remarks. Both original formulations are retained here as the claims refuted, rather than silently replaced by primitive estimates.

The additive Integrated pressure recovery and MINI pressure-primitive convergence now gives a unique mean-zero time-distribution pressure and bounded weakly continuous pressure primitive for the limiting velocity, plus uniformly bounded discrete primitives and weak convergence of reconstructed pressures as time distributions. It also proves the exact criterion: the regular $L_t^{4\mathrm{/}3}{L}_0^2$ pressure exists precisely when the actual full distributional derivative $u_t\in L^{4\mathrm{/}3}(0,T;V^{\ast })$. The current limit argument establishes the derivative only after restriction to $V_{\sigma }$. Extending that restricted functional by a divergence lift does not prove that the extension is the actual full derivative. The sufficient condition $u\in W^{1,4\mathrm{/}3}(0,T;L^2(D{)}^d)$ would imply the criterion, but this regularity is not supplied by the stated energy hypotheses. The primitive result does not establish it. The square counterexample proves that this extra regularity can actually fail for the original energy data, so the generic expectation cannot be supplied without a genuine additional hypothesis. The regular-pressure Definition remains a valid conditional solution class.

The additive First-order smooth pressure-primitive error for backward Euler MINI gives a first-order estimate for the accumulated pressure $P(t)={\int }_0^{t}p(s)ds$, its piecewise-affine discrete primitive, and the specified negative time norm. It does not bound the instantaneous $L_t^{4\mathrm{/}3}{L}_x^2$ pressure error. Applying the discrete inf-sup inequality to the unintegrated pressure equation leaves the full-test derivative residual $d_n={\sup }_{v_h\mathrm{\ne }0}\frac{\mathrm{\mid }(D_{\tau }{e}_n,v_h{)}_{L^2}\mathrm{\mid }}{\mathrm{\parallel }{v}_h{\mathrm{\parallel }}_V},\mathrm{\Delta }={\left({\sum }_n{\tau }_n{d}_n^{4\mathrm{/}3}\right)}^{3\mathrm{/}4}\mathrm{.}$ The nodal velocity estimate does not itself control this derivative norm uniformly as the smallest time step tends to zero, and the primitive estimate cannot be differentiated to supply it. The exact-scope smooth counterexample now shows that the original unrestricted instantaneous rate cannot be recovered by another argument under the unchanged hypotheses. This is a refutation of that universal rate, not a claim that every stronger initialization or more restricted mesh class fails.

The formerly proposed uniform $H^1$ bound for the $L^2$-orthogonal projector $P_{Z_h}:V_h\to Z_h$ is false. On the square, the continuous orthogonal projection of an explicit smooth no-slip velocity onto the Leray space has a nonzero tangential boundary trace, so it is not in $H_0^1$. If the proposed discrete bound held on a refining MINI family, composition with the actual uniformly $H^1$-stable Fortin maps and weak compactness would force that false continuous mapping property. The full-test and restricted-$V_h$ versions are therefore unavailable. The projector obstruction is a refutation of that proposed supplier only; it proves neither that the original pressure claims are impossible nor that another route cannot prove them.

Affected claims: the PE06 pressure-bearing regularity clause following MINI backward Euler converges subsequentially to a solenoidal energy weak velocity, including $p\in L^{4\mathrm{/}3}(0,T;L_0^2(D))$, and the instantaneous pressure-rate clause paired with Smooth-branch velocity error for backward Euler MINI.

Source limits: Süli–Tscherpel explicitly use a pressure-free weak formulation and discuss a distributional-in-time pressure expectation for their Lipschitz no-slip setting. Their problem has a more general stress law and a different approximation/limit route, so that statement is not a proof of the exact Newtonian, $L^2$-forcing PE06 claim or of its impossibility. Guermond and Boffi–Brezzi–Fortin supply static divergence lifting and mixed inf-sup context, not either missing evolutionary estimate.

Current decision: retain the original generic formulation and record its exact-scope refutation. Preserve the complete velocity and distributional/primitive pressure results and their exact full-derivative criterion. Retain the original smooth instantaneous formulation and its exact-scope counterexample; the primitive estimate is an independent positive result and does not substitute for that formulation. Do not classify either authoring obligation as an open problem or use the false projector route.

The contextual An energy Navier–Stokes velocity on the square without endpoint-integrable pressure ↗ gives an actual two-dimensional square no-slip trajectory with $u_0\in H_{\sigma }$ and $f\in L_t^2{L}_x^2$, including the nonlinear force, whose unique mean-zero pressure lies in $L_t^1{L}_x^2$ but not $L_t^{4\mathrm{/}3}{L}_x^2$. Its energy equality and pressure-free two-dimensional uniqueness identify every MINI velocity limit for those data. Thus it refutes the original generic pressure-bearing conclusion rather than merely a proposed proof route. It is contextual counterexample evidence, not a logical premise of this remark or of any pressure-recovery theorem. Its singular initial profile does not meet the original smooth-branch assumptions, so the smooth clause needs the independent construction below.

The contextual Graded MINI mass initialization can destroy the smooth instantaneous pressure rate ↗ refutes the separate instantaneous smooth pressure-error assertion on one fixed convex polygon and one fixed-shape family of arbitrarily graded actual MINI meshes. Its smooth exact data have uniformly bounded original norms, its initialization is the actual weak-kernel orthogonal mass projection, and its pressures arise from genuine backward Euler roots with arbitrarily small steps satisfying the original upper-step condition. The proof retains the actual symmetric-gradient operator, full nonlinear force, mean-zero gauge and arbitrary-root quantifier. It does not infer pressure growth from arbitrary fractional velocity roughness: a positive pressure observation for four specifically realized coarse sources and a finite real-amplitude nonlinear argument supply the actual pressure channel. This contextual generated counterexample is not a logical dependency of a positive theorem.