No-Slip Vanishing-Viscosity Limit in the Energy Norm

No-Slip Vanishing-Viscosity Limit in the Energy Norm

Definitions and mathematical statement

For fixed smooth divergence-free $u_0$ vanishing on $\partial\Omega$, let $u^\nu$ be any no-slip Leray--Hopf weak solution of Navier--Stokes and let $u$ be the smooth Euler solution with $u\cdot n=0$. A Leray--Hopf solution has finite-energy weak evolution and satisfies the energy inequality. For each interval $[0,T]$ of smooth Euler existence, the target is \[ \lim_{\nu\downarrow0}\operatorname*{ess\,sup}_{0\le t\le T} \|u^\nu(t)-u(t)\|_{L^2(\Omega)}=0. \] Both $d=2$ and $d=3$ are included. The same initial data are used at every viscosity, and convergence is asserted for every allowed weak-solution family, not a specially chosen subsequence.

Formulation and concept references: [S1].

Short English statement

Do all no-slip viscous flows converge in the uniform-in-time energy norm to the smooth inviscid flow as viscosity vanishes?

Research attempt

Current frontier.

Kato's criterion identifies the missing boundary estimate: in the fixed-data, smooth-Euler setting, the requested convergence is equivalent to \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-36-3a-6b-61-74-6f} D_\nu(c):=\nu\int_0^T\int_{d(x,\partial\Omega)<c\nu} |\nabla u^\nu|^2\,dx\,dt\longrightarrow0 \end{equation} for fixed $c>0$. It is an equivalence applicable to energy-inequality solutions, not a theorem that this expression always vanishes. The supplied survey states it as Theorem 3.7. Instability constructions using viscosity-dependent initial boundary layers do not automatically meet this question's single fixed datum; see the discussion after its Theorem 3.6. [S1]

Attack route.

Testing the viscous equation against Euler works if its full boundary trace remains zero. A boundary corrector works under (30006746:kato). Stronger integrability would force that condition if its viscosity loss is below an explicit threshold. We investigate the first and third routes, and test the proposed shortcut that initially no-slip Euler flow stays no-slip.

Attempt: the trace-preserving special case.

Assume, in addition to the actual hypotheses, that $u|_{\partial\Omega}=0$ for the whole smooth interval. A Leray--Hopf solution with the same initial value can then be tested weakly against $u$. Combining that identity, Euler energy conservation and the energy inequality for $u^\nu$ gives, for $w=u^\nu-u$, \[ \frac12\|w(t)\|_2^2+\nu\int_0^t\|\nabla u^\nu\|_2^2 \le\int_0^t\|\nabla u\|_\infty\|w\|_2^2 +\nu\int_0^t\langle\nabla u^\nu,\nabla u\rangle. \] Time approximation justifies the smooth time-dependent test; no uniqueness of $u^\nu$ is used. Young's inequality and Gronwall give \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-36-3a-74-72-61-63-65} \operatorname*{ess\,sup}_{t\le T}\|u^\nu(t)-u(t)\|_2^2 \le\nu\exp\!\left(2\int_0^T\|\nabla u\|_\infty dt\right) \int_0^T\|\nabla u\|_2^2dt. \end{equation} This proves a uniform $O(\sqrt\nu)$ energy-norm convergence estimate for this restricted class and every one of its allowed weak-solution families. It is a direct relative-energy calculation, not a claim of novelty.

A fixed smooth datum that immediately loses the full Euler trace.

In the unit disk set $s=x^2+y^2$, $\psi=x(1-s)^2$, and $u_0=(-\partial_y\psi,\partial_x\psi)$. Both components vanish on $s=1$, and $\nabla\cdot u_0=0$. The initial pressure solves \[ \Delta p_0=-\operatorname{tr}((Du_0)^2),\qquad \partial_n p_0=0 \quad(s=1). \] An explicit polynomial solution, up to an irrelevant constant, is \begin{align*} p_0={}&-\frac35s^4+\frac73s^3-\frac{10}3s^2+\frac{31}{15}s\\ &+x^2\left(\frac{16}5s^3-10s^2+\frac{32}3s-\frac{62}{15}\right). \end{align*} Differentiation verifies the equations exactly; the companion script checks them in ${\mathbb{Q}}[x,y]$. On the circle, \[ p_0=\frac15+\frac4{15}y^2,\qquad (-y\partial_x+x\partial_y)p_0=\frac8{15}xy. \] Since $(u_0\cdot\nabla)u_0=0$ at the boundary, Euler's equation gives \[ \partial_tu(0)\cdot(-y,x)=-\frac8{15}xy, \] which is not identically zero. Local smooth Euler flow from this polynomial datum therefore immediately loses its tangential zero trace, while keeping its normal trace zero. This is not an inviscid-limit counterexample; it refutes the extra trace-persistence lemma needed to apply (30006746:trace) universally.

A quantitative boundary-integrability reduction.

For $p>2$, the smooth tubular-neighborhood volume estimate and spatial Hölder inequality give \[ D_\nu(c)\le C_{\Omega,c,p}\nu^{2-2/p} \int_0^T\|\nabla u^\nu(t)\|_p^2dt. \] Hence the additional, uniform estimate \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-36-3a-6d-69-73-73-69-6e-67} \int_0^T\|\nabla u^\nu\|_p^2dt=O(\nu^{-\alpha}), \qquad \alpha<2-2/p, \end{equation} for every allowed family would prove the exact target by Kato's criterion. The strip $L^p$ norm would suffice. At $p=2$, the energy inequality gives $O(\nu^{-1})$, precisely the nondecaying threshold, not a strict improvement. Thus the available energy estimate cannot be substituted into this step.

Stress test.

A boundary layer with order-one velocity jump across thickness $\nu$ has squared $L^2$ mass of order $\nu$ but order-one contribution to (30006746:kato). This is a dimensional test, not a Navier--Stokes solution. Small layer mass therefore cannot by itself exclude boundary dissipation. The pressure calculation concerns actual fixed smooth data but says nothing about how much its viscous boundary layer dissipates. No unproved uniqueness or compactness of arbitrary three-dimensional weak solutions was used in (30006746:trace). The $p$-integrability in (30006746:missing) is an extra hypothesis, not a property deduced from the Leray--Hopf definition.

Outcome.

Outcome: Conditional reduction. The attempt gives a sufficient integrability threshold, a relative-energy proof when Euler retains no-slip, and an exact polynomial countertest to automatic trace persistence. For the full problem, one still needs Kato boundary dissipation, an estimate such as (30006746:missing), or a genuine fixed-data family violating convergence. None is established here.

Sources

• S1 Y. Maekawa and A. Mazzucato, The Inviscid Limit and Boundary Layers for Navier-Stokes Flows $2016$. https://arxiv.org/html/1610.05372. Catalogue formulation source.