The inviscid limit problem for Navier-Stokes equations

A longstanding open problem is to establish the inviscid limit of classical solutions to the incompressible Navier-Stokes equations for smooth initial data on a domain with boundaries. The question is of great physical and mathematical interest, and it deeply links to the transition to turbulence in fluids that may possibly take place faster than expected due to the presence of a boundary. In this article, I shall give a quick overview of this subject, and then highlight some recent works with my former student, Trinh T. Nguyen, (currently a Van Vleck Assistant Professor at University of Wisconsin, Madison), whose main results establish the inviscid limit for smooth data that are only required to be analytic locally near the boundary. This may be the best possible type of positive results that one can hope for, given the known violent instabilities at the boundary, which I shall discuss below. Before getting on, this picture should already hint at the great delicacy in studying boundary layers (source internet):

1. The inviscid limit problem

Consider the classical Navier-Stokes equations with small viscosity ${\nu>0}$

$\displaystyle \partial _t u^\nu+u^\nu\cdot\nabla u^\nu+\nabla p^\nu=\nu\Delta u^\nu, \qquad \nabla\cdot u^\nu=0 \ \ \ \ \ (1)$

modeling the dynamics of an incompressible fluid in a spatial domain ${\Omega\subset \mathbb{R}^d}$ , ${d\ge 2}$ , with boundary ${\partial \Omega}$ on which we impose the no-slip condition

$\displaystyle u^\nu_{\vert_{\partial \Omega}}=0.\ \ \ \ \ (2)$

Of great interest is the asymptotic behavior of solutions to (1) in the small viscosity limit. One may take smooth initial data, say in the Sobolev spaces ${H^s}$ for ${s>d/2+1}$ , so that the existence of such a smooth solution is ensured at least locally in time (noting the maximal time of existence is independent of the small viscosity ${\nu}$ . In the two dimensional case ${d=2}$ , the solution exists and remains smooth globally in time). The question is how the solution behaves asymptotically in the limit ${\nu \rightarrow 0}$ . At the zero viscosity limit ${\nu=0}$ , the equations (1) formally reduce to the Euler equations

$\displaystyle \partial _tu^0+u^0\cdot\nabla u^0+\nabla p^0=0,\qquad \nabla\cdot u^0=0 \ \ \ \ \ (3)$

now with the non-penetration boundary condition ${u^0\cdot n=0}$ on ${\partial \Omega}$ (noting only the normal component of (2) is imposed, which is sufficient for the wellposedness of the equation (3)). The mismatch between the boundary conditions for Euler and Navier-Stokes equations will dynamically generate some transition layers near the boundary, or famously known as boundary layers, correcting possibly nonzero traces of Euler solutions to the zero velocity of Navier-Stokes solutions imposed by (2). This traditional belief comes from the fact that solutions to Navier-Stokes equations remain sufficiently smooth, and so ${\nu \Delta u^\nu}$ should be neglected in the interior of the fluid domain ${\Omega}$ and would only be dominant in a thin vanishing region close to the boundary. As a result (of this belief), one is tempted to prove

Conjecture 1 (Inviscid Limit Conjecture) For general smooth initial data, there is a positive time ${T}$ , independent of ${\nu}$ , so that solutions to Navier-Stokes equations converge strongly to that of Euler equations in ${L^\infty(0,T;L^2(\Omega))}$ in the inviscid limit of ${\nu \rightarrow 0}$ .

Note that since ${T}$ may be sufficiently small, solutions to both Euler and Navier-Stokes equations remain regular. The question is whether the presence of a boundary may cause any anomalous dissipation in an arbitrarily short time. Let ${v = u^\nu - u^0}$ be the difference between the solutions to Euler and Navier-Stokes equations (for simplicity, starting from the same initial data). Subtracting (3) from (1) leads to

$\displaystyle \partial_t v + u^\nu \cdot \nabla v + v\cdot \nabla u^0 + \nabla p = \nu \Delta u^\nu , \qquad \nabla \cdot v=0$

which gives the following fundamental energy identity (noting ${v\cdot n=0}$ on the boundary)

$\displaystyle \frac12 \frac{d}{dt}\| v\|^2_{L^2} = - \int_\Omega (v\cdot \nabla u^0)\cdot v \; dx + \nu \int_\Omega \Delta u^\nu \cdot (u^\nu - u^0) \; dx.$

Integrating by parts the last integral term and using the Hölder and then Young’s inequality, we get

$\displaystyle \frac{d}{dt}\| v\|^2_{L^2} + \nu \|\nabla u^\nu\|_{L^2}^2 \le 2 \| \nabla u^0\|_{L^\infty} \| v\|_{L^2}^2 + \nu \| \nabla u^0\|_{L^2}^2 - 2\nu \int_{\partial \Omega} \partial_n u^\nu \cdot u^0 \; d\sigma .\ \ \ \ \ (4)$

Since both Euler and Navier-Stokes equations start from the same initial data, we have ${v(0,x)=0}$ initially. If there were no boundary ${\partial \Omega =\emptyset}$ , the standard Grönwall’s inequality yields ${\| v(t)\|_{L^2} \le \sqrt{\nu} e^{\| \nabla u^0\|_{L^1_tL^\infty_x}} \| \nabla u^0\|_{L_{t,x}^2}}$ for all ${t \in [0,T]}$ , where ${T}$ is the maximal time of existence of the solutions to Euler equations (3). In particular, the inviscid limit trivially holds for classical solutions (note however for less regular data on the whole space, there remain some interesting open questions, see, e.g., this paper of mine with Trinh T. Nguyen).

In case of a boundary, the last boundary term in (4) no longer vanishes, since the tangential component of Euler velocity ${u^0}$ may not vanish on the boundary. In fact, since ${u^\nu =0}$ and ${u^0\cdot n=0}$ on the boundary, one observes that ${\partial_n u^\nu \cdot u^0 = (\omega^\nu \times n)_\tau \cdot u^0_\tau}$ , where ${u_\tau }$ denotes the tangential component of ${u}$ , and ${\omega^\nu = \nabla \times u^\nu}$ denotes the fluid vorticity. That is, again by the Grönwall’s inequality, the inviscid limit holds on a time interval ${[0,T]}$ if and only if

$\displaystyle \nu \int_0^T \int_{\partial \Omega} (\omega^\nu \times n)_\tau \cdot u_\tau^0 \; d\sigma dt \rightarrow 0\ \ \ \ \ (5)$

in the limit of ${\nu \rightarrow 0}$ (in the two dimensional case, ${(\omega^\nu \times n)_\tau}$ reduces to the scalar vorticity ${\omega^\nu = \nabla^\perp \cdot u^\nu}$ ). As the Euler solution ${u^0}$ remains regular, the inviscid limit holds iff ${\nu \omega^\nu \rightharpoonup 0}$ weakly in ${L^2([0,T]\times \partial\Omega)}$ . That is, one needs to quantitatively control the vorticity on the boundary. Equivalently, Kato in his celebrated work shows that the inviscid limit holds if and only if the energy dissipation term satisfies

$\displaystyle \nu \int_0^T\int_{\{d(x,\partial \Omega)\lesssim \nu\}} | \nabla u(t,x)|^2\;dxdt\rightarrow 0 \ \ \ \ \ (6)$

in the limit of ${\nu \rightarrow 0}$ . See also here in a previous blog post of mine where I discussed more details of the proof. Remarkably, for the inviscid limit to hold, one only needs to control vorticity in the vanishing region ${\{d(x,\partial \Omega)\lesssim \nu\}}$ near the boundary.

2. Boundary vorticity conjecture

As seen in the previous section, one needs to quantitatively control the vorticity on the boundary: namely, (5) or equivalently ${\nu \omega^\nu \rightharpoonup 0}$ on ${(0,T)\times \partial \Omega}$ in the inviscid limit. This is however notoriously difficult to prove or disprove due to the extremely rich underlying physics at the boundary. To complicate the story, let me propose another conjecture heading in the opposite direction of the traditional belief laid out in the previous section.

Conjecture 2 (Boundary vorticity conjecture) There are ${C^\infty}$ smooth initial data ${u_0}$ so that the Navier-Stokes vorticity ${\omega^\nu}$ satisfies for any positive ${T}$ ,

$\displaystyle \liminf_{\nu \rightarrow 0}\sup_{t\in [0,T]} \|\nu t \omega^\nu(t)\|_{L^\infty(\partial \Omega)} >0.$

The weight in time is simply to account for the parabolic singularity at ${t=0}$ . The conjecture asserts that within any short time the vorticity ${\omega^\nu(t)}$ becomes significantly large of order ${\nu^{-1}}$ (as will be seen below, this is much larger than expected from the classical Prandtl’s boundary layer theory; see also a previous blog article of mine). The conjecture is widely open. The failure of Conjecture 2 would imply the validity of the inviscid limit, Conjecture 1, however its validity would indicate that the inviscid limit is unlikely to hold.

Let us start with the simple picture proposed by L. Prandtl in his classical work asserting that solutions to Navier-Stokes equations can be approximated by an Euler solution, plus a boundary layer whose thickness is of order ${\sqrt \nu}$ (namely, the boundary region of relevance is where ${\nu \Delta u^\nu \sim 1}$ ), leading to his famous Prandtl’s boundary layer theory. The Prandtl’s theory has a great practical advantage of approximating the real fluid flows by a nonlocal scalar equation with a known pressure law that has explicit self similar solutions. The idea also forms the basic principle of the singular perturbation theory or the multi-scale asymptotic analysis. Mathematically speaking, the well-posedness theory of the Prandtl boundary layer equations is well established: the Cauchy problem is well-posed in a monotonic setting (i.e. vorticity must have a definite sign, e.g. Oleink’s works), else the problem is dynamically ill-posed, e.g., Gerard-Varet and Dormy’s important paper on JAMS 2010. The Prandtl’s asymptotic analysis of Navier-Stokes flows, if valid, would in particular yield the inviscid limit (i.e. Conjecture 1), since the vorticity of the Prandtl’s boundary layers is of order ${\nu^{-1/2}}$ , which verifies (5).

In their celebrated work, Sammartino-Caflisch proved the Prandtl’s theory for data that are analytic. The results were extended by Gérard-Varet-Maekawa-Masmoudi for data that have Gevrey regularity and are close to some concave boundary layer Ansatz. The theory is however false in general for ${C^\infty}$ smooth initial data. Indeed, in a recent work of mine with E. Grenier, we constructed an asymptotic solution to the Navier-Stokes problem that involves three scalings: one for Euler solutions, one for the classical Prandtl’s boundary layers, and yet another sublayer whose thickness is of order ${\nu^{3/4}}$ , much smaller than the classical one of order ${\nu^{1/2}}$ predicted by L. Prandtl. This boundary sublayer is proven to form dynamically and grow reaching to order one in its amplitude independent of the smallness of ${\nu}$ (Grenier-Toan, Annals of PDEs 2019). In particular, the vorticity of this boundary sublayer is of order ${\nu^{-3/4}}$ , much larger than the size predicted by the Prandtl’s theory, however (5) is still verified.

The formation of boundary sublayers not only invalidates the classical boundary layer theory, but also reveals the rich underlying structure of fluids near the boundary in the inviscid limit. Indeed, one may further compute the local Reynolds number of this first boundary sublayer ${Re_1 = \frac{UL}{\nu} = \nu^{-1/4}U_{sub}}$ , recalling ${L = \nu^{3/4}}$ is the thickness of the sublayer and ${U_{sub}}$ is the amplitude of the velocity field of the sublayer. Since ${U_{sub}}$ is proven to reach to order one, ${Re_1 \sim \nu^{-1/4}}$ remains sufficiently large and therefore the sublayer itself is spectrally unstable. This latter instability lies at the heart of the hydrodynamics stability theory in fluid dynamics: namely, all physical shear flows are spectrally unstable at a sufficiently high Reynolds number, including those monotone shear flows that are stable to Euler equations. This is known as the Heisenberg’s viscous destabilization – adding viscosity will destabilize the flow. The complete proof of this linear instability theory is given in my earlier joint work, Grenier-Toan-Guo’s paper, while the nonlinear instability is established in this paper by Grenier and myself. In particular, the instability gives rise to yet another smaller sublayer, now of order ${\nu^{7/8}}$ , thinner than the first sublayer of order ${\nu^{3/4}}$ and much thinner than the classical Prandtl’s layer thickness of order ${\nu^{1/2}}$ . This smaller scaled boundary sublayer would give a much larger vorticity, at order ${\nu^{-7/8}}$ near the boundary, provided that the instability can continue to grow and reach to order one in its amplitude. The order one instability of stable profiles remains open (i.e. whether Grenier-Toan’s result for unstable profiles remains valid for stable profiles), and therefore the possible cascade to the thinest boundary layer of thickness of order ${\nu}$ is widely open. This rich underlying physics is the main obstruction in establishing the inviscid limit of Navier-Stokes flows in the presence of a boundary.

3. State of the art

I shall now focus on the positive direction towards proving the Conjecture 1. First, the classical work by Asano and Sammartino-Caflisch on the boundary layer expansion for analytic data in particular establishes the inviscid limit for such an analytic data. Then, in his remarkable work, Maekawa was able to establish the boundary layer expansion (again in particular, the inviscid limit) for Sobolev data whose vorticity is compactly supported away from the boundary (i.e. initial vorticity vanishes in a vicinity of the boundary). These works rely on the matched asymptotic expansion analysis with Euler flows in the interior and Prandtl’s boundary layers near the boundary.

Together with Trinh T. Nguyen, we revived this research direction by giving a direct proof of the inviscid limit for analytic data. Unlike the previous classical works, this direct proof formulates the problem via a boundary vorticity formulation and treats the nonlinear problem as a perturbation of the linear Stokes problem. As a byproduct, we give a detailed linear analysis of the Stokes problem in the small viscosity regime which turns out to be useful in other contexts as well (e.g., see the recent nice article by Jang and Kim on the hydrodynamics limit from Boltzmann equations). In a recent work, Kukavica, Vicol and Wang was able to extend this work of ours and that of Maekawa to include initial data that are only analytic near the boundary and Sobolev regular in the interior. All these mentioned results are on the half-space.

Most recently, building upon the recent advances, including the boundary vorticity formulation, the analyticity estimates on the Green function for the linear Stokes problem, and the Sobolev-analytic iterative scheme, we are able to establish the inviscid limit for data on curved bounded domains, in a joint work with C. Bardos, E. Titi and Trinh himself, and for data on an exterior circular domain with Trinh. To my knowledge, these works are the first to deal with curved domains. Incidentally, the result on an exterior circular domain was announced 25 years ago, however the proof appears missing in the literature. The extension to an exterior domain faces a fundamental difficulty that the Stokes semigroup may not be contractive in analytic spaces as was the case on the half-space, and the issue was circumvented in my recent paper with Trinh.

The main results in this positive direction are roughly as follows.

Theorem 3 For any initial data in ${H^3}$ Sobolev spaces whose vorticity is locally analytic near the boundary, there is a positive time ${T}$ , independent of ${\nu}$ , so that the unique solution ${u^\nu(t)}$ to the Navier-Stokes problem (1)–(2) exists on ${[0,T]}$ and has vorticity ${\omega^\nu(t)}$ satisfying

$\displaystyle \|\omega^\nu(t)\|_{L^\infty(\partial \Omega)} \le C_0(\nu t)^{-1/2} . \ \ \ \ \ (7)$

Of course, as discussed, the inviscid limit is a direct consequence of (7) and (5): namely, in the inviscid limit as ${\nu \rightarrow 0}$ , the Navier-Stokes solutions ${u^\nu}$ converge strongly in ${L^\infty([0,T];L^p(\Omega))}$ , for any ${2\le p<\infty}$ , to the corresponding solution ${u}$ of the Euler equations (3) with the same initial data. The difficulty is to prove the vorticity bound (7), which is only possible for data that are locally analytic (else much larger for Sobolev data as discussed). The theorem is the current best result towards establishing the Conjecture 1, and in fact probably the best possible type of the positive results in the inviscid limit that one can get, see the discussion in the previous section. Here, the locally analytic data are in the following sense.

Definition 4 Let ${\delta_0>0}$ and ${p\ge 1}$ . An ${L^p}$ function ${f(z)}$ defined on ${\{z\ge 0\}}$ is said to be locally analytic near the boundary ${z=0}$ if it can be extended analytically to the pencil-like complex domain

$\displaystyle \begin{aligned} R_{\rho}=& \{z\in \mathbb{C}:\quad 0\le {\mathbb R}e z\le \delta_0,\quad |\Im z|\le \rho {\mathbb R}e z \} \end{aligned}$

for some positive analyticity radius ${\rho}$ with a finite norm ${\|f\|_{L ^p_\rho}=\sup_{0 \le \eta <\rho}\| f \|_{L^p(\partial R_\eta)} }$ .

Note that a locally near boundary analytic function needs not to be analytic on the boundary, but only has bounded derivatives ${z\partial_z}$ . For general bounded domains ${\Omega\subset \mathbb{R}^2}$ , we assume that

${\Omega}$ is a simply-connected bounded domain in ${\mathbb{R}^2}$ whose boundary ${\partial\Omega}$ is an analytic curve, defined by an analytic map: ${\theta \in \mathbb{T} \mapsto x(\theta) =(x_1(\theta),x_2(\theta) ) \in \partial\Omega\,.}$

The analyticity of the boundary naturally extends to an analytic map which maps the near-boundary part of the domain ${\{x\in \Omega: \,\,d(x,\partial \Omega)<\delta\}}$ to the case of half-plane ${(\theta,z)\in \mathbb{T}\times (0,\delta)}$ , where ${z}$ is the distance function from the boundary. Therefore, by definition, a function is said to be locally analytic near the boundary ${\partial\Omega}$ , if it is locally analytic in ${z}$ and analytic in ${\theta}$ (for each ${|z|\le \delta_0}$ ). Here, for sake of presentation, we have chosen to consider the case of simply-connected domain ${\Omega}$ . The results of this paper apply to the general setting of multi-connected bounded domains whose boundaries consist of closed analytic curves, i.e., including domains with holes.

4. Sketch of the proof

The proof of Theorem 3 relies on several important ideas. I only sketch a few details, and refer the readers to the paper with C. Bardos, E. Titi and Trinh as well as the paper with Trinh for the complete proof.

4.1. Global geodesic coordinates

First, we introduce well adapted coordinates of ${\Omega}$ to effectively localize the problem near the boundary and reduce the analysis to that of the half-space problem (recalling the main issue of the inviscid limit problem lies at the boundary). For this, for each ${x = (x_1,x_2) \in \partial \Omega}$ , we parametrize it by

$\displaystyle \theta \in \mathbb{T} \mapsto x(\theta) =(x_1(\theta),x_2(\theta) ) \in \partial\Omega$

which, being global, preserves the analyticity hypothesis. Let ${\vec \tau(\theta) }$ and ${\vec n(\theta) }$ be the unit tangent and interior normal vectors at the boundary. We then extend it to a neighborhood of ${\partial \Omega}$ through the mapping

$\displaystyle (\theta,z)\mapsto X(\theta,z) = x(\theta) + z\vec n( x(\theta)), \ \ \ \ \ (8)$

which defines a global ${C^2}$ diffeomorphisme of ${\mathbb{T}\times [-\delta ,\delta ]}$ on ${\{ d(x,\partial\Omega)< {\delta}\}}$ for sufficiently small ${\delta}$ . We also denote by ${\gamma(\theta)}$ the curvature of the boundary ${\partial \Omega}$ . It then follows by a direct calculation that

$\displaystyle \nabla_X = \nabla_{\theta,z} + \mathcal{O}_1(\gamma ), \qquad \Delta_X = \Delta_{\theta,z} + \mathcal{O}_2(\gamma)$

where ${\mathcal{O}_j(\gamma)}$ are of order ${\gamma}$ , but may contain ${\partial_\theta^j}$ derivatives (of course, a precise form of the above identities is easily derived and done in the paper, but I skip for sake of presentation). Since we are dealing with the issue near the boundary, we may treat the problem as a flat boundary ${\{ z\ge 0\}}$ by making use of the smallness of ${\gamma}$ , treating ${\mathcal{O}_1(\gamma), \mathcal{O}_2(\gamma)}$ as a perturbation. It’s crucial to note that the errors contain only top derivatives in ${\theta}$ , but not in ${z}$ .

4.2. Boundary vorticity formulation

The next important idea is to work with the boundary vorticity formulation (for simplicity, we work only work with the two dimensional case). Precisely, let ${\omega = \nabla^\bot \cdot u}$ be the fluid vorticity. Then, the Navier-Stokes problem is equivalently to solving

$\displaystyle \begin{aligned} \partial_{t}\omega+ u\cdot\nabla\omega&=\nu\Delta\omega, \end{aligned} \ \ \ \ \ (9)$

together with the boundary condition for the vorticity

$\displaystyle \nu (\partial_n + DN)\omega_{\vert_{\partial \Omega}} = [\partial_n \Delta^{-1} ( u \cdot \nabla \omega)] _{\vert_{\partial \Omega}}, \ \ \ \ \ (10)$

in which the fluid velocity is recovered through the Biot-Savart law ${u = \nabla^\perp \Delta^{-1} \omega}$ , and ${DN}$ denotes the Dirichlet-Neumann operator on ${\Omega}$ (defined just below). The equation (9) is obtained by taking the curl of (1). We shall now derive (10). Here, note that ${\Delta^{-1}}$ is understood as the inverse of Laplace operator with the zero Dirichlet boundary condition, which in particular ensures that ${u\cdot n =0}$ on the boundary by construction. Now, to ensure the full no-slip boundary condition, i.e., that ${u\cdot \tau=0}$ on the boundary ${\partial \Omega}$ , where ${\tau}$ in the unit tangent vector to the boundary, we first require that the initial data satisfy the no-slip boundary condition (2), and then we impose in addition that ${\partial_t u\cdot \tau= 0}$ on the boundary, ${\partial \Omega}$ , for all positive time. This leads to the boundary condition

$\displaystyle 0 =\tau\cdot \partial_t u = \tau\cdot \nabla^\perp \Delta^{-1} \partial_t \omega = \partial_n [\Delta^{-1} (\nu \Delta \omega - u \cdot \nabla \omega)] \ \ \ \ \ (11)$

on the boundary. Introduce ${\omega^*}$ to be the solution of the nonhomogeneous Dirichlet boundary-value problem

$\displaystyle \left\{ \begin{aligned}\Delta \omega^* &= 0, \qquad \mbox{in}\quad \Omega \\ {\omega^*} &=\omega, \qquad \mbox{on }\quad \partial \Omega. \end{aligned} \right. \ \ \ \ \ (12)$

and define the Dirichlet-Neumann operator by

$\displaystyle DN \omega = -\partial_n \omega^*, \qquad \mbox{on}\quad \partial \Omega, \ \ \ \ \ (13)$

where ${\omega^*}$ solves (12). Observe that ${\partial_n [\Delta^{-1}\Delta \omega] =\partial_n [\Delta^{-1}\Delta (\omega - \omega^*)]= (\partial_n + DN)\omega}$ , which gives (10).

The boundary vorticity formulation (9)–(10) plays an important role in our analysis. Among other things, it reveals rather precisely the localization structure of boundary layers and their interaction with the interior vorticity.

4.3. Near boundary analytic spaces

Next, we introduce the near boundary analytic norm used to control the vorticity that is analytic near the boundary, but however only has Sobolev regularity away from the boundary. Precisely, let ${\delta_0>0}$ be the size of the analytic domain for our solution near the boundary. Fix ${\rho_0\ge \delta_0}$ , and take ${\rho\in(0,\rho_0)}$ . We define the complex domain

$\displaystyle \begin{aligned} \Omega_{\rho}=& \{y\in \mathbb{C}:\quad 0\le {\mathbb R}e y\le \delta_0,\quad |\Im y|\le \rho{\mathbb R}e y\}\\ &\bigcup \{y\in \mathbb{C}:\quad \delta_0\le {\mathbb R}e y\le \delta_0+\rho,\quad |\Im y|\le \delta_0+\rho-{\mathbb R}e y\}. \end{aligned}$

We note that the domain ${\Omega_\rho}$ only contains ${y}$ with ${0\le {\mathbb R}e y\le \delta_0+\rho}$ . For a complex valued function ${f}$ defined on ${\Omega_\rho}$ , let

$\displaystyle \|f\|_{L ^1_\rho}=\sup_{0 \le \eta <\rho}\| f \|_{L^1(\partial \Omega_\eta)} , \qquad \|f\|_{L ^\infty_\rho}=\sup_{0 \le \eta <\rho}\| f \|_{L^\infty(\partial \Omega_\eta)}$

where the integration is taken over the two directed paths along the boundary of the domain ${\Omega_{\eta}}$ . Now for an analytic function ${f(x,y)}$ defined on ${(x,y)\in \mathbb{T}_{2\pi}\times \Omega_\rho}$ , we define

$\displaystyle \begin{aligned} \|f\|_{\mathcal L^1_\rho} &=\sum_{\alpha\in \lambda \mathbb{Z}} \|e^{\epsilon _0(\delta_0+\rho- {\mathbb R}e y)|\alpha|}f_\alpha \|_{L^1_\rho} , \\ \|f\|_{\mathcal L^\infty_\rho} &=\sum_{\alpha\in \lambda \mathbb{Z}}\|e^{\epsilon _0(\delta_0+\rho- {\mathbb R}e y)|\alpha|}f_\alpha\|_{L^\infty_\rho}. \end{aligned} \ \ \ \ \ (14)$

The function spaces ${\mathcal{L}^1_\rho}$ and ${\mathcal{L}_\rho^\infty}$ are to control the scaled vorticity and velocity, respectively. We stress that the analyticity weight is identically zero on ${{\mathbb R}e y \ge \delta_0+\rho}$ . The advantage of working with analytic norms is that it controls all the derivatives. In particular, we have the following simple algebra

$\displaystyle \| fg \|_{\mathcal{L}^1_\rho} \le \| f\|_{\mathcal{L}^\infty_\rho } \| g\|_{\mathcal{L}^1_\rho} \ \ \ \ \ (15)$

and for any ${0<\rho'<\rho}$ ,

$\displaystyle \| \partial_{x} f \|_{\mathcal{L}^1_{\rho'}} + \| y \partial_{y} f\|_{\mathcal{L}^1_{\rho'}} \lesssim \frac{1}{\rho - \rho'} \| f\|_{\mathcal{L}^1_\rho}. \ \ \ \ \ (16)$

Note that the last estimate is crucial in controlling the loss of derivatives. We next need to check that the analytic norms are compatible with both the Biot-Savart law and the linear Stokes problem.

4.4. Biot-Savart law

We need to study how the local analytic norms commute with the Biot-Savart law ${u = \nabla^\perp \Delta^{-1} \omega}$ . Precisely,

Proposition 5 The velocity field ${u= \nabla^\perp \Delta^{-1} \omega}$ satisfies

$\displaystyle \begin{aligned} \|u\|_{\mathcal{L}^\infty_\rho } & \lesssim \| \omega\|_{\mathcal{L}^1_\rho} + \|\omega\|_{H^1(\{z\ge \delta_0+\rho\})} \\ \|\frac{1}{ z} u_2 \|_{\mathcal{L}^\infty_\rho } &\lesssim \| \omega\|_{\mathcal{L}^1_\rho} + \|\partial_{ \theta}\omega\|_{\mathcal{L}^1_\rho} + \|\omega\|_{H^1(\{z\ge \delta_0 + \rho\})} \end{aligned} \ \ \ \ \ (17)$

for ${k\ge 0}$ .

Proof: We study the elliptic problem ${\Delta \phi = \omega}$ , which in Fourier reads ${(\partial_z^2 -\alpha^2) \phi_\alpha = \omega_\alpha}$ for the Fourier transform ${\phi_\alpha}$ . The solution is explicitly given by

$\displaystyle \phi_\alpha( z)=\int_{0}^\infty G(y, z) \omega_\alpha(y)dy$

with the Green function given by ${G(y,z) = -\frac{1}{2\alpha} ( e^{-\alpha |y-z| } - e^{-\alpha (y+z)} ), }$ which is also valid for complex variables ${y,z}$ . In particular, we have ${ |G(y,z)|\le \alpha^{-1} e^{-\alpha |y-z|} .}$ This proves that

$\displaystyle \begin{aligned} | \phi_\alpha( z)| &\le \int_{\partial \Omega_\theta} \alpha^{-1} e^{-\alpha |y- z|}|\omega_\alpha(y)| |dy|. \end{aligned}\ \ \ \ \ (18)$

By definition of ${\mathcal{L}^1_\rho}$ norm, we only need to consider the case when ${0\le {\mathbb R}e z \le \delta_0 + \rho}$ . Now, for ${0\le {\mathbb R}e y \le \delta_0 + \rho}$ , we bound

$\displaystyle e^{-\alpha |{\mathbb R}e y- {\mathbb R}e z|} e^{-\epsilon _0(\delta_0+\rho- {\mathbb R}e y)\alpha} \le e^{-\epsilon_0(\delta_0+\rho- {\mathbb R}e z)\alpha} e^{-(1-\epsilon_0)\alpha |{\mathbb R}e y -{\mathbb R}e z|}$

noting ${\epsilon_0 \le 1/2}$ . On the other hand, for ${{\mathbb R}e y \ge \delta_0 + \rho}$ (recalling ${\delta_0 + \rho \ge {\mathbb R}e z}$ ), we bound

$\displaystyle e^{-\alpha |{\mathbb R}e y- {\mathbb R}e z|} \le e^{- \epsilon_0(\delta_0+\rho- {\mathbb R}e z) \alpha } e^{- (1-\epsilon_0)\alpha |{\mathbb R}e y- {\mathbb R}e z|}.$

Therefore, we bound

$\displaystyle \begin{aligned} \int_{{\mathbb R}e y \le \delta_0 + \rho} \alpha^{-1} e^{-\alpha |y- z|} | \omega_\alpha(y)| |dy| &\lesssim \alpha^{-1} e^{-\epsilon _0(\delta_0+\rho- {\mathbb R}e z)\alpha} \| e^{\epsilon _0(\delta_0+\rho- {\mathbb R}e y)\alpha}\omega_\alpha\|_{L^1_\rho}, \\ \int_{{\mathbb R}e y \ge \delta_0 + \rho} \alpha^{-1} e^{-\alpha |y- z|} | \omega_\alpha(y)| |dy| &\lesssim \alpha^{-3/2} e^{-\epsilon _0(\delta_0+\rho- {\mathbb R}e z)\alpha} \| \omega_\alpha\|_{L^2(y \ge \delta_0 + \rho)}. \end{aligned}$

Similarly, we also have

$\displaystyle \begin{aligned} \int_{{\mathbb R}e y \le \delta_0 + \rho} \alpha^{-1} e^{-\alpha |y- z|} | \omega_\alpha(y)| |dy| &\lesssim \alpha^{-2} e^{-\epsilon _0(\delta_0+\rho- {\mathbb R}e z)\alpha} \| e^{\epsilon _0(\delta_0+\rho- {\mathbb R}e y)\alpha}\omega_\alpha\|_{L^\infty_\rho}, \end{aligned}$

which gains an extra factor of ${\alpha}$ . This proves

$\displaystyle \begin{aligned} \| e^{\epsilon _0(\delta_0+\rho- {\mathbb R}e z)\alpha}(\alpha,\partial_z) \phi_\alpha\|_{L^\infty_\rho} &\le \| e^{\epsilon _0(\delta_0+\rho- {\mathbb R}e y)\alpha}\omega_\alpha\|_{L^1_\rho} + \alpha^{-1/2}\| \omega_\alpha\|_{L^2(y \ge \delta_0 + \rho)} . \end{aligned}$

Taking the summation in ${\alpha \in \mathcal{Z}}$ yields the first estimate in (17). As for the second, we use the fact that the Green function vanishes on the boundary ${z=0}$ , and so ${ |G(y,z)|\le z e^{-\alpha |y-z|} .}$ The proposition follows. $\Box$

4.5. The Stokes problem in the half-plane

The next important advance is the precise pointwise estimates on the semigroup of the linear Stokes problem on the half-space. These estimates play a crucial role in propagating the locally analytic norm of the solutions. This was first derived in my earlier work with Trinh T. Nguyen. Precisely, we study the linear Stokes problem with the boundary vorticity formulation,

$\displaystyle \begin{aligned} (\partial_t-\nu \Delta_{\theta,z} ) \omega &=0 \\ \nu (\partial_z + |\partial_{\theta}|)\omega_{\vert_{z=0}} &=0 \end{aligned}\ \ \ \ \ (19)$

on ${\mathbb{T} \times \mathbb{R}_+}$ . Denote by ${e^{\nu t S}}$ the Stokes semigroup of (19). Taking the Fourier transform of the semigroup ${e^{\nu t S}}$ in variable ${\theta}$ , we obtain the integral representation of the semigroup

$\displaystyle (e^{\nu t S} f)_\alpha (z) = \int_0^\infty G_\alpha(t,y;z) f_{\alpha} (y) \; dy \ \ \ \ \ (20)$

for each Fourier variable ${\alpha \in \mathcal{Z}}$ , where ${f_\alpha}$ denotes the Fourier transform of ${f(x,y)}$ in variable ${x}$ , and ${G_\alpha(t,y;z)}$ is the corresponding Green function. One of the main results from our ARMA paper are the precise estimates on the Green function, namely

$\displaystyle G_\alpha(t,y;z) = H_\alpha(t,y;z) + R_\alpha (t,y;z), \ \ \ \ \ (21)$

where

$\displaystyle \begin{aligned} H_\alpha(t,y;z) & = \frac{1}{\sqrt{\nu t}} ( e^{-\frac{|y-z|^{2}}{4\nu t}} + e^{-\frac{|y+z|^{2}}{4\nu t}} ) e^{-\alpha^{2}\nu t}, \\ | R_\alpha (t,y;z)| &\lesssim \mu_f e^{-\theta_0\mu_f |y+z|} + (\nu t)^{-\frac{1}{2}}e^{-\theta_0\frac{|y+z|^{2}}{\nu t}} e^{-\frac18 \alpha^{2}\nu t} , \end{aligned}$

for ${y,z\ge 0}$ , and for some ${\theta_0>0}$ and for ${\mu_f = |\alpha| + \frac{1}{\sqrt \nu}}$ . In particular, ${\|G_\alpha(t,y;\cdot)\|_{L^1_\rho} \lesssim 1}$ , for each fixed ${y,t}$ . The result asserts that the Green function of the linear vorticity is exactly that of the heat kernel with the Neumann boundary condition, plus a boundary layer contribution, namely ${\mu_f e^{-\theta_0\mu_f |y+z|} }$ . Note that this is stationary and exponentially localized near the boundary, a property that plays a crucial role in studying the localization of boundary layers for analytic data. Using the pointwise estimates, we easily obtain the propagation of the local analyticity for the semigroup.

Proposition 6 Then, for any ${t\ge 0}$ and ${\rho>0}$ , there hold

$\displaystyle \begin{aligned} \| e^{\nu t S} f\|_{\mathcal{L}^{1}_\rho} &\le C_0 \| f\|_{\mathcal{L}^1_\rho} + \| z f\|_{H^1(z\ge \delta_0 + \rho)} \end{aligned}\ \ \ \ \ (22)$

uniformly in the inviscid limit.

Proof: The proof is now direct, using the integral formulation (20). Indeed, for ${z, y \le \delta_0 + \rho}$ , we note that

$\displaystyle \begin{aligned} e^{- a|y\pm z|} e^{-\epsilon_0 (\delta_0 + \rho- y) |\alpha|} &= e^{-a|y\pm z| + \epsilon_0 |\alpha| (y - z)} e^{-\epsilon_0 (\delta_0 + \rho- z) |\alpha|} \\ &\le e^{- (a - \epsilon_0 |\alpha|) |y\pm z| } e^{-\epsilon_0 (\delta_0 + \rho- z) |\alpha|} \end{aligned} \ \ \ \ \ (23)$

for any real number ${a}$ and for ${\epsilon_0}$ sufficiently small. Taking ${a = \frac12 \theta_0 \mu_f}$ , we have ${a \ge \epsilon_0 |\alpha| }$ and so

$\displaystyle e^{-\theta_0\mu_f |y+z|}e^{-\epsilon_0 (\delta_0 + \rho- y) |\alpha|} \le e^{-\epsilon_0 (\delta_0 + \rho- z) |\alpha|} e^{- \frac12\theta_0\mu_f |y+z|}$

On the other hand, taking ${a =\frac12\theta_0 \frac{|y\pm z|}{\nu t}}$ in (23), we have either ${a \ge \epsilon_0 |\alpha|}$ or ${\frac12 \theta_0 \alpha^2 \nu t \ge \epsilon_0 |\alpha| |y\pm z|}$ . Therefore, we have

$\displaystyle e^{-\theta_0\frac{|y+z|^{2}}{\nu t}} e^{-\theta_0 \alpha^{2}\nu t} e^{-\epsilon_0 (\delta_0 + \rho- y) |\alpha|} \le e^{-\frac12 \theta_0\frac{|y+z|^{2}}{\nu t}} e^{-\epsilon_0 (\delta_0 + \rho- z) |\alpha|} .$

This proves that for ${z\le \delta_0 + \rho}$ ,

$\displaystyle \begin{aligned} &e^{\epsilon_0 (\delta_0 + \rho- z) |\alpha|} \int_0^{\delta_0 + \rho} |G_\alpha(t,y;z) f_{\alpha} (y)| \; dy \le \int_0^{\delta_0 + \rho} [ (\nu t)^{-\frac{1}{2}}e^{- \frac12\theta_0\frac{|y\pm z|^{2}}{\nu t}} + \mu_f e^{- \frac12\theta_0\mu_f |y+z|} ] |e^{\epsilon_0 (\delta_0 + \rho- y) |\alpha|} f_{\alpha} (y) |\; dy . \end{aligned}$

Since the term in the bracket is bounded in ${L^1_z}$ norm, we have

$\displaystyle \begin{aligned} \|e^{\epsilon_0 (\delta_0 + \rho- z) |\alpha|} \int_0^{\delta_0 + \rho} G_\alpha(t,y;z) f_{\alpha} (y) \; dy\|_{\mathcal{L}^1_\rho} \lesssim \|e^{\epsilon_0 (\delta_0 + \rho- y) |\alpha|} f_{\alpha} \|_{\mathcal{L}^1_\rho} . \end{aligned}$

Taking the summation in ${\alpha}$ yields the stated bounds for this term.

Next, consider the case when ${y\ge \delta_0 + \rho \ge z}$ . In this case, we simply use

$\displaystyle e^{- \epsilon_0 |\alpha| |y-z|} \le e^{-\epsilon_0 |\alpha| (\delta_0 + \rho - z)},$

giving the right analyticity weight in ${z}$ . The control of the weight ${e^{\epsilon_0 |\alpha| |y-z|} }$ is done exactly as above, yielding

$\displaystyle \begin{aligned} &e^{\epsilon_0 (\delta_0 + \rho- z) |\alpha|} \int_{\delta_0 + \rho}^\infty |G_\alpha(t,y;z) f_{\alpha} (y)| \; dy \le \int_{\delta_0 + \rho}^\infty [ (\nu t)^{-\frac{1}{2}}e^{- \frac12\theta_0\frac{|y\pm z|^{2}}{\nu t}} + \mu_f e^{- \frac12\theta_0\mu_f |y+z|} ] | f_{\alpha} (y) |\; dy . \end{aligned}$

Therefore,

$\displaystyle \begin{aligned} \sum_\alpha \| e^{\epsilon_0 (\delta_0 + \rho- z) |\alpha|} \int_{\delta_0 + \rho}^\infty |G_\alpha(t,y;z) f_{\alpha} (y)| \; dy \|_{\mathcal{L}^1_\rho} &\lesssim \sum_\alpha \| f_{\alpha} \|_{L^1(z\ge \delta_0 + \rho)} \\ &\lesssim \| z f \|_{H^1(z\ge \delta_0 + \rho)}. \end{aligned}$

The proposition follows. $\Box$

4.6. Sobolev-analytic iterative nonlinear scheme

Finally, we close the nonlinear problem via a Sobolev-analytic iterative scheme. We construct the solutions to the Navier-Stokes equation via the vorticity formulation

$\displaystyle \begin{aligned} \partial_{t}\omega + u\cdot\nabla\omega =\nu\Delta\omega \end{aligned}\ \ \ \ \ (24)$

together with the nonlocal vorticity boundary condition (10). To localize the problem near the boundary, we introduce the smooth cutoff function ${\phi^b}$ that is equal to one near the boundary and vanishes away from the boundary, and write

$\displaystyle \omega = \omega^b + \omega^i, \qquad \omega^b = \phi^b \omega, \qquad \omega^i = (1-\phi^b)\omega.\ \ \ \ \ (25)$

We also define the corresponding velocity field through the Biot-Savart law

$\displaystyle u = u^b + u^i, \qquad u^b = \nabla^\perp \Delta^{-1} \omega^b, \qquad u^i = \nabla^\perp \Delta^{-1} \omega^i. \ \ \ \ \ (26)$

This yields

$\displaystyle \left\{ \begin{aligned} \partial_t \omega^b + u\cdot\nabla \omega^b &= \nu\Delta \omega^b \\ \nu (\partial_n + DN)\omega^b_{\vert_{\partial \Omega}} &= [\partial_n \Delta^{-1} ( u \cdot \nabla \omega)] _{\vert_{\partial \Omega}} \end{aligned}\right. \ \ \ \ \ (27)$

for the vorticity near the boundary, and

$\displaystyle \left\{ \begin{aligned} \partial_t \omega^i + u\cdot \nabla \omega^i &=\nu\Delta \omega^i \\ \omega^i_{\vert{\partial\Omega}} & = 0 \end{aligned}\right. \ \ \ \ \ (28)$

for the vorticity away from the boundary. Here, we note that the boundary condition on ${\omega^i}$ follows directly from the definition (25), while the boundary condition on ${\omega^b}$ was due to the fact that ${DN \omega^i = 0}$ . We also note that the velocity field ${u}$ that appears in both the systems is the full velocity, which is the summation of ${u^b}$ and ${u^i}$ generated by ${\omega^b}$ and ${\omega^i}$ , respectively.

The nonlinear iteration for the Sobolev-analytic norms now mimics the standard iteration for the Cauchy-Kovalevskaya theorem that uses analyticity to treat the loss of one derivatives in the vorticity equation (one can also follow the alternative framework introduced in a Grenier-Toan’s paper to construct analytic solutions through generator functions – I discuss it here in a previous blog article). Precisely, let us fix positive numbers ${\rho_0, \delta_0,}$ and ${\zeta\in (0,1)}$ , and introduce the following family of nonlinear iterative norms for vorticity:

$\displaystyle \begin{aligned} A(\beta) : = \sup_{0<\lambda^2\beta t< \rho_0} [\sup_{0<\rho<\rho_0-\beta \lambda^2 t} ( \| \omega(t)\|_{\mathcal{W}^{1,1}_\rho} + \|\omega(t)\|_{\mathcal{W}^{2,1}_\rho}(\rho_0 - \rho- \lambda^2\beta t)^{\zeta} ) + \|\omega(t)\|_{H^4(\{\lambda d(x,\partial\Omega)\ge \delta_0/2\})} ] \end{aligned} \ \ \ \ \ (29)$

for a parameter ${\beta>0}$ , with recalling

$\displaystyle \| \omega(t)\|_{\mathcal{W}^{k,1}_\rho} = \sum_{j+\ell \le k} \|\partial_{ \theta}^j ( z\partial_{ z})^\ell \omega(t)\|_{\mathcal{L}^1_\rho}.$

Here, ${\lambda}$ is some sufficiently small parameter. Note that by definition the norm ${\|\cdot\|_{\mathcal{W}^{k,1}_\rho}}$ controls the analyticity of the vorticity near the boundary, precisely in the region ${\lambda d(x,\partial\Omega)\le \delta_0+\rho,}$ while the ${H^4}$ norm is to control the Sobolev regularity away from the boundary. We shall show that the vorticity norm remains finite for sufficiently large ${\beta}$ . The weight ${(\rho_0-\rho-\lambda^2\beta t)^\zeta}$ , with a small ${\zeta>0}$ , is standard in the literature to avoid time singularity when recovering the loss of derivatives (\cite{Asano, Caflisch}). The goal is then to prove the following key proposition.

Proposition 7 For ${\beta>0}$ , there holds

$\displaystyle A(\beta) \le C_0 \|\omega_0\|_{\mathcal{W}^{2,1}_\rho} + C_0\|\omega_0\|_{H^4(\{\lambda d(x,\partial\Omega)\ge \delta_0/2\})} + C_0 \beta^{-1} A(\beta)^2 .$

The main result then follows directly from Proposition 7, upon taking ${\beta}$ sufficiently large and ${T = \beta^{-1}\lambda^{-2}\rho_0}$ . To prove the Proposition 7, we bound the near boundary vorticity solving (27) through the semigroup of the Stokes problem. Indeed, we have the following standard Duhamel’s integral representation, written in the scaled variables,

$\displaystyle \omega^b( t)=e^{\nu t S} \omega_{0} +\int_{0}^{ t}e^{\nu( t- t')S} f( t') \; d t' + \int_0^{ t} \Gamma(\nu ( t- t'))g( t') \;d t' \ \ \ \ \ (30)$

where

$\displaystyle \begin{aligned} f( t) = - u \cdot \nabla \omega^b , \qquad g( t )& = [\partial_n \Delta^{-1} ( u \cdot \nabla \omega)] _{\vert_{\partial \Omega}} . \end{aligned}\ \ \ \ \ (31)$

Here, ${e^{\nu t S}}$ denotes the semigroup of the corresponding Stokes problem and ${\Gamma(\nu t)}$ being its trace on the boundary. The nonlinear iteration for the analytic component ${\omega^b}$ now follows directly from the linear propagation estimates from Proposition 6, upon deriving the bilinear estimates of ${u \cdot \nabla \omega^b}$ , which are a direct consequence of the algebra property of the analytic norms and the elliptic estimates from Proposition 5. On the other hand, the vorticity away from the boundary ${\omega^i}$ is estimated through the standard energy estimates for the Navier-Stokes equations, since ${\omega^i}$ vanishes in a neighborhood of the boundary.

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The inviscid limit problem for Navier-Stokes equations

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