In this work, we establish the convergence of 2D, stationary Navier-Stokes flows, $(u^\epsilon, v^\epsilon)$ to the classical Prandtl boundary layer, $(\bar{u}_p, \bar{v}_p)$, posed on the domain $(0, \infty) \times (0, \infty)$: \begin{equation*} \| u^{\epsilon} - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrt{\epsilon} \langle x \rangle^{- \frac 1 4 + \delta}, \qquad \| v^{\epsilon} - \sqrt{\epsilon} \bar{v}_p \|_{L^\infty_y} \lesssim \sqrt{\epsilon} \langle x \rangle^{- \frac 1 2}. \end{equation*} This validates Prandtl's boundary layer theory \textit{globally} in the $x$-variable for a large class of boundary layers, including the entire one parameter family of the classical Blasius profiles, with sharp decay rates. The result demonstrates asymptotic stability in two senses simultaneously: (1) asymptotic as $\epsilon \rightarrow 0$ and (2) asymptotic as $x \rightarrow \infty$. In particular, our result provides the first rigorous confirmation for the Navier-Stokes equations that the boundary layer cannot "separate" in these stable regimes, which is very important for physical and engineering applications.

Global-in-$x$ Stability of Steady Prandtl Expansions for 2D Navier-Stokes Flows

We study the asymptotic behavior of the Oleinik’s solution to the steady Prandtl equation when the outer flow $$U(x)=1$$ . Serrin proved that the Oleinik’s solution converges to the famous Blasius solution $${\bar{u}}$$ in $$L^\infty _y$$ sense as $$x\rightarrow +\infty $$ . The explicit decay estimates of $$u-{\bar{u}}$$ and its derivatives were proved by Iyer (ARMA 237,2020) when the initial data is a small localized perturbation of the Blasius profile. In this paper, we prove the explicit decay estimate of $$u-\bar{u}$$ for general initial data with exponential decay. We also prove the decay estimates of its derivatives when the initial data has an additional concave assumption. Our proof is based on the maximum principle techniques. The key ingredient is to find a series of barrier functions. 

/pdf/asymptotic-behavior-of-the-steady-prandtl-equation-5dck5nuk.pdf

Asymptotic behavior of the steady Prandtl equation

In this paper, we study the back flow of the two-dimensional unsteady Prandtl boundary layer under an adverse pressure gradient. The occurrence of back flow is an important physical event in the evolution of boundary layer, which eventually leads to separation. For the two-dimensional unsteady Prandtl boundary layer equations, when the initial tangential velocity is strictly monotonic with respect to the normal variable, and the pressure gradient of the outer flow is adverse, we obtain that the first critical point of the tangential velocity profile with respect to the normal variable, if exists when the boundary layer evolves in time, must appear on the boundary. Moreover, we give a condition on the growth rate of the initial tangential velocity such that there is a back flow point of the Prandtl boundary layer under the adverse pressure gradient. In the appendix, we introduce two examples showing that back flow occurs either when the flow distance is long in the streamwise direction for a given initial monotonic tangential velocity field, or when the initial tangential velocity grows slowly in a large neighborhood of the boundary for a fixed flow distance in the streamwise direction.

Separation of the two-dimensional unsteady Prandtl boundary layers under adverse pressure gradient

This is the first part of a two paper sequence in which we prove the global-in-x stability of the classical Prandtl boundary layer for the 2D, stationary Navier-Stokes equations. In this part, we provide a construction of an approximate Navier-Stokes solution, obtained by a classical Euler- Prandtl asymptotic expansion. We develop here sharp decay estimates on these quantities. Of independent interest, we establish \textit{without} using the classical von-Mise change of coordinates, proofs of global in x regularity of the Prandtl system. The results of this paper are used in the second part of this sequence, [IM20] (arXiv:2008.12347), to prove the asymptotic stability of the boundary layer as $\eps \rightarrow 0$ and $x \rightarrow \infty$.

Boundary Layer Expansions for the Stationary Navier-Stokes Equations

Motivated by \cite{DG19}, we prove the global existence and large time behavior of small solutions to 2-D Prandtl system for data with Gevrey 2 regularity in the $x$ variable and Sobolev regularity in the $y$ variable. In particular, we extend the global well-posedness result in \cite{PZ5} for 2-D Prandtl system with analytic data to data with optimal Gevery regularity in the sense of \cite{Ger1}.

On the global small solution of 2-D Prandtl system with initial data in the optimal Gevrey class

In the case of favorable pressure gradient, Oleinik proved the global existence of classical solution for the 2-D steady Prandtl equation for a class of positive data. In the case of adverse pressure gradient, an important physical phenomena is the boundary layer separation. In this paper, we prove the boundary layer separation for a large class of Oleinik's data and confirm Goldstein's hypothesis concerning the local behavior of the solution near the separation, which gives a partial answer to open problem 5 proposed by Oleinik and Samokin in \cite{Olei}.

Boundary layer separation and local behavior for the Steady Prandtl equation

Let $X$ be a compact strictly pseudoconvex embeddable CR manifold and let $T_P$ be the Toeplitz operator on $X$ associated with some first order pseudodifferential operator $P$. We consider $\chi_k(T_P)$ the functional calculus of $T_P$ by any rescaled cut-off function $\chi$ with compact support in the positive real line. In this work, we show that $\chi_k(T_P)$ admits a full asymptotic expansion as $k\to+\infty$. As applications, we obtain several CR analogous of results concerning high power of line bundles in complex geometry but without any group action assumptions on the CR manifold. In particular, we establish a Kodaira type embedding theorem, Tian's convergence theorem and a perturbed spherical embedding theorem for strictly pseudoconvex CR manifolds.

pdf/semi-classical-spectral-asymptotics-of-toeplitz-operators-on-2cszusm4.pdf

Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

Blow-up sets of Ricci curvatures of complete conformal metrics

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Papers

Boundary layer separation and local behavior for the Steady Prandtl equation

Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds

Boundary layer separation and local behavior for the Steady Prandtl equation

Blow-up sets of Ricci curvatures of complete conformal metrics