Chen Gao, Liqun Zhang
In this paper, we consider the zero-viscosity limit of the 2D steady Navier-Stokes equations in $(0,L)\times\mathbb{R}^+$ with non-slip boundary conditions. By estimating the stream-function of the remainder, we justify the validity of the Prandtl boundary layer expansions.
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In this paper, we prove the global $C^\infty$ regularity of the Oleinik's solution for the steady Prandtl equation with favorable pressure.
By establishing an invariant set (1.11) for the Prandtl equation in Crocco transformation, we prove orbital and asymptotic stability of Blasius-like steady states against Oleinik's monotone solutions.
In this article we apply the machinery developed in Guo-Iyer[1] together with a new compactness estimate and an object called the degree in order to prove validity of steady Prandtl layer expansions with external forcing.
This paper concerns the validity of the Prandtl boundary layer theory for steady, incompressible Navier-Stokes flows over a rotating disk. We prove that the Navier Stokes flows can be decomposed into Euler and Prandtl flows in the inviscid limit. In so doing, we develop a new set of function spaces and prove several embedding theorems which capture the interaction between the Prandtl scaling and the geometry of our domain.
Due to degeneracy near the boundary, the question of high regularity for solutions to the steady Prandtl equations has been a longstanding open question since the celebrated work of Olenick. We settle this open question in affirmative in the absence of an external pressure. Our method is based on energy estimates for the quotient, $q = \frac{v}{\bar{u}}$, $\bar{u}$ being the classical Prandtl solution, via the linear Derivative Prandtl Equation (LDP). As a consequence, our regularity result leads to the construction of Prandtl layer expansion up to any order.
In this paper, we prove the stability of shear flows of Prandtl type as $ \big(U(y/\sqrtν),0\big)$ for the steady Navier-Stokes equations under a natural spectral assumption on the linearized NS operator. We develop a direct energy method combined with compact method to solve the Orr-Sommerfeld equation.
We consider a steady, geophysical 2D fluid in a domain, and focus on its western boundary layer, which is formally governed by a variant of the Prandtl equation. By using the von Mises change of variables, we show that this equation is well-posed under the assumption that the trace of the interior stream function has large variations, and that the variations in the coastline profile are moderate.
This paper is concerned with the validity of the Prandtl boundary layer theory in the inviscid limit of the steady incompressible Navier-Stokes equations, which is an extension of the pioneer paper (Y. Guo et al., 2017, Ann. PDE) from a domain of $[0,L]\times\mathbb{R}_+$ to $[0,L]\times[0,2]$. Under the symmetry assumption, we establish the validity of the Prandtl boundary layer expansions and the error estimates. The convergence rate as $\e\rightarrow 0$ is also given.
In this article we establish the validity of Prandtl layer expansions around Euler flows which are not shear. The presence of non-shear flows at the leading order creates a singularity of $o(\frac{1}{\sqrtε})$. A new $y$-weighted positivity estimate is developed to control this leading-order growth at the far field.
Steady Prandtl Layer for Compressible Fluids
针对可压缩流体稳态Prandtl层，结合速度场商估计与温度场伪熵估计，建立局部适定性并构造强Sobolev范数下的展开。
In this paper, we obtain the global-in-$x$ Sobolev stability of Prandtl layer expansions for 2-D steady incompressible MHD flows with shear outer ideal MHD flows $(1,0,σ,0)$ ($σ\geq 0$) on a moving plate. It is worth noticing that in the Sobolev sense, the degeneracy of the tangential magnetic field in our result is allowed, which is much different from both the unsteady case in [25,26,27] and the steady case in [3,4].
Validity of Prandtl Expansion for Steady Compressible Navier-Stokes-Fourier Flows
本文研究了在粘度趋于零的极限下，普朗特层展开在稳态可压缩纳维-斯托克斯-傅里叶流中的有效性，通过建立统一的余项估计框架证明了其有效性。
In this three-part monograph, we prove that steady, incompressible Navier-Stokes flows posed over the moving boundary, $y = 0$, can be decomposed into Euler and Prandtl flows in the inviscid limit globally in $[1,\infty) \times [0,\infty)$, assuming a sufficiently small velocity mismatch. Sharp decay rates and self-similar asymptotics are extracted for both Prandtl and Eulerian layers. We then develop a functional framework to capture precise decay rates of the remainders, and prove the corresponding embedding theorems by establishing weighted estimates for their higher order tangential deriva…
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}.
Let the viscosity $\varepsilon \rightarrow 0$ for the 2D steady Navier-Stokes equations in the region $0\leq x\leq L$ and $0\leq y<\infty$ with no slip boundary conditions at $y=0$. For $L<<1$, we justify the validity of the steady Prandtl layer expansion for scaled Prandtl layers, including the celebrated Blasius boundary layer. Our uniform estimates in $\varepsilon$ are achieved through a fixed-point scheme: \begin{equation*} [u^{0}, v^0] \overset{\text{DNS}^{-1}}{\longrightarrow }v\overset{\mathcal{L}^{-1}}{ \longrightarrow }[u^{0}, v^0] \label{fixedpoint} \end{equation*} for solving the Na…
We continue the study of the validity of the Prandtl boundary layer expansions in \cite{GZ}, where by estimating the stream-function of the remainder, we proved if the Euler flow is perturbation of shear flows when the width of domain is small. In this paper, we obtain a new derivatives estimate of stream-function away from the boundary layer and then prove the validity of expansions for any non-shear Euler flow, provided the width of domain is small.
We show the $H^1$ stability of shear flows of Prandtl type: $U^ν= (U_s(y/\sqrtν),0)$, in the steady two-dimensional Navier-Stokes equations, under the natural assumptions that $U_s(Y) > 0$ for $Y > 0$, $U_s(0) = 0$, and $U_s'(0) > 0$. Our result is in sharp contrast with the unsteady ones, in which at most Gevrey stability can be obtained, even under global monotonicity and concavity hypotheses. It provides the first positive answer to the inviscid limit problem in Sobolev regularity for a non-trivial class of steady Navier-Stokes flows with no-slip boundary condition.
Sharp Asymptotic Behavior of the Steady Pressure-free Prandtl system
针对稳态无压 Prandtl 系统，本文利用修正 von Mises 变换、加权能量法和锐最大值原理，证明解远场收敛于 Blasius 流，并得到两类初始数据下的最优收敛速率。
This paper is concerned with the vanishing viscosity and magnetic resistivity limit for the two-dimensional steady incompressible MHD system on the half plane with no-slip boundary condition on velocity field and perfectly conducting wall condition on magnetic field. We prove the nonlinear stability of shear flows of Prandtl type with nondegenerate tangential magnetic field, but without any positivity or monotonicity assumption on the velocity field. It is in sharp contrast to the steady Navier-Stokes equations and reflects the stabilization effect of magnetic field. Unlike the unsteady MHD sy…
In this paper, we validate the boundary layer theory for 2D steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain $\{(X, Y)\in[0, L]\times\mathbb{R}_+\}$ under the assumption of a moving boundary at $\{Y=0\}$. The validity of the boundary layer expansion and the convergence rates are established in Sobolev sense. We extend the results for the case with the shear outer ideal MHD flows [3] to the case of the nonshear flows.
Prandtl-Batchelor flows on an annulus
本文研究圆环上受迫稳态Navier-Stokes方程的广义Prandtl-Batchelor理论，证明当壁面速度与同向刚性旋转略有不同时，存在具有嵌套闭合流线且无粘极限为旋转剪切流的Prandtl-Batchelor流。
Asymptotic Stability of Steady Prandtl Equations with Suction
本文证明了壁面吸气下稳态Prandtl方程的全局渐近稳定性，通过von Mises变换和Hardy型估计，实现了任意代数衰减率的下游衰减。
In this paper, we are concerned with the validity of Prandtl boundary layer expansion for the solutions to two dimensional (2D) steady viscous incompressible magnetohydrodynamics (MHD) equations in a domain $\{(X, Y)\in[0, L]\times\mathbb{R}_+\}$ with a moving flat boundary $\{Y=0\}$. As a direct consequence, even though there exist strong boundary layers, the inviscid type limit is still established for the solutions of 2D steady viscous incompressible MHD equations in Sobolev spaces provided that the following three assumptions hold: the hydrodynamics and magnetic Reynolds numbers take the s…
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 $\|u(x,y)-\bar{u}(x,y)\|_{L^\infty_y}$ for general initial data with exponential decay. We also prove the …
Global stability of three dimensional steady Prandtl equation
针对三维稳态Prandtl方程这一公开难题，提出新方法通过曲率型交换子和向量场最大原理建立包含Blasius解的背景剖面全局稳定性。
In this work, we establish the convergence of 2D, stationary Navier-Stokes flows, $(u^ε, v^ε)$ to the classical Prandtl boundary layer, $(\bar{u}_p, \bar{v}_p)$, posed on the domain $(0, \infty) \times (0, \infty)$: \begin{equation*} \| u^ε - \bar{u}_p \|_{L^\infty_y} \lesssim \sqrtε \langle x \rangle^{- \frac 1 4 + δ}, \qquad \| v^ε - \sqrtε \bar{v}_p \|_{L^\infty_y} \lesssim \sqrtε \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…
We provide an optimal Gevrey stability result for general boundary layer expansions, under a mild concavity condition on the boundary layer profile. Our result generalizes (and even improves in the non strictly concave case) the one obtained in [Gerard-Varet et al, Duke Math. J. 167 (2018)], restricted to expansions of shear flow type.
This paper concerns the validity of the Prandtl boundary layer theory in the inviscid limit for steady incompressible Navier-Stokes flows. The stationary flows, with small viscosity, are considered on $[0,L]\times \mathbb{R}_{+}$, assuming a no-slip boundary condition over a moving plate at $y=0$. We establish the validity of the Prandtl boundary layer expansion and its error estimates.
In this paper, we prove the $L^\infty\cap L^2$ stability of Prandtl expansions of shear flow type as $\big(U(y/\sqrtν),0\big)$ for the initial perturbation in the Gevrey class, where $U(y)$ is a monotone and concave function and $ν$ is the viscosity coefficient. To this end, we develop the direct resolvent estimate method for the linearized Orr-Sommerfeld operator instead of the Rayleigh-Airy iteration method introduced by Grenier, Guo and Nguyen.
We establish the validity of the Euler$+$Prandtl approximation for solutions of the Navier-Stokes equations in the half plane with Dirichlet boundary conditions, in the vanishing viscosity limit, for initial data which are analytic only near the boundary, and Sobolev smooth away from the boundary. Our proof does not require higher order correctors, and works directly by estimating an $L^{1}$-type norm for the vorticity of the error term in the expansion Navier-Stokes$-($Euler$+$Prandtl$)$. An important ingredient in the proof is the propagation of local analyticity for the Euler equation, a re…
Sharp Asymptotic Stability of Blasius Profile in the Steady Prandtl Equation
本文研究稳态Prandtl方程中Blasius剖面的渐近稳定性，通过发现近守恒低频量和边界退化结构并利用迭代技术，证明了最优的 $(x+1)^{-1}$ 收敛率。
For steady two-dimensional flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. In this paper, by constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is …
As a continuation of \cite{LXY}, the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on velocity field and perfect conducting boundary condition on magnetic field. Under the assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give a $L^\infty$ estimate on the error by multi-scale analysis.
We investigate the stability of boundary layer solutions of the two-dimensional incompressible Navier-Stokes equations. We consider shear flow solutions of Prandtl type : $$ u^ν(t,x,y) \, = \, \big (U^E(t,y) + U^{BL}(t,\frac{y}{\sqrtν})\,, \, 0 \big )\, , \quad 0<ν\ll 1\,. $$ We show that if $U^{BL}$ is monotonic and concave in $Y = y /\sqrtν$ then $u^ν$ is stable over some time interval $(0,T)$, $T$ independent of $ν$, under perturbations with Gevrey regularity in $x$ and Sobolev regularity in $y$. We improve in this way the classical stability results of Sammartino and Caflisch in analytic c…
In this paper, we prove global existence of solutions with analytic regularity to the 2D MHD boundary layer equations in the mixed Prandtl and Hartmann regime derived by formal multi-scale expansion in \cite{GP}. The analysis shows that the combined effect of the magnetic diffusivity and transveral magnetic field on the boundary leads to a linear damping on the tangential velocity field near the boundary. And this damping effect yields the global in time analytic norm estimate in the tangential space variable on the perturbation of the classical steady Hartmann profile.
Prandtl boundary layer expansion with strong boundary layers for inhomogeneous incompressible magnetohydrodynamics equations in Sobolev framework
本文在Sobolev框架下，针对粘性和电阻率系数同阶趋于零且存在强边界层的情况，证明了非齐次不可压缩MHD方程初边值问题解的Prandtl边界层展开在$L^\infty$意义下的有效性。
This note concerns a nonlinear ill-posedness of the Prandtl equation and an invalidity of asymptotic boundary-layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear ill-posedness results established by Gérard-Varet and Dormy [2], and an analysis in Guo and Tice [5]. We show that the asymptotic boundary-layer expansion is not valid for non-monotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well-posedness and prove that the nonlinear Prandtl equation is not well-posed in this sense near non-stationa…
Prandtl-Batchelor flows with a point vortex on a disk
研究高雷诺数下带中心强迫和近刚性旋转边界的圆盘内二维不可压缩纳维-斯托克斯流动，构造含点涡的定常解，通过对数变换与傅里叶分析克服奇异性带来的困难并完成稳定性估计。
In this paper, the infinite limit of the Prandtl number is justified for the two-dimensional incompressible magneto-convection, which describes the nonlinear interaction between the Rayleigh-B$\rm\acute{e}$nard convection and an externally magnetic field. Both the convergence rates and the thickness of initial layer are obtained. Moreover, based on the method of formal asymptotic expansions, an effective dynamics is constructed to simulate the motion within the initial layer.
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…
The aim of this paper is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: $ν\to 0$. In \cite{Grenier}, one of the authors proved that there exists no asymptotic expansion involving one Prandtl's boundary layer with thickness of order $\sqrtν$, which describes the inviscid limit of Navier-Stokes equations. The instability gives rise to a viscous boundary sublayer whose thickness is of order $ν^{3/4}$. In this paper, we point out how the stability of the classical Prandtl's layer is linked to the stability of this sublayer. In particular, we prove that th…
We present our recent results on the Prandtl-Meyer reflection for supersonic potential flow past a solid ramp. When a steady supersonic flow passes a solid ramp, there are two possible configurations: the weak shock solution and the strong shock solution. Elling-Liu's theorem (2008) indicates that the steady supersonic weak shock solution can be regarded as a long-time asymptotics of an unsteady flow for a class of physical parameters determined by certain assumptions for potential flow. In this paper we discuss our recent progress in removing these assumptions and establishing the stability t…
Prandtl Boundary Layers in An Infinitely Long Convergent Channel
本文证明了在曲率递减的无限长收缩喷管中，小黏性Navier-Stokes流可由Euler流与Prandtl流叠加一致逼近，并揭示了质量通量对顶点及无穷远处奇异渐近行为的决定作用。
Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems
本研究针对Prandtl-Meyer反射中定常弱/强激波的物理可容许性这一长期未决问题，通过将问题转化为含跨声速激波的自由边界问题，结合先验估计与Leray-Schauder度论证，证明了定常弱激波解作为非定常势流反射构型大时间渐近态的稳定性。
Prandtl--Batchelor and flux-expulsion selection for steady MHD flows in a disk
该研究针对圆盘内稳态不可压缩MHD流动，在特定条件下构造收敛解，实现了普朗特-巴彻勒选择与通量驱逐的完全耦合，还通过论证解释了单涡族的核心结构。
Small scales in inviscid limits of steady fluids
针对二维通道内定常 Navier-Stokes 方程的无粘极限，构造 Euler 剪切流附近的解，发现垂直速度中除经典 Prandtl 层外还存在量级为 $\varepsilon$ 的新小尺度。
In this paper we show how the stability of Prandtl boundary layers is linked to the stability of shear flows in the incompressible Navier Stokes equations. We then recall classical physical instability results, and give a short educational presentation of the construction of unstable modes for Orr Sommerfeld equations. We end the paper with a conjecture concerning the validity of Prandtl boundary layer asymptotic expansions.
In this paper we shall consider the Navier-Stokes equations in the half plane with Euler-type initial conditions, i.e. initial conditions which have a non-zero tangential component at the boundary. Under analyticity assumptions for the data, we shall prove that the solution exists for a small time independent of the viscosity. The solution is constructed through a composite asymptotic expansion involving the solutions of the Euler and Prandtl equations, plus an error term. The norm of the error goes to zero with the square root of the viscosity. The Prandtl solution contains a singular term, w…
Zero-viscosity limit of the compressible Naiver-Stokes equations in the analytic setting
本文在解析设定下证明了半空间无滑移边界可压缩Navier-Stokes方程零粘性极限的Prandtl边界层展开，即远离边界由Euler方程、近边界由Prandtl方程近似。
Validity of Prandtl's boundary layer from the Boltzmann theory
本研究首次从Boltzmann方程的流体力学极限严格论证了Prandtl方程及高阶展开，通过交换子估计和局部守恒律等关键技术，获得了带扩散边界条件的线性化Boltzmann方程在解析空间中的估计。
We revisit the diffusive limit of a steady neutron transport equation in a $2$-D unit disk with one-speed velocity. We show the classical result in [4] with Milne expansion is incorrect in $L^{\infty}$ and we give the right answer in studying the $ε$-Milne expansion with geometric correction.
In their classical work Caflisch and Sammartino proved the inviscid limit of the incompressible Navier-Stokes equations for well-prepared data with analytic regularity in the half-space. Their proof is based on the detailed construction of Prandtl's boundary layer asymptotic expansions. In this paper, we give a direct proof of the inviscid limit for general analytic data without having to construct Prandtl's boundary layer correctors. Our analysis makes use of the boundary vorticity formulation and the abstract Cauchy-Kovalevskaya theorem on analytic boundary layer function spaces that capture…
In this paper, we establish the mathematical validity of the Prandtl boundary layer theory for a class of nonlinear plane parallel flows of viscous incompressible magnetohydrodynamic (MHD) flow with no-slip boundary condition of velocity and perfectly conducting wall for magnetic fields. The convergence is shown under various Sobolev norms, including the physically important space-time uniform norm $L^\infty(H^1)$. In addition, the similar convergence results are also obtained under the case with uniform magnetic fields. This implies the stabilizing effects of magnetic fields. Besides, the hig…
On the classical solution for the steady triple-deck equations
本文通过分解粗糙度效应和引入加权索伯列夫范数，确立了稳态三叠方程经典解的存在性和唯一性，并证明了耦ette流的局部唯一性。
Boundary layer expansions of the steady MHD equations in a bounded domain
该研究考察矩形区域内定常 MHD 系统的边界层展开，在切向磁场与切向速度场强弱悬殊的两种情形下，将解刻画至任意阶，并扩展了已有结果。
The present paper is about a famous extension of the Prandtl equation, the so-called Interactive Boundary Layer model (IBL). This model has been used intensively in the numerics of steady boundary layer flows, and compares favorably to the Prandtl one, especially past separation. We consider here the unsteady version of the IBL, and study its linear well-posedness, namely the linear stability of shear flow solutions to high frequencyperturbations. We show that the IBL model exhibits strong unrealistic instabilities, that are in particular distinct from the Tollmien-Schlichting waves. We also e…
A selection principle for 2D steady Euler flows via the vanishing viscosity limit
本文研究了二维稳态欧拉流的消粘性极限，发现有限域内仅存在常涡量流，且在无限长带状域中，周期性粘性速度的极限仅包括常流、Couette流和Poiseuille流，证明依赖于对粘性和无粘流流线的细致分析和一个强大的刚性定理。
Starting from the local-in-time classical solution to the compressible Euler system with impermeable boundary condition in half-space, by employing the coupled weak viscous layers (governed by linearized compressible Prandtl equations with Robin boundary condition) and linear kinetic boundary layers, and the analytical tools in \cite{Guo-Jang-Jiang-2010-CPAM} and some new boundary estimates both for Prandtl and Knudsen layers, we proved the local-in-time existence of Hilbert expansion type classical solutions to the scaled Boltzmann equation with Maxwell reflection boundary condition with acco…
On nonlinear instability of Prandtl's boundary layers: the case of Rayleigh's stable shear flows
本文研究 Prandtl 边界层展开，证明对 Sobolev 正则解该展开在小黏度下失效，且 Rayleigh 稳定的单调剪切流在 ν>0 时具有非线性不稳定性。
Intrinsic expansions in large Grashof numbers for the steady states of the Navier-Stokes equations
本文通过引入嵌套空间中的渐近展开概念，克服无穷维空间缺乏局部紧性的困难，为Navier-Stokes方程稳态解建立了大Grashof数下的内在渐近展开理论，并在二维周期边界条件下进一步刻画了展开项的性质。
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