Well-Posedness of the Nonlinear Unsteady Prandtl Equations with Robin Boundary Condition in Weighted Sobolev Spaces

Global-in-$x$ stability of Prandtl layer expansions for steady magnetohydrodynamics flows over a moving plate

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].

Global Well-posedness and Regularity of Weak Solutions to the Prandtl's System

We continue our study on the global solution to the two-dimensional Prandtl's system for unsteady boundary layers in the class considered by Oleinik provided that the pressure is favorable. First, by using a different method from [13], we gave a direct proof of existence of a global weak solution by a direct BV estimate. Then we prove the uniqueness and continuous dependence on data of such a weak solution to the initial-boundary value problem. Finally, we show the smoothness of the weak solutions and then the global existence of smooth solutions.

High frequency analysis of the unsteady Interactive Boundary Layer model

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…

Sobolev stability of Prandtl expansions for the steady Navier-Stokes equations

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.

Validity of Prandtl expansions for steady MHD in the Sobolev framework

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…

二维非齐次Prandtl方程解的局部存在性与唯一性：能量方法

Local existence and uniqueness of solution to the two-dimensional inhomogeneous Prandtl equations by energy method

本文用能量方法解决二维非齐次Prandtl方程局部存在唯一性，克服密度引起的切向导数丢失，并借助“商”建立定常情形高正则空间局部x适定性。

High Reynolds number Navier-Stokes solutions and boundary layer separation induced by a rectilinear vortex

We compute the solutions of Prandtl's and Navier-Stokes equations for the two dimensional flow induced by a rectilinear vortex interacting with a boundary in the half plane. For this initial datum Prandtl's equation develops, in a finite time, a separation singularity. We investigate the different stages of unsteady separation for Navier-Stokes solution at different Reynolds numbers $Re=10^3-10^5$, and we show the presence of a large-scale interaction between the viscous boundary layer and the inviscid outer flow. We also see a subsequent stage, characterized by the presence of a small-scale i…

Zero-viscosity limit of the Navier-Stokes equations with the Navier friction boundary condition

In this paper, we consider the zero-viscosity limit of the Navier-Stokes equations in a half space with the Navier friction boundary condition $$ (βu^{\varepsilon}-\varepsilon^γ\partial_y u^{\varepsilon})|_{y=0}=0, $$ where $β$ is a constant and $γ\in (0,1]$. In the case of $γ=1$, the convergence to the Euler equations and the Prandtl equation with the Robin boundary condition is justified for the analytic data. In the case of $γ\in (0,1)$, the convergence to the Euler equations and the linearized Prandtl equation is justified for the data in the Gevrey class $\frac 1 γ$.

Prandtl-Meyer 反射构型、跨声速激波与自由边界问题

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。

Compressible Euler limit from Boltzmann equation with Maxwell reflection boundary condition in half-space

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…

Multidimensional Transonic Shock Waves and Free Boundary Problems

We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for …