On the Prandtl Equation

The unique solvability of the airfoil (Prandtl) integro-differential equation on the semi-axis ⁺ = [0, ) is proved in the Sobolev space W _p ¹ and Bessel potential spaces H _p ^s under certain restrictions on p and s.     

一类Zakharov-Kuznestov方程解的正则性

In this paper, we consider the regularity of solution in S for Zakharov-Kuznestov equation in Hs(s > 2). Meanwhile, by method of undetermined coefficient we prove that there don't exist and conservative integral include 2 order of higher order derived functions.     

Regularity - Properties of the Solution of the Homogeneous Wave Equation

The aim of this paper is to prove regularity-properties for solutions of the wave equation in B with p < 1. Up to now no such result is known. For p ≥ 1 a large amount of work has been done, cf. for instance [9], [7], [8], [10]. The results in this paper for p> 1 are not optimal, which can be seen comparing them with [9]. Whether or not our results are optimal for p ≤ 1 is open.     

Constant Scalar Curvature Equation and Regularity of Its Weak Solution

In this paper we study the constant scalar curvature equation (CSCK), a nonlinear fourth-order elliptic equation, and its weak solutions on Kähler manifolds. We first define the notion of a weak solution of CSCK for an L^∞ Kähler metric. The main result is to show that such a weak solution (with uniform L^∞ bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of K-energy minimizers. The new technical ingredient is a W^{2, 2} regularity result for the Laplacian equation Δ_gu=f on Kähler manifolds, where the metric has only L^∞ coefficients. It is well-known that such a W^{2, 2} regularity (W^{2, p} regularity for any p > 1) fails in general (except for dimension 2) for uniform elliptic equations of the form for a^ij ∊ L^∞ without certain smallness assumptions on the local oscillation of a^ij. We observe that the Kähler condition plays an essential role in obtaining a W^{2, 2} regularity for elliptic equations with only L^∞ elliptic coefficients on compact manifolds. © 2017 Wiley Periodicals, Inc.     

一个四阶抛物方程整体弱解的正则性

本文研究一个四阶抛物方程的非负大初值混合Dirichlet-Neumann边值问题.使用半离散化解的精细熵估计与插值技巧,得到了正则性更好的整体弱解.     

The Boundary Regularity of a Weak Solution of the Navier-Stokes Equation and its Connection to the Interior Regularity of Pressure

We assume that v is a weak solution to the non-steady Navier-Stokes initial-boundary value problem that satisfies the strong energy inequality in its domain and the Prodi-Serrin integrability condition in the neighborhood of the boundary. We show the consequences for the regularity of v near the boundary and the connection with the interior regularity of an associated pressure and the time derivative of v.     

Regularity of Solution for Fully Nonlinear Second Order Elliptic Equation with Gradient Constraint

The existence and regularity of the solution for the fully nonlinear second order elliptic equation with gradient constraint are discussed in this paper. We prove the existence of the strong solution by choice of special penalty function, and obtain the local W^{2,∞} estimate of the solution by means of the theory of viscosity solution with the auxiliary function method.     

在边界附近蜕化的椭圆型Monge-Ampère方程解的正则性

我们证明了在边界附近蜕化的椭圆型Monge-Ampère方程Dirichlet问题解的整体C1,1正则性,并举例说明了我们的结果是最佳的.     

组合流形上的Poisson方程广义解的正则性

In this Paper, as a model, we consider the following problem(?).We prove that the solution is sufficiently smooth on Ω_i (i=1,2), if curves ?Ω,Γ and data g, f_i(i = 1,2) are sufficiently smooth too.     

On an Algorithm for the Solution of Generalized Prandtl Equations

The aim of this paper is to present and discuss an algorithm related to a numerical model, based on the modified quasi-interpolatory splines approximation, to solve generalized Prandtl integro-differential equations, with particular singular kernels.     

On Another Kind of Parabolic Monge－Ampere Equation：the Existence，Uniqueness and Regularity of the Vi

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A New H 2 Regularity Condition of the Solution to Dirichlet Problem of the Poisson Equation and Its Applications

We study the regularity of the solution of Dirichlet problem of Poisson equations over a bounded domain.A new sufficient condition,uniformly positive reach is introduced.Under the assumption that the closure of the underlying domain of interest has a uniformly positive reach,the H^2 regularity of the solution of the Poisson equation is established.In particular,this includes all star-shaped domains whose closures are of positive reach,regardless if they are Lipschitz domains or non-Lipschitz domains.Application to the strong solution to the second order elliptic PDE in non-divergence form and the regularity of Helmholtz equations will be presented to demonstrate the usefulness of the new regularity condition.     

Local Regularity for the Modified SQG Patch Equation

We study the patch dynamics on the whole plane and on the half‐plane for a family of active scalars called modified surface quasi‐geostrophic (SQG) equations. These involve a parameter α that appears in the power of the kernel in their Biot‐Savart laws and describes the degree of regularity of the equation. The values α =0 and α =½ correspond to the two‐dimensional Euler and SQG equations, respectively. We establish here local‐in‐time regularity for these models, for all α ? (0,½) on the whole plane and for all small α > 0 on the half‐plane. We use the latter result in [16], where we show existence of regular initial data on the half‐plane that lead to a finite‐time singularity.© 2016 Wiley Periodicals, Inc.     

Conditions Implying Regularity of the Three Dimensional Navier-Stokes Equation

We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac-like inequalities. As part of our methods, we give a different approach to a priori estimates of Foias, Guillope and Temam. The author was partially supported by an NSF grant.     

On a Quasilinear Degenerate Parabolic Equation from Prandtl Boundary Layer Theory

The equation arising from Prandtl boundary layer theory $$\frac{\partial u}{\partial t} -\frac{\partial }{\partial x_i}\left( a(u,x,t)\frac{\partial u}{\partial x_i}\right)-f_i(x)D_iu+c(x,t)u=g(x,t)$$ is considered. The existence of the entropy solution can be proved by BV estimate method. The interesting problem is that, since $a(\cdot,x,t)$ may be degenerate on the boundary, the usual boundary value condition may be overdetermined. Accordingly, only dependent on a partial boundary value condition, the stability of solutions can be expected. This expectation is turned to reality by Kružkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, if $a_{x_i}(\cdot,x,t)\mid_{x\in \partial \Omega}=a(\cdot,x,t)\mid_{x\in \partial \Omega}=0$ and $f_i(x)\mid_{x\in \partial \Omega}=0$, the stability can be proved even without any boundary value condition.     

非线性应变波耦合组的吸引子的正则性

本文研究了非线性应变波方程与Schr(?)dinger方程耦合系统Cauchy问题吸引 子的正则性.获得了该系统在空间Eo中存在整体吸引子Ao,并且Ao与E1中的强吸 引子A1相等.     

RESEARCH ANNOUNCEMENTS：Morrey Regularity of the Solution to the Diri

To study the local regularity of solutions to second orderelliptic partial differential equations, Morrey in ［1］ introduced somefunction spaces, which are called the Morrey spaces today. Since then, manymathematicians have studied regularities of solutions to some kinds of secondorder elliptic equations in Morrey spaces.