Smoluchowski方程解的存在性和结构

研究在Onsager势下Smeluchowski方程Cauchy问题的古典解的存在性及其结构．通过线性化方法构造叠代序列，对叠代序列得到一致估计，应用Arzela-Ascoli定理从而获得非线性问题的局部解．然后借助于极值原理可以把局部解拓展到整体解．同时也获得解的非负性、周期性和归一性等基本性质。     

一类热传导方程的整体解及解的逐点估计

研究长柱体区域中的热传导方程.通过构造辅助函数,利用Hopf极值原理,得到整体解的充分条件,并给出解的逐点估计与空间衰减估计.     

Vacuum states on compressible Navier-Stokes equations with general density-dependent viscosity and general pressure law

In this paper,we study the one-dimensional motion of viscous gas with a general pres- sure law and a general density-dependent viscosity coefficient when the initial density connects to the vacuum state with a jump.We prove the global existence and the uniqueness of weak solutions to the compressible Navier-Stokes equations by using the line method.For this,some new a priori estimates are obtained to take care of the general viscosity coefficientμ(ρ)instead ofρ~θ.     

Global regularity for modified critical dissipative quasi-geostrophic equations

We consider the n-dimensional modified quasi-geostrophic (SQG) equations

t_∂θ + u . ∇θ + κΛαθ = 0,${\partial }_t\theta \hspace{0.17em}+\hspace{0.17em}u\hspace{0.17em}.\hspace{0.17em}\nabla \theta \hspace{0.17em}+\hspace{0.17em}\kappa {\Lambda }^{\alpha }\theta \hspace{0.17em}=\hspace{0.17em}0,$

u = Λα-1 R^⊥θ$u\hspace{0.17em}=\hspace{0.17em}{\Lambda }^{\alpha -1}\hspace{0.17em}{R}^{\perp }\theta$

with κ > 0, α ∈ (0,1] and θ₀ ∈ W1, ∞ (?ⁿ). In this paper, we establish a different proof for the global regularity of this system. The original proof was given by Constantin, Iyer, and Wu [5], who employed the approach of Besov space techniques to study the global existence and regularity of strong solutions to modified critical SQG equations for two dimensional case. The proof provided in this paper is based on the nonlinear maximum principle as well as the approach in Constantin and Vicol [2].     

The Cauchy Problem for a Class of Coupled Systems Containing a Convolution Operator

Some results on the invariant regions ,existence and uniqueness of solutions to a class of integrodifferential systems are established. Applying these results to integrodifferential systems with a small parameter ε > 0, we obtain,in particular, some estimates of solutions uniform in ε > 0.     

非线性Klein-Gordon方程柯西问题解的整体存在性与Blow-up

研究非线性Klein-Gordon方程的柯西问题u_(tt)-Δu+u=u|u|~(p-1),x∈R~n,t>0;u(x,0)=u_0(x),u_t(x,0)=u_1(x),x∈R~n.通过引进一族位势井,得到了解的整体存在性与不存在的门槛结果.     

Initial boundary value problem for modified Zakharov equations

In this paper the authors consider the existence and uniqueness of the solution to the initial boundary value problem for a class of modified Zakharov equations, prove the global existence of the solution to the problem by a priori integral estimates and Galerkin method.     

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.     

关于海洋动力学中二维的大尺度原始方程组(Ⅰ)

考虑地球物理学中大尺度海洋运动的二维原始方程组的初边值问题．先假定海洋的深度为正的常数．首先,当初始数据是平方可积时,应用Faedo-Galerkin方法,得到了这一问题整体弱解的存在性．其次,当初始数据及其它们关于垂直方向的导数均为平方可积时,应用Faedo-Galerkin方法和各向异性不等式,得到了上述初边值问题的整体弱强解的存在、唯一性．     

Global smooth solution for nonlinearly damped p-system with large initial data

In this paper, we prove the existence of global smooth solution for the Cauchy problem of nonlinearly damped p-system with large initial data. The analysis is based on several key a prioriestirmates. which are obtained by the tnaxirnum principle. Our results extend the corresponding results in [l0,11].     

半导体模型中退化漂移-扩散问题解的存在性

研究半导体物理中出现的漂移—扩散模型,这是一个关于带电粒子浓度n,p,静电位ψ的抛物—椭圆耦合方程组,并带有混合初边值条件.对初值在L2+(Ω)时弱解的存在性展开讨论,利用正则化方法和适当的函数变换,使抛物型方程的解具有正下界n,p≥ε>0,同时得到一系列先验估计.然后通过紧性引理和Schauder不动点定理,得到初值在L2+(Ω)时弱解的存在性.     

On the Solutions of the Coupled Nonlinear Parabolic Equations

By means of the fixed point technique and integral estimation method, we study the solutions of periodic boundary value problem and initial value problem for the coupled nonlinear parabolic equations. The global classical solutions of the mentioned problems are shown to exists.     

The Cauchy problem for non-autonomous nonlinear Schrdinger equations

In this paper we study the Cauchy problem for cubic nonlinear Schrodinger equation with space- and time-dependent coefficients on Rm and Tm. By an approximation argument we prove that for suitable initial values, the Cauchy problem admits unique local solutions. Global existence is discussed in the cases of m = 1,2.