一般无界区域中带有阻尼的三维可压缩欧拉方程

考虑在一般的三维无界区域中的具有滑移边界条件的带有阻尼的可压缩欧拉方程.当初始值接近平衡态时,获得了全局存在性和唯一性.同时,研究了在半空间情形下系统的衰减率.证明了经典解的L~2范数以(1+t)~(-3/4)衰减到常值背景解.     

The Zero-Viscosity Limit of the 2D Navier–Stokes Equations

It is shown that in a radially symmetric configuration, the zero-viscosity limit of solutions of the Navier–Stokes equations satisfies the associated Euler equations. An ancillary result on continuous dependence of solutions on ν is also established.     

On the Two-Dimensional Navier—Stokes Equations with the Free Boundary Condition

In this article we consider the two-dimensional Navier—Stokes equations with free boundary condition (open surface), and derive a number of different results: a new orthogonal property for the nonlinear term, improved a priori estimates on the solution, an upper bound on the dimension of the attractor which agrees with the conventional theory of turbulence; finally, for elongated rectangular domains, an improved Lieb—Thirring (collective Sobolev) inequality leads to an upper bound on the dimension of the attractor which might be optimal. Accepted 11 July 1996     

Similarity Reductions and Exact Solutions for the Two-Dimensional Incompressible Navier–Stokes Equations

We study similarity reductions and exact solutions of the (2+1)-dimensional incompressible Navier–Stokes equations using the direct method originally developed by Clarkson and Kruskal [37]. The Navier–Stokes equations are reduced to their conventional stream function form, and it is shown that there exist essentially five reductions into lower-order partial differential equations. Furthermore, we study the possibilities for reducing each of these five forms to ordinary differential equations, some of which can be solved analytically, while others necessitate numerical treatment. In particular we exhibit several new reductions that are not obtained using the classical Lie group method of infinitesimal transformations, and thus we generate new exact solutions of the governing equations. Some of our solutions admit physical interpretations, and many of them contain well-known Navier–Stokes solutions as special examples.     

Existence of Solution for a Coupled System of Fractional Integro-Differential Equations on an Unbounded Domain

We present the existence of solution for a coupled system of fractional integro-differential equations. The differential operator is taken in the Caputo fractional sense. We combine the diagonalization method with Arzela-Ascoli theorem to show a fixed point theorem of Schauder.     

Existence and Regularity of a Class of Weak Solutions to the Navier–Stokes Equations

We construct a class of weak solutions to the Navier–Stokes equations, which have second order spatial derivatives and one order time derivatives, ofppower summability for 1 < p ≤ 5/4. Meanwhile, we show thatu L^s(0, T; W^{2, r}(Ω)) with 1/s + 3/2r = 2 for 1 < r ≤ 5/4.rcan be relaxed not to exceed 3/2 if we consider only in the interior of Ω. In the end, we extend the classical regularity theorem. Our results show thatuis a regular solution if u L^s(0, T; L^r(Ω)) with 1/s + 3/2r = 1 for Ω satisfying (1.3), with 1/s + 1/r = 5/6 for arbitrary domain inR³and 1 < s ≤ 2. For Ω = Rⁿwithn ≥ 3, this result was previously obtained byH. Beirão da Veiga (Chinese Ann. Math. Ser. B16, 1995, 407–412).     

无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

Approximate Solution of the Kuramoto-Shivashinsky Equation on an Unbounded Domain

The main goal of this paper is to approximate the Kuramoto-Shivashinsky (K-S for short) equation on an unbounded domain near a change of bifurcation, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider PDEs with quadratic nonlinearities and derive rigorously the modulation equation, which is called the Ginzburg-Landau (G-L for short) equation, for the amplitudes of the dominating modes.     

On the Solution of a Mixed Type Equation in an Unbounded Domain

Differential Equations - Methods for the numerical solution of a problem for a mixed type equation in an unbounded domain are developed. A method for setting artificial boundary conditions is...     

On Navier–Stokes Equations with Slip Boundary Conditions in an Infinite Pipe

The paper examines steady Navier–Stokes equations in a two-dimensional infinite pipe with slip boundary conditions. At both inlet and outlet, the velocity of flow is assumed to be constant. The main results show the existence of weak and regular solutions with no restrictions of smallness of the flux vector, also simply connectedness of the domain is not required.     

Global Attractor for Weakly Damped Nonlinear Schrodinger-Boussinesq Equations in an Unbounded Domain

In this paper the authors consider the Cauchy problem of nonlinear Schrödinger-Boussinesq equations in R and prove the existence of the maximal attractor.     

Global well posedness for the Maxwell–Navier–Stokes system in 2D

We prove global existence of regular solutions to the full MHD system (or more precisely the Maxwell–Navier–Stokes system) in 2D. We also provide an exponential growth estimate for the Hs norm of the solution when the time goes to infinity.     

一类广义长短波方程组在无界区域上的整体吸引子

研究了一类广义长短波方程组在无界区域上的解的长时间行为.通过引进加权空间,应用内插不等式和在加权空间的先验估计,证明了长短波方程组整体吸引子的存在性.     

Operator Estimates for the Steklov Problem in an Unbounded Domain with Rapidly Changing Conditions on the Boundary

Doklady Mathematics - A spectral problem of the Steklov type for the Laplacian in an unbounded domain with a smooth boundary is considered. The Steklov condition rapidly alternates with the...     

一类广义耦合的非线性波动方程组在无界区域上的整体吸引子

该文研究了一类广义耦合的非线性波动方程组在无界区域上的 整体吸引子问题. 利用插值不等式和加权空间, 得到了一系列关于时间的一致先验估计，证 明了整体吸引子的存在性.