Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds

Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation D _t u _t  = D _x u _x  + (X _u  + λ₀(u)u _t  + λ₁(u)u _x )[Wdot] where X is a continuous vector field on M, λ₀ and λ₁ are continuous vector bundles homomorphisms from TM to TM, and W is a spatially homogeneous Wiener process on ? with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.     

Existence of Weak Solutions Up to Collision for Viscous Fluid‐Solid Systems with Slip

We study in this paper the movement of a rigid solid inside an incompressible Navier‐Stokes flow within a bounded domain. We consider the case where slip is allowed at the fluid/solid interface through a Navier condition. Taking into account slip at the interface is very natural within this model, as classical no‐slip conditions lead to unrealistic collisional behavior between the solid and the domain boundary. We prove for this model existence of weak solutions of Leray type, up to collision, in three dimensions. The key point is that, due to the slip condition, the velocity field is discontinuous across the fluid/solid interface. This prevents obtaining global H¹ bounds on the velocity, which makes many aspects of the theory of weak solutions for Dirichlet conditions inappropriate. © 2014 Wiley Periodicals, Inc.     

Global Weak Solutions for the Cauchy Problem to One-Dimensional Heat-Conductive MHD Equations of Viscous Non-resistive Gas

We investigate the dynamics of the Vlasov-Poisson system in the presence of radiation damping. A propagation result for velocity moments of order \(k>3\) is established in (Kunze and Rendall in Ann. Henri Poincaré 2:857–886, 2001). In this paper, we prove existence of global solutions propagating velocity and velocity-spatial moments of order \(k>2\) and establish an explicit polynomially growing in time bound on the moments.

     

Variational Characterizations of Weak Solutions to the Dirichlet Problem for Mixed‐Type Equations

For linear and nonlinear second‐order partial differential equations of mixed elliptic‐hyperbolic type, we prove that weak solutions to the Dirichlet problem are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functional restricted to suitable infinite‐dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights associated to the linear operator. This spectral theory has been recently developed by the authors, which in turn exploits weak well‐posedness results obtained by Morawetz and the authors. © 2015 Wiley Periodicals, Inc.     

Weak Formulation of Free-Surface Wave Equations

An alternative method for deriving water wave dispersion relations and evolution equations is to use a weak formulation. The free-surface displacement η is written as an eigenfunction expansion, [equation] where the an(t) are time-dependent coefficients. For a tank with vertical sides the E_n are eigenfunctions of the eigenvalue problem, [equation] on the tank side walls. Evolution equations for the a_n(t) can be obtained by taking inner products of the lin-earised equation of motion, [equation] with the normal irrotational wave modes. For unforced waves each evolution equation is a simple harmonic oscillator, but the method is most useful when the body force F is something more exotic than gravity. It can always be represented by a forcing term in the SHM evolution equation, and it is not necessary to assume F irrotational. Several applications are considered: the Faraday experiment, generation of surface waves by an unsteady magnetic field, and the metal-pad instability in aluminium reduction cells.     

Strong and Weak Property of Travelling Wave Front Solutions for a Class of Degenerate Parabolic Equations

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｉｎｔｈｉｓｐａｐｅｒｗｅｓｔｕｄｙｔｈｅｓｔｒｏｎｇａｎｄｗｅａｋｐｒｏｐｅｒｔｙｏｆｔｒａｖｅｌｌｉｎｇｗａｖｅｆｒｏｎｔｓｏｌｕｔｉｏｎｓ (ＴＷＦＳ)ｆｏｒｔｈｅｆｏｌｌｏｗｉｎｇｄｅｇｅｎｅｒａｔｅｐａｒａｂｏｌｉｃｅｑｕａｔｉｏｎ : ψ(ｕ) ｔ = 2 ｕ ｘ2 +ｆ(ｕ) . ( 1 .1 )　Ｗｅｌｏｏｋｆｏｒｓｏｌｕｔｉｏｎｓｏｆ( 1 .1 )ｏｆｔｈｅｆｏｒｍｕ(ｘ,ｔ) =ｑ(ｘ -ｃｔ) (ｃｉｓｔｈｅｗａｖｅｓｐｅｅｄ) ,ｗｈｅｒｅｑ( ξ) ,ξ=ｘ -ｃｔ,ａｓａｆｕｎｃｔｉｏｎｏｆａｓ…     

Weak Solutions of Monge-Ampére Type Equations in Optimal Transportation

This paper concerns the weak solutions of some Monge-Ampére type equations in the optimal transportation theory. The relationship between the Aleksandrov solutions and the viscosity solutions of the Monge-Ampére type equations is discussed. A uniform estimate for solution of the Dirichlet problem with homogeneous boundary value is obtained.     

对称正则长波方程的行波解分岔

通过平面动力系统的方法讨论了对称正则长波方程的分岔问题.得到了该方程的分岔条件,在一些参数的具体值的情况下给出相图并通过微分方程的数值模拟方法模拟出了该方程的周期行波解、孤立行波解及无界行波解.     

非线性波方程新的显式精确解

利用Darboux和一个可化为标准Bernoulli方程的4阶常微分方程,统一地处理了三个著名方程KdV方程,Kadomtsev-Petviashvili(KP)方程和Hirota-Satsuma(HS)方程的求解问题.给出了这些方程一批新的具有更为丰富形式的精确解,其中包括孤波解和行波解.     

非线性演化方程的孤立波解

用齐次平衡原则和辅助微分方程方法得到了6个重要的n次非线性演化方程的孤立波解.辅助微分方程方法的主要思想是借助简单的可解微分方程的解去构造复杂的非线性演化方程的行进波解.这里简单的可解微分方程称为辅助微分方程.本文使用的辅助方程有双曲正割幂型解或双曲正切幂型解.     

变更Boussinesq方程和Kupershmidt方程的多孤子解

使用王明亮引进的齐次平衡方法,求出了变更Boussinesq方程和Kupershmidt方程的多孤子解,而王明亮给出的变更的Boussinesq方程的单孤子解仅是上述结果的一种特殊情况.     

一类非线性抛物方程弱解的L~2 衰减(英文)

本文研究了非线性抛物方程弱解的L2 衰减 ,证明了若u是非线性抛物方程的一个弱解 ,则u满足‖u‖L2 ≤C( 1 t) -n/ 4,其中C是常数且仅依赖于初始值u0 的范数 .     

Jaulent-Miodek方程的行波解分支

利用平面动力系统分支理论研究了耦合的Jaulent-Miodek方程的孤立波及周期波的存在性,求出了分支参数集．在给定的参数条件下,得到了该方程光滑孤立波解及周期行波解的所有可能的显式表达式．     

On Weak Solutions of Stochastic Differential Equations

A new proof of existence of weak solutions to stochastic differential equations with continuous coefficients based on ideas from infinite-dimensional stochastic analysis is presented. The proof is fairly elementary, in particular, neither theorems on representation of martingales by stochastic integrals nor results on almost sure representation for tight sequences of random variables are needed.     

Klein-Gordon-Zakharov方程组的周期波解

通过使用符号计算系统Mathematica,并借助于推广的F-展开法,我们得到了Klein- Gordon-Zakharov方程组的用不同Jacobi椭圆函数表示的一系列周期波解．在极限情况下,还求出了对应的孤立波解．     

一类次椭圆方程弱解的正则性

在区域Ω上考虑一类由退化向量场形成的Schrodinger方程： ∑i,j=1^mXi^＊（aij（x）Xju）-vu=0 其中X1,…,Xm为R^n（n≥）3上满足Hormander条件的实C^∞向量场,Xi^＊为Xi的形式共轭,v属于Kato类的某一类比Kη^loc（Ω）.并得到以下结果：若u为以上方程的弱解,则｜Xu｜^2w=∑i=1^m｜Xiu｜^2w∈Kη^loc（Ω）.     

粘性流体运动自型问题的分析解

本文用逐步逼近法得到了粘性流体运动的自型问题的微分方程(1.1～1.4)的分析解Проснак(1969)用小参数法也得到了这些方程的解.但他把控制方程变换成为一组线性变系数微分方程.本文则把控制方程变换成为线性常系数微分方程.     

非线性波方程准确孤立波解的符号计算

该文将机械化数学方法应用于偏微分方程领域，建立了构造一类非线性发展方程孤立波解的一种统一算法，并在计算机数学系统上加以实现，推导出了一批非线性发展方程的精确孤立波解．算法的基本原理是利用非线性发展方程孤立波解的局部性特点，将孤立波表示为双曲正切函数的多项式．从而将非线性发展方程（组）的求解问题转化为非线性代数方程组的求解问题．利用吴文俊消元法在计算机代数系统上求解非线性代数方程组，最终获得非线性发展方程（组）的准确孤立波解.