Time Discrete Approximation of Weak Solutions to Stochastic Equations of Geophysical Fluid Dynamics and Applications

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans,the time discretization of these equations by an implicit Euler scheme is studied.From the deterministic point of view,the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions.From the probabilistic viewpoint,this paper deals with a wide class of nonlinear,state dependent,white noise forcings which may be interpreted in either the It6 or the Stratonovich sense.The proof of convergence of the Euler scheme,which is carried out within an abstract framework,covers the equations for the oceans,the atmosphere,the coupled oceanic-atmospheric system as well as other related geophysical equations.The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense,a result which is new by itself to the best of our knowledge.     

Singular Solutions to Conformal Hessian Equations

The authors show that for any e ∈]0,1[,there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B C R5 which belongs to C1'ε(B) \ C1'ε+(B).     

MBIUS-TODA HIERARCHY AND ITS INTEGRABILITY

In this paper, we construct a new integrable equation called Mbius-Toda equation which is a generalization of q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the Mbius-Toda equation and a whole integrable Mbius-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the Mbius-Toda hierarchy are given and this leads to the existence of the tau function.     

Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces

We consider a transmission problem consisting of two semilinear parabolic equations involving fractional diffusion operators of different orders in a general non-smooth setting with emphasis on Lipschitz interfaces and transmission conditions along the interface. We give a unified framework for the existence and uniqueness of strong and mild solutions, and their global regularity properties.     

抽象空间中非线性算子方程变号解的存在性研究

利用Banach空间中的锥理论和不动点定理讨论了非线性算子方程变号解的存在性,给出了E_u_0空间下非线性算子方程变号解至少有一个变号解、一个正解和一个负解的条件,并讨论了仅通过一个上解条件得出非线性算子方程变号解的存在性定理.     

一类具有收获率的脉冲Lotka-Volterra竞争合作系统的八个正概周期解

研究了一类具有收获率的脉冲Lotka-Volterra竞争合作系统的正概周期解.通过利用重合度理论延拓定理、概周期理论和不等式分析技巧,获得了系统至少存在8个正概周期解的充分条件,推广和改进了早期文献的相关结果.     

Degasperis-Procesi方程的类孤子新解

利用辅助方程与函数变换相结合的方法,构造了Degasperis-Procesi(D-P)方程的无穷序列类孤子新解.首先,通过两种函数变换,把D-P方程化为常微分方程组.然后,利用常微分方程组的首次积分,把D-P方程的求解问题化为几种常微分方程的求解问题.最后,利用几种常微分方程的Bcklund变换等相关结论,构造了D-P方程的无穷序列类孤子新解.这里包括由Riemannθ函数、Jacobi椭圆函数、双曲函数、三角函数和有理函数组成的无穷序列光滑孤立子解、尖峰孤立子解和紧孤立子解.     

三维空间中带非线性阻尼项的等熵欧拉方程组轴对称解的爆破

研究了三维空间中带非线性阻尼项的可压缩等熵欧拉方程组Dirichlet初边值问题.采用泛函方法,定义几种不同的泛函,当初始速度足够大时分别得到了经典解在某一时间内必定爆破的结论.由于出现了非线性阻尼项,较之线性阻尼的情形,经典解爆破的难度随之增加.     

Entropy Unilateral Solution for Some Noncoercive Nonlinear Parabolic Problems Via a Sequence of Penalized Equations

We give an existence result of the obstacle parabolic equations(b(x,u))/(t)-div(a(x,t,u,▽u))+div(φ(x,t,u))=f in Q_T,where b(x,u) is bounded function of u,the term-div(a(x,t,u,▽u)) is a Leray-Lions type operator and the function φ is a nonlinear lower order and satisfy only the growth condition.The second term f belongs to L~1(Q_T).The proof of an existence solution is based on the penalization methods.     

GLOBAL EXISTENCE OF WEAK SOLUTIONS FOR GENERALIZED QUANTUM MHD EQUATION

We prove the existence of a weak solution for a generalized quantum MHD equation in a 2-dimensional periodic box for large initial data. The existence of a global weak solution is established through a three-level approximation,energy estimates, and weak convergence for the adiabatic exponent γ 1.