The Exact Limits and Improved Decay Estimates for All Order Derivatives of Global Weak Solutions of Three Incompressible Fluid Dynamics Equations

First of all, the author accomplishes the exact limits for all order derivatives of the global weak solutions of the $n$-dimensional incompressible magnetohydrodynamics equations, the $n$-dimensional incompressible Navier-Stokes equations and the two-dimensional incompressible dissipative quasi-geostrophic equation. Secondly, by making use of the exact limits, he establishes the improved decay estimates with sharp rates for all order derivatives of the global weak solutions, for all sufficiently large $t$. The author proves these results by making use of existing ideas, existing results and several new, novel ideas.     

On Well-Posedness of 2D Dissipative Quasi-Geostrophic Equation in Critical Mixed Norm Lebesgue Spaces

We establish local and global well-posedness of the 2D dissipative quasi-geostrophic equation in critical mixed norm Lebesgue spaces. The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation. The phenomenon is a priori nontrivial due to the nonlocal structure of the equation. Our approach is based on Kato's method using Picard's iteration, which can be adapted to the multi-dimensional case and other nonlinear non-local equations. We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.     

Asymptotic profile of solutions to the two-dimensional dissipative quasi-geostrophic equation

This paper is concerned with the asymptotic behavior of the two-dimensional dissipative quasi-geostrophic equation. Based on the spectral decomposition of the Laplacian operator and iterative techniques, we obtain improved L ² decay rates of weak solutions and derive more explicit upper bounds of higher order derivatives of solutions. We also prove the asymptotic stability of the subcritical quasi-geostrophic equation under large initial and external perturbations.     

多孔介质中两相不可压缩流体的时间局部网格加密有限差分格式

引入Charent压力变量,对于多孔介质中两相不可压缩流体的非混溶驱动问题,其模型表现为耦合的非线性偏微分方程组,一个是压力方程,另一个为饱和度方程．文中考虑一维问题且假定达西速度“已知,建立了在时间上进行局部加密的有限差分格式,给出了饱和度的最大模误差估计．最后给出了数值算例．     

Sobolev multipliers,maximal functions and parabolic equations with a quadratic nonlinearity

We develop a general framework to describe global mild solutions to a Cauchy problem with small initial values concerning a general class of semilinear parabolic equations with a quadratic nonlinearity. This class includes the Navier–Stokes equations, the subcritical dissipative quasi-geostrophic equation and the parabolic–elliptic Keller–Segel system.     

Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation

In this article we apply the method used in the recent elegant proof by Kiselev, Nazarov and Volberg of the well-posedness of critically dissipative 2D quasi-geostrophic equation to the super-critical case. We prove that if the initial value satisfies for some small number cs>0, where s is the power of the fractional Laplacian, then no finite time singularity will occur for the super-critically dissipative 2D quasi-geostrophic equation.     

Gevrey regularity for a class of dissipative equations with applications to decay

In this paper, following the techniques of Foias and Temam, we establish Gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional Laplacian. The initial data is taken to be in Besov type spaces defined via “caloric extension”. We apply our result to the Navier–Stokes equations, the surface quasi-geostrophic equations, the Kuramoto–Sivashinsky equation and the barotropic quasi-geostrophic equation. Consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the Navier–Stokes equations. In the 3D case, we extend the class of initial data where such decay holds while in 2D we provide a new class for such decay. Similar decay result, and uniform analyticity band on the attractor, is also proven for the sub-critical 2D surface quasi-geostrophic equation.     

Decay of Fourier modes of solutions to the dissipative surface quasi-geostrophic equations on a finite domain

We consider the two dimensional dissipative surface quasi-geostrophic equation on the unit square with mixed boundary conditions. Under some suitable assumptions on the initial stream function, we obtain existence and uniqueness of solutions in the form of a fast converging trigonometric series. We prove that the Fourier coefficients of solutions have a non-uniform decay: in one direction the decay is exponential and along the other direction it is only power like. We establish global wellposedness for arbitrary large initial data.     

不可压混流驱动问题的最小二乘混合元方法

1引言考虑多孔介质中两相不可压缩可混溶渗流驱动问题,它是由一组非线性耦合的椭园型压力方程和抛物型浓度方程组成：dVV。—一山人V什）gVV却）一q,VEn,（．1）＆,,。＿．、。。—一。x）＿＋u·grade－dlv（D（u）grade）一（1－c）q－,xEn,tEJ,（1．2）＆”－－’”””‘”－”””——－’——,、—’一其中a（）一a（x,c）一是（x）／卢（c）,J一［0,Ti,DcyR‘为水平油藏区域．方程式（1．l）一（1．2）中各物理量的意义如下：广为流体压力,c为流体的浓度,u为流体的Darer速度,叶为源汇项,／一—。x（q,O）,…     

Averaging of acoustic equation for partially perforated viscoelastic material with channels filled by a liquid

Acoustic equations for combined media consisting of partially perforated viscoelastic material and viscous incompressible liquid filling pores are considered. An averaged model is constructed for the model under consideration, and boundary conditions connecting equations of the obtained averaged model on the boundary between solid viscoelastic material and porous viscoelastic material filled by a viscous incompressible liquid are found. The convergence of limit problems to the solution of corresponding averaged problem with respect to the norm of the space L ² is proved.     

Global Small Solutions to an MHD‐Type System: The Three‐Dimensional Case

In this paper, we consider the global well‐posedness of a three‐dimensional incompressible MHD type system with smooth initial data that is close to some nontrivial steady state. It is a coupled system between the Navier‐Stokes equations and a free transport equation with a universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system due to the fact that there is a free transport equation of ? in the coupled equations and only the horizontal derivatives of ? is dissipative with respect to time. To achieve this, we first employ anisotropic Littlewood‐Paley analysis to establish the key L¹(? ⁺ ; Lip(?³)) estimate to the third component of the velocity field. Then we prove the global well‐posedness to this system by the energy method, which depends crucially on the divergence‐free condition of the velocity field. © 2014 Wiley Periodicals, Inc.     

A FINITE ELEMENT COLLOCATION METHOD FOR TWO-PHASE INCOMPRESSIBLE IMMISCIBLE PROBLEMS

Two-phase,incompressible,immiscible flow in porous media is governed by a coupled system of nonlinear partial differential equations.The pressure equation is elliptic, whereas the concentration equation is parabolic,and both are treated by the collocation scheme.Existence and uniqueness of solutions of the algorithm are proved.A optimal convergence analysis is given for the method.     

Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces

In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove a global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces Lp, with p∈[1,∞]. Local results for arbitrary initial data are also given.     

Dissipativity of delay functional differential equations with bounded lag

Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ=1. One numerical example is given to confirm our theoretical result.     

Existence of solutions to various rock types odel model of two-phase flow in porous media

We study the flow of two immiscible and incompressible fluids through a porous media c,onsisting of different rock types: capillary pressure and relative permeablities curves are different in each type of porous media. This process can be formulated as a coupled system of partial differential equations which includes an elliptic pressurevelocity equation and a nonlinear degenerated parabolic saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. A change of unknown leads to a new formulation of this problem. We derive a weak form for this new problem, which is a combination of a mixed formulation for the elliptic pressure-velocity equation and a standard variational formulation for the new parabolic equation. Under some realistic assumptions, we prove the existence of weak solutions to the implicit system given by time discretization.     

SPDE in Hilbert space with locally monotone coefficients

The aim of this paper is to extend the usual framework of SPDE with monotone coefficients to include a large class of cases with merely locally monotone coefficients. This new framework is conceptually not more involved than the classical one, but includes many more fundamental examples not included previously. Thus our main result can be applied to various types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier-Stokes equation, stochastic p-Laplace equation and stochastic porous media equation with non-monotone perturbations.     

Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system

A well-known diffuse interface model consists of the Navier–Stokes equations nonlinearly coupled with a convective Cahn–Hilliard type equation. This system describes the evolution of an incompressible isothermal mixture of binary fluids and it has been investigated by many authors. Here we consider a variant of this model where the standard Cahn–Hilliard equation is replaced by its nonlocal version. More precisely, the gradient term in the free energy functional is replaced by a spatial convolution operator acting on the order parameter φ, while the potential F may have any polynomial growth. Therefore the coupling with the Navier–Stokes equations is difficult to handle even in two spatial dimensions because of the lack of regularity of φ. We establish the global existence of a weak solution. In the two-dimensional case we also prove that such a solution satisfies the energy identity and a dissipative estimate, provided that F fulfills a suitable coercivity condition.     

The Exact Limits and Improved Decay Estimates for All Order Derivatives of the Global Weak Solutions to a Two-Dimensional Incompressible Dissipative Quasi-Geostrophic Equation

We will accomplish the exact limits for all order derivatives of the global weak solutions to a two-dimensional incompressible dissipative quasi-geostrophic equation. We will also establish the improved decay estimates with sharp rates for all order derivatives. We will consider two cases for the initial function and the external force and prove the optimal results for both cases. We will couple together existing ideas (including the Fourier transformation and its properties, Parseval''s identity, iteration technique, Lebesgue''s dominated convergence theorem, Gagliardo-Nirenberg-Sobolev interpolation inequality, squeeze theorem, Cauchy-Schwartz''s inequality, etc) existing results (the existence of global weak solutions, the existence of local smooth solution on $(T,\infty)$ and the elementary decay estimate with a sharp rate) and a few novel ideas to obtain the main results.     

A NEW NUMERICAL METHOD FOR TWO-PHASE IMMISCIBLE INCOMPRESSIBLE PROBLEM

Two-phase, immiscible, incompressible flow in porous media is governed by a system of nonlinear partial differential equations. In most practical applications convection physically dominates diffusion, and the object of this paper is to develop a finite difference method combined with the method of characteristics and the lumped mass method to treat the parabolic equation of the differential system. This method is shown satisfy the maximum principle and its error analysis is presented.     

不可压饱和多孔弹性杆动力响应的多辛方法

研究了不可压饱和多孔弹性杆的一维动力响应问题.基于多孔介质理论,在流相和固相微观不可压、固相骨架小变形的假定下,建立了不可压流体饱和多孔弹性杆一维轴向动力响应的数学模型.利用Hamilton空间体系的多辛理论,构造了不可压饱和多孔弹性杆轴向振动方程的多辛形式及其多种局部守恒律.采用中点Box离散方法得到轴向振动方程的多辛离散格式和局部能量守恒律以及局部动量守恒律的离散格式;数值模拟了不可压饱和多孔弹性杆的轴向振动过程,记录了每一时间步的局部能量数值误差和局部动量数值误差.结果表明,已构造的多辛离散格式具有很高的精确性和较长时间的数值稳定性,这为解决饱和多孔介质的动力响应问题提供了新的途径.