The exponential behavior of the stochastic three-dimensional primitive equations with multiplicative noise

In this article, we study the stability of weak solutions to the stochastic three-dimensional (3D) primitive equations (PEs) with multiplicative noise. In particular, we prove that under some conditions on the forcing terms, the weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions. We also prove a result related to the stabilization of these equations.     

A global existence result for the heat flow of higher dimensional H-systems

We prove the existence of a global “small” weak solution to the flow of the H-system with initial–boundary conditions. We also analyze its time asymptotic behavior. Finally we give a stability result for weak solutions to the heat flow of higher dimensional H-systems.     

Optimal control of a quasi-variational obstacle problem

We consider an optimal control where the state-control relation is given by a quasi-variational inequality, namely a generalized obstacle problem. We give an existence result for solutions to such a problem. The main tool is a stability result, based on the Mosco-convergence theory, that gives the weak closeness of the control-to-state operator. We end the paper with some examples.     

Stability and regularity of weak solutions for a generalized thin film equation

In this paper we establish a stability result and an error estimate of weak solutions for the initial-boundary value problem of a generalized thin film equation and also obtain some higher regularity results for weak solutions.     

A new stability result for viscosity solutions of nonlinear parabolic equations with weak convergence in time

We present a new stability result for viscosity solutions of fully nonlinear parabolic equations which allows to pass to the limit when one has only weak convergence in time of the nonlinearities. To cite this article: G. Barles, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Dynamical behavior of a stochastic model of gene expression with distributed delay and degenerate diffusion

This article is concerned with a stochastic model of gene expression with distributed delay and degenerate diffusion. We transform the model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is of degenerate type, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the solutions can converge in L¹ to an invariant density. The existence of the stationary distribution implies stochastic weak stability. Numerical simulation is introduced to illustrate the analytical result.     

Existence of global weak solutions to one-component vlasov-poisson and fokker-planck-poisson systems in one space dimension with measures as initial data

We consider Cauchy problems for the 1-D one component Vlasov-Poisson and Fokker-Planck-Poisson equations with the initial electron density being in the natural space of arbitrary non-negative finite measures. In particular, the initial density can be a Dirac measure concentrated on a curve, which we refer to as “electron sheet” initial data. These problems resemble both structurally and functional analytically Cauchy problems for the 2-D Euler and Navier-Stokes equations (in vorticity formulation) with vortex sheet initial data. Here, we need to define weak solutions more specifically than usual since the product of a finite measure with a function of bounded variation is involved. We give a natural definition of the product, establish its weak stability, and existence of weak solutions follows. Our concept of weak solutions through the newly defined product is justified since solutions to the Fokker-Planck-Poisson equation, the analogue of Navier-Stokes equation, are shown to converge to weak solutions of the Vlasov-Poisson equation as the Fokker-Planck term vanishes. The main difficulty is the aforementioned weak stability which we establish through a careful analysis of the explicit structure of these equations. This is needed because the problem studied here is beyond the range of applicability of the “velocity averaging” compactness methods of DiPerna-Lions. © 1994 John Wiley & Sons, Inc.     

Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles

Dissipative solutions [12] of the Euler equations of incompressible fluids are obtained as the hydrodynamic limit of a properly scaled BGK equation. This stability result comes from refined entropy and entropy dissipation bounds. It uses in a crucial way the local conservation laws which are known to hold for weak solutions of this simplified model of the Boltzmann equation.     

Weak–strong uniqueness for a class of generalized dissipative weak solutions for non-homogeneous,non-Newtonian and incompressible fluids

We consider a system of PDEs describing a non-homogeneous, non-Newtonian and incompressible fluid in a space-periodic domain. We define the generalized dissipative weak solutions as well as give a proof of its existence, when the boundary data is provided. The main result of this work is the weak–strong uniqueness of the defined solutions.     

R~3中各向异性Landau-Lifshitz方程的部分正则解

证明了当三维空间中各向异性Landau-Lifshitz方程的弱解满足稳定性条件时,其解具有部分正则性,并且对易面类型的方程,利用Ginzburg-Landau逼近构造了一个整体的部分正则解。     

RESEARCH ANNOUNCEMENTS —— On the conditions for the Orbitally Asymptot

This paper studies the behaviors of the solutions in the vicinity of a givenalmost periodic solution of the autonomous system x′=f(x), x Rn , (1) where f C1 (Rn ,Rn ). Since the periodic solutions of the autonomous system are not Liapunov asymptotic stable, we consider the weak orbitally stability. 　　For the planar autonomous systems (n=2), the classical result of orbitally stability about its periodic solution with period w belongs to Poincare, i.e.     

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.     

BMO and the regularity criterion for weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity criterion for weak solutions to the incompressible magnetohydrodynamic equations. We derive the regularity of weak solutions in the marginal class. Moreover, our result demonstrates that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations.     

New approach to the incompressible Maxwell-Boussinesq approximation: Existence, uniqueness and shape sensitivity

The Boussinesq approximation to the Fourier-Navier-Stokes (F-N-S) flows under the electromagnetic field is considered. Such a model is the so-called Maxwell-Boussinesq approximation. We propose a new approach to the problem. We prove the existence and uniqueness of weak solutions to the variational formulation of the model. Some further regularity in W1,2+δ, δ>0, is obtained for the weak solutions. The shape sensitivity analysis by the boundary variations technique is performed for the weak solutions. As a result, the existence of the strong material derivatives for the weak solutions of the problem is shown. The result can be used to establish the shape differentiability for a broad class of shape functionals for the models of Fourier-Navier-Stokes flows under the electromagnetic field.     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

Dissipativity and stability for semilinear anomalous diffusion equations involving delays

We analyze the dissipativity and stability of solutions to a class of semilinear anomalous diffusion equations involving delays. The existence of absorbing set, the stability, and weak stability will be shown under suitable assumptions on the nonlinearity. Our analysis is based on new Halanay-type inequality, local estimates, and fixed-point arguments.     

Results for a turbulent system with unbounded viscosities: Weak formulations,existence of solutions,boundedness and smoothness

We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solution are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solution.     

Saddle-point solutions to Dirichlet problems on the Sierpiński gasket

We provide sufficient conditions for the existence of saddle-point solutions to a system driven by the weak Laplacian on the Sierpiński gasket. We analyze also its stability by proving its continuous dependence on parameters.     

Convergence of a Mixed Finite Element–Finite Volume Scheme for the Isentropic Navier–Stokes System via Dissipative Measure-Valued Solutions

We study convergence of a mixed finite element–finite volume numerical scheme for the isentropic Navier–Stokes system under the full range of the adiabatic exponent. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solutions of the limit system. In particular, using the recently established weak–strong uniqueness principle in the class of dissipative measure-valued solutions we show that the numerical solutions converge strongly to a strong solutions of the limit system as long as the latter exists.     

Stability of Solutions of Stochastic Functional-Differential Equations with Locally Lipschitz Coefficients in Hilbert Spaces

We obtain sufficient conditions for the stability of weak solutions of nonlinear stochastic functional-differential equations in Hilbert spaces with random coefficients satisfying the nonlocal Lipschitz condition.