Martingale and stationary solutions for stochastic Navier-Stokes equations

Summary We prove the existence of martingale solutions and of stationary solutions of stochastic Navier-Stokes equations under very general hypotheses on the diffusion term. The stationary martingale solutions yield the existence of invariant measures, when the transition semigroup is well defined. The results are obtained by a new method of compactness.     

Non-monotone stochastic generalized porous media equations

By using the Nash inequality and a monotonicity approximation argument, existence and uniqueness of strong solutions are proved for a class of non-monotone stochastic generalized porous media equations. Moreover, we prove for a large class of stochastic PDE that the solutions stay in the smaller L²-space provided the initial value does, so that some recent results in the literature are considerably strengthened.     

Stochastic generalized porous media and fast diffusion equations

We present a generalization of Krylov-Rozovskii's result on the existence and uniqueness of solutions to monotone stochastic differential equations. As an application, the stochastic generalized porous media and fast diffusion equations are studied for σ-finite reference measures, where the drift term is given by a negative definite operator acting on a time-dependent function, which belongs to a large class of functions comparable with the so-called N-functions in the theory of Orlicz spaces.     

Large deviations for the Boussinesq equations under random influences

A Boussinesq model for the Bénard convection under random influences is considered as a system of stochastic partial differential equations. This is a coupled system of stochastic Navier–Stokes equations and the transport equation for temperature. Large deviations are proved, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion.     

Degenerate two-phase incompressible flow II: regularity, stability and stabilization

In this paper, we analyze a coupled system of highly degenerate elliptic-parabolic partial differential equations for two-phase incompressible flow in porous media. This system involves a saturation and a global pressure (or a total flow velocity). First, we show that the saturation is Hölder continuous both in space and time and the total velocity is Hölder continuous in space (uniformly in time). Applying this regularity result, we then establish the stability of the saturation and pressure with respect to initial and boundary data, from which uniqueness of the solution to the system follows. Finally, we establish a stabilization result on the asymptotic behavior of the saturation and pressure; we prove that the solution to the present system converges (in appropriate norms) to the solution of a stationary system as time goes to infinity. An example is given to show typical regularity of the saturation.     

On the method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations

3D stochastic Navier-Stokes equations with a suitable nondegenerate noise are considered. Following a method introduced by Da Prato and Debussche, it is proved that every Markov process associated to the equations has a Strong Feller like continuity property with respect to initial conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Stochastic Navier-Stokes equations

We construct a solution to stochastic Navier-Stokes equations in dimension n4 with the feedback in both the external forces and a general infinite-dimensional noise. The solution is unique and adapted to the Brownian filtration in the 2-dimensional case with periodic boundary conditions or, when there is no feedback in the noise, for the Dirichlet boundary condition. The paper uses the methods of nonstandard analysis.The research of this author was supported by an SERC Grant.     

2D backward stochastic Navier-Stokes equations with nonlinear forcing

The paper is concerned with the existence and uniqueness of a strong solution to a two-dimensional backward stochastic Navier-Stokes equation with nonlinear forcing, driven by a Brownian motion. We use the spectral approximation and the truncation and variational techniques. The methodology features an interactive analysis on the basis of the regularity of the deterministic Navier-Stokes dynamics and the stochastic properties of the Itô-type diffusion processes.     

Ergodicity of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noise

We prove that any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e. all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.     

Dispersive blow-up for nonlinear Schrödinger equations revisited

The possibility of finite-time, dispersive blow-up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. In dimension one, the fact that dispersive blow up does occur for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey–Stewartson and Gross–Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.     

Random modulation of solitons for the stochastic Korteweg–de Vries equation

We study the asymptotic behavior of the solution of a Korteweg–de Vries equation with an additive noise whose amplitude ε   tends to zero. The noise is white in time and correlated in space and the initial state of the solution is a soliton solution of the unperturbed Korteweg–de Vries equation. We prove that up to times of the order of 1/ε²$1/{\epsilon }^2$, the solution decomposes into the sum of a randomly modulated soliton, and a small remainder, and we derive the equations for the modulation parameters. We prove in addition that the first order part of the remainder converges, as ε tends to zero, to a Gaussian process, which satisfies an additively perturbed linear equation.     

Degenerate two-phase incompressible flow III. Sharp error estimates

Summary. This is the third paper of a series in which we analyze mathematical properties and develop numerical methods for a degenerate elliptic-parabolic partial differential system which describes the flow of two incompressible, immiscible fluids in porous media. In this paper we consider a finite element approximation for this system. The elliptic equation for the pressure and velocity is approximated by a mixed finite element method, while the degenerate parabolic equation for the saturation is approximated by a Galerkin finite element method. A fully discrete approximation is analyzed. Sharp error estimates in energy norms are obtained for this approximation. The error analysis does not use any regularization of the saturation equation; the error estimates are derived directly from the degenerate equation. Also, the analysis does not impose any restriction on the nature of degeneracy. Finally, it respects the minimal regularity on the solution of the differential system. Received March 9, 1998 / Revised version received July 17, 2000 / Published online May 30, 2001     

Effective velocity in compressible Navier-Stokes equations with third-order derivatives

A formulation of certain barotropic compressible Navier-Stokes equations with third-order derivatives as a viscous Euler system is proposed by using an effective velocity variable. The equations model, for instance, viscous Korteweg or quantum Navier-Stokes flows. The formulation in the new variable allows for the derivation of an entropy identity, which is known as the BD (Bresch-Desjardins) entropy equation. As a consequence of this estimate, a new global-in-time existence result for the one-dimensional quantum Navier-Stokes equations with strictly positive particle densities is proved.     

On the regularity of a class of generalized quasi-geostrophic equations

In this article we consider the following generalized quasi-geostrophic equation

     

Homogenization of a three-phase flow model in porous media

We give homogenization results for an immiscible and incompressible three-phase flow model in a heterogeneous petroleum reservoir with periodic structure, including capillary effects. We consider a model which leads to a coupled system of partial differential equations which includes an elliptic equation and two nonlinear degenerate parabolic equations of convection–diffusion types. Using two-scale convergence, we get an homogenized model which governs the global behavior of the flow. The determination of effective properties require the numerical resolution of local problems in a standard cell.     

A global existence result for the equations describing unsaturated flow in porous media with dynamic capillary pressure

In this paper we investigate the pseudoparabolic equation

     

A free boundary problem for the Navier–Stokes equations with measure data

The existence of a weak solution of an initial boundary-value problem for the plane nonstationary Navier–Stokes equations with Radon measure data on the free boundary, is established. The problem may be considered as a model of the blood flow around the heart valves. An inverse problem is studied, it allows us to find the boundary forces acting on the valve from the observed values of the velocity of the fluid in a fixed subregion.     

Stochastic partial differential equations in Hölder spaces

Summary We prove the existence and regularity of solutions to stochastic partial differential equations of parabolic Itô type in Hölder spaces under the usual sublinear growth and local Lipschitz conditions. Some examples are given to which our main theorems apply.The work of the first author was supported in part by the NSF grant DMS-91-01360