Ergodicity for the 3D stochastic Navier–Stokes equations

We consider the Kolmogorov equation associated with the stochastic Navier–Stokes equations in 3D, we prove existence of a solution in the strict or mild sense. The method consists in finding several estimates for the solutions u_m of the Galerkin approximations of u and their derivatives. These estimates are obtained with the help of an auxiliary Kolmogorov equation with a very irregular negative potential. Although uniqueness is not proved, we are able to construct a transition semigroup for the 3D Navier–Stokes equations. Furthermore, this transition semigroup has a unique invariant measure, which is ergodic and strongly mixing.     

Optimal control for 3D stochastic Navier–Stokes equations

Loeb space methods are used to prove existence of an optimal control for general 3D stochastic Navier–Stokes equations with multiplicative noise. The possible non-uniqueness of the solutions mean that it is necessary to utilize the notion of a non-standard approximate solution developed in the paper by N.J. Cutland and Keisler H.J. 2004, Global attractors for 3-dimensional stochastic Navier–Stokes equations, Journal of Dynamics and Differential Equations, pp. 16205–16266, for the study of attractors.     

Malliavin calculus for highly degenerate 2D stochastic Navier–Stokes equations

This Note mainly presents the results from “Malliavin calculus and the randomly forced Navier–Stokes equation” by J.C. Mattingly and E. Pardoux. It also contains a result from “Ergodicity of the degenerate stochastic 2D Navier–Stokes equation” by M. Hairer and J.C. Mattingly. We study the Navier–Stokes equation on the two-dimensional torus when forced by a finite dimensional Gaussian white noise. We give conditions under which the law of the solution at any time $t>0$, projected on a finite dimensional subspace, has a smooth density with respect to Lebesgue measure. In particular, our results hold for specific choices of four dimensional Gaussian white noise. Under additional assumptions, we show that the preceding density is everywhere strictly positive. This Note's results are a critical component in the ergodic results discussed in a future article. To cite this article: M. Hairer et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

On unbiased stochastic Navier–Stokes equations

A random perturbation of a deterministic Navier?CStokes equation is considered in the form of an SPDE with Wick type nonlinearity. The nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term ${u{\nabla}u}$ . This perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic Navier?CStokes equation. The perturbed equation is solved in the space of generalized stochastic processes using the Cameron?CMartin version of the Wiener chaos expansion. It is shown that the generalized solution is a Markov process and scales effectively by Catalan numbers.     

Approximations of stochastic Navier–Stokes equations

In this paper we show that solutions of two-dimensional stochastic Navier–Stokes equations driven by Brownian motion can be approximated by stochastic Navier–Stokes equations forced by pure jump noise/random kicks.     

Uniform global attractors for the nonautonomous 3D Navier–Stokes equations

We obtain the existence and the structure of the weak uniform (with respect to the initial time) global attractor and construct a trajectory attractor for the 3D Navier–Stokes equations (NSE) with a fixed time-dependent force satisfying a translation boundedness condition. Moreover, we show that if the force is normal and every complete bounded solution is strongly continuous, then the uniform global attractor is strong, strongly compact, and solutions converge strongly toward the trajectory attractor. Our method is based on taking a closure of the autonomous evolutionary system without uniqueness, whose trajectories are solutions to the nonautonomous 3D NSE. The established framework is general and can also be applied to other nonautonomous dissipative partial differential equations for which the uniqueness of solutions might not hold. It is not known whether previous frameworks can also be applied in such cases as we indicate in open problems related to the question of uniqueness of the Leray–Hopf weak solutions.     

On the uniqueness of weak solutions for the 3D Navier–Stokes equations

In this paper, we improve some known uniqueness results of weak solutions for the 3D Navier–Stokes equations. The proof uses the Fourier localization technique and the losing derivative estimates.     

A criterion for the existence of strong solutions for the 3D Navier–Stokes equations

In this note we provide a criterion for the existence of globally defined solutions for any regular initial data for the 3D Navier–Stokes system in Serrin’s classes.     

Axisymmetric solutions of the 3D Navier–Stokes equations for compressible isentropic fluids

We prove the existence of global weak solutions to the Navier–Stokes equations for compressible isentropic fluids for any γ>1 when the Cauchy data are axisymmetric, where γ is the specific heat ratio. Moreover, we obtain a new integrability estimate of the density in any neighborhood of the symmetric axis (the singularity axis).     

The approximation of a Crank–Nicolson scheme for the stochastic Navier–Stokes equations

In this paper we prove the convergence of stochastic Navier–Stokes equations driven by white noise. A linearized version of the implicit Crank–Nicolson scheme is considered for the approximation of the solutions to the N–S equations. The noise is defined as the distributional derivative of a Wiener process and approximated by using the generalized _L2$L_2$-projection operator. Optimal strong convergence error estimates in the _L2$L_2$ norm are obtained.     

2D constrained Navier–Stokes equations

We study 2D Navier–Stokes equations with a constraint forcing the conservation of the energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier–Stokes equation on ${\mathbb{R}}^2$ and ${\mathbb{T}}^2$, by a fixed point argument. We also show that the solution of the constrained equation converges to the solution of the Euler equation as the viscosity ν vanishes.     

Unique solvability for the density-dependent Navier–Stokes equations

In this paper we consider the incompressible Navier–Stokes equations with a density-dependent viscosity in a bounded domain $\Omega$ of ${\mathbf{R}}^n(n=2,3)$. We prove the local existence of unique strong solutions for all initial data satisfying a natural compatibility condition. This condition is also necessary for a very general initial data. Moreover, we provide a blow-up criterion for the regularity of the strong solution. For these results, the initial density need not be strictly positive. It may vanish in an open subset of $\Omega$.     

Existence of global stochastic flow and attractors for Navier–Stokes equations

For 2-D stochastic Navier-Stokes equations on the torus with multiplicative noise we construct a perfect cocycle and show the existence of global random compact attractors. The equations considered do not admit a pathwise method of solution. Received: 9 June 1998 / Revised version: 17 December 1998     

Existence and uniqueness of solutions to the backward 2D stochastic Navier–Stokes equations

The backward two-dimensional stochastic Navier–Stokes equations (BSNSEs, for short) with suitable perturbations are studied in this paper, over bounded domains for incompressible fluid flow. A priori estimates for adapted solutions of the BSNSEs are obtained which reveal a pathwise L^∞(H)$L^{\infty }\left(H\right)$ bound on the solutions. The existence and uniqueness of solutions are proved by using a monotonicity argument for bounded terminal data. The continuity of the adapted solutions with respect to the terminal data is also established.