Dynamics of stochastic 2D Navier-Stokes equations
The authors study mainly two aspects of solutions of 2\(D\) stochastic Navier-Stokes equations driven by a multiplicative white noise with Dirichlet boundary conditions, namely existence of a perfect locally compacting \(C^{1,1}\) cocycle generated by all solutions of the equation, and the characterization of the local behavior of solutions near an equilibrium or a stationary point. The results are presented in Sections 3 and 4. In Section 3, the authors address the first problem and give in Theorem 3.2 the main result of this section i.e., that strong solutions of the stochastic Navier-Stokes equation generates a Fréchet \(C^{1,1}\) locally compacting cocycle. This result is obtained by combining a variational technique which allows one to transform the equation into a random Navier-Stokes equation and by using Galerkin approximations and a priori estimates. Section 4 is devoted to the study of the local behavior of solutions near an equilibrium or near a stationary point. First, the later notion is defined in Definition 4.1 and then, in section 4.1, the dynamics near a general equilibrium are described using a countable non-random Lyapunov spectrum of the associated Osledec operator (see Theorem 4.1 and Theorem 4.2 for hyperbolic stationary random solutions). Finally Section 4.2 is devoted to the same study but when the equilibrium is identically zero. Among other results, the authors provide the Lyapunov spectrum (see Theorem 4.3), as well as sufficient conditions on the parameters of the coefficients and on the geometry of the domain to obtain that zero is the unique equilibrium (see Theorem 4.6), and so-called local and global invariant manifolds Theorems for the stochastic Navier-Stokes equation (we refer to Theorem 4.6).
scientific article; zbMATH DE number 5708618
scientific article