Long-time dynamics of stochastic 2D hydrodynamic-type evolution equations driven by multiplicative L\'{e}vy noise

TL;DR

This paper studies the long-term behavior of solutions to stochastic hydrodynamic equations driven by Lévy noise, including existence of attractors and invariant measures, with applications to models like 2D Navier-Stokes.

Contribution

It establishes the existence of random attractors and invariant measures for a broad class of stochastic hydrodynamic models driven by Lévy noise, extending previous results.

Findings

Proved global well-posedness under Lipschitz and growth conditions.

Established existence and uniqueness of weak pullback mean random attractors.

Analyzed the asymptotic behavior of invariant measures as noise intensities vary.

Abstract

This paper investigates the long-time dynamics of solutions for an abstract nonlinear stochastic hydrodynamic-type equation driven by multiplicative L\'{e}vy noise. The framework encompasses several key hydrodynamical models, including the stochastic 2D Navier-Stokes equations, magnetohydrodynamic equations, the magnetic B\'{e}rnard problem, as well as various stochastic shell models of turbulence. Under the assumption that the nonlinear noise coefficients satisfy local Lipschitz and linear growth conditions, we first establish global well-posedness using a truncation technique. Then, by introducing a mean random dynamical system, we prove the existence and uniqueness of weak pullback mean random attractors for the system. Furthermore, when the external force is time-independent, we study the existence of invariant measures for the corresponding autonomous system, as well as the double limiting behavior of invariant measures with respect to the intensities of Gaussian and L\'{e}vy noise. Finally, under additional assumptions on the bilinear nonlinear term (e.g., as in the Navier-Stokes equations), we examine the existence and uniqueness of pullback measure attractors, along with the asymptotically autonomous stability of such attractors as the time parameter tends to negative infinity. It is worth noting that the results of this paper are new even for the single stochastic 2D Navier-Stokes equations.