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In this article, we first prove some sufficient conditions guaranteeing the existence of invariant sample measures for random dynamical systems via the approach of global random attractors. Then we consider the two-dimensional incompressible Navier-Stokes equations with additive white noise as an example to show how to check the sufficient conditions for concrete stochastic partial differential equations. Our results generalize the Liouville type theorem to the random case and reveal that the invariance of the sample measures is a particular situation of the random Liouville type theorem

This paper studies the impulsive discrete Klein-Gordon-Schrödinger-type equations. We first prove that the problem of the discrete Klein-Gordon-Schrödinger-type equations with initial and impulsive conditions is global well-posedness. Then we establish that the solution operators form a continuous process and that this process possesses a pullback attractor and a family of invariant Borel probability measures. Further, we prove that this family of Borel probability measures satisfies the Liouville type theorem piecewise and is a statistical solution of the impulsive discrete Klein-Gordon-Schrödinger-type equations. Finally, we formulate the concept of Kolmogorov entropy for the statistical solution and estimate its upper bound.

We first prove the existence and regularity of the trajectory attractor for a three-dimensional system of globally modified Navier–Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.

We consider the nonautonomous modified Swift-Hohenberg equation $$u_t+\Delta^2u+2\Delta u+au+b|\nabla u|^2+u^3=g(t,x)$$ on a bounded smooth domain $\Omega\subset\R^n$ with $n\leqslant 3$. We show that, if $|b

We first verify the global well-posedness of the impulsive reaction-diffusion equations on infinite lattices. Then we establish that the generated process by the solution operators has a pullback attractor and a family of Borel invariant probability measures. Furthermore, we formulate the definition of statistical solution for the addressed impulsive system and prove the existence. Our results show that the statistical solution of the impulsive system satisfies merely the Liouville type theorem piecewise, and the Liouville type equation for impulsive system will not always hold true on the interval containing any impulsive point.

In this paper, we are interested in proving the existence of a weak martingale solution of the stochastic smectic-A liquid crystal system driven by a pure jump noise in both 2D and 3D bounded domains. We prove the existence of a global weak martingale solution under some non-Lipschitz assumptions on the coefficients. The construction of the solution is based on a Faedo–Galerkin approximation, a compactness method and the Skorokhod representation theorem. In the two-dimensional case, we prove the pathwise uniqueness of the weak solution, which implies the existence of a unique probabilistic strong solution.

In this article, the authors investigate the system of Schr odinger and Klein-Gordon equations with Yukawa coupling. They rst prove the existence of pullback attractor and construct a family of invariant Borel probability measures. Then they establish that this family of probability measures satis es a Liouville type theorem and is indeed a statistical solution for the coupling equations. Further, they reveal that the invariant property of the statistical solution is a particular situation of the Liouville type theorem. Ministerio de Ciencia e Innovación (MCIU). España PGC2018-096540-B-I00 Agencia Estatal de Investigación (AEI). España PGC2018-096540-B-I00 Fondo Europeo de Desarrollo Regional (FEDER) PGC2018-096540-B-I00

This article studies the probability distributions of solutions in the phase space for the discrete Zakharov equations. The authors first prove that the generated process of the solutions operators possesses a pullback-${\mathcal D}$ attractor, and then they establish that there exists a unique family of invariant Borel probability measures supported by the pullback attractor.

This paper studies the trajectory asymptotic behavior of a non-autonomous incompressible non-Newtonian fluid in 3D bounded domains. In appropriate topologies, the authors prove the existence of the uniform trajectory attractor for the translation semigroup acting on the united trajectory space.

This article studies the three-dimensional regularized Magnetohydrodynamics (MHD) equations. Using the approach of energy equations, the authors prove that the associated process possesses a pullback attractor. Then they establish the unique existence of the family of invariant Borel probability measures which is supported by the pullback attractor.

In this article, we first provide a sufficient and necessary condition for the existence of a pullback- \begin{document}$ {\mathcal D} $\end{document} attractor for the process defined on a Hilbert space of infinite sequences. As an application, we investigate the non-autonomous discrete Klein-Gordon-Schrodinger system of equations, prove the existence of the pullback- \begin{document}$ {\mathcal D} $\end{document} attractor and then the existence of a unique family of invariant Borel probability measures associated with the considered system.

In this article, we first prove, from the viewpoint of infinite dynamical system, sufficient conditions ensuring the existence of trajectory statistical solutions for autonomous evolution equations. Then we establish that the constructed trajectory statistical solutions possess invariant property and satisfy a Liouville type equation. Moreover, we reveal that the equation describing the invariant property of the trajectory statistical solutions is a particular situation of the Liouville type equation. Finally, we study the equations of three-dimensional incompressible magneto-micropolar fluids in detail and illustrate how to apply our abstract results to some concrete autonomous evolution equations.

This article investigates the three-dimensional globally modified Navier–Stokes equations with unbounded variable delays. Firstly, we prove the global well-posedness of the solutions, and give the existence of the pullback attractor for the associated process. Then, we construct a family of invariant Borel probability measures, which is supported by the pullback attractor.

In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics (MHD) equations in ℝ 3 . Then we establish that the solutions with initial data belonging to H m ( ℝ 3 ) ∩ L 1 ( ℝ 3 ) have the following time decay rate: ‖ ∇ m u ( x , t ) ‖ 2 + ‖ ∇ m b ( x , t ) ‖ 2 + ‖ ∇ m + 1 u ( x , t ) ‖ 2 + ‖ ∇ m + 1 b ( x , t ) ‖ 2 ≤ c ( 1 + t ) - 3 / 2 - m for large t , where m = 0, 1.