Statistical solutions and Liouville theorem for the second order lattice systems with varying coefficients

Abstract

In this article, we first verify the global well-posedness of the second order lattice systems with varying coefficients. Then we prove that the solution mappings form a continuous process on the time-dependent phase spaces and the process has a time-dependent pullback attractor. Afterwards, we establish that there exists a family of Borel probability measures carried by the time-dependent pullback attractor which possesses invariant property under the action of the process. Further, we formulate the definition of statistical solution for the addressed evolution equations on time-dependent phase spaces and prove its existence. Our results reveal that the statistical solution of the second order lattice systems with varying coefficients satisfies the Liouville theorem in Statistical Mechanics. Finally, we propose some interesting issues concerning the singular limiting behavior as the varying coefficients tend to zero.