Invariant measures and statistical solutions for a nonautonomous nonlocal Swift–Hohenberg equation

Abstract

This paper investigates a two-dimensional nonautonomous nonlocal Swift–Hohenberg equation with two kinds of kernels and studies the existence of invariant measures and statistical solutions, which are important research objects in the area of turbulence for fluid systems. The existence of weak solutions guarantees a norm-to-weak continuous process associated with the nonautonomous equation. We first prove the existence of the pullback attractor for the process via the pullback flattening. Then the unique existence of invariant measures is obtained by appropriate construction, so that the invariant measure is supported by this pullback attractor. This invariant measure is turned out to be exactly a statistical solution of the original nonlocal Swift–Hohenberg equation.