Pullback dynamics and statistical solutions for dissipative non-autonomous Zakharov equations

This article studies the pullback dynamics and statistical solutions for the dissipative non-autonomous Zakharov equations in a bounded interval. The main obstacle in our investigating comes from the geometric constrain of global regularity of solutions for the Zakharov equations. This geometric constrain is caused essentially by its particular structure which is a special coupling of a hyperbolic equation with a dispersive nonlinear Schrodinger equation. The nonlinear term A|E|2 in the hyperbolic equation is the core ingredient leading to this geometric constrain. Firstly we show the global existence and uniqueness of solutions, and prove the continuous dependence of solutions on the initial data by exploiting a certain coercivity of the operator corresponding to the linear principle part of the Zakharov equations. Then we establish that the generated process of solution mappings possesses a bounded pullback absorbing set and pullback asymptotic compactness by using a delicate decomposition of the original problem, and obtain the existence of a pullback attractor. Next we formulate an appropriate definition of v-continuity for the process and employ the calculus of variations to prove this v-continuity. Afterwards, we construct a family of invariant Borel probability measures for the process via the pullback attractor and the notion of generalized Banach limit. We also propose an appropriate definition of statistical solutions for the addressed Zakharov equations, and establish that the constructed family of invariant Borel probability measures is indeed a statistical solution which satisfies the Liouville theorem of Statistical Mechanics. Our definitions and approach present here can overcome effectively the difficulty caused by the geometric constrain of the Zakharov equations. Finally, we propose some open problems related to the existence and singular limiting behavior of the pullback attractors and statistical solutions for the Zakharov equations with varying coefficient. (c) 2024 Published by Elsevier Inc.