Quantum Zakharov Model in a Bounded Domain

Abstract

We consider an initial boundary value problem for a quantum version of the Zakharov system arising in plasma physics. We prove the global well-posedness of this problem in some Sobolev type classes and study properties of solutions. This result confirms the conclusion recently made in physical literature concerning the absence of collapse in the quantum Langmuir waves. In the dissipative case the existence of a finite dimensional global attractor is established and regularity properties of this attractor are studied. For this we use the recently developed method of quasi-stability estimates. In the case when external loads are $C^\infty$ functions we show that every trajectory from the attractor is $C^\infty$ both in time and spatial variables. This can be interpret as the absence of sharp coherent structures in the limiting dynamics.