Quantum Zakharov model in a bounded domain

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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 means that the quantum Zakharov model is coherent with the observation of an absence of collapse of (quantum) Langmuir waves, hence might be a valid model for the description of electronic plasma 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 forces are C ∞ functions, we show that every trajectory in the attractor is C ∞ in both time and spatial variables. This can be interpreted as the absence of sharp coherent structures in the limiting dynamics.

Mathematics Subject Classification (2010)

Primary 35Q40 Secondary 35B40 37L30 

Keywords

Quantum Zakharov equation Well-posedness Global attractor Finite fractal dimension 
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© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsKharkov National UniversityKharkovUkraine

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