Multilinear Weighted Estimates and Quantum Zakharov System

Abstract

We consider the compact case of one-dimensional quantum Zakharov system, as an initial-value problem with periodic boundary conditions. We apply the Bourgain norm method to show low regularity local well-posedness for a certain class of Sobolev exponents that are sharp up to the boundary, under the condition that Schrödinger Sobolev regularity is non-negative. Using the conservation law and energy method, we show global well-posedness for sufficiently regular initial data, without any smallness assumption. Lastly we show the semi-classical limit as $\epsilon \to 0$ on a compact time interval, whereas the quantum perturbation proves to be singular on an infinite time interval.

18 pages

References in corpus (3)

• Modified Zakharov equations for plasmas with a quantum correction
• Arrest of Langmuir wave collapse by quantum effects
• On one dimensional Quantum Zakharov system