Communications in Mathematical Sciences

Volume 18 (2020)

Number 5

Local well-posedness for the quantum Zakharov system

Pages: 1383 – 1411

DOI: https://dx.doi.org/10.4310/CMS.2020.v18.n5.a9

Authors

Yung-Fu Fang (Department of Mathematics, National Cheng Kung University, Tainan City, Taiwan)

Kuan-Hsiang Wang (Department of Applied Mathematics, National University of Kaohsiung, Taiwan)

Abstract

We consider the quantum Zakharov system in spatial dimensions greater than $1$. The local well-posedness is obtained for initial data of the electric field and of the ion density lying in some Sobolev spaces with certain regularities. For higher dimensions, the results cover the subcritical region. We get major part of the subcritical region for lower dimensions. For the quantum Zakharov system with initial data possessing the critical regularities, the local well-posedness is also proved for spatial dimensions greater than $7$. As the quantum parameter approaches zero, we prove the local well-posedness for Zakharov system which improves the known result.

Keywords

quantum Zakharov system, Zakharov system, local well-posedness, quantum parameter, Strichartz estimates, Fourier restriction norm method

2010 Mathematics Subject Classification

Primary 35L30. Secondary 35L05, 35Q55.

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The first author was partially supported by MOST, MRPC, TIMS, and NCTS (Taiwan).

The second author was partially supported by MOST, Taiwan (Grant No. 107-2811-M-390-501).

Received 5 September 2019

Accepted 1 March 2020

Published 23 September 2020