Communications in Mathematical Sciences
Volume 18 (2020)
Local well-posedness for the quantum Zakharov system
Pages: 1383 – 1411
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.
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.
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