Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data

Xueke Pu; Min Li

doi:10.3934/dcdsb.2019055

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2019, Volume 24, Issue 9: 5149-5181. Doi: 10.3934/dcdsb.2019055

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Asymptotic behaviors for the full compressible quantum Navier-Stokes-Maxwell equations with general initial data

Xueke Pu^1, and
Min Li^2, ,

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China
2.
Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, China

^* Corresponding author: Min Li
^* Corresponding author: Min Li

Received: November 2016

Revised: January 2019

Early access: April 2019

Published: July 2019

The first author is supported by NSFC grant 11871172

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In this paper, we study the asymptotic behaviors for the quantum Navier-Stokes-Maxwell equations with general initial data in a torus $\mathbb{T}^{3}$. Based on the local existence theory, we prove the convergence of strong solutions for the full compressible quantum Navier-Stokes-Maxwell equations towards those for the incompressible e-MHD equations plus the fast singular oscillating in time of the sequence of solutions as the Debye length goes to zero. We also mention that similar arguments can be applied to the Euler-Maxwell system. Remarkably, we eliminate the highly oscillating terms produced by the general initial data by using the formal two-timing method. Moreover, using the curl-div decomposition and elaborate energy estimates, we derive uniform (in the Debye length) estimates for the remainder system.

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Mathematics Subject Classification: Primary: 35B40, 35C20; Secondary: 35Q40.

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