Communications in Mathematical Sciences

Volume 20 (2022)

Number 7

Asymptotic behavior of solutions to the outflow problem for the compressible Navier–Stokes–Maxwell equations

Pages: 1995 – 2027



Huancheng Yao (School of Mathematics, South China University of Technology, Guangzhou, China)

Changjiang Zhu (School of Mathematics, South China University of Technology, Guangzhou, China)


We investigate the large-time behavior of solutions toward the combination of the boundary layer and $3$‑rarefaction wave to the outflow problem for the compressible non-isentropic Navier–Stokes equations coupled with the Maxwell equations through the Lorentz force (called the Navier–Stokes–Maxwell equations) on the half line $\mathbb{R}_{+}$. It includes the electrodynamic effects into the dissipative structure of the hyperbolic-parabolic system and turns out to be more complicated than that in the simpler compressible Navier–Stokes equations. We prove that this typical composite wave pattern is time-asymptotically stable with the composite boundary condition of the electromagnetic fields, under some smallness conditions and the assumption that the dielectric constant is bounded. This can be viewed as the first result regarding the nonlinear stability of the combination of two different wave patterns for the IBVP of the non-isentropic Navier–Stokes–Maxwell equations.


non-isentropic Navier–Stokes–Maxwell equations, outflow problem, electromagnetic fields, dielectric constant, boundary layer, rarefaction wave

2010 Mathematics Subject Classification

35Q30, 35Q61

The authors’ research was supported by the National Natural Science Foundation of China grants 12171160, 11771150, 11831003; and by Guangdong Basic and Applied Basic Research Foundation grant 2020B1515310015.

Received 23 November 2021

Accepted 11 February 2022

Published 21 October 2022