Asymptotic limits of solutions to the initial boundary value problem for the relativistic Euler–Poisson equations

• Published in: Journal of Differential Equations 2014-12, Vol.257 (12), p.4319-4368
• Main Authors: Mai, La-Su, Li, Hai-Liang, Zhang, Kaijun
• Format: Article
• Language: English
• Subjects: Combined zero-relaxation and non-relativistic limit; Global smooth solution; Non-relativistic limit; Relativistic Euler–Poisson equations
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2014.08.010

We study the asymptotic limit problem on the relativistic Euler–Poisson equations. Under the assumptions of both the initial data being the small perturbation of the given steady state solution and the boundary strength being suitably small, we have the following results: (i) the global smooth solution of the relativistic Euler–Poisson equation converges to the solution of the drift-diffusion equations provided the light speed c and the relaxation time τ satisfying c=τ−1/2 when the relaxation time τ tends to zero; (ii) the global smooth solution of the relativistic Euler–Poisson equations converges to the subsonic global smooth solution of the unipolar hydrodynamic model for semiconductors when the light speed c→∞. In addition, the related convergence rate results are also obtained.