Stability of non‐constant steady‐state solutions for non‐isentropic Euler–Maxwell system with a temperature damping term

Abstract

This work is concerned with the periodic problem for compressible non-isentropic Euler-Maxwell systems with a temperature damping term arising in plasmas. For this problem, we prove the global in time existence of a smooth solution around a given non-constant steady state with the help of an induction argument on the order of the mixed time-space derivatives of solutions in energy estimates. Moreover, we also show the convergence of the solution to this steady state as the time goes to the infinity. This phenomenon on the charge transport shows the essential relation of the systems with the non-isentropic Euler-Maxwell and the isentropic Euler-Maxwell systems.

Stability of non-constant equilibrium solutions for two-fluid non-isentropic Euler-Maxwell systems arising in plasmas

“…Very recently, Liu-Peng [16] considered the stability of the one-fluid non-isentropic models for s ≥ 3. It should be pointed out that the techniques of choosing a non-diagonal symmetrizers and making a change of unknown variables in [16] can replace the help of diffusion effects of temperature used in [9]. We remark that the two-fluid non-isentropic…”

“…These techniques, firstly employed by Peng [25] in the one-fluid isentropic case and then extended by Feng-Peng-Wang [7] to the two-fluid isentropic case, can release the difficulty due to the appearance of non-constant equilibrium solutions. Besides of these techniques, by using diffusion effects of temperature, Feng-Wang-Li [9,15] proved the stability of non-constant equilibrium solution of the periodic problems to non-isentropic models for s ≥ 6. Very recently, Liu-Peng [16] considered the stability of the one-fluid non-isentropic models for s ≥ 3.…”

“…When b(x) is large, such a stability problem is much more complicated than before. Recently, motivated by the Guo-Strauss's work on the study of the damped Euler-Poisson system on the general bounded domain [12], by employing an induction argument on the order of the derivatives of solutions, the stabilities of non-constant equilibrium solutions for the isentropic Euler-Maxwell systems [7,25] and non-isentropic Euler-Maxwell systems with temperature diffusion terms [9,15], respectively. Very recently, with the help of choosing a new symmetrizer matrix, the stability of the one-fluid nonisentropic Euler-Maxwell systems is considered [16].…”

Stability of nonconstant steady‐state solutions for 2‐fluid nonisentropic Euler‐Poisson equations in semiconductor

“…Next, motivated by the work in Guo and Strauss, Peng used an induction argument on the order of the mixed time‐space derivatives of solutions in energy estimates and then overcome the difficulties due to appearance of nonconstant steady‐state solutions for isentropic Euler‐Poisson system in his study . After that, by the similar way, Feng, Peng, and Wang; Feng, Wang, and Li; and Li, Wang, and Feng considered the stability problems for the 2‐fluid isentropic case, 1‐fluid, and 2‐fluid nonisentropic cases with temperature diffusion terms, respectively. Recently, with the help of choosing a new symmetrizer matrix, the stability of the 1‐fluid nonisentropic Euler‐Poisson system is considered .…”

“…These techniques, firstly employed by Peng in the 1‐fluid isentropic case and then extended by Feng‐Peng‐Wang to the 2‐fluid isentropic case, can remove the difficulty due to the appearance of nonconstant equilibrium solutions. On the other hand, the techniques of choosing a nondiagonal symmetrizers and making a change of unknown variables in Liu and Peng can replace the help of diffusion effects of temperature that was used in Feng et al Moreover, we remark that the 2‐fluid nonisentropic Euler‐Poisson equations are much more complex than both the 2‐fluid isentropic and the 1‐fluid nonisentropic Euler‐Poisson equations because they contain 2 different charged fluids energy equations besides the equations of mass and momentum. Different from the 1‐fluid nonisentropic Euler‐Poisson systems in Liu and Peng, we shall overcome the difficulties caused by the coupling of 2 fluids when we establish the energy estimates.…”