Decay estimates of solutions to the bipolar non-isentropic compressible Euler–Maxwell system

Abstract

We study the large time behavior of solutions near a constant equilibrium to the compressible Euler-Maxwell system in R 3 . We first refine a global existence theorem by assuming that the H 3 norm of the initial data is small, but the higher order derivatives can be arbitrarily large. If the initial data belongs toḢ −s (0 ≤ s < 3/2) orḂ −s 2,∞ (0 < s ≤ 3/2), by a regularity interpolation trick, we obtain the various decay rates of the solution and its higher order derivatives. As an immediate byproduct, the usual L p -L 2 (1 ≤ p ≤ 2) type of the decay rates follow without requiring that the L p norm of initial data is small.2010 Mathematics Subject Classification. 83C22; 82D37; 76N10; 35Q35; 35B40.

The optimal decay rate of strong solution for the compressible Navier–Stokes equations with large initial data

“…Specifically, they combined energy estimates with the interpolation between negative and positive Sobolev norms and obtained the time decay rates for the isentropic compressible Navier-Stokes equations and Boltzmann equation. The new method developed in [5] has a wide range of applications recently, see [38][39][40][41][42][43][44]. It should be noticed that all of these decay results are established under the assumption that the initial data is a small perturbation of constant equilibrium state.…”

The optimal time‐decay estimate of solutions to two‐fluid Euler‐Maxwell equations in the critical Besov space

“…Also, we would like to mention the recent works. [17,18] Tan, Wang and Tong [17] considered the bipolar non-isentropic compressible Euler-Maxwell system and established the optimal decay estimates, as the initial data belong to (ℝ 3 )( ≥ 5). In fact, their result shows that ≤ 5, so our paper can be regarded as the decay improvement of [17] in the isentropic case.…”

“…[17,18] Tan, Wang and Tong [17] considered the bipolar non-isentropic compressible Euler-Maxwell system and established the optimal decay estimates, as the initial data belong to (ℝ 3 )( ≥ 5). In fact, their result shows that ≤ 5, so our paper can be regarded as the decay improvement of [17] in the isentropic case. In [18], Ueda found a linear coupled hyperbolic-parabolic system, where the dissipative structure is weaker than the present Euler-Maxwell system at the low frequencies, since the decay rate is slower than one of heat kernel.…”

“…Although an accurate long-time asymptopic description is still out of reach, a number of results concerning the time-decay rates of global solutions, sometimes referred to as optimal decay rates of -, are available. See for example [8,9,13,[15][16][17] and references therein.…”

Asymptotic behavior of solutions to the compressible bipolar Euler–Maxwell system in $\mathbb{R}^3$

“…For the unipolar case, Chen, Jerome, and Wang [2] showed the global existence of entropy weak solutions to the initial-boundary value problem for arbitrarily large initial data in L ∞ (R), Guo and Tahvildar-Zadeh [11] showed a blow-up criterion for a spherically symmetric Euler-Maxwell system. Recently, there have been some results on the global existence and the asymptotic behavior of smooth solutions with small amplitudes, see [3,26,30,31]. For the asymptotic limits that derive simplified models starting from the Euler-Maxwell system, we refer to [13,22,35] for the relaxation limit, and to [35] for the non-relativistic limit, [20,21] for the quasineutral limit, [28,29] for WKB asymptotics and the references therein.…”