Global Well-Posedness in Critical Besov Spaces for Two-Fluid Euler--Maxwell Equations

Abstract

In this paper, we study two-fluid compressible Euler-Maxwell equations in the whole space or periodic space. In comparison with the one-fluid case, we need to deal with the difficulty mainly caused by the nonlinear coupling and cancelation between electrons and ions. Precisely, the expected dissipation rates of densities for two carriers are no longer available. To capture the weaker dissipation, we develop a continuity for compositions, which is a natural generalization from Besov spaces to Chemin-Lerner spaces (space-time Besov spaces). An elementary fact that indicates the relation between homogeneous Chemin-Lerner spaces and inhomogeneous Chemin-Lerner spaces will been also used.

Global solutions to nonisentropic hydrodynamic models for two-carrier plasmas

“…As a matter of fact, we need another continuity result for composition to capture the dissipation rate from the cancellation of two densities of plasmas, which was first shown by [18].…”

“…To do this, Chemin-Lerner spaces (mixed space-time Besov spaces) in [16], which is a refinement of the usual spaces L ρ T (X), are introduced to construct new a priori estimates. In addition, the continuity for compositions in Chemin-Lerner spaces in [18] is also used to evaluate the cancellation of densities of two carriers, which is a natural generalization on the corresponding case in [19,Corollary 2.91,P. Precisely, we make the best use of an elementary fact in the localized energy approaches, which indicates the connection between homogeneous and inhomogeneous Chemin-Lerner spaces (see, e.g., [17]).…”

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

“…Peng [16] obtained the dissipation of the difference of densities in whole space and periodic space. The third author, Xiong & Kawashima [1] developed a continuity for compositions in space-time mixed Besov spaces (which is sometimes referred as Chemin-Lerner's spaces). Furthermore, the dissipation with respect to variables (∇ , ∇ ) and − was captured, which leads to the global-in-time existence of solutions in critical spaces.…”

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We obtain the optimal time-decay rate of classical solutions to two-fluid Euler-Maxwell equations in ℝ ( = 2, 3), which is a remaining question in the framework of critical Besov space (see [1]). The system is of regularity-loss, so it is difficult to get decay rates in the solution space.

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The frequency-localization technique and minimal decay-regularity for Euler–Maxwell equations

“…The first author [38] made the best use of the coupling structure of each equation in (1.5) and constructed the global classical solutions in spatially critical Besov spaces. So far there are a number of efforts on the Euler-Maxwell system (1.1) with or without dissipation, see [9,11,12,23,24,32,34,36,43] and therein references.…”