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Darcy’s law and diffusion for a two-fluid Euler-Maxwell system with dissipation. (English) Zbl 1330.35318

Summary: This paper is concerned with the large-time behavior of solutions to the Cauchy problem on the two-fluid Euler-Maxwell system with dissipation when initial data are around a constant equilibrium state. The main goal is the rigorous justification of diffusion phenomena in fluid plasma at the linear level. Precisely, motivated by the classical Darcy’s law for the nonconductive fluid, we first give a heuristic derivation of the asymptotic equations of the Euler-Maxwell system in large time. It turns out that both the density and the magnetic field tend time-asymptotically to the diffusion equations with diffusive coefficients explicitly determined by given physical parameters. Then, in terms of the Fourier energy method, we analyze the linear dissipative structure of the system, which implies the almost exponential time-decay property of solutions over the high-frequency domain. The key part of the paper is the spectral analysis of the linearized system, exactly capturing the diffusive feature of solutions over the low-frequency domain. Finally, under some conditions on initial data, we show the convergence of the densities and the magnetic field to the corresponding linear diffusion waves with the rate \((l+t)^{-{\frac{5}{4}}}\) in \(L^{2}\)-norm and also the convergence of the velocities and the electric field to the corresponding asymptotic profiles given in the sense of the generalized Darcy’s law with the faster rate \((l+t)^{-{\frac{7}{4}}}\) in \(L^{2}\)-norm. Thus, this work can be also regarded as the mathematical proof of the Darcy’s law in the context of collisional fluid plasma.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
76S05 Flows in porous media; filtration; seepage
35Q60 PDEs in connection with optics and electromagnetic theory
76K05 Hypersonic flows
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