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Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion. (English) Zbl 1412.35248

Summary: This paper studies the asymptotic behavior of smooth solutions to the generalized Hall-magneto-hydrodynamics system (1.1) with one single diffusion on the whole space \(\mathbb{R}^3\). We establish that, in the inviscid resistive case, the energy \(\| b(t) \|_2^2\) vanishes and \(\| u(t) \|_2^2\) converges to a constant as time tends to infinity provided the velocity is bounded in \(W^{1 - \alpha, \frac{3}{\alpha}}(\mathbb{R}^3)\); in the viscous non-resistive case, the energy \(\| u(t) \|_2^2\) vanishes and \(\| b(t) \|_2^2\) converges to a constant provided the magnetic field is bounded in \(W^{1 - \beta, \infty}(\mathbb{R}^3)\). In summary, one single diffusion, being as weak as \((- {\Delta})^\alpha b\) or \((- {\Delta})^\beta u\) with small enough \(\alpha, \beta\), is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs
35Q85 PDEs in connection with astronomy and astrophysics
76W05 Magnetohydrodynamics and electrohydrodynamics
35B65 Smoothness and regularity of solutions to PDEs
35R11 Fractional partial differential equations
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References:

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