The Hall-magnetohydrodynamics (Hall-MHD) equations, rigorously derived from kinetic models, are useful in describing many physical phenomena in geophysics and astrophysics. This paper studies the local well-posedness of classical solutions to the Hall-MHD equations with the magnetic diffusion given by a fractional Laplacian operator, (−∆) α . Due to the presence of the Hall term in the Hall-MHD equations, standard energy estimates appear to indicate that we need α ≥ 1 in order to obtain the local well-posedness. This paper breaks the barrier and shows that the fractional Hall-MHD equations are locally well-posed for any α > 1 2 . The approach here fully exploits the smoothing effects of the dissipation and establishes the local bounds for the Sobolev norms through the Besov space techniques. The method presented here may be applicable to similar situations involving other partial differential equations.2010 Mathematics Subject Classification. 35Q35, 35B65, 35Q85, 76W05.

In this paper, we obtain global existence and energy decay for 3D Hall-magnetohydrodynamics (Hall-MHD) system with − u and − B. Besides the classical energy method and Besov space techniques, the interpolating inequalities are crucial in the proof of decay estimates. Then two Osgood type blow-up criteria are established. Our results improve the corresponding theorems in [3] and [4]. In addition, we establish two Beale-Kato-Majda blow-up criterion for the generalized version of Hall-MHD with − u and (− ) β B, β > 1.

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In this paper, we obtain the uniqueness of the 2D MHD equations, which fills the gap of recent work [2] by Chemin et al.

In this paper, we prove global well-posedness of smooth solutions to the twodimensional incompressible Boussinesq equations with only a velocity damping term when the initial data is close to an nontrivial equilibrium state (0, x2). As a by-product, under this equilibrium state, our result gives a positive answer to the question proposed by [1] (see P.3597).2010 Mathematics Subject Classification. 35Q35, 76B03.