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Abstract: In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small initial data, the smallness conditions of which are given by the suitable Sobolev and the Besov norms respectively. Although the Sobolev space version is already an improvement of the corresponding result in [4], the optimality in terms of the scaling property is achieved via the Besov space estimate. The special property of the energy estimate in terms of norm is essential for this result. Contrary to the usual MHD the global well-posedness in the dimensional Hall-MHD is wide open. [Copyright &y& Elsevier]

Abstract: We prove local existence of smooth solutions for large data and global smooth solutions for small data to the incompressible, resistive, viscous or inviscid Hall-MHD model. We also show a Liouville theorem for the stationary solutions. [Copyright &y& Elsevier]

Abstract: Let be a smooth solution pair of the velocity and the pressure for the Navier–Stokes (Euler) equations on , . We set the Bernoulli function . Under suitable decay conditions at infinity for we prove that for almost all and defined on there holds where is the vorticity. This shows that, in each region squeezed between two levels of the Bernoulli function, besides the energy dissipation due to the enstrophy, the energy flows into the region through the level hypersurface having the higher level, and the energy flows out of the region through the level hypersurface with the lower level. Passing and , we recover the well-known energy equality, . A weaker version of the above equality under the weaker decay assumption of the solution at spatial infinity is also derived. The stationary version of the equality implies the previous Liouville type results on the Navier–Stokes equations. [Copyright &y& Elsevier]

Abstract: In this paper we prove the Liouville type theorem for the stationary Navier–Stokes equations on . More specifically, if a solution to the stationary Navier–Stokes system satisfies additional conditions characterized by the decays near infinity and by the oscillation, then we show that . [Copyright &y& Elsevier]

Abstract: This paper investigates the global (in time) regularity of solutions to a system of equations that generalize the vorticity formulation of the 2D Boussinesq–Navier–Stokes equations. The velocity in this system is related to the vorticity through the relations and , which reduces to the standard velocity–vorticity relation when . When either or , the velocity is more singular. The “quasi-velocity” determined by satisfies an equation of very special structure. This paper establishes the global regularity and uniqueness of solutions for the case when and . In addition, the vorticity is shown to be globally bounded in several functional settings such as for in a suitable range. [Copyright &y& Elsevier]

Abstract: We prove Liouville type theorems for weak solutions of the Navier–Stokes and the Euler equations. In particular, if the pressure satisfies with , then the corresponding velocity should be trivial, namely on . In particular, this is the case when , where , , the Hardy space. On the other hand, we have equipartition of energy over each component, if with . Similar results hold also for the magnetohydrodynamic equations. [Copyright &y& Elsevier]

We prove the local in time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler equations of inviscid incompressible fluid flows in Rn, n≥2. As a corollary we obtain the persistence of Besov space regularity for the solutions of the 2‐D Euler equations with initial velocity belonging to the Besov spaces. For the proof of the results we establish a logarithmic inequality of the Beale–Kato–Majda type and a Moser type of inequality in the Besov spaces. [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville type theorem for the stationary Navier–Stokes equations in R 3 . Let V = (V ij) be a potential function of a smooth solution u, which means u = ∇ · V . We show that if there exists 3 < s < + ∞ such that the L s mean oscillation of the potential function has certain growth condition near infinity, namely 1 | B (r) | ∫ B (r) | V - V B (r) | s d x ≤ C r min { s - 3 3 , s 6 } ∀ 1 < r < + ∞ , then u ≡ 0 . [ABSTRACT FROM AUTHOR]