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In this paper we show that the incompressible Hall-MHD system without resistivity is not globally in time well-posed in any Sobolev space H m ( R 3 ) for any m > 7 2 . Namely, either the system is locally ill-posed in H m ( R 3 ) , or it is locally well-posed, but there exists an initial data in H m ( R 3 ) , for which the H m ( R 3 ) norm of solution blows-up in finite time if m > 7 2 . In the latter case we choose an axisymmetric initial data u 0 ( x ) = u 0 r ( r , z ) e r + b 0 z ( r , z ) e z and B 0 ( x ) = b 0 θ ( r , z ) e θ , and reduce the system to the axisymmetric setting. If the convection term survives sufficiently long time, then the Hall term generates the singularity on the axis of symmetry and we have lim ⁡ sup t → t ⁎ ⁡ sup z ∈ R ⁡ | ∂ z ∂ r b θ ( r = 0 , z ) | = ∞ for some t ⁎ > 0 , which will also induce a singularity in the velocity field. [ABSTRACT FROM AUTHOR]

Abstract: We establish temporal decay estimates for weak solutions to the Hall-magnetohydrodynamic equations. With these estimates in hand we obtain algebraic time decay for higher order Sobolev norms of small initial data solutions. [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]

Abstract: We study the dynamics along the particle trajectories for the 3D axisymmetric Euler equations. In particular, by rewriting the system of equations we find that there exists a complex Riccati type of structure in the system on the whole of , which generalizes substantially the previous results in [5] (D. Chae, On the blow-up problem for the axisymmetric 3D Euler equations, Nonlinearity 21 (2008) 2053–2060). Using this structure of equations, we deduce the new blow-up criterion that the radial increment of pressure is not consistent with the global regularity of classical solution. We also derive a much more refined version of the Lagrangian dynamics than that of [6] (D. Chae, On the Lagrangian dynamics for the 3D incompressible Euler equations, Comm. Math. Phys. 269 (2) (2007) 557–569) in the case of axisymmetry. [Copyright &y& Elsevier]

Abstract: We obtain new continuation principle of the local classical solutions of the 3D Euler equations, where the regularity condition of the direction field of the vorticiy and the integrability condition of the magnitude of the vorticity are incorporated simultaneously. The regularity of the vorticity direction field is most appropriately measured by the Triebel–Lizorkin type of norm. Similar result is also obtained for the inviscid 2D quasi-geostrophic equation. [Copyright &y& Elsevier]

Abstract: In this paper, we prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity (viscosity) tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion (viscosity) equations. Our result for the zero diffusion system, in particular, solves the Problem no. 3 posed by Moffatt in [R.L. Ricca, (Ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 3–10]. [Copyright &y& Elsevier]

Abstract: In this paper we study 1D equations with nonlocal flux. These models have resemblance of the 2D quasi-geostrophic equation. We show the existence of singularities in finite time and construct explicit solutions to the equations where the singularities formed are shocks. For the critical viscosity case we show formation of singularities and global existence of solutions for small initial data. [Copyright &y& Elsevier]