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We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $${\mathbb{R}^2}$$ . We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations. [ABSTRACT FROM AUTHOR]

In this paper we prove three different Liouville type theorems for the steady Navier–Stokes equations in R 3 . In the first theorem we improve logarithmically the well-known L 9 2 ( R 3 ) result. In the second theorem we present a sufficient condition for the trivially of the solution ( v = 0 ) in terms of the head pressure, Q = 1 2 | v | 2 + p . The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee v = 0 . [ABSTRACT FROM AUTHOR]

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]

We present a simple, short, and elementary proof that if v is a Beltrami flow with a finite energyin R3,then v=0. In the case of the Beltrami flows satisfying v∈Lloc∞(R3)∩Lq(R3) with q ∈[2,3),or [ABSTRACT FROM AUTHOR]

We study a nonlinear parabolic system for a time dependent solenoidal vector field on ℝ3. The nonlinear term of these new model equations is obtained slightly modifying that of the Navier-Stokes equations. The system has the same scaling property and Galilean invariance as the Navier-Stokes equations. For such a system we prove the global regularity for smooth initial data. [ABSTRACT FROM AUTHOR]

We study a nonlinear parabolic system for a time dependent solenoidal vector field on ℝ3. The nonlinear term of these new model equations is obtained slightly modifying that of the Navier-Stokes equations. The system has the same scaling property and Galilean invariance as the Navier-Stokes equations. For such a system we prove the global regularity for smooth initial data. [ABSTRACT FROM AUTHOR]