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Register a new userClassification and Liouville-type theorems for semilinear elliptic equations in unbounded domains
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We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.
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In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
This note is devoted to investigating Liouville type properties of the two dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, under smallness conditions only on the magnetic field, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some $L^p$ norm of the velocity-magnetic fields are finite. In particular, these results generalize the corresponding Liouville type properties for the 2D Navier-Stokes equations, such as Gilbarg-Weinberger cite{GW1978} and Koch-Nadirashvili-Seregin-Sverak cite{KNSS}, to the MHD setting.
We prove that, in a two-dimensional strip, a steady flow of an ideal incompressible fluid with no stationary point and tangential boundary conditions is a shear flow. The same conclusion holds for a bounded steady flow in a half-plane. The proofs are based on the study of the geometric properties of the streamlines of the flow and on one-dimensional symmetry results for solutions of some semilinear elliptic equations. Some related rigidity results of independent interest are also shown in n-dimensional slabs in any dimension n.
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition $u to 0$ at infinity with respect to the spacial variables to a polynomial growth on $u$ by constructing auxiliary functions.Then we derive monotonicity for the solutions in a half space $mathbb{R}_+^n times mathbb{R}$ and obtain some new connections between the nonexistence of solutions in a half space $mathbb{R}_+^n times mathbb{R}$ and in the whole space $mathbb{R}^{n-1} times mathbb{R}$ and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the non-locality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of non-local parabolic problems.
In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of cite{kpr15} in the case of zero swirl, where we replaced the Dirichlet integrability condition by mild decay conditions. Then we prove some Liouville theorems under the assumption $|f{u_r}{r}{bf 1}_{{u_r< -f 1r}}|_{L^{3/2}(mbR^3)}< C_{sharp}$ where $C_{sharp}$ is a universal constant to be specified. In particular, if $u_r(r,z)geq -f1r$ for $forall (r,z)in[0,oo)timesmbR$, then ${bf u}equiv 0$. Liouville theorems also hold if $displaystylelim_{|x|to oo}Ga =0$ or $Gain L^q(mbR^3)$ for some $qin [2,oo)$ where $Ga= r u_{th}$. We also established some interesting inequalities for $Omco f{p_z u_r-p_r u_z}{r}$, showing that $ aOm$ can be bounded by $Om$ itself. All these results are extended to the axially symmetric MHD and Hall-MHD equations with ${bf u}=u_r(r,z){bf e}_r +u_{th}(r,z) {bf e}_{th} + u_z(r,z){bf e}_z, {bf h}=h_{th}(r,z){bf e}_{th}$, indicating that the swirl component of the magnetic field does not affect the triviality. Especially, we establish the maximum principle for the total head pressure $Phi=f {1}{2} (|{bf u}|^2+|{bf h}|^2)+p$ for this special solution class.
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