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Register a new userLiouville type theorems for 3D stationary Navier-Stokes equations in weighted mixed-norm Lebesgue spaces
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Tuoc Phan
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This work studies the system of $3D$ stationary Navier-Stokes equations. Several Liouville type theorems are established for solutions in mixed-norm Lebesgue spaces and weighted mixed-norm Lebesgue spaces. In particular, we show that, under some sufficient conditions in mixed-norm Lebesgue spaces, solutions of the stationary Navier-Stokes equations are identically zero. This result covers the important case that solutions may decay to zero with different rates in different spatial directions, and some these rates could be significantly slow. In the un-mixed norm case, the result recovers available results. With some additional geometric assumptions on the supports of solutions, this work also provides several other important Liouville type theorems for solutions in weighted mixed-norm Lebesgue spaces. To prove the results, we establish some new results on mixed-norm and weighted mixed-norm estimates for Navier-Stokes equations. All of these results are new and could be useful in other studies.
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We study the Cauchy problem in $n$-dimensional space for the system of Navier-Stokes equations in critical mixed-norm Lebesgue spaces. Local well-posedness and global well-posedness of solutions are established in the class of critical mixed-norm Lebesgue spaces. Being in the mixed-norm Lebesgue spaces, both of the initial data and the solutions could be singular at certain points or decaying to zero at infinity with different rates in different spatial variable directions. Some of these singular rates could be very strong and some of the decaying rates could be significantly slow. Besides other interests, the results of the paper particularly show an interesting phenomena on the persistence of the anisotropic behavior of the initial data under the evolution. To achieve the goals, fundamental analysis theory such as Youngs inequality, time decaying of solutions for heat equations, the boundedness of the Helmholtz-Leray projection, and the boundedness of the Riesz tranfroms are developed in mixed-norm Lebesgue spaces. These fundamental analysis results are independently topics of great interests and they are potentially useful in other 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.
We consider Stokes systems in non-divergence form with measurable coefficients and Lions-type boundary conditions. We show that for the Lions conditions, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm $L_{s,q}$-estimates of the spatial second-order derivatives of solutions hold, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with $s=q=2$, the result is still new and provides local boundary Caccioppoli-type estimates, which are important in applications. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.
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.
Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $ abla_{h}{u}$ (or $ abla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; dot{B}_{p,r}^{s}(mathbb{R}^{3}))$, where $ abla_{h}=(partial_{x_{1}},partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $partial_3u_3$.
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