Global Tangentially Analytical Solutions of the 3D Axially Symmetric Prandtl Equations

In this paper,the authors will prove the global existence of solutions to the three dimensional axially symmetric Prandtl boundary layer equations with small initial data,which lies in H1Sobolev space with respect to the normal variable and is analytical with respect to the tangential variables.The main novelty of this paper relies on careful constructions of a tangentially weighted analytic energy functional and a specially designed good unknown for the reformulated system.The result extends that of Paicu-Zhang in[Paicu,M.and Zhang,P.,Global existence and the decay of solutions to the Prandtl system with small analytic data,Arch.Ration.Mech.Anal.,241(1),2021,403–446].from the two dimensional case to the three dimensional axially symmetric case,but the method used here is a direct energy estimates rather than Fourier analysis techniques applied there.

On Leray's problem in an infinitely long pipe with the Navier-slip boundary condition

The original Leray’s problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe,which approach the Poiseuille flow subject to the no-slip boundary condition at spatial infinity.In this paper,the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition,is addressed.Due to the complexity of the boundary condition,some new ideas,presented as follows,are introduced to handle the extra difficulties caused by boundary terms.First,the Poiseuille flow in the semi-infinite straight pipe with the Navier-slip boundary condition will be introduced,which will serve as the asymptotic profile of the solution to the generalized Leray’s problem at spatial infinity.Second,a solenoidal vector function defined in the whole pipe,satisfying the Navierslip boundary condition,having the designated flux and equalling the Poiseuille flow at a large distance,will be carefully constructed.This plays an important role in reformulating our problem.Third,the energy estimates depend on a combined L2-estimate of the gradient and the stress tensor of the velocity.

A Review of Results on Axially Symmetric Navier-Stokes Equations,with Addendum by X.Pan and Q.Zhang

In this paper,we give a brief survey of recent results on axially symmetric Navier-Stokes equations(ASNS)in the following categories:regularity criterion,Liouville property for ancient solutions,decay and vanishing of stationary solutions.Some discussions also touch on the full 3 dimensional equations.Two results,closing of the scaling gap for ASNS and vanishing of homogeneous D solutions in 3 dimensional slabs will be described in more detail.In the addendum,two new results in the 3rd category will also be presented,which are generalizations of recently published results by the author and coauthors.