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Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is axi-symmetric and features fast growth of vorticity near a ring of hyperbolic points of the flow located at the boundary of a cylinder containing the fluid. An important role is played by a small boundary layer where intense growth is observed. Several simplified models of the scenario have been considered, all leading to finite time blow up cite{CKY15,CHKLVY17,HORY,KT1,HL15,KY1}. In this paper, we propose two models that are designed specifically to gain insight in the evolution of fluid near the hyperbolic stagnation point of the flow located at the boundary. One model focuses on analysis of the depletion of nonlinearity effect present in the problem. Solutions to this model are shown to be globally regular. The second model can be seen as an attempt to capture the velocity field near the boundary to the next order of accuracy compared with the one-dimensional models such as cite{CKY15,CHKLVY17}. Solutions to this model blow up in finite time.
Inspired by the numerical evidence of a potential 3D Euler singularity cite{luo2014potentially,luo2013potentially-2}, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in cite{luo2014potentially,luo2013p
We study the weak boundary layer phenomenon of the Navier-Stokes equations in a 3D bounded domain with viscosity, $epsilon > 0$, under generalized Navier friction boundary conditions, in which we allow the friction coefficient to be a (1, 1) tensor o
As a continuation of cite{LXY}, the paper aims to justify the high Reynolds numbers limit for the MHD system with Prandtl boundary layer expansion when no-slip boundary condition is imposed on velocity field and perfect conducting boundary condition
In this article we find the solution of the Burger equation with viscosity applying the boundary layer theory. In addition, we will observe that the solution of Burger equation with viscosity converge to the solution of Burger stationary equation in the norm of $L_{2}([-1,1])$.
In the case of favorable pressure gradient, Oleinik proved the global existence of classical solution for the 2-D steady Prandtl equation for a class of positive data. In the case of adverse pressure gradient, an important physical phenomena is the b