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Register a new userBoundary layer models of the Hou-Luo scenario
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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.
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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,luo2013potentially-2} for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in cite{luo2014potentially,luo2013potentially-2} share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in cite{chen2019finite} to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the $C^gamma$ norm of the density $theta$ with $gammaapprox 1/3$ is uniformly bounded up to the singularity time.
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 on the boundary. When the tensor is a multiple of the identity we obtain Navier boundary conditions, and when the tensor is the shape operator we obtain conditions in which the vorticity vanishes on the boundary. By constructing an explicit corrector, we prove the convergence of the Navier-Stokes solutions to the Euler solution as the viscosity vanishes. We do this both in the natural energy norm with a rate of order $epsilon^{3/4}$ as well as uniformly in time and space with a rate of order $epsilon^{3/8 - delta}$ near the boundary and $epsilon^{3/4 - delta}$ in the interior, where $delta, delta$ decrease to 0 as the regularity of the initial velocity increases. This work simplifies an earlier work of Iftimie and Sueur, as we use a simple and explicit corrector (which is more easily implemented in numerical applications). It also improves a result of Masmoudi and Rousset, who obtain convergence uniformly in time and space via a method that does not yield a convergence rate.
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 on magnetic field. Under the assumption that the viscosity and resistivity coefficients are of the same order and the initial tangential magnetic field on the boundary is not degenerate, we justify the validity of the Prandtl boundary layer expansion and give a $L^infty$ estimate on the error by multi-scale analysis.
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 boundary layer separation. In this paper, we prove the boundary layer separation for a large class of Oleiniks data and confirm Goldsteins hypothesis concerning the local behavior of the solution near the separation, which gives a partial answer to open problem 5 proposed by Oleinik and Samokin in cite{Olei}.
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