极粗糙剪切流的瞬时总增强耗散

Mixing and diffusion for rough shear flows

This article addresses mixing and diffusion properties of passive scalars advected by rough ($C^α$) shear flows. We show that in general, one cannot expect a rough shear flow to increase the rate of inviscid mixing to more than that of a smooth shear without critical points. On the other hand, diffusion may be enhanced at a much faster rate. This shows that in the setting of low regularity, the interplay between inviscid mixing properties and enhanced dissipation is more intricate, and in fact contradicts some of the natural heuristics that are valid in the smooth setting.

Boundary Layer for a Non-Newtonian Flow over a Rough Surface

We analyze the effect of a rough surface on shear thinning and shear thickening fluids, modeled by power law stress tensors. The roughness is modeled by a small wavelength and small amplitude oscillation, parametrized by eps. We study the boundary layer corrector near the rough surface, and provide error estimates in terms of eps for the associated wall laws. With regards to previous studies, Mathematical difficulties arise from the nonlinearity of the corrector equations.

一般粗糙剪切流的混合

Mixing for generic rough shear flows

本研究针对一般粗糙剪切流驱动的被动标量，证明了无粘混合速率$t^{1/(2α)}$和增强耗散速率$ν^{α/(α+2)}$的精确上下界，并基于$ρ$-不规则性提供了新见解。

修正型和三次广义Korteweg-de Vries（gKdV）方程的粗糙数据的瞬时解析光滑化

Instantaneous analytic smoothing of rough data for the modified and cubic gKdV equations

该研究针对修正型和三次gKdV方程，证明满足Nelson型条件的低正则奇异初始数据对应的解，在t≠0时瞬时实解析，扩展了KdV的光滑效应，采用Lorentz空间细化避免伪微分演算。

Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary

We consider a non-Newtonian fluid flow in a thin domain with thickness $η_\varepsilon$ and an oscillating top boundary of period $\varepsilon$. The flow is described by the 3D incompressible Navier-Stokes system with a nonlinear viscosity, being a power of the shear rate (power law) of flow index $p$, with $9/5\leq p<+\infty$. We consider the limit when the thickness tends to zero and we prove that the three characteristic regimes for Newtonian fluids are still valid for non-Newtonian fluids, i.e. Stokes roughness ($η_\varepsilon\approx \varepsilon$), Reynolds roughness ($η_\varepsilon\ll \var…

Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations

A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of solutions of Euler equations may, in some cases, fail to be a solution of Euler equations. We use this shear flow example to provide non-generic, yet nontrivial, examples concerning the loss of smoothness of solutions of the three-dimensional Euler equations, for initial data that do not belong to $C^{1,α}$. Moreover, we show by means of this shear flow exa…