Backward stochastic differential equations associated with the vorticity equations

In this paper we derive a non-linear version of the Feynman–Kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions. We prove the existence (global in time) and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2. A particular class of terminal values provide, via these probabilistic methods, solutions for the vorticity equation.     

爆炸金属管绝热剪切断裂的细观研究

介绍了正火及调质后45号钢、TC4钛合金和钨合金四种金属管在内部高能炸药加载下,破片宏观剪切断口的细观断裂形貌;观察到这四种金属管不同的细观剪切断裂机制。根据金相研究及物理分析表明,金属材料的热导率越低,晶粒越细,则越容易产生绝热剪切,生成相变带。     

Interactions between vortices and flexible walls

We give two fundamental solutions for the motion of a point vortex near a flexible wall, up to first order in wall deflection, using computational methods. For a point vortex near an infinite horizontal wall, the deformation of the wall intensifies the flow at the wall near the vortex, and increases the speed of the vortex. Near a circular wall there is a strong mutual amplification of the deflection of the wall and the pressure force induced by the deflection, as the point vortex approaches the wall. The total force on the wall diverges as the inverse cube of the distance to the point vortex, and the induced speed of the point vortex diverges as the inverse fourth power of distance to the wall.     

Adaptive panel representation for oblique collision of two vortex rings

In this article, we use a hierarchical panel method for representing vortex sheet surface motion in 3D flow to investigate the oblique collision of two vortex rings. The particles representing the sheet are advected by a regularized Biot-Savart integral with smoothed Rosenhead-Moore kernel. The particle velocities are evaluated by an adaptive treecode algorithm based on Taylor expansions in Cartesian coordinates. The method allowed us to consider late stages of a vortex rings collision, producing a funnel region. Vorticity iso-surfaces evolution is also investigated.     

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.     

Global well-posedness for the 3D Navier–Sokes equations with a large component of vorticity

In this paper, we give a new approach to improve the Leray's result concerning the cauchy problem to the 3D Navier–Stokes equations. In particular, global well-posedness with a large component of the initial vorticity is obtained. Our idea is considering the vorticity equations and using some suitable function spaces.     

Kink band propagation in layered structures

Confined layered structures in layer-parallel compression exhibit kink band deformation that has an associated severe initial instability. However, once the first kink band forms the system restabilizes and the bands then propagate in two mechanisms: band broadening and band progression. Previous work on triggering the initial instability and band broadening is now extended to include band progression. A new model for this propagation mechanism that successfully accounts for the modulating restabilizing response is presented and quantitative comparisons with the physical experiments yield excellent results.     

Geometric constrains for global regularity of 2D quasi-geostrophic flows

We study the two-dimensional quasi-geostrophic equations (2D QG) in Sobolev spaces. We first prove a new analytic condition for global regularity which is both sufficient and necessary. We then prove several new results on the geometric constraints on the 2D QG active scalar which suppress the development of singularity from the nonlinear stretching mechanism. We focus mainly on the case with critical dissipation. Our results are also relevant to the inviscid case.     

A class on non-local linear operators for vorticity waves

A steady longitudinal current in the nearshore can, in some conditions, support oscillations known as vorticity waves or shear waves. In this article, we consider a family of nonlinear evolution equations derived by Shrira and Voronovitch to describe the dynamics of vorticity waves near the coastal line and make the study of the dispersion and smoothing properties of the associated nonlocal free problems. More precisely, after establishing long and short time uniform estimates for a certain class of oscillatory integrals, we derive “L ^p ?L ^q ” and Strichartz-type estimates for the solutions of the linearized equations.