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We propose the viscous Camassa-Holm equations as a closure approximation for the Reynolds-averaged equations of the incompressible Navier-Stokes fluid. This approximation is tested on turbulent channel flows with steady mean. Analytical…
We establish existence and uniqueness results for the modified binormal curvature flow equation that generalizes the binormal curvature flow equation for a curve in $\R^3.$ In this generalization, the velocity of the curve is still directed…
We describe a formal procedure to obtain and specify the general form of a marginal distribution for the Lagrangian acceleration of fluid particle in developed turbulent flow using Langevin type equation and the assumption that velocity…
In this paper, the Liouville-type theorems for the steady Navier-Stokes system are investigated. First, we prove that any bounded smooth helically symmetric solution in $\mathbb{R}^3$ must be a constant vector. Second, for steady…
In this article we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral…
In this paper, we introduce a class of new logarithmic curvature flow. The flows are designed to embrace the monotonicity of the related functional, and the convergence of this flow would tackle the solvability of the weighted…
In this paper we study the initial boundary value problem for the system $\Delta v= u_{x_1},\ u_t-\mbox{div}\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\right)=-\nabla…
Consider an integral Brakke flow $(\mu_t)$, $t\in [0,T]$, inside some ball in Euclidean space. If $\mu_{0}$ has small height, its measure does not deviate too much from that of a plane and if $\mu_{T}$ is non-empty, then Brakke's local…
In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $u^\alpha f^{-\beta}$, where $u$ is the support function of the hypersurface, $f$ is a…
Continuous normalizing flows are known to be highly expressive and flexible, which allows for easier incorporation of large symmetries and makes them a powerful computational tool for lattice field theories. Building on previous work, we…
We prove that the spacetime Brakke flow constructed by Buet et al. is non-trivial as long as the initial varifold is a union of boundaries of domains of finite perimeter. In the codimension 1 setting, we show that, starting from a smooth…
In this paper, we provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the…
We consider the fractional mean curvature flow of entire Lipschitz graphs. We provide regularity results, and we study the long time asymptotics of the flow. In particular we show that in a suitable rescaled framework, if the initial graph…
Graphical flows add further structure to normalizing flows by encoding non-trivial variable dependencies. Previous graphical flow models have focused primarily on a single flow direction: the normalizing direction for density estimation, or…
We consider the evolution of hypersurfaces on the unit sphere $\mathbb{S}^{n+1}$ by their mean curvature. We prove a differential Harnack inequality for any weakly convex solution to the mean curvature flow. As an application, by applying…
We use the vorticity transportation equation as the start point--with the help of stream function for two-dimensional planar incompressible flows--to obtain exact solutions that characterize evolution and dynamics of the flows. These…
We prove differential Harnack inequalities for flows of strictly convex hypersurfaces by powers $p$, $0<p<1$, of the mean curvature in Einstein manifolds with a positive lower bound on the sectional curvature. We assume that this lower…
In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time…
Generalizing results of Chou and Wang \cite{1} we study the flows of the leaves $(M_{\Theta})_{\Theta>0}$ of a foliation of $\mathbb{R}^{n+1}\setminus \{0\}$ consisting of uniformly convex hypersurfaces in the direction of their outer…
In this paper, we mainly study the mean curvature flow in K\"ahler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2,…