Related papers: Asymptotics for the level set equation near a maxi…

We estimate from above the rate at which a solution to the rescaled mean curvature flow on a closed hypersurface may converge to a limit self-similar solution, i.e. a shrinker. Our main result implies that any solution which converges to a…

A detailed investigation is presented of a simple unidirectional finite-energy solution of the 3D wave equation. Its asymptotics as a spatial point runs to infinity with the wave propagations speed is a standard spherical wave as z < 0,…

We present a technique for approximating generic normalization constants subject to constraints. The method is then applied to derive the exact asymptotics for the conditional normalization constant of constrained exponential random graphs.

We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower…

Wall-bounded turbulent shear flows are known to exhibit universal small-scale dynamics that are modulated by large-scale flow structures. Strong pressure gradients complicate this characterization, however; they can cause significant…

We show that the mean curvature flow of generic closed surfaces in $\mathbb{R}^{3}$ avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces…

We consider a nonlinear parabolic model that forces solutions to stay on a $L^2$-sphere through a nonlocal term in the equation. We study the local and global well-posedness on a bounded domain and the whole Euclidean space in the energy…

We study the asymptotics of solutions to a particular class of systems of linear wave equations, namely, of silent equations. We obtain asymptotic estimates of all orders for the solutions, and show that solutions are uniquely determined by…

We prove that the limit hypersurfaces of converging curvature flows are stable, if the initial velocity has a weak sign, and give a survey of the existence and regularity results.

Consider neutron transport equations in 3D convex domains with in-flow boundary. We mainly study the asymptotic limits as the Knudsen number $\epsilon\rightarrow 0^+$. Using Hilbert expansion, we rigorously justify that the solution of…

The existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case…

In this paper, we study the skew mean curvature flow. The results are threefold. First, we prove the global regularity of solutions with initial data which are small perturbations of planes in Sobolev spaces. Second, we prove the modified…

Assume that a stochastic processes can be approximated, when some scale parameter gets large, by a fluid limit (also called "mean field limit", or "hydrodynamic limit"). A common practice, often called the "fixed point approximation"…

In this paper, we first investigate a new locally constrained mean curvature flow (1.5) and prove that if the initial hypersurface is of smoothly compact starshaped, then the solution of the flow (1.5) exists for all time and converges to a…

We consider an anisotropic curvature flow $V= A(\mathbf{n})H + B(\mathbf{n})$ in a band domain $\Omega :=[-1,1]\times R$, where $\mathbf{n}$, $V$ and $H$ denote the unit normal vector, normal velocity and curvature, respectively, of a…

In this paper we look for the convex hull of a set using the geometric evolution by minimal curvature of a hypersurface that surrounds the set. To find the convex hull, we study the large time behavior of solutions to an obstacle problem…

The equation governing the streaming of a quantity down its gradient superficially looks similar to the simple constant velocity advection equation. In fact, it is the same as an advection equation if there are no local extrema in the…

By variational methods, for a kind of Webster scalar curvature problems on the CR sphere with cylindrically symmetric curvature, we construct some multi-peak solutions as the parameter is sufficiently small under certain assumptions. We…

Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in a domain $\Omega$ and with prescribed contact angle on $\partial\Omega$, we prove that a graphical solution to the mean curvature flow with the same prescribed…

It is a fundamental open problem for the mean curvature flow, and in fact for many partial differential equations, whether or not all blowup limits are selfsimilar. In this short note, we prove that for the mean curvature flow of mean…