Related papers: A flow approach to the $L_{-2}$ Minkowski problem

We find a local solution to the Ricci flow equation under a negative lower bound for many known curvature conditions. The flow exists for a uniform amount of time, during which the curvature stays bounded below by a controllable negative…

In this paper a generalized Gauss curvature flow about a convex hypersurface in the Euclidean $n$-space is studied. This flow is closely related to the Orlicz-Minkowski problem, which involves Gauss curvature and a function of support…

Studying the geometric flow plays a powerful role in mathematics and physics. In this paper, we introduce the mean curvature flow on Finsler manifolds and give a number of examples of the mean curvature flow. For Minkowski spaces, a special…

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 study families of smooth, embedded, regular planar curves $ \alpha : \left [-1,1 \right ]\times \left [0,T \right )\to \mathbb{R}^{2}$ with generalised Neumann boundary conditions inside cones, satisfying three variants of the…

In this paper, inspired by the work of Guan and Li (2015), we introduce a fourth-order centro-equiaffine invariant curve flow via the affine Minkowski formula. Without any smallness assumptions on the initial curve, we establish the…

We are concerned with a model describing the motion of two compressible, immiscible fluids with density-dependent viscosity in the whole $\mathbb R^3$. The phases of the flow may have different pressure and viscosity laws and are separated…

A method of solving perfect fluid Einstein equations with two commuting spacelike Killing vectors is presented. Given a spacelike 2-dimensional surface in the 3-dimensional nonphysical Minkowski space the field equations reduce to a single…

We prove the $C^0$ estimate for the $L_p$ $q$th dual Minkowski problem on $S^2$ under fairly general conditions; namely, when $p$ lies in [0,1) and $q>2+p$, and the $L_p$ $q$th dual curvarture is bounded and bounded away from zero. We note…

We analytically solve for the time dependent solutions of various density evolution models. With specific forms of the diffusion, drift and sink coefficients, the eigenfunctions can be expressed in terms of hypergeometric functions. We…

We prove existence and uniqueness of solution of a class of semi-linear wave equations with initial data prescribed on the light-cone with vertex the origin of a Minkowski space-time. The nonlinear term is assumed to satisfy a nullity…

For a class of fully nonlinear equations having second order operators which may be singular or degenerate when the gradient of the solutions vanishes, and having first order terms with power growth, we prove the existence and uniqueness of…

It has long been conjectured that for nonlinear wave equations which satisfy a nonlinear form of the null condition, the low regularity well-posedness theory can be significantly improved compared to the sharp results of Smith-Tataru for…

We consider a normalization of the Ricci flow on a closed Riemannian manifold given by the evolution equation $\partial_{t}g(t)=-2(Ric(g(t))-\frac{1}{2\tau}g(t))$ where $\tau$ is a fixed positive number. Assuming that a solution for this…

Using nonstandard methods, we show that the time dependent Fourier series of any smooth function F, solving the wave equation, on a finite closed interval, with vanishing boundary conditions, converges uniformly to F.

We consider the global regularity problem for defocusing nonlinear wave systems $$ \Box u = (\nabla_{{\bf R}^m} F)(u) $$ on Minkowski spacetime ${\bf R}^{1+d}$ with d'Alambertian $\Box := -\partial_t^2 + \sum_{i=1}^d \partial_{x_i}^2$, the…

In this paper, we study fully nonlinear curvature flows of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show…

We establish the unique solvability of a coupling problem for entire functions which arises in inverse spectral theory for singular second order ordinary differential equations/two-dimensional first order systems and is also of relevance…

We give a very simple method for finding the exact analytical solution for the problem of a particle undergoing diffusive motion on a flat potential in the presence of a gaussian sink function. The diffusion process is modelled by using one…

Can the Minkowski sum of two compact convex bodies be made smoother by rotating one of them? We construct two infinitely differentiable strictly convex plane bodies such that after any generic rotation (in the Baire category sense) of one…