Related papers: Regularized $n$-Conformal heat flow and global smo…
We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures on $M$ for…
This paper revisits the problem of heat conduction in relativistic fluids, associated with issues concerning both stability and causality. It has long been known that the problem requires information involving second order deviations from…
In this paper we investigate a kind of generalized Ricci flow which possesses a gradient form. We study the monotonicity of the given function under the generalized Ricci flow and prove that the related system of partial differential…
This article considers nonlocal heat flows into a singular target space. The problem is the parabolic analogue of a stationary problem that arises as the limit of a singularly perturbed elliptic system. It also provides a gradient flow…
We consider the Cauchy problem to the two-dimensional incompressible liquid crystal equation and the heat flows of harmonic maps equation. Under a natural geometric angle condition, we give a new proof of the global well-posedness of smooth…
We develop estimates for the solutions and derive existence and uniqueness results of various local boundary value problems for Dirac equations that improve all relevant results known in the literature. With these estimates at hand, we…
A solution to the heat equation between Riemannian manifolds, where the domain is compact and possibly has boundary, will not leave a compact and locally convex set before the image of the boundary does.
We introduce and study generalized $1$-harmonic equations (1.1). Using some ideas and techniques in studying $1$-harmonic functions from [W1] (2007), and in studying nonhomogeneous $1$-harmonic functions on a cocompact set from [W2, (9.1)]…
We consider a thermodynamically consistent model for the evolution of thermally conducting two-phase incompressible fluids. Complementing previous results, we prove additional regularity properties of solutions in the case when the…
In the past years, there has been a new light shed on the harmonic map problem with free boundary in view of its connection with nonlocal equations. Here we fully exploit this link, considering the harmonic map flow with free boundary…
We consider the Navier--Stokes equations for compressible heat-conducting ideal polytropic gases in a bounded annular domain when the viscosity and thermal conductivity coefficients are general smooth functions of temperature. A…
We consider the Landau-Lifshitz equations of ferromagnetism (including the harmonic map heat-flow and Schroedinger flow as special cases) for degree m equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal energy…
J.-M. Coron proved in [5] that the global weak solutions of the heat flow from $M$ to $N$, starting at non-stationary weakly harmonic maps, are not unique when $M = B^3$ and $N = S^2$. Hence, the semigroup property of the solution map does…
Let $M$ be a Riemannian manifold and $\Omega$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $\Omega$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of…
In this paper we study the parabolic evolution equation $\partial_t u=(|Du|^{2}+2|\det Du|)^{-1} \Delta u$, where $u : M\times[0,\infty) \to N$ is an evolving map between compact flat surfaces. We use a tensor maximum principle for the…
Many generative models originally developed in finite-dimensional Euclidean space have functional generalizations in infinite-dimensional settings. However, the extension of rectified flow to infinite-dimensional spaces remains unexplored.…
Recently a new no-global-recollapse argument was given for some inhomogeneous and anisotropic cosmologies that utilizes surface deformation by the mean curvature flow. In this paper we discuss important properties of the mean curvature flow…
We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is…
We derive a normal form for a near-integrable, four-dimensional symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately…
We establish effective existence and uniqueness for the heat flow on time-dependent Riemannian manifolds, under minimal assumptions tailored towards the study of Ricci flow through singularities. The main point is that our estimates only…