arXiv research

The paper proves well-posedness of nonlocal PDEs related to stochastic control problems.

Study well-posedness of SPDE on Riemannian manifolds with rough initial conditions.

(Working Paper) Using a purely probabilistic argument, we prove the global well-posedness of multidimensional superquadratic backward stochastic differential equations (BSDEs) without Markovian assumption. The key technique is the interplay between the local well-posedness of fully coupled path-dependent forward backwa…

We establish the global well-posedness of the initial value problem for the Schrodinger map flow for maps from the real line into Kahler manifolds and for maps from the circle into Riemann surfaces. This partially resolves a conjecture of W.-Y. Ding.

Study well-posedness of Faraday tensor problem on specific spacetime manifolds.

We establish both local and global well-posedness for the heat flow of polyharmonic maps from $R^n$ to a compact Riemannian manifold without boundary for initial data with small BMO norms.

This paper establish the local (or global, resp.) well-posedness of the heat flow of biharmonic maps from $R^n$ to a compact Riemannian manifold without boundary with small local BMO (or BMO, resp.) norms.

Study proves well-posedness and scattering for wave equations on hyperbolic spaces with singular data.

Global well-posedness and asymptotic convergence for vacuum Einstein's equations proved.

Model strategic interactions between market makers and traders to optimize execution.

We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniquene…

Proves well-posedness of the Cauchy problem for the Dirac operator on non-compact spacetimes.

In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical ph…

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

Global solutions found for a wave-Klein-Gordon system with strong couplings in divergence form.

Introduces conformal Bach flow and proves its well-posedness and backward uniqueness.

Study global solutions for Boussinesq systems on curved manifolds.

Study well-posedness of fast diffusion equation on noncompact manifolds.

New method shortens and straightens curves, proving convergence and well-posedness.

Sharp heat kernel estimates on manifolds lead to solutions of the Parabolic Anderson model.

Extends Atiyah-Singer Dirac operator study to non-compact spacetimes.

In this paper, a type of integrable evolution equation--the generalized Landau-Lifshitz equation into $S^n$ is considered. We deal with this equation from a geometric point of view by rewriting it in a geometric form. Through the geometric energy method, we show the global well-posedness of the corresponding Cauchy pro…

Paper solves local well-posedness for Schrödinger flow into sphere with natural boundary conditions.

We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out tha…

Develop intrinsic consensus-based optimization framework on Riemannian manifolds with bounded curvature.

Study on symmetric hyperbolic systems with nonlocal potentials, proving well-posedness and existence of solutions.

Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa--Holm equations are well-studied examples.A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description.Geometrically it corresponds …

Study on Navier-Stokes equations on non-compact manifolds, proving existence and decay of solutions.

We study the question of well-posedness of the Cauchy problem for Schrödinger maps from $\rone \times \rtwo$ to the sphere $\stwo$ or to ${\mathbb H^2}$, the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equa…

This book provides a detailed introduction to linear wave equations on Lorentzian manifolds (for vector-bundle valued fields). After a collection of preliminary material in the first chapter one finds in the second chapter the construction of local fundamental solutions together with their Hadamard expansion. The third…

Gradient flow on diffeomorphisms for image registration, with well-posedness proven.

Study on dynamic curves with elastic energy and spontaneous curvature.

The subject of this article is the introduction of a new concept of well-posedness of Bayesian inverse problems. The conventional concept of (Lipschitz, Hellinger) well-posedness in [Stuart 2010, Acta Numerica 19, pp. 451-559] is difficult to verify in practice and may be inappropriate in some contexts. Our concept sim…

We study a class of nonlocal, energy-driven dynamical models that govern the motion of closed, embedded curves from both an energetic and dynamical perspective. Our energetic results provide a variety of ways to understand physically motivated energetic models in terms of more classical, combinatorial measures of compl…

In this paper, we study the well-posedness of boundary value problems for a special class of degenerate elliptic equations coming from geometry. Such problems is intimately tied to rigidity problem arising in infinitesimal isometric deformation, The characteristic form of this class of equations is changing its signs i…

Proves well-posedness of gradient solitons on bundle gerbe.

The commuting vector fields approach, devised for strichartz estimates in [13], was developed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\mathbb R}^{1+3}$ by Klainerman and Rodnianski. Via this approach they obtained the l…

Study well-poses Dirac operator problem with APS boundary conditions.

We consider an integro-differential equation derived from a system of coupled parabolic PDE and an ODE which describes an European option pricing with liquidity shocks. We study the well-posedness and prove comparison principle for the corresponding initial value problem.

Proves well-posedness for hard phase model in general relativity.

In this paper we describe the behavior of solutions of the Klein-Gordon equation, (Box_g+lambda)u=f, on Lorentzian manifolds (X^o,g) which are anti-de Sitter-like (AdS-like) at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces, in the…

Proves well-posedness for Einstein equations with specific boundary conditions.

Novel framework for portfolio selection considering utility and risk.

The paper studies boundedness of pseudo-differential operators on smooth manifolds.

We investigate the well-posedness of (i) the heat flow of harmonic maps from $R^n$ to a compact Riemannian manifold without boundary for initial data in BMO; and (ii) the hydrodynamic flow $(u,d)$ of nematic liquid crystals on $\mathbb R^n$ for initial data in $BMO^{-1}\times BMO$.

Proves well-posedness for Einstein equations with specific boundary data.

Noise stabilizes solutions to transport equations, preventing blow-up.

We extend Eardley and Moncrief's $L^\infty$ estimates for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localise their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates to…