arXiv research

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

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

(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…

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

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

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…

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

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 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.

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…

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.

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.

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.

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…

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

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

Novel framework for portfolio selection considering utility and risk.

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

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.

Investigates optimal execution under time-varying liquidity, preventing price manipulation.

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

The paper examines utility maximization in markets with hidden Gaussian drift, finding restrictions on model parameters.

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

In this work, we extend existing well-posedness by noise results for the stochastic transport and continuity equations by treating them as special cases of the linear advection equation of $k$-forms, which arises naturally in geometric fluid dynamics. In particular, we prove the existence and uniqueness of weak $L^p$-s…

We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data.

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…

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

This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of degenerate non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. In a semi-Markov modulated GBM model the locally risk minimizing price function satisfies a…

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

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

Local well-posedness proved for Bartnik static extension near Schwarzschild spheres.

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

Paper analyzes GANs training difficulties and proposes a control framework.

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

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

Paper develops a framework for hyperbolic Monge-Ampère equation on strips, proving well-posedness and stability.

Given a compact manifold $M$ and a Riemannian manifold $N$ of bounded geometry, we consider the manifold ${\rm Imm} (M,N)$ of immersions from $M$ to $N$ and its subset ${\rm Imm}_μ(M,N)$ of those immersions with the property that the volume-form of the pull-back metric equals $μ$. We first show that the non-minimal ele…

Extends static vacuum metrics with specific boundary conditions.

Study well-posedness of generalized Stokes operator on cylindrical domains.

Regularizes 3D inverse scattering with tangent-point energy for better solutions.

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

Proves well-posedness for Einstein equations with totally geodesic timelike boundary condition.

We prove local well-posedness of the Schrödinger flow from $R^n$ into a compact K\{"a}hler manifold $N$ with initial data in $H^{s+1}(R^n, N)$ for $s\geq n/2+4$.

We recall the notions of Frölicher and diffeological spaces and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of the Mulase factorization of infinite dimensional groups based on formal pseud…