Related papers: Well-posedness properties of geometric variational…
This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former,…
This two-part paper details a theory of solvability for the power flow equations in lossless power networks. In Part I, we derive a new formulation of the lossless power flow equations, which we term the fixed-point power flow. The model is…
We prove exponential instability properties for the fractional Calder\'on problem and the conductivity formulation of the fractional Calder\'on problem in the regime of fractional powers $s\in (0,1)$. We particularly focus on two settings:…
The primary objective of this paper is to investigate the well-posedness theories associated with the discrete nonlinear Schr\"odinger equation and Klein-Gordon equation. These theories encompass both local and global well-posedness, as…
In this paper, we study the well-posedness of Poisson-Nernst-Planck system with no-flux boundary condition and singular permanent charges in two dimension. The main difficulty comes from the lack of integrability of singular permanent…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
In this paper, we consider the well-posedness of the Cauchy problem for a physical model of the extrusion process, which is described by two systems of conservation laws with a free boundary. By suitable change of coordinates and fixed…
We introduce and investigate properties of a variant of the semi-discrete optimal transport problem. In this problem, one is given an absolutely continuous source measure and cost function, along with a finite set which will be the support…
We study functional stochastic differential equations with a locally unbounded, functional drift focusing on well-posedness, stability and the strong Feller property. Following the non-functional case, we only consider integrability…
In this paper we consider the following non-linear stochastic partial differential equation (SPDE): \begin{align*} \begin{cases} \mathrm{d}u(s,x)=\sum^n_{i=1} \mathscr{L}_i u(s,x)\circ \mathrm{d}W_i(s)+\left(V(x)+\mu\Delta…
In this article, we study the strong well-posedness, stability and optimal control of an incompressible magneto-viscoelastic fluid model in two dimensions. The model consists of an incompressible Navier--Stokes equation for the velocity…
Within the global setting of singular drifts in Morrey-Campanato spaces presented in [6], we study now the H{\"o}lder regularity properties of the solutions of a transport-diffusion equation with nonlinear singular drifts that satisfy a…
We study a free boundary problem which arises as the continuum version of a stochastic particles system in the context of Fourier law. Local existence and uniqueness of the classical solution are well known in the literature of free…
We give a characterization of completely regular topological spaces. Applying some recent results for supinf problems in completely regular topological spaces we establish a variational principle for saddle points. Well-posedness of saddle…
Perturbation theory in geometric theories of gravitation is a gauge theory of symmetric tensors defined on a Lorentzian manifold (the background spacetime). The gauge freedom makes uniqueness problems in perturbation theory particularly…
Pathwise uniqueness for stochastic PDEs with drift in differential form is a main open problem in the recent literature on regularisation by noise. This paper establishes a self-contained theory in the framework of stochastic evolution…
In this paper we study solitary traveling wave solutions to a damped shallow water system, which is in general quasilinear and of mixed type. We develop a small data well-posedness theory and prove that traveling wave solutions are a…
Fractional differential equation (FDE) provides an accurate description of transport processes that exhibit anomalous diffusion but introduces new mathematical difficulties that have not been encountered in the context of integer-order…
We discuss initial-boundary value problems of arbitrary spatial order subject to arbitrary boundary conditions. We formalise the concept of the conditioning of such a problem and show that it represents a necessary criterion for…
The aim of this paper is to obtain quantitative bounds for solutions to the optimal matching problem in dimension two. These bounds show that up to a logarithmically divergent shift, the optimal transport maps are close to be the identity…