Related papers: Dimension-free Euler estimates of rough differenti…

We consider a class of $n^{\text{th}}$-order linear ordinary differential equations with a large parameter $u$. Analytic solutions of these equations can be described by (divergent) formal series in descending powers of $u$. We demonstrate…

We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs…

This notes gives several criteria which exclude the existence of discretely self-similar solutions of the three dimensional incompressible Euler equations.

We prove a uniform Fourier extension-restriction estimate for a certain class of curves in d-dimensional Euclidean space.

We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial…

Motivated by problems in contact mechanics, we propose a duality approach for computing approximations and associated a posteriori error bounds to solutions of variational inequalities of the first kind. The proposed approach improves upon…

We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit…

In this work we study fractal properties of rough differential equations driven by a fractional Brownian motions with Hurst parameter $H>\frac{1}{4}$. In particular, we show that the Hausdorff dimension of the sample paths of the solution…

A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. For this, a mathematical model is developed to incorporate homogeneous Dirichlet and Neumann type boundary conditions. The…

In this paper we propose a new point of view on weak solutions of the Euler equations, describing the motion of an ideal incompressible fluid in $\mathbb{R}^n$ with $n\geq 2$. We give a reformulation of the Euler equations as a differential…

The non-linear sewing lemma constructs flows of rough differential equations from a braod class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential…

We show that a certain error estimate for a fully discrete finite element approximation of the solution of the heat equation which is defined in a two-dimensional Euclidean domain carries over to the case of a general linear parabolic…

Motivated by applications to fluid dynamics, we study rough differential equations (RDEs) and rough partial differential equations (RPDEs) with non-Lipschitz drifts. We prove well-posedness and existence of a flow for RDEs with Osgood…

We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such…

The strong convergence of Euler approximations of stochastic delay differential equations is proved under general conditions. The assumptions on drift and diffusion coefficients have been relaxed to include polynomial growth and only…

This paper establishes the existence and uniqueness of solutions for rough differential equations driven by reduced rough paths with low regularity, specifically in the roughness regime $\frac{1}{3} < \alpha \leq \frac{1}{2}$. While the…

In in this paper we show how using D.A. it is found a simple change of variables (c.v.) that brings us to obtain differential equations simpler than the original one. In a pedagogical way (at least we try to do that) and in order to make…

We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different…

Let $(M^n,g)$ be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\Delta u+au\log u=0,$$ where $a$ is a nonzero…

In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is…