Related papers: On the well-posedness of (nonlinear) rough continu…
One-dimensional stochastic differential equations with additive L\'evy noise are considered. Conditions for existence and uniqueness of a strong solution are obtained. In particular, if the noise is a L\'evy symmetric stable process with…
We extend the theory of viscosity solutions to treat scalar-valued doubly-nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for…
We consider density solutions for gradient flow equations of the form $u_t = \nabla \cdot ( \gamma(u) \nabla \mathrm N(u))$, where $\mathrm N$ is the Newtonian repulsive potential in the whole space $\mathbb R^d$ with the nonlinear convex…
We study the motion of charged liquid drop in three dimensions where the equations of motions are given by the Euler equations with free boundary with an electric field. This is a well-known problem in physics going back to the famous work…
We study a dissipative system of nonlinear and nonlocal equations modeling the flow of electrohydrodynamics. The existence, uniqueness and regularity of solutions is proven for general $\mathbf{L}^2$ initial data in two space dimensions and…
We show that given an initial vorticity which is bounded and $m$-fold rotationally symmetric for $m \ge 3$, there is a unique global solution to the 2D Euler equation on the whole plane. That is, in the well-known $L^1 \cap L^\infty$ theory…
In this article we show a robustness theorem for controlled stochastic differential equations driven by approximations of Brownian motion. Often, Brownian motion is used as an idealized model of a diffusion where approximations such as…
In this paper we prove full local well-posedness for the Cauchy problem for the compressible 3D Euler equation, i.e. local existence, uniqueness, and continuous dependence on initial data, with initial velocity, density and vorticity…
We prove probabilistic well-posedness for a 2D viscous nonlinear wave equation modeling fluid-structure interaction between a 3D incompressible, viscous Stokes flow and nonlinear elastodynamics of a 2D stretched membrane. The focus is on…
We investigate the inviscid 2D Boussinesq equations driven by rough transport noise of Kraichnan type with regularity index $\alpha\in (0,1/2)$. For all $1<p<\infty$, we establish the existence and uniqueness of probabilistic strong…
We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are…
In the present work, we investigate stochastic third grade fluids equations in a $d$-dimensional setting, for $d = 2, 3$. More precisely, on a bounded and simply connected domain $\mathcal{D}$ of $\mathbb{R}^d$, $d = 2,3$, with a…
A study of regularity estimate for weak solution to generalized stationary Stokes-type systems involving $p$-Laplacian is offered. The governing systems of equations are based on steady incompressible flow of a Newtonian fluids. This paper…
We develop first-principles theory of relativistic fluid turbulence at high Reynolds and P\'eclet numbers. We follow an exact approach pioneered by Onsager, which we explain as a non-perturbative application of the principle of…
Conserved or dissipated quantities, like energy or entropy, are at the heart of the study of many classes of time-dependent PDEs in connection with fluid mechanics. This is the case, for instance, for the Euler and Navier-Stokes equations,…
We consider uniformly rotating incompressible Euler and Navier-Stokes equations. We study the suppression of vertical gradients of Lagrangian displacement ("vertical" refers to the direction of the rotation axis). We employ a formalism that…
We study the rheology of a suspension of soft deformable droplets subjected to a pressure-driven flow. Through computer simulations, we measure the apparent viscosity as a function of droplet concentration and pressure gradient, and provide…
The purpose of this work is twofold. First, we construct probabilistically strong solutions to the three-dimensional Euler equations perturbed by additive noise that are $\mathbb{P}$-almost surely continuous in time, H\"older in space, and…
We present two criteria to conclude that a stochastic partial differential equation (SPDE) posseses a unique maximal strong solution. This paper provides the full details of the abstract well-posedness results first given in…
Rough stochastic differential equations (rough SDEs), recently introduced by Friz, Hocquet and L\^e in arXiv:2106.10340, have emerged as a versatile tool to study "doubly" SDEs under partial conditioning (with motivation from pathwise…