Related papers: The non-linear sewing lemma I : weak formulation
We investigate the existence of weak solutions to a certain system of partial differential equations, modelling the behaviour of a compressible non-Newtonian fluid for small Reynolds number. We construct the weak solutions despite the lack…
This is a survey of some of the recent developments on the geometric and analytic aspects of the Anomaly flow. It is a flow of $(2,2)$-forms on a $3$-fold which was originally motivated by string theory and the need to preserve the…
We investigate diffusion-type partial differential equations that are irregular in the sense that they admit weak solutions which are nowhere smooth, even for prescribed smooth data. By reformulating these equations as first-order partial…
A basic example of shear flow was introduced by DiPerna and Majda to study the weak limit of oscillatory solutions of the Euler equations of incompressible ideal fluids. In particular, they proved by means of this example that weak limit of…
In this paper, we consider the stochastic %equations of incompressible non-Newtonian fluids driven by a cylindrical Wiener process $W$ with shear rate dependent on viscosity in a bounded Lipschitz domain $D\in \mathbb{R}^n$ during the time…
The effect of strong shear flow on highly fluctuating lamellar systems stabilized by intermembrane collisions via the Helfrich interaction is studied. Advection enters the microscopic equation of motion for a single membrane via a…
In this article we examine the interaction of incompressible 2D flows with compact material boundaries. Our focus is the dynamic behavior of the circulation of velocity around boundary components and the possible exchange between flow…
The equations of stationary compressible flows of active liquid crystals are considered in a bounded three-dimensional domain. The system consists of the stationary Navier-Stokes equations coupled with the equation of Q-tensors and the…
The purpose of this contribution is to show that some of the basic ideas of turbulence can be addressed in a deterministic setting instead of introducing random realizations of the fluid. Weak limits of oscillating sequences of solutions…
New exact solutions are obtained for several nonlinear physical equations, namely the Navier-Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinear Schroedinger equation. The solution methods make use of…
This paper presents a unified Least-Squares framework for solving nonlinear partial differential equations by recasting the governing system as a residual minimisation problem. A Least-Squares functional is formulated and the corresponding…
In this paper, a novel fully-explicit weakly compressible solver is developed for solving incompressible two-phase flows. The two-phase flow is modelled by coupling the general pressure equation, momentum conservation equations and the…
We study a class of semilinear diffusion equations on infinite, connected, weighted graphs, focusing on two types of nonlinearities: monotone decreasing and Lipschitz continuous. Under minimal structural assumptions on the graph, we…
We consider the problem of the approximation of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as $x^\alpha$, with $\alpha>1$. We propose an (semi-explicit) exponential-Euler…
We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in $\mathbb R^N$ with no ellip\-ticity nor smoothness requirements, in order to…
We prove the existence of weak solutions for the one obstacle problem associated with a class of quasilinear wave equations in one space dimension, extending previous results obtained in the linear case, and we also address the two…
In this paper, we extend our previous result from [16]. We prove that transport equations with rough coefficients do possess a uniqueness property. Our method relies strongly on duality and bears a strong resemblance with the well-known…
We study porous medium equations with a divergence form of drift terms in a bounded domain with no-flux lateral boundary conditions. We establish $L^q$-weak solutions for $ 1\leq q < \infty$ in Wasserstein space under appropriate conditions…
We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure…
The shear shallow water model is a higher order model for shallow flows which includes some shear effects that are neglected in the classical shallow models. The model is a non-conservative hyperbolic system which can admit shocks,…