Showing papers by "Peter Constantin published in 2017"
TL;DR: In this paper, the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law, is considered.
82 citations
TL;DR: In this paper, the inviscid limit of the Navier-Stokes equations in a half-plane with Dirichlet boundary conditions was shown to hold in the energy norm.
Abstract: We consider the vanishing viscosity limit of the Navier--Stokes equations in a half-plane, with Dirichlet boundary conditions. We prove that the inviscid limit holds in the energy norm if the product of the components of the Navier--Stokes solutions are equicontinuous at $x_2=0$. A sufficient condition for this to hold is that the tangential Navier--Stokes velocity remains uniformly bounded and has a uniformly integrable tangential gradient near the boundary.
32 citations
TL;DR: In this article, it was shown that weak space-time inviscid limits of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second order structure function.
Abstract: We prove that any weak space-time $L^2$ vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of ${\mathbb{R}}^2$ satisfies the Euler equation if the solutions' local enstrophies are uniformly bounded. We also prove that $t-a.e.$ weak $L^2$ inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.
25 citations
TL;DR: In this paper, the authors consider a model of electroconvection motivated by studies of the motion of a two-dimensional annular suspended smectic film under the influence of an electric potential maintained at the boundary by two electrodes.
Abstract: We consider a model of electroconvection motivated by studies of the motion of a two-dimensional annular suspended smectic film under the influence of an electric potential maintained at the boundary by two electrodes. We prove that this electroconvection model has global in time unique smooth solutions.
7 citations
TL;DR: In this article, the authors review results about nonlocal advection-diffusion equations based on lower bounds for the fractional Laplacian and show that the lower bound is tight.
Abstract: The author reviews some results about nonlocal advection-diffusion equations based on lower bounds for the fractional Laplacian.
6 citations
TL;DR: In this paper, a lubrication approximation model of the interface between two immiscible fluids in a Hele-Shaw cell, derived in CDGKSZ93, was discussed.
Abstract: We discuss a lubrication approximation model of the interface between two immiscible fluids in a Hele-Shaw cell, derived in \cite{CDGKSZ93} and widely studied since. The model consists of a single one dimensional evolution equation for the thickness $2h = 2h(x,t)$ of a thin neck of fluid, \[ \partial_t h + \partial_x( h \, \partial_x^3 h) = 0\, , \] for $x\in (-1,1)$ and $t\ge 0$. The boundary conditions fix the neck height and the pressure jump: \[ h(\pm 1,t) = 1, \qquad \partial_{x}^2 h(\pm 1,t) = P>0. \] We prove that starting from smooth and positive $h$, as long as $h(x,t) >0$, for $x\in [-1,1], \; t\in [0,T]$, no singularity can arise in the solution up to time $T$. As a consequence, we prove for any $P>2$ and any smooth and positive initial datum that the solution pinches off in either finite or infinite time, i.e., $\inf_{[-1,1]\times[0,T_*)} h = 0$, for some $T_* \in (0,\infty]$. These facts have been long anticipated on the basis of numerical and theoretical studies.
1 citations