Related papers: Splash singularities for the free boundary Navier-…
In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down…
In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. We prove that for $d$-dimensional flows, $d=2$ or $3$, the free-surface of a viscous water wave, modeled by the…
We exhibit smooth initial data for the 2D water wave equation for which we prove that smoothness of the interface breaks down in finite time. Moreover, we show a stability result together with numerical evidence that there exist solutions…
In this paper we analyze a 2D free-boundary viscoelastic fluid model of Oldroyd-B type at infinite Weissenberg number. Our main goal is to show the existence of the so-called splash singularities, namely points where the boundary remains…
In this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity…
We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time "splash" (or "splat") singularity first introduced in [9], wherein the…
We consider a system of two incompressible fluids separated by a free interface. The first fluid is inviscid, governed by the Euler system, while the second fluid has positive viscosity and is governed by the Navier-Stokes system. We…
In this paper we construct two families of initial data being arbitrarily large under any scaling-invariant norm for which their corresponding weak solution to the three-dimensional Navier-Stokes equations become smooth on either $[0,T_1]$…
In fluid dynamics, an interface splash singularity occurs when a locally smooth interface self-intersects in finite time. By means of elementary arguments, we prove that such a singularity cannot occur in finite time for vortex sheet…
In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in $ \mathbb{R}^{3}$, based on a construction of a…
In this proceeding we expose a particular case of a recent result obtained by the authors regarding the incompressible Navier-Stokes equations in a smooth bounded and simply connected bounded domain, either in 2D or in 3D, with a Navier…
Consider an exterior space-time domain where the incompressible Navier-Stokes equation and continuity equation hold with no bodies or force fields present, and smooth velocity at initial time. This is equivalent to the velocity being…
We investigate the formation of singularities in the incompressible Navier-Stokes equations in $d\geq 2$ dimensions with a fractional Laplacian $|\nabla |^\alpha$. We derive analytically a sufficient but not necessary condition for…
This paper establishes finite-time splash singularity formation for 3D viscous incompressible neo-Hookean elastodynamics with free boundaries. The system features mixed stress-kinematic conditions where viscous-elastic stresses balance…
In this paper we consider the Cauchy problem for the 3D Navier-Stokes equations for incompressible flows. The initial data are assumed to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solutions can…
We analyze the two-dimensional incompressible Navier-Stokes equations on a smooth, bounded domain with Navier boundary conditions. Starting from an initial vorticity in $L^p$ with $p>2$, we show strong convergence of the vorticity in the…
In this paper we are concerned with the initial boundary value problem of the 2, 3-D Navier-Stokes equations with mixed boundary conditions including conditions for velocity, static pressure, stress, rotation and Navier slip condition…
In this paper, we investigate the existence of a unique global smooth solution to the three-dimensional incompressible Navier-Stokes equations and provide a concise proof. We establish a new global well-posedness result that allows the…
This article examines the smoothness of the solution to the Navier-Stokes equation from a novel perspective. Here, the existence of the smoother solution relative to x and to the time t was shown only for a finite time. Moreover, for each…
We prove unique existence of local-in-time smooth solutions of the Navier-Stokes equations for initial data in $L^{p}$ and $p \in [3, \infty)$ in an infinite cylinder, subject to the Neumann boundary condition.