Related papers: Primitive Equations with half horizontal viscosity

We consider solutions of the Navier-Stokes equations in $3d$ with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth…

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations \begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,&v(x,0)=\phi(x),\\ \partial_tw +…

This paper is dedicated to the study of viscous compressible barotropic fluids in dimension $N\geq2$. We address the question of well-posedness for {\it large} data having critical Besov regularity. %Our sole additional assumption is that…

We revisit the local well-posedness theory of nonlinear Schr\"odinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data…

In this article, we prove the local well-posedness, for arbitrary initial data with certain regularity assumptions, of the equations of a Viscoelastic Fluid of Johnson-Segalman type with a free surface. More general constitutive laws can be…

The three-dimensional quasi-geostrophic equation is considered over a cylindrical domain with a multiply connected horizontal cross-section. Homogeneous Neumann boundary conditions, tantamount to homogeneous density fields, are imposed on…

We prove energy estimates for a relativistic free liquid body with sufficiently small fluid velocity in a general Einstein spacetime. These estimates control Sobolev norms of the fluid velocity and enthalpy in the interior as well as…

This paper is devoted to studying the well-posedness, conservation of magnetic helicity, inviscid limit and asymptotic stability of the generalized Navier-Stokes-Maxwell (NSM) equations with the standard Ohm's law in $\mathbb{R}^d$ for $d…

In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics.

A linear stochastic vector advection equation is considered. The equation may model a passive magnetic field in a random fluid. The driving velocity field is a integrable to a certain power and the noise is infinite dimensional. We prove…

Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove…

This paper is concerned with the 1-D compressible Euler-Poisson equations with moving physical vacuum boundary condition. It is usually used to describe the motion of a self-gravitating inviscid gaseous star. The local well-posedness of…

We consider the 3D viscous primitive equations with periodic boundary conditions. These equations arise in the study of ocean dynamics and generate a dynamical system in a Sobolev H^1 type space. Our main result establishes the so-called…

We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic…

This paper is concerned with the Cauchy problem of the modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove the sharp estimate which implies local in time well-posedness in the Sobolev space $H^s(\mathbb{R}^2)$ for $s…

Well-posedness is proved for the stochastic viscous Cahn-Hilliard equation with homogeneous Neumann boundary conditions and Wiener multiplicative noise. The double-well potential is allowed to have any growth at infinity (in particular,…

In this paper we consider the initial value problem of the Benjamin equation $$ \partial_{t}u+\nu \H(\partial^2_xu) +\mu\partial_{x}^{3}u+\partial_xu^2=0, $$ where $u:\R\times [0,T]\mapsto \R$, and the constants $\nu,\mu\in \R,\mu\neq0$. We…

In this article we consider the stability threshold of the 2D magnetohydrodynamics (MHD) equations near a combination of Couette flow and large constant magnetic field. We study the partial dissipation regime with full viscous and only…

We study the well-posedness of the initial value problem for fully nonlinear evolution equations, $u_{t}=f[u],$ where $f$ may depend on up to the first three spatial derivatives of $u.$ We make three primary assumptions about the form of…

In this paper, we prove the existence of viscosity solutions to complex Hessian equations on compact Hermitian manifolds, assuming the existence of a strict subsolution in the viscosity sense. The results cover the complex Hessian quotient…