Related papers: Two dimensional gravity waves at low regularity II…
In this paper, we study the global regularity problem for the 2D Rayleigh-B\'{e}nard equations with logarithmic supercritical dissipation. By exploiting a combined quantity of the system, the technique of Littlewood-Paley decomposition and…
We prove local well-posedness results for the semi-linear wave equation for data in $H^\gamma$, $0 < \gamma < \frac{n-3}{2(n-1)}$, extending the previously known results for this problem. The improvement comes from an introduction of a…
We establish the local existence and uniqueness of solutions to the two-dimensional compressible Euler equations with initial velocity $\bv_0$, logarithmic density $\rho_0$, and specific vorticity \(w_0\), which satisfy $(\bv_0, \rho_0,…
We prove global existence and asymptotic behavior of classical solutions for two dimensional inviscid Rotating Shallow Water system with small initial data subject to the zero-relative-vorticity constraint. One of the key steps is a…
In this paper we prove global existence and global behavior of solutions to quasilinear wave-Klein-Gordon systems in $\mathbb{R}^{1+2}$ with quadratic nonlinearities satisfying the null condition. We consider small, regular and compactly…
We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posdness of the system in both space…
In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to…
We study the regularity properties of solutions to the initial value problem for the magnetohydrodynamic equations in $\mathbb{R}^N, N\geq 3$. We obtain a global in-time strong solution without any smallness assumptions on the initial data.
This is the final paper in the series \cite{tao:heatwave}, \cite{tao:heatwave2}, \cite{tao:heatwave3}, \cite{tao:heatwave4} that establishes global regularity for two-dimensional wave maps into hyperbolic targets. In this paper we establish…
We consider the gravity-capillary water waves equations for a bi-dimensional fluid with a periodic one-dimensional free surface. We prove a rigorous reduction of this system to Birkhoff normal form up to cubic degree. Due to the possible…
In certain models of conformal gravity, the propagation of gravitational waves is governed by a fourth order scalar partial differential equation. We study the initial value problem for a generalization of this equation, and derive a…
We consider a coupled Wave-Klein-Gordon system in 3D, and prove global regularity and modified scattering for small and smooth initial data with suitable decay at infinity. This system was derived by Wang and LeFloch-Ma as a simplified…
The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity…
In this paper we study global nonlinear stability for a system of semilinear wave and Klein-Gordon equations with quadratic nonlinearities. We consider nonlinearities of the type of wave-Klein-Gordon interactions where there are no…
A novel mathematical nonlinear theory of surface gravity waves in deep water is presented, in which analytical analysis of the classical nonlinear equations of fluid dynamics is performed under less restrictive assumptions than those…
The nonlinear two-dimensional problem, describing periodic steady waves on water of finite depth is considered in the absence of surface tension. It is reduced to a single pseudo-differential operator equation (Babenko's equation), which is…
In this paper, we will prove a new result that guarantees the global existence of solutions to the Navier--Stokes equation in three dimensions when the initial data is sufficiently close to being two dimensional. This result interpolates…
The global characteristic initial value problem for linear wave equations on globally hyperbolic Lorentzian manifolds is examined, for a class of smooth initial value hypersurfaces satisfying favourable global properties. First it is shown…
We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data $(u_0,\theta_0)$ are required to be only in the space $X=\{f\in L^2(\mathbb R^2)\,|\,\partial_xf\in L^2(\mathbb…
We study the global regularity, for all time and all initial data in $H^{1/2}$, of a recently introduced decimated version of the incompressible 3D Navier-Stokes (dNS) equations. The model is based on a projection of the dynamical evolution…