Related papers: Almost sure existence of global solutions for gene…
We prove the global strong solvability of a quasilinear initial-boundary value problem with fractional time derivative of order less than one. Such problems arise in mathematical physics in the context of anomalous diffusion and the…
The initial value problem for the Vlasov-Poisson system is by now well understood in the case of an isolated system where, by definition, the distribution function of the particles as well as the gravitational potential vanish at spatial…
We consider the initial value problem for the inviscid Primitive and Boussinesq equations in three spatial dimensions. We recast both systems as an abstract Euler-type system and apply the methods of convex integration of De Lellis and…
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…
We study the global existence of Einstein-Maxwell(EM) equations on $\mathbb{R}^4$. We use the method, which relies on wave and Lorentzian gauge conditions, to obtain some exquisite estimates. Our main conclusion is that if the initial data…
This paper is concerned with the original Kirchhoff equation $$\left\{\begin{aligned} & \pa_{tt}u-\Big(1+\int_{0}^{\pi}|\pa_xu|^2 dx\Big)\pa_{xx}u=0, \\&u(t,0)=u(t,\pi)=0. \end{aligned}\right.$$ We obtain almost global existence and…
Assuming initial data have small weighted $H^4\times H^3$ norm, we prove global existence of solutions to the Cauchy problem for systems of quasi-linear wave equations in three space dimensions satisfying the null condition of Klainerman.…
We prove an abstract result of almost global existence of small solutions to semi-linear Hamiltonian partial differential equations satisfying very weak non resonance conditions and basic multilinear estimates. Thanks to works by…
We consider a modified Euler equation on $\mathbb R^2$. We prove existence of weak global solutions for bounded (and fast decreasing at infinity) initial conditions and construct Gibbs-type measures on function spaces which are…
We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding…
It is well known that for the quasilinear Klein-Gordon equation with quadratic nonlinearity and sufficiently decaying small initial data, there exists a global smooth solution if the space dimensions $d\geq2$. When the initial data are of…
In recent years, the global existence of classical solutions to the Cauchy problem for 2D incompressible viscous MHD equations without magnetic diffusion has been proved in \cite{Ren,TZhang}, under the assumption that initial data is close…
This article represents the second installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on global solutions for small and localized data. Such…
A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely…
We consider the initial value problem for the 2D quasi-geostrophic equation with weak dissipation term $\kappa(-\Delta)^{\alpha/2}\theta\ (0<\alpha\leqslant 1)$ and dispersive forcing term $Au_2$. We establish a unique global solution for a…
We study the existence of global solutions to semilinear wave equations on exterior domains $\mathbb{R}^n\setminus\mathcal{K}$, $n\geq2$, with small initial data and nonlinear terms $F(\partial u)$ where $F\in C^\kappa$ and…
We study the initial value problem for a kind of Euler equation with a source term. Our main result is the existence of a globally-in-time weak solution whose total variation is bounded on the the domain of definition, allowing the…
We prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the product of the maximal and minimal slopes is strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$…
For 3-D quadratic quasilinear wave equations with or without null conditions in exterior domains, when the compatible initial data and Dirichlet boundary values are given, the global existence or the maximal existence time of small data…
We prove existence of global-in-time weak solutions of the incompressible Navier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data in a weighted space that allow non-uniformly locally square integrable functions that…