Related papers: Global smooth solutions to 4D quasilinear wave equ…

In this paper, we consider the semilinear damped wave equation with nonlinearities of derivative type $|u_t|^p$. We observe that this problem admits a unique global (in time) solution with small initial data for all $p > 1$ in low spatial…

We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density. The central feature of the solutions is the…

There is an interesting open question: for the $n$-D ($n\ge 1$) semilinear wave equation with scale-invariant damping $\partial_t^2u-\Delta u+\frac{\mu}{t}\partial_tu=|u|^p$, where $t\ge 1$, $p>1$ and $\mu>0$, the global small data weak…

The motion of a compressible inviscid radiative flow can be described by the radiative Euler equations, which consists of the Euler system coupled with a Poisson equation for the radiative heat flux through the energy equation. Although…

We study global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case. We prove the global well-posedness for small data in the energy space for the…

Short pulse initial datum is referred to the one supported in the ball of radius $\delta$ and with amplitude $\delta^{\frac12}$ which looks like a pulse. It was first introduced by Christodoulou to prove the formation of black holes for…

In this paper, we prove almost global existence of solutions to certain quasilinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides with Neumann boundary conditions. We use a Galerkin method to expand the…

In this paper, we give a criterion on the Cauchy data for the semilinear wave equations satisfying the null condition in $\mathbb{R}^+\times\mathbb{R}^{3}$ such that the energy of the data can be arbitrarily large while the solution is…

We study a class of $3D$ quadratic Schr\"odinger equations as follows, $(\partial_t -i \Delta) u = Q(u, \bar{u})$. Different from nonlinearities of the $uu$ type and the $\bar{u}\bar{u}$ type, which have been studied by…

This paper is concerned with the global existence of small solutions to pure-power nonlinear Schroedinger equations subject to radially symmetric data with critical regularity. Under radial symmetry we focus our attention on the case where…

In this paper, we study the global wellposedness of a radiation hydrodynamics model with viscosity and thermal conductivity. It is now well-understood that, unlike the compressible Euler equations whose smooth solutions must blow up in…

We prove global in time existence of solutions of the Euler compressible equations for a Van der Waals gas when the density is small enough in $\H{m}$, for $m$ large enough. To do so, we introduce a specific symmetrisation allowing areas of…

We study both of the scattering and Cauchy problems for the semilinear wave equation with null quadratic form on the Schwarzschild background. Prescribing the scattering data that are given by the short pulse data on the future null…

In this paper, we prove global regularity for all smooth, axisymmetric, swirl-free solutions of the Euler equation in four dimensions. Previous works establishing global regularity for certain axisymmetric, swirl-free solutions of the Euler…

We are interested in the "almost" global-in-time existence of classical solutions in the general theory for nonlinear wave equations. All the three such cases are known to be sharp due to blow-up results in the critical case for model…

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…

The goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size $\epsilon$, is almost…

This paper concerns the global in time existence of solutions for a semilinear heat equation \begin{equation} \tag{P} \label{eq:P} \begin{cases} \partial_t u = \Delta u + f(u), &x\in \mathbb{R}^N, \,\,\, t>0, \\[3pt] u(x,0) = u_0(x) \ge 0,…

We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely…

In his 2007 monograph, D. Christodoulou proved a breakthrough result giving a detailed description of the formation of shocks in solutions to the relativistic Euler equations in three spatial dimensions. He assumed that the data have small…