Related papers: Global smooth solutions to 4D quasilinear wave equ…
In this paper, we consider the three dimensional Cauchy problem of the compressible micropolar viscous flows, we prove the existence of unique global classical solution for smooth initial data with small initial energy but possibly large…
We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\"odinger equation (NLS), $i \partial_t u+\Delta u+ |u|^{p-1}u=0, $ with initial data $u_0\in H^1,\; x \in R^n$. The nonlinearity power…
We establish the global-in-time existence of solutions of the Cauchy problem for the full Navier-Stokes equations for compressible heat-conducting flow in multidimensions with initial data that are large, discontinuous, spherically…
This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant $(x,t)\in \mathbb{R}^+\times\mathbb{R}^+$, \begin{equation}\notag \partial_t v - \partial_x u=0, \qquad…
For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, we consider the Cauchy problem. The initial conditions have discontinuity of the first kind at one point. We construct the solution using the method of…
We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu…
In this thesis we explore S. Klainerman's proof on the global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, as established in his paper from 1985. We consider initial data with…
In this paper we prove that for a certain class of initial data, smooth solutions of the hydrostatic Euler equations blow up in finite time.
We consider a class of quasilinear wave equations in $3+1$ space-time dimensions that satisfy the "weak null condition" as defined by Lindblad and Rodnianski \cite{LR1}, and study the large time behavior of solutions to the Cauchy problem.…
It is well-known that in dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data. It is also known that in dimension two such result still holds for a certain class of…
In this paper, we discuss a new nonlinear phenomenon. We find that in $n\geq 2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) =…
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…
This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. By constructing a special test function, we first present a simpler proof of the main result in…
In this paper, we consider Cauchy problem of simplified Ericksen-Leslie system in dimension three. We establish the unique existence of global smooth solution under some nonlinear conditions on initial data. However, we do not need small…
We consider the Euler-Poincar\'e equation on $\mathbb R^d$, $d\ge 2$. For a large class of smooth initial data we prove that the corresponding solution blows up in finite time. This settles an open problem raised by Chae and Liu \cite{Chae…
Global existence for small data Cauchy problem of semilinear wave equations with scaling invariant damping in 3-D is established in this work, assuming that the data are radial and the constant in front of the damping belongs to $[1.5, 2)$.…
Global existence and boundedness of solutions to the Cauchy problem for the four dimensional fully parabolic chemotaxis system with indirect signal production are studied. We prove that solutions with initial mass below $(8\pi)^2$ exist…
First of all, we get the global existence of classical and strong solutions of the full compressible Navier-Stokes equations in three space dimensions with initial data which is large and spherically or cylindrically symmetric. The…
Our interest itself of this paper is strongly inspired from an open problem in the paper [1] published by D'Abbicco. In this article, we would like to study the Cauchy problem for a weakly coupled system of semi-linear structurally damped…
We prove global existence of solutions to quasilinear wave equations with quadratic nonlinearities exterior to nontrapping obstacles in spatial dimensions four and higher. This generalizes a result of Shibata and Tsutsumi in spatial…