arXiv subjects

Xuecheng Wang

Publications and source records attributed to Xuecheng Wang.

24 records · Page 2Linked to original sources

arXiv2016

Global infinite energy solutions for the 2D gravity water waves system

We prove global existence and modified scattering property for the solutions of the $2D$ gravity water waves system in the infinite depth setting for a class of initial data, which is only required to be small above the level $\dot{H}^{1/5}\times \dot{H}^{1/5+1/2}$. No assumption is assumed below this level, therefore, it allows to have infinite energy. As a direct consequence, the momentum condition assumed on the physical velocity in all previous small energy results by Ionescu-Pusateri, Alazard-Delort and Ifrim-Tataru is removed.

math.AP↗

arXiv2016

Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u})$ type

We study a class of $3D$ quadratic Schrödinger equations as follows, $(\partial_t -i Δ) u = Q(u, \bar{u})$. Different from nonlinearities of the $uu$ type and the $\bar{u}\bar{u}$ type, which have been studied by Germain-Masmoudi-Shatah, the interaction of $u$ and $\bar{u}$ is very strong at the low frequency part, e.g., $1\times 1 \rightarrow 0$ type interaction (the size of input frequency is "$1$" and the size of output frequency is "$0$"). It creates a growth mode for the Fourier transform of the profile of solution around a small neighborhood of zero. This growth mode will again cause the growth of profile in the medium frequency part due to the $1\times 0\rightarrow 1$ type interaction. The issue of strong $1\times 1\rightarrow 0$ type interaction makes the global existence problem very delicate. In this paper, we show that, as long as there are "$ε$" derivatives inside the quadratic term $Q (u, \bar{u})$, there exists a global solution for small initial data. As a byproduct, we also give a simple proof for the almost global existence of the small data solution of $(\partial_t -i Δ)u = |u|^2 = u\bar{u}$, which was first proved by Ginibre-Hayashi. Instead of using vector fields, we consider this problem purely in Fourier space.

math.AP↗

arXiv2016

Global regularity for the 3D finite depth capillary water waves

In this paper, we prove global regularity, scattering, and the non-existence of small traveling waves for the $3D$ finite depth capillary waves system for small initial data. The non-existence of small traveling waves shows a fundamental difference between the capillary waves ($σ=1, g=0$) and the gravity-capillary waves ($σ=1,$ $ 0< g< 3$) in the finite depth setting. As, for the later case, there exists arbitrary small $L^2$ traveling waves. Different from the water waves system in the infinite depth setting, the quadratic terms of the same system in the finite depth setting are worse due to the absence of null structure inside the Dirichlet-Neumann operator. In the finite depth setting, the capillary waves system has the worst quadratic terms among the water waves systems with all possible values of gravity effect constant and surface tension coefficient. It loses favorable cancellations not only in the High $\times$ Low type interaction but also in the High $\times $ High type interaction. In the worst scenario, the best decay rate of the nonlinear solution that one could expect is $ (1+t) ^ {-1/2} $, because the $3D$ finite depth capillary waves system lacks null structures and there exists $Q(u, \bar{u})$ type quadratic term, which causes a very large time resonance set and the definite growth of the associated profile. As a result, the problematic terms are not only the quadratic terms but also the cubic terms. To prove global regularity for the capillary waves system, we identify a good variable to prove the dispersion estimate, fully exploit the hidden structures inside the capillary waves system, and use a novel method to control the weighted norms.

math.AP↗

arXiv2016

On 3D water waves system above a flat bottom

As a starting point of studying the long time behavior of the $3D$ water waves system in the flat bottom setting, in this paper, we try to improve the understanding of the Dirichlet-Neumann operator in this setting. As an application, we study the $3D$ gravity waves system and derive a new $L^2-L^\infty$ type energy estimate, which has a good structure in the $L^\infty-$type space. In our second paper, base on the results we obtained in this paper, we prove the global regularity of the $3D$ gravity waves system for suitably small initial data.

math.AP↗

arXiv2014

On Global Existence of 3D Charge Critical Dirac-Klein-Gordon system

In this paper, we prove the global well-posedness property of charge critical Dirac-Klein-Gordon (DKG) system in $\mathbb{R}^{3+1}$ for small initial data in a space of scale invariant data which has extra weighted regularity in the angular variables. Therefore, by finite speed propagation, we could also derive the local well-posedness property for large initial data in the same space. To author's knowledge, our result appears to be the first result on the critical 3D DKG.

math.AP↗

arXiv2013

Global regularity for the energy-critical NLS on $\mathbb{S}^3$

We establish global existence for the energy-critical nonlinear Schrödinger equation on $\mathbb{S}^3$. This follows similar lines to the work on $\mathbb{T}^3$ but requires new extinction results for linear solutions and bounds on the first nonlinear iterate at a Euclidean profile that are adapted to the new geometry.

math.AP↗