arXiv subjects

Kuijie Li

Publications and source records attributed to Kuijie Li.

10 recordsLinked to original sources

arXiv2025

Scattering for the Klein-Gordon-Zakharov system in two dimensions

We study the Klein-Gordon-Zakharov system in two spatial dimensions, an important model in plasma physics. For small, smooth, and spatially localized initial data, we establish the global existence of solutions and characterize their sharp long-time behavior, including sharp time decay and scattering properties. A particularly interesting phenomenon is that the Klein-Gordon component exhibits modified scattering for certain initial data, while for others it undergoes linear scattering-a dichotomy highlighting delicate long-range interaction effects. The major obstacles are lack of symmetry and weak decay of the solution in two dimensions. To overcome these, we introduce a novel nonlinear transformation of the wave component and reinterpret the nonlinear coupling as a perturbation of the mass term in the Klein-Gordon equation. The proof employs a combination of physical space and frequency space methods.

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arXiv2024

Large data global existence for coupled massive-massless wave-type systems

We consider 3D Klein-Gordon-Zakharov (KGZ) and Dirac-Klein-Gordon (DKG) systems, where a common feature is that there exist both massless and massive fields in each system. We establish global existence and asymptotic behavior for both systems with a class of large data. More precisely, in the KGZ system, we allow the massless field to be large, while in the DKG system we allow the massive field to be large.

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arXiv2023

Cubic Dirac equations with a class of large data

We are interested in massless cubic Dirac equations in two and three space dimensions, known as the Soler model. The solution to this model is known as a wave function, which has the unit $L^2$ norm. We aim to show global existence and asymptotic behavior for the cubic Dirac model with a class of initial data that can be large in $L^2$.

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arXiv2022

Global solution to the 3D Dirac--Klein-Gordon system with uniform energy bounds

On the (1+3) dimensional Minkowski spacetime, for small, regular initial data, it is well-known that the Dirac-Klein-Gordon system admits a global solution. In the present paper, we aim to establish the uniform boundedness of the total energy of the solution for this system. The proof relies on Klainerman's vector field and Alinhac's ghost weight methods. The main difficulty originates from the slow decay nature of the Dirac and wave components in three space dimensions. To overcome the difficulty, a sharp understanding of the structure for this system, and a new weighted conformal energy estimate are required. In addition, we also provide a few scattering results for the system.

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arXiv2022

Global Behavior of Small Data Solutions for The 2D Dirac-Klein-Gordon Equations

In this paper, we are interested in the two-dimensional Dirac-Klein-Gordon system, which is a basic model in particle physics. We investigate the global behaviors of small data solutions to this system in the case of a massive scalar field and a massless Dirac field. More precisely, our main result is twofold: 1) we show sharp time decay for the pointwise estimates of the solutions which imply the asymptotic stability of this system; 2) we show the linear scattering result of this system which is a fundamental problem when it is viewed as dispersive equations. Our result is valid for general small, high-regular initial data, in particular, there is no restriction on the support of the initial data.

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arXiv2021

Global solution to the cubic Dirac equation in two space dimensions

We are interested in the cubic Dirac equation with mass $m \in [0, 1]$ in two space dimensions, which is also known as the Soler model. We conduct a thorough study on this model with initial data sufficiently small in high regularity Sobolev spaces. First, we show the global existence of the model, which is uniform-in-mass. In addition, we derive a unified pointwise decay result valid for all $m \in [0, 1]$. Last but not least, we prove the cubic Dirac equations scatter linearly with an explicit scattering speed. When the mass $m=0$, we can show an improved pointwise decay result.

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arXiv2019

Navier-Stokes Equation in Super-Critical Spaces $E^s_{p,q}$

In this paper we develop a new way to study the global existence and uniqueness for the Navier-Stokes equation (NS) and consider the initial data in a class of modulation spaces $E^s_{p,q}$ with exponentially decaying weights $(s<0, \ 1<p,q<\infty)$ for which the norms are defined by $$ \|f\|_{E^s_{p,q}} = \left(\sum_{k\in \mathbb{Z}^d} 2^{s|k|q}\|\mathscr{F}^{-1} χ_{k+[0,1]^d}\mathscr{F} f\|^q_p \right)^{1/q}. $$ The space $E^s_{p,q}$ is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding $H^σ\subset E^s_{2,1}$ for all $σ<0$ and $s<0$. It is known that $H^σ$ ($σ<d/2-1$) is a super-critical space of NS, it follows that $ E^s_{2,1}$ ($s<0$) is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to $E^s_{2,1}$ ($s<0$) and their Fourier transforms are supported in $ \mathbb{R}^d_I:= \{ξ\in \mathbb{R}^d: \ ξ_i \geq 0, \, i=1,...,d\}$. Similar results hold for the initial data in $E^s_{r,1}$ with $2< r \leq d$. Our results imply that NS has a unique global solution if the initial value $u_0$ is in $L^2$ with ${\rm supp} \, \widehat{u}_0 \, \subset \mathbb{R}^d_I$.

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arXiv2019

Remarks on the well-posedness of the Euler equations in the Triebel-Lizorkin spaces

We prove the continuous dependence of the solution maps for the Euler equations in the (critical) Triebel-Lizorkin spaces, which was not shown in the previous works(\cite{Ch02, Ch03, ChMiZh10}). The proof relies on the classical Bona-Smith method as \cite{GuLiYi18}, where similar result was obtained in critical Besov spaces $B^1_{\infty,1}$.

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arXiv2018

Blowup criterion for Navier-Stokes equation in critical Besov space with spatial dimensions $d \geq 4$

This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions $d \geq 4$. By establishing an $ε$ regularity criterion, we show that if the mild solution $u$ with initial data in $\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d) $, $d<p,\,q<\infty$ becomes singular at a finite time $T_*$, then $$ \limsup_{t\to T_*} \|u(t)\|_{\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d)} = \infty. $$ The corresponding result in 3D case has been obtained by I.Gallagher, G.S.KochandF.Planchon. As a by-product, we also prove a regularity criterion for the Leray-Hopf solution in the critical Besov space, which generalizes the results in~\cite{DoDu09}, where blowup criterion in critical Lebesgue space $L^d(\mathbb{R}^d)$ is obtained.

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arXiv2016

Dynamical Behavior for the Solutions of the Navier-Stokes Equation

We study the Cauchy problem for the incompressible Navier-Stokes equations (NS) in three and higher spatial dimensions: \begin{align} u_t -Δu+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0(x). \label{NSa} \end{align} Leray and Giga obtained that for the weak and mild solutions $u$ of NS in $L^p(\mathbb{R}^d)$ which blow up at finite time $T>0$, respectively, one has that for $d 0$, then \eqref{NSa} has a unique global solution $u\in C(\mathbb{R}_+, L^\infty)$. Finally, if the blowup rate is of type I: $$ \|u(t)\|_p \sim ( T-t )^{-(1-d/p)/2}, \ for \ 0< t<T<\infty, \ d<p<\infty $$ in 3 dimensional case, then we can obtain a minimal blowup solution $Φ$ for which $$ \inf \{\limsup_{t \to T}(T-t)^{(1-3/p)/2}\|u(t)\|_{L^p_x}: \ u\in C([0,T); L^p) \mbox{\ solves \eqref{NSa}}\} $$ is attainable at some $Φ\in L^\infty (0,T; \ \dot B^{-1+6/p}_{p/2,\infty})$.

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