arXiv subjects
Publications and source records attributed to Zonglin Jia.
In this article, we study two Hamiltonian type flows: Yang-Mills-Higgs-Schrödinger flow and $A$-Schrödinger flow. For the first one, we only obtain local existence. However, the uniqueness follows from classical tricks for the second one.
In the article we obtain almost global existence for Dirac Equations with high regularity and small initial datum on Tori. Besides, the global existence with low regularity and small initial datum is gotten. The approaches are mainly Gagliardo-Nirenberg-Moser estimates and Bernstein-Type Lemma.
Classical results of second order parabolic quasi-linear equations always require that the nonlinear terms are controlled by a power of the unknown functions and their first derivatives. We improve the previous results. More precisely, in the present article the upper bound of nonlinear term can be extended to a power series.
We get decay rate of higher derivatives of nonlinear massless Dirac equations with a kind of "good" spin null form. The method we rely on is similar to that of Li and Zang. However, they only give the decay rate of solution itself to nonlinear massless Dirac system.
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 is small enough, then the EM system has a global in time solution.
The problem involved in this paper is the global existence of the solution to the $\mathfrak{su}(2)$-Einstein-Yang-Mills-Higgs(EYMH) equation. The approach we employ stems from H. Lindblad and I. Rodnianski and is dependent of wave coordinates and Lorentzian gauge conditions. Our main conclusion is that the EYMH system admits global existence provided the initial datum are sufficiently small. To the best of our knowledge, there is no similar result in the area of EYMH equations.
In this paper, we consider the high order geometric flows of a submanifolds $M$ in a complete Riemannian manifold $N$ with $\dim(N)=\dim(M)+1=n+1$, which were introduced by Mantegazza in the case the ambient space is an Euclidean space, and extend some results due to Mantegazza to the present situation under some assumptions on $N$. Precisely, we show that if $m\in\mathbb{N}$ is strictly larger than the integer part of $n/2$ and $φ(t)$ is a immersion for all $t\in[0,T)$ and if $\mathfrak{F}_m(φ_0)$ is bounded by a constant which relies on the injectivity radius $\bar{R}>0$ and sectional curvature $\bar{K}_π(\bar{K}_π\leqslant1)$ of $N$ , then $T$ must be $\infty$.
In this paper, we consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra $\mathfrak{g}$, which can be viewed as the extension of Landau-Lifshtiz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such Landau-Lifshtiz equations from an $n$-dimensional closed Riemannian manifold $\mathbb{T}$ or a bounded domain in $\mathbb{R}^n$ into a unit sphere $S_\mathfrak{g}(1)$ in $\mathfrak{g}$. In particular, we consider the Hamiltonian system associated with the nonlocal energy--{\it micromagnetic energy} defined on a bounded domain of $\mathbb{R}^3$ and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.
In this article, we bring in Landau-Lifshitz-Bloch(LLB) equation on $m$-dimensional closed Riemannian manifold and prove that it admits a unique local solution. In addition, if $m\geqslant3$ and $L^{\infty}-$norm of initial data is sufficiently small, the solution can be extended globally. Moreover, if $m=2$, we can prove that the unique solution is global without assuming small initial data.
$\,\,\,\,\,\,$In this paper, we prove that the nonautonomous Schrödinger flow from a compact Riemannian manifold into a Kähler manifold admits a local solution. Under some certain conditions, the solution is unique and has higher regularity.
In this paper, we bring in General Landau-Lifshitz-Bloch equation and prove that it admits a local strong solution.