Related papers: Global oscillatory solutions for the Yang-Mills he…
We investigate the long time behaviour of the Yang-Mills heat flow on the bundle $\mathbb{R}^4\times SU(2)$. Waldron \cite{Waldron2019} proved global existence and smoothness of the flow on closed $4-$manifolds, leaving open the issue of…
We consider the heat flow of Yang-Mills-Higgs functional where the base manifold is a Riemannian surface and the fiber is a compact symplectic manifold. We show that the corresponding Cauchy problem admits a global weak solution for any…
In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an…
We consider the $SO(d)$-equivariant Yang-Mills heat flow \begin{equation*} \partial_t u-\partial_r^2 u-\frac{(d-3)}{r}\partial_r u+\frac{(d-2)}{r^2}u(1-u)(2-u)=0 \end{equation*} in dimensions $d>10.$ We construct a family of…
We consider a spherically symmetric (purely magnetic) SU(2) Yang-Mills field propagating on an ultrastatic spacetime with two asymptotically hyperbolic regions connected by a throat of radius $\alpha$. Static solutions in this model are…
This is the first part of the four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the energy critical hyperbolic Yang--Mills equation in the (4 + 1)-dimensional Minkowski…
In this paper, we consider the Yang-Mills heat flow on $\mathbb R^d \times SO(d)$ with $d \ge 11$. Under a certain symmetry preserved by the flow, the Yang-Mills equation can be reduced to: $$ \partial_t u =\partial_r^2 u +\frac{d+1}{r}…
In this paper, we use the modified potential well method to study the long time behaviors of solutions to the heat flow of H-system in a bounded smooth domain of $R^2$. Global existence and finite time blowup of solutions are proved when…
In this paper, we construct an infinite-dimensional family of solutions for the Yang-Mills flow on $\mathbb{R}^n \times SO(n)$ for $5 \leq n \leq 9$, which converge to $SO(n)$-equivariant homothetically shrinking solitons, modulo the gauge…
In this work, along with the companion work Oh (2012), we propose a novel approach to the problem of gauge choice for the \emph{Yang-Mills equations} on the Minkowski space $\mathbb{R}^{1+3}$. A crucial ingredient is the associated…
We consider the global dynamics of finite energy solutions to energy-critical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose…
We explore the small-time behavior of solutions to the Yang-Mills heat equation with rough initial data. We consider solutions $A(t)$ with initial value $A_0\in H_{1/2}(M)$, where $M$ is a bounded convex region in $\mathbb{R}^3$ or all of…
We study the heat flow for Yang-Mills connections on $\mathbb R^d \times SO(d)$. It is well-known that in dimensions $5 \leq d \leq 9$ this model admits homothetically shrinking solitons, i.e., self-similar blowup solutions, with an…
We continue our work \cite{Glo22a} on the analysis of spatially global stability of self-similar blowup profiles for semilinear wave equations in the radial case. In this paper we study the Yang-Mills equations in $(1+d)$-dimensional…
The set of all possible spherically symmetric magnetic static Einstein-Yang-Mills field equations for an arbitrary compact semi-simple gauge group $G$ was classified in two previous papers. Local analytic solutions near the center and a…
We study both analytically and numerically a coupled system of spherically symmetric SU(2) Yang-Mills-dilaton equation in 3+1 Minkowski space-time. It has been found that the system admits a hidden scale invariance which becomes transparent…
We consider corotational wave maps from Minkowski spacetime into the sphere and the equivariant Yang-Mills equation for all energy-supercritical dimensions. Both models have explicit self-similar finite time blowup solutions, which continue…
This paper develops Yang-Mills flow on Riemannian manifolds with special holonomy. By analogy with the second-named author's thesis, we find that a supremum bound on a certain curvature component is sufficient to rule out finite-time…
In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon…
In this paper, we shall prove that, on a non-flat Riemannian vector bundle over a compact Riemannian manifold, the smooth solution of the Yang-Mills flow will blow up in finite time if the energy of the initial connection is small enough.…