Related papers: Bubble decomposition for the harmonic map heat flo…
We study $O(d)$-equivariant biharmonic maps in the critical dimension. A major consequence of our study concerns the corresponding heat flow. More precisely, we prove that blowup occurs in the biharmonic map heat flow from $B^4(0, 1)$ into…
In this paper we introduce the channel of energy argument to the study of energy critical wave maps into the sphere. More precisely, we prove a channel of energy type inequality for small energy wave maps, and as an application we show that…
Terminal velocity reached by bubble walls in first order phase transitions is an important parameter determining both primordial gravitational-wave spectrum and production of baryon asymmetry in models of electroweak baryogenesis. We…
In the context of an Abelian gauge symmetry, spontaneously broken at a first-order transition, we discuss the evolution of the phase difference between the Higgs fields in colliding bubbles. We show that the effect of dissipation,…
We investigate the harmonic map heat flow from a compact Riemannian manifold \( M \) into the moduli space \( \mathcal{M}_1 \) of unit-area flat tori, which carries a natural hyperbolic structure as the quotient \( \mathrm{SL}(2,\mathbb{Z})…
When an asymmetric bubble collapses it generally produces a well defined high velocity jet. This is remarkable because one might expect such a collapse to produce a complex or chaotic flow rather than an ordered one. I present a dimensional…
Building upon the recent findings regarding inverse phase transitions in the early universe, we present the first natural realisation of this phenomenon within a supersymmetry-breaking sector. We demonstrate that inverse hydrodynamics,…
We develop a set of controlled, analytic approximations to study the effects of bubble collisions on cosmology. We expand the initial perturbation to the inflaton field caused by the collision in a general power series, and determine its…
For any $k$-dimensional smooth, compact Riemannian manifold $(N, h)\subset\mathbb R^L$ without boundary, there exists an $\varepsilon_0>0$ such that for any homogeneous of degree zero map $u_0(x)=\phi_0(\frac{x}{|x|}):\mathbb R^n\to N$…
Consider the dynamics of a gas bubble in an inviscid, compressible liquid with surface tension. Kinematic and dynamic boundary conditions couple the bubble surface deformation dynamics with the dynamics of waves in the fluid. This system…
In this paper we consider the polyharmonic heat flow of a closed curve in the plane. Our main result is that closed initial data with initially small normalised oscillation of curvature and isoperimetric defect flows exponentially fast in…
Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition.…
The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in a previous paper by the authors for scalar valued functions, or zero-forms, and represents a new…
In this paper we focus on the uniqueness question for (expanding) solutions of the Harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On…
We study the Ricci flow for initial metrics which are C^0 small perturbations of the Euclidean metric on R^n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and…
The nucleation and evolution of bubbles are investigated in the model of an $O(3)$-symmetric scalar field coupled to gravity in the high temperature limit. It is shown that, in addition to the well-known bubble of which the inside region is…
We investigate the control problem of harmonic map heat flow by means of an external magnetic field. In contrast to the situation of a parabolic system with internal or boundary control, the magnetic field acts as the coefficients of the…
In this paper, we prove that the solution of the Landau-Lifshitz flow $u(t,x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map as $t\to\infty$. The essential observation is that although there exist infinite numbers of…
We study harmonic maps from degenerating Riemann surfaces with uniformly bounded energy and show the so-called generalized energy identity. We find conditions that are both necessary and sufficient for the compactness in $W^{1,2}$ and…
We propose a many-particle-inspired theory for granular outflows from a hopper and for the escape dynamics through a bottleneck based on a continuity equation in polar coordinates. If the inflow is below the maximum outflow, we find an…