Related papers: Quantization of time-like energy for wave maps int…

We consider the Cauchy problem for wave maps u: \R times M \to N for Riemannian manifolds, (M, g) and (N, h). We prove global existence and uniqueness for initial data that is small in the critical Sobolev norm in the case (M, g) = (\R^4,…

We propose a generalization of the so-called rational map ansatz on the Euclidean space $\mathbb{R}^3$, for any compact simple Lie group $G$ such that $G/{\widehat K}\otimes U(1)$ is an Hermitian symmetric space, for some subgroup…

Given a 3-dimensional Riemannian manifold $(M,g)$, we prove that if $(\Phi_k)$ is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy bounded above uniformly strictly by $8 \pi$, and…

We calculate the net energy per unit time exchanged between two sets of modes in a generic system governed by a three-wave kinetic equation. Our calculation is based on the property of detailed energy conservation of the triadic resonant…

We prove the local energy decay for the wave equation in a wave guide with dissipation at the boundary. It appears that for large times the dissipated wave behaves like a solution of a heat equation in the unbounded directions. The proof is…

We establish the decay and Strichartz estimates for the wave equation with large scaling-critical electromagnetic potentials on a conical singular space $(X,g)$ with dimension $n\geq3$, where the metric $g=dr^2+r^2 h$ and…

We establish small energy H\"{o}lder bounds for minimizers $u_\varepsilon$ of \[E_\varepsilon (u):=\int_\Omega W(\nabla u)+ \frac{1}{\varepsilon^2} \int_\Omega f(u),\] where $W$ is a positive definite quadratic form and the potential $f$…

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation $\partial_t^2 u - \Delta u = - |u|^{p -1} u$ in the 3-dimensional space ($3\leq p<5$) whose initial data are radial and come with a finite energy. In…

Energy minimizing maps (E.M.M.s) play a central role in the calculus of variations, partial differential equations (PDEs), and geometric analysis. These maps are often embedded into $C^\infty$ Riemannian manifolds to minimize the Dirichlet…

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of…

This paper establishes that on the domain of outer communications of a general class of stationary and asymptotically flat Lorentzian manifolds of dimension $d+1$, $d\ge3$, the local energy of solutions to the scalar wave equation…

Using the concept of real tunneling configurations (classical signature change) and nucleation energy, we explore the consequences of an alternative minimization procedure for the Euclidean action in multiple-dimensional quantum cosmology.…

For 2 + 1 dimensional wave maps with $\mathbb{S}^2$ as the target, we show that for all positive numbers $T_0 > 0$ and $E_0 > 0$, there exist Cauchy initial data with energy at least $E_0$, so that the solution's life-span is at least…

We develop a general theory for the existence, uniqueness, and higher regularity of solutions to wave-type equations on Lorentzian manifolds with timelike curves of cone-type singularities. These singularities may be of geometric type (cone…

We study the wave equation in a bounded domain or on a compact Riemannian manifold with boundary. Assume that we are given the hyperbolic Neumann-to-Dirichlet map on the boundary corresponding to physical boundary measurements. We consider…

Using recent results in the theory of $C_0$-semigroups due to Batty, Chill and Tomilov (J. Eur. Math. Soc. 18(4):853-929, 2016) we study energy decay in a one-dimensional coupled wave-heat system with finite wave part and infinite heat…

This work makes analytic progress in the deterministic study of turbulence in Hamiltonian systems by identifying two types of energy cascade solutions and the corresponding large- and small-scale structures they generate. The first cascade…

We consider the energy-critical (corotational) 1-equivariant wave maps into the two-sphere. By the seminal work [53] of Rapha\"el and Rodnianski, there is an open set of initial data whose forward-in-time development blows up in finite time…

We prove uniqueness of solutions to the wave map equation in the natural class, namely $ (u, \partial_t u) \in C([0,T); \dot{H}^{d/2})\times C^1([0,T); \dot{H}^{d/2-1})$ in dimensions $d\geq 4$. This is achieved through estimating the…

We study singularity formation in spherically symmetric solitons of the charge one sector of the (2+1) dimensional S^2 sigma model, also known as $\IC P^1$ wave maps, in the adiabatic limit. These equations are non-integrable, and so…