arXiv subjects

Casey Rodriguez

Publications and source records attributed to Casey Rodriguez.

23 records · Page 2Linked to original sources

arXiv2017

Conditional stable soliton resolution for a semi-linear Skyrme equation

We study a semi-linear version of the Skyrme system due to Adkins and Nappi. The objects in this system are maps from $(1+3)$-dimensional Minkowski space into the $3$-sphere and 1-forms on $\mathbb{R}^{1+3}$, coupled via a Lagrangian action. Under a co-rotational symmetry reduction we establish the existence, uniqueness, and unconditional asymptotic stability of a family of stationary solutions $Q_n$, indexed by the topological degree $n \in \mathbb{N} \cup \{0\}$ of the underlying map. We also prove that an arbitrarily large equivariant perturbation of $Q_n$ leads to a globally defined solution that scatters to $Q_n$ in infinite time as long as the critical norm for the solution remains bounded on the maximal interval of existence given by the local Cauchy theory. We remark that the evolution equations are super-critical with respect to the conserved energy.

math.AP↗

arXiv2016

Soliton resolution for equivariant wave maps on a wormhole: II

In this paper, we continue our study of equivariant \emph{wave maps on a wormhole} initiated in our companion paper. More precisely, we study finite energy $\ell$--equivariant wave maps from the (1+3)-dimensional spacetime $\mathbb R \times (\mathbb R \times \mathbb{S}^2) \rightarrow \mathbb{S}^3$ where the metric on $\mathbb R \times (\mathbb R \times \mathbb{S}^2)$ is given by \begin{align*} ds^2 = -dt^2 + dr^2 + (r^2 + 1) \left ( d θ^2 + \sin^2 θd φ^2 \right ), \quad t,r \in \mathbb R, (θ,φ) \in \mathbb{S}^2. \end{align*} The constant time slices are each given by a Riemannian manifold $\mathcal M$ with two asymptotically Euclidean ends at $r = \pm \infty$ that are connected by a 2--sphere at $r = 0$. The spacetime $\mathbb R \times (\mathbb R \times \mathbb{S}^2)$ has appeared in the general relativity literature as a prototype wormhole geometry (but is not expected to exist in nature). Each $\ell$--equivariant finite energy wave map can be indexed by its topological degree $n$. For each $\ell$ and $n$, there exists a unique, linearly stable energy minimizing $\ell$--equivariant harmonic map $Q_{\ell,n} : \mathcal M \rightarrow \mathbb{S}^3$ of degree $n$. In this work, we prove the soliton resolution conjecture for this model. More precisely, we show that modulo a free radiation term every $\ell$--equivariant wave map of degree $n$ converges strongly to $Q_{\ell,n}$. This fully resolves a conjecture made by Bizon and Kahl. In the companion paper, we showed this for the corotational case $\ell = 1$ and established many preliminary results that are used in the current work.

math.AP↗

arXiv2016

Soliton resolution for equivariant wave maps on a wormhole: I

In this paper, we initiate the study of finite energy equivariant wave maps from the (1+3)-dimensional spacetime $\mathbb R \times (\mathbb R \times \mathbb{S}^2) \rightarrow \mathbb{S}^3$ where the metric on $\mathbb R \times (\mathbb R \times \mathbb{S}^2)$ is given by ds^2 = -dt^2 + dr^2 + (r^2 + 1) \left ( d θ^2 + \sin^2 θd φ^2 \right ), \quad t,r \in \mathbb{R}, (θ,φ) \in \mathbb{S}^2. The constant time slices are each given by the Riemannian manifold $\mathcal M := \mathbb R \times \mathbb{S}^2$ with metric ds^2 = dr^2 + (r^2 + 1) \left ( d θ^2 + \sin^2 θd φ^2 \right ). The Riemannian manifold $\mathcal M$ contains two asymptotically Euclidean ends at $r \rightarrow \pm \infty$ that are connected by a spherical throat of area $4 π^2$ at $r = 0$. The spacetime $\mathbb R \times \mathcal M$ is a simple example of a wormhole geometry in general relativity. In this work we will consider 1--equivariant or corotational wave maps. Each corotational wave map can be indexed by its topological degree $n$. For each $n$, there exists a unique energy minimizing corotational harmonic map $Q_{n} : \mathcal M \rightarrow \mathbb{S}^3$ of degree $n$. In this work, we show that modulo a free radiation term, every corotational wave map of degree $n$ converges strongly to $Q_{n}$. This resolves a conjecture made by Bizon and Kahl in the corotational case.

math.AP↗

arXiv2016

A partial data result for less regular conductivities in admissible geometries

We consider the Calderón problem with partial data in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that measuring the Dirichlet-to-Neumann map on roughly half of the boundary determines a conductivity that has essentially 3/2 derivatives. As a corollary, we strengthen a partial data result due to Kenig, Sjöstrand, and Uhlmann.

math.AP↗

arXiv2014

Profiles for the radial focusing energy-critical wave equation in odd dimensions

In this paper we consider global and non-global radial solutions of the focusing energy--critical wave equation on $\mathbb{R} \times \mathbb{R}^N$ where $N \geq 5$ is odd. We prove that if the solution remains bounded in the energy space as you approach the maximal forward time of existence, then along a sequence of times converging to the maximal forward time of existence, the solution decouples into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume a bound on the evolution that rules out formation of multiple solitons, then this decoupling holds for all times approaching the maximal forward time of existence.

math.AP↗