Global finite energy solutions of the Maxwell-scalar field system on the Einstein cylinder

Abstract

We prove the existence and uniqueness of global finite energy solutions of the Maxwell-scalar field system in Lorenz gauge on the Einstein cylinder. Our method is a combination of a conformal patching argument, the finite energy existence theorem in Lorenz gauge on Minkowski space of Selberg and Tesfahun, a careful localization of finite energy data, and null form estimates of Foschi-Klainerman type. Although we prove that the energy-carrying components of the solution maintain regularity, due to the incompleteness of the null structure in Lorenz gauge and the nature of our foliation-change arguments we find small losses of regularity in both the scalar field and the potential.

Figures (5)

• Figure 1: A depiction of two copies of Minkowski space (for positive times), $\mathbb{M}_+ = \{ t \geqslant 0\} \cap \mathbb{M}$ and $\mathbb{M}'_+ = \{t' \geqslant 0\} \cap\mathbb{M}'$, conformally embedded in $\mathbb{R} \times \mathbb{S}^3$ in a way that situates the origin $O$ of $\mathbb{M}$ at the spatial infinity $(i^0)'$ of $\mathbb{M}'$, and vice-versa. The Einstein cylinder $\mathbb{R} \times \mathbb{S}^3$ is represented by the surface of the cylinder and the two future-pointing halves $\mathbb{M}_+$, $\mathbb{M}_+'$ of Minkowski space are represented by the half-diamonds wrapped around the cylinder. Two spherical dimensions are suppressed. Via the Hopf fibration $\mathbb{S}^1 \hookrightarrow \mathbb{S}^3 \to \mathbb{S}^2$, each point on the surface of the depicted cylinder may be thought of as a 2-sphere. The future timelike infinities $i^+$ and $(i^+)'$, and spatial infinities $i^0$ and $(i^0)'$, are points rather than spheres in the usual Penrose conformal compactification of $\mathbb{M}$ (resp. $\mathbb{M}'$), and are therefore not included in $\mathbb{M}_+$, $\mathbb{M}_+'$. The future null infinities $\mathscr{I}^+$ and $(\mathscr{I}^+)'$ are (open) backward lightcones of $i^+$ and $(i^+)'$, with topology $\mathbb{R} \times \mathbb{S}^2$, forming the future null boundaries of $\mathbb{M}_+$ and $\mathbb{M}'_+$ in $\mathbb{R} \times \mathbb{S}^3$. The initial surface $\Sigma \simeq \mathbb{S}^3 \subset \mathbb{R} \times \mathbb{S}^3$ on the cylinder is the one-point compactification of each of the Minkowskian initial surfaces $\tilde{\Sigma} = \{t = 0\}$ and $\tilde{\Sigma}' = \{ t' = 0 \}$, $\Sigma \simeq \tilde{\Sigma} \cup i^0 \simeq \tilde{\Sigma}' \cup (i^0)'$.
• Figure 2: Change of local foliation of the solution on Minkowski space.
• Figure 3: By construction, the new foliation agrees with the standard foliation of $\mathbb{R} \times \mathbb{S}^3$ in $D^+(B_R)$ after the conformal embedding of $\mathbb{M}$ into $\mathbb{R} \times \mathbb{S}^3$.
• Figure 4: The image of $\tilde{\mathcal{R}}_{R,\tau_*}$ in $\mathbb{R} \times \mathbb{S}^3$ under the conformal embedding $\mathbb{M} \hookrightarrow \mathbb{R} \times \mathbb{S}^3$.
• Figure 5: The overlap of the two (future halves of) antipodal copies of Minkowski space $\mathbb{M}_+$ and $\mathbb{M}'_+$ conformally embedded in $\mathbb{R} \times \mathbb{S}^3$, unwrapped. The picture should be read as being horizontally periodic. The North and South poles of the $3$-sphere are denoted by $N$ and $S$ respectively.

Theorems & Definitions (52)

• Remark 2.1
• Lemma 4.1: Theorem 2, §5.3.6, RunstSickel1996
• Lemma 4.2: Theorem 3, §5.3.6, RunstSickel1996
• Lemma 4.3: Proposition 4.1, Taylor2007, or Theorem 1 and Remark 2, §5.3.6, RunstSickel1996
• Lemma 4.4
• Lemma 4.5
• Definition 5.1
• Definition 5.2
• Lemma 5.3
• proof