[Paper Review] Low regularity local well-posedness for the Chern-Simons-Higgs system in temporal gauge

Hartmut Pecher|arXiv (Cornell University)|Dec 31, 2013

Advanced Mathematical Physics Problems10 references3 citations

TL;DR

This paper establishes low regularity local well-posedness for the Chern-Simons-Higgs system in (2+1)D Minkowski space under the temporal gauge, using $X^{s,b}$-type spaces and bilinear wave-Sobolev estimates. It proves well-posedness for initial data in $H^{s+3/4}$ for $A_j$ and $H^{s+1} \times H^s$ for $(\phi_0, \phi_1)$ with $s > \frac{1}{4}$, improving prior results by Huh and others through exploitation of a null condition in the nonlinearity.

ABSTRACT

The Cauchy problem for the Chern-Simons-Higgs system in the (2+1)-dimensional Minkowski space in temporal gauge is locally well-posed for low regularity initial data improving a result of Huh. The proof uses the bilinear space-time estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg and takes advantage of a null condition.

Motivation & Objective

• To establish local well-posedness for the Chern-Simons-Higgs system in the temporal gauge with low regularity initial data.
• To improve upon prior results by Huh, particularly for the temporal gauge, by lowering the required regularity threshold.
• To leverage the null condition in the system's nonlinearity to achieve well-posedness at lower regularity.
• To apply advanced bilinear $X^{s,b}$-type estimates from d'Ancona, Foschi, and Selberg to control nonlinear interactions.

Proposed method

• Uses a contraction argument in $X^{s,b}$-type spaces adapted to the wave and transport phase functions $\tau \pm |\xi|$ and $\tau = 0$.
• Applies bilinear space-time estimates in wave-Sobolev spaces developed by d'Ancona, Foschi, and Selberg to control nonlinear terms.
• Employs frequency decomposition of the gauge field $A^{cf}$ into low and high-frequency parts to estimate nonlinear interactions.
• Utilizes the null condition in the nonlinearity to gain additional decay and improve regularity thresholds.
• Applies Sobolev embeddings and $L^p$-type estimates in time and space to control $L^2$ norms of nonlinear terms.
• Relies on duality and interpolation techniques to bound products of fields in $X^{s,b}$-spaces, especially for $\nabla A^{cf} \phi$ and $A^{cf}A^{cf}\phi$.

Experimental results

Research questions

• RQ1 Can local well-posedness be established for the Chern-Simons-Higgs system in the temporal gauge with initial data below the energy regularity threshold?
• RQ2 What is the optimal regularity threshold for local well-posedness in the temporal gauge, given the structure of the nonlinearity?
• RQ3 How can the null condition in the Chern-Simons-Higgs system be exploited to improve regularity requirements?
• RQ4 To what extent can bilinear estimates in wave-Sobolev spaces be adapted to systems with mixed hyperbolic and elliptic components?

Key findings

• Local well-posedness is established for initial data $a_j \in H^{s+3/4}$, $\phi_0 \in H^{s+1}$, $\phi_1 \in H^s$ with $s > \frac{1}{4}$, improving Huh's result for the temporal gauge.
• The proof relies on a contraction argument in $X^{s,b}$-type spaces tailored to the phase functions of the wave and transport equations.
• The null condition in the nonlinearity is crucial for gaining the necessary regularity gain to reach $s > \frac{1}{4}$.
• Bilinear estimates from d'Ancona, Foschi, and Selberg are adapted and applied to control the nonlinear terms $A_j \partial_j \phi$, $A_j A_j \phi$, and $\phi V'(|\phi|^2)$.
• The estimate $\|A^{cf} A^{cf} \phi\|_{X^{s,0}_{|\tau|=|\xi|}}$ is bounded by $\|\nabla A^{cf}\|_{X^{s-1/2-,1/2+}_{\tau=0}}^2 \|\phi\|_{X^{s+1,1/2+}_{|\tau|=|\xi|}}$ and lower-order terms, ensuring control in the contraction.
• The compatibility condition $\partial_1 a_2 - \partial_2 a_1 = 2\operatorname{Im}(\bar{\phi}_0 \phi_1)$ is required for the initial data to be consistent with the system.