[Paper Review] Low regularity local well-posedness for the (N+1)-dimensional Maxwell-Klein-Gordon equations in Lorenz gauge

Hartmut Pecher|arXiv (Cornell University)|May 1, 2017

Advanced Mathematical Physics Problems14 references3 citations

TL;DR

This paper establishes low regularity local well-posedness for the (N+1)-dimensional Maxwell-Klein-Gordon equations in Lorenz gauge, leveraging the null structure of nonlinear terms and refined product estimates in wave-Sobolev spaces. It proves well-posedness down to regularity thresholds of $ s > 3/4 $ in 3D and $ s > 1/2 $ in 2D, even without finite energy, extending prior results in Coulomb gauge to a more general gauge setting.

ABSTRACT

The Cauchy problem for the Maxwell-Klein-Gordon equations in Lorenz gauge in $n$ space dimensions ($n \ge 2$) is locally well-posed for low regularity data, in two and three space dimensions even for data without finite energy. The result relies on the null structure for the main bilinear terms which was shown to be not only present in Coulomb gauge but also in Lorenz gauge by Selberg and Tesfahun, who proved global well-posedness for finite energy data in three space dimensions. This null structure is combined with product estimates for wave-Sobolev spaces given systematically by d'Ancona, Foschi and Selberg.

Motivation & Objective

• To establish local well-posedness for the Maxwell-Klein-Gordon system in Lorenz gauge with low regularity initial data, including data without finite energy.
• To extend the applicability of null structure techniques beyond Coulomb gauge to the Lorenz gauge, where the nonlinearities lack obvious null conditions.
• To generalize well-posedness results in higher dimensions (n ≥ 4) to lower regularity thresholds using refined bilinear estimates.
• To demonstrate that the null structure present in the equations for the scalar field and electromagnetic field in Lorenz gauge enables control of nonlinear interactions even at low regularity.
• To provide a framework for low regularity well-posedness in Lorenz gauge that parallels and extends existing results in Coulomb gauge.

Proposed method

• Utilizes the Fourier restriction norm method to analyze low regularity solutions in Sobolev spaces.
• Applies product estimates for wave-Sobolev spaces developed by d’Ancona, Foschi, and Selberg to control bilinear terms.
• Relies on the null structure of key nonlinear terms in Lorenz gauge, previously identified by Selberg and Tesfahun, to achieve improved estimates.
• Employs frequency-localized estimates and Littlewood-Paley decomposition to handle low- and high-frequency interactions in the nonlinearities.
• Combines the null structure with Sobolev embedding and interpolation techniques to close the contraction argument in the function space framework.
• Adapts results from Klainerman and Tataru for high-dimensional cases (n ≥ 4) to derive necessary bilinear estimates in the absence of full null condition.

Experimental results

Research questions

• RQ1 Can local well-posedness be established for the (N+1)-dimensional Maxwell-Klein-Gordon system in Lorenz gauge with initial data below the energy space?
• RQ2 Does the null structure present in the Lorenz gauge formulation allow for control of nonlinear interactions at low regularity, similar to the Coulomb gauge case?
• RQ3 What is the optimal regularity threshold for local well-posedness in Lorenz gauge across different space dimensions?
• RQ4 How do product estimates in wave-Sobolev spaces contribute to controlling the nonlinear terms in low-regularity regimes?
• RQ5 Can the framework used in 2D and 3D be extended to higher dimensions (n ≥ 4) with comparable regularity thresholds?

Key findings

• Local well-posedness is established for the (N+1)-dimensional Maxwell-Klein-Gordon system in Lorenz gauge for initial data with regularity $ s > 3/4 $ in three space dimensions.
• In two space dimensions, local well-posedness holds for $ s > 1/2 $, with data in a slightly different function space, and for $ s eq 3/4 $ under specific regularity assumptions.
• For $ n o rac{n}{2} - rac{3}{4} $, local well-posedness is proven in higher dimensions ($ n o 4 $), extending the range of applicability.
• The null structure of the nonlinear terms in Lorenz gauge enables the use of bilinear estimates that are sufficient to close the contraction argument in the function space framework.
• The result holds even for data without finite energy, demonstrating that the regularity threshold is below the energy-critical level.
• The proof relies on refined product estimates in wave-Sobolev spaces and frequency localization techniques to control low-regularity interactions.