[Paper Review] Local well-posedness for the Maxwell-Schrödinger equation

Makoto Nakamura, Takeshi Wada|ArXiv.org|Apr 29, 2003

Advanced Mathematical Physics Problems6 references4 citations

TL;DR

This paper establishes local well-posedness for the Maxwell-Schrödinger system in the Coulomb gauge within optimal Sobolev regularity spaces using the contraction mapping principle. It proves unique, continuous solutions exist for initial data in $X^{s, au}$ with $s \geq 5/3$ and $\sigma$ in a specified range, overcoming derivative loss via self-adjointness of the Schrödinger Hamiltonian and projection techniques, and extends results to Lorentz and temporal gauges via gauge transformations.

ABSTRACT

Time local well-posedness for the Maxwell-Schrödinger equation in the coulomb gauge is studied in Sobolev spaces by the contraction mapping principle. The Lorentz gauge and the temporal gauge cases are also treated by the gauge transform.

Motivation & Objective

• To establish time local well-posedness of the Maxwell-Schrödinger system in the Coulomb gauge for initial data in Sobolev spaces.
• To resolve the gauge ambiguity inherent in the system by fixing the Coulomb gauge condition $\mathrm{div}\,\mathbf{A} = 0$.
• To minimize regularity requirements by overcoming derivative loss in the nonlinear terms $\mathbf{A} \cdot \nabla u$ and $\mathbf{J}(u,\mathbf{A})$.
• To extend the well-posedness result to the Lorentz and temporal gauges via gauge transformation techniques.
• To achieve the widest possible range of Sobolev regularity indices $s$ and $\sigma$ for which unique, continuous solutions exist.

Proposed method

• Applies the contraction mapping principle in a fixed-time interval to prove existence and uniqueness of solutions in $C([0,T];X^{s, au})$.
• Uses the Coulomb gauge to decouple the scalar potential $\phi$ via $\phi = (-\Delta)^{-1}|u|^2$, eliminating the need for initial data on $\phi$.
• Employs the projection operator $P = 1 - \nabla \mathrm{div} \Delta^{-1}$ to handle the vector potential $\mathbf{A}$ in the wave equation.
• Overcomes derivative loss by estimating $\|\mathcal{H}u; H^{s-2}\|$ instead of $\|u; H^s\|$, leveraging the self-adjointness of $\mathcal{H}(\mathbf{A})$.
• Applies gauge transformations to extend results from the Coulomb gauge to the Lorentz and temporal gauges.
• Uses the wave propagator $K(t) = \sin(t\omega)/\omega$ and $\dot{K}(t) = \cos(t\omega)$ to solve the scalar field $\lambda$ in the gauge transformation, ensuring regularity propagation.

Experimental results

Research questions

• RQ1 What is the minimal Sobolev regularity $s$ and $\sigma$ for which the Maxwell-Schrödinger system in the Coulomb gauge admits a unique local solution?
• RQ2 How can derivative loss in the nonlinear terms $\mathbf{A} \cdot \nabla u$ and $\mathbf{J}(u,\mathbf{A})$ be controlled in low-regularity regimes?
• RQ3 Can the well-posedness result in the Coulomb gauge be extended to other gauges such as Lorentz or temporal gauge?
• RQ4 What role does the projection $P$ play in weakening the regularity requirement $\sigma \leq s$ for the vector potential?
• RQ5 Under what conditions is the solution map continuous in the initial data in the $X^{s,\sigma}$ topology?

Key findings

• Local well-posedness of the Maxwell-Schrödinger system in the Coulomb gauge holds for $s \geq 5/3$ and $\sigma$ satisfying $\max\{4/3, s-2, (2s-1)/4\} \leq \sigma \leq \min\{s+1, (5s-2)/3\}$, excluding the pairs $(5/2,7/2)$ and $(7/2,3/2)$.
• The solution satisfies $(u, \mathbf{A}, \partial_t \mathbf{A}) \in C([0,T]; X^{s,\sigma})$ for some $T > 0$ depending only on $s$, $\sigma$, and the norm of the initial data.
• Continuous dependence on initial data holds when $\sigma \geq \max\{s-1, (2s+1)/4\}$, excluding the pair $(5/2, 3/2)$.
• The Lorentz gauge case is reduced to the Coulomb gauge via a gauge transformation, with the scalar field $\lambda$ solving a wave equation with data in $\dot{H}^1 \cap \dot{H}^{\sigma+1}$.
• The solution in the Lorentz gauge satisfies $(u^\mathrm{L}, \phi^\mathrm{L}, \mathbf{A}^\mathrm{L}, \partial_t \mathbf{A}^\mathrm{L}) \in C([0,T]; Y^{s,\sigma})$ under the same regularity assumptions.
• The result fills a gap between the global $X^{1,1}$ result of Guo-Nakamitsu-Strauss and the $X^{s,s}$ result of Nakamitsu-Tsutsumi for $s > 5/2$, achieving optimal regularity in the local regime.