<p>We consider the Maxwell–Schrödinger equations (MS) in the Lorenz gauge, in a bounded or exterior domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d=2,3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. On <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>, we impose the Dirichlet condition on the Schrödinger function, and the conditions associated with the perfect electric conductor boundary conditions on the electro-magnetic potentials. We prove that the system (MS) has a unique time-local solution in the class <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^2(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and that the solution exists time globally if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$d=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Initial-boundary value problems for Maxwell–Schrödinger equations

期刊论文
发表日期：

2025年7月29日

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Takashi Sakane,
Takeshi Wada

摘要

We consider the Maxwell–Schrödinger equations (MS) in the Lorenz gauge, in a bounded or exterior domain $\Omega \subset \mathbb {R}^d$ $Ω \subset R^{d}$ , $d=2,3$ $d = 2, 3$ . On $\partial \Omega $ $\partial Ω$ , we impose the Dirichlet condition on the Schrödinger function, and the conditions associated with the perfect electric conductor boundary conditions on the electro-magnetic potentials. We prove that the system (MS) has a unique time-local solution in the class $H^2(\Omega )$ $H^{2} (Ω)$ , and that the solution exists time globally if $d=2$ $d = 2$ .