<p>We develop a complete local wellposedness theory for a Maxwell system on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> and a large class of nonlinear material laws which are nonlocal in time. Such constitutive relations are typical for nonlinear optics. The problem was treated before in the Sobolev space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H^s$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>s</mi> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$s>3/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> by means of energy methods. Using a recently shown Strichartz estimate, we can lower this level of regularity to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$s>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this context ’charge-type’ terms would spoil the analysis. We avoid them by the Helmholtz projection for the divergence operator with coefficients, which requires mapping properties of the projection also in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$H^{\alpha , q}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>q</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$q\ne 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Local wellposedness of Maxwell systems with retarded material laws in low regularity

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2025年11月7日

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Christopher Bresch,
Roland Schnaubelt

摘要

We develop a complete local wellposedness theory for a Maxwell system on $\mathbb {R}^3$ $R^{3}$ and a large class of nonlinear material laws which are nonlocal in time. Such constitutive relations are typical for nonlinear optics. The problem was treated before in the Sobolev space $H^s$ $H^{s}$ for $s>3/2$ $s > 3 / 2$ by means of energy methods. Using a recently shown Strichartz estimate, we can lower this level of regularity to $s>1$ $s > 1$ . In this context ’charge-type’ terms would spoil the analysis. We avoid them by the Helmholtz projection for the divergence operator with coefficients, which requires mapping properties of the projection also in $H^{\alpha , q}$ $H^{α, q}$ with $q\ne 2$ $q \neq 2$ .