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<p>We establish Morawetz-type estimates for solutions to the elastic wave equation with singular weights of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(|x|^{-\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(|(x,t)|^{-\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">|</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>. In particular, we show that space-time weights <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(|(x,t)|^{-\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">|</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> admit stronger singularities and require weaker regularity assumptions on the initial data compared to purely spatial weights <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(|x|^{-\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>.</p>

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On Morawetz estimates for the elastic wave equation

  • 期刊论文

  • 发表日期：

    2026年3月17日

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  • Seongyeon Kim,
  • Ihyeok Seo

摘要

We establish Morawetz-type estimates for solutions to the elastic wave equation with singular weights of the form \(|x|^{-\alpha }\) | x | - α or \(|(x,t)|^{-\alpha }\) | ( x , t ) | - α . In particular, we show that space-time weights \(|(x,t)|^{-\alpha }\) | ( x , t ) | - α admit stronger singularities and require weaker regularity assumptions on the initial data compared to purely spatial weights \(|x|^{-\alpha }\) | x | - α .