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<p>In a fractional Sobolev space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(H^s(\mathbb {R}^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(s\le \frac{7}{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>≤</mo> <mfrac> <mn>7</mn> <mn>4</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>, we prove the low-regularity ill-posedness for the 2D compressible Euler equations and the 2D ideal compressible MHD system. Our ill-posedness results match the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H^\frac{7}{4}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mfrac> <mn>7</mn> <mn>4</mn> </mfrac> </msup> </math></EquationSource> </InlineEquation> regularity threshold for the 2D compressible Euler system with respect to the fluid velocity and density.</p>

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The Cauchy Problems for the 2D Compressible Euler Equations and Ideal MHD System are Ill-Posed in \(H^\frac{7}{4}(\mathbb {R}^2)\)

  • 期刊论文

  • 发表日期：

    2026年4月4日

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  • Xinliang An,
  • Haoyang Chen,
  • Silu Yin

摘要

In a fractional Sobolev space \(H^s(\mathbb {R}^2)\) H s ( R 2 ) with \(s\le \frac{7}{4}\) s ≤ 7 4 , we prove the low-regularity ill-posedness for the 2D compressible Euler equations and the 2D ideal compressible MHD system. Our ill-posedness results match the \(H^\frac{7}{4}\) H 7 4 regularity threshold for the 2D compressible Euler system with respect to the fluid velocity and density.