<p>In Buck and Modena (Journal of Mathematical Fluid Mechanics, 2024. 10.1007/s00021-024-00860-9), we constructed by convex integration examples of energy dissipating solutions to the 2D Euler equations on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathbb {R}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> with vorticity in the real Hardy space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H^p({\mathbb {R}}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>p</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>. In the present paper, we develop tools that significantly improve the result in Buck and Modena (Journal of Mathematical Fluid Mechanics, 2024. 10.1007/s00021-024-00860-9) in two ways: Firstly, we achieve vorticities in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> in the optimal range <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p\in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> compared to (2/3, 1) in our previous work. Secondly, the solutions constructed here possess compact support and in particular preserve linear and angular momenta.</p>

Compactly supported anomalous weak solutions for 2D Euler equations with vorticity in Hardy spaces

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发表日期：

2026年2月12日

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Miriam Buck,
Stefano Modena

摘要

In Buck and Modena (Journal of Mathematical Fluid Mechanics, 2024. 10.1007/s00021-024-00860-9), we constructed by convex integration examples of energy dissipating solutions to the 2D Euler equations on ${\mathbb {R}}^2$ $R^{2}$ with vorticity in the real Hardy space $H^p({\mathbb {R}}^2)$ $H^{p} (R^{2})$ . In the present paper, we develop tools that significantly improve the result in Buck and Modena (Journal of Mathematical Fluid Mechanics, 2024. 10.1007/s00021-024-00860-9) in two ways: Firstly, we achieve vorticities in $H^p$ $H^{p}$ in the optimal range $p\in (0,1)$ $p \in (0, 1)$ compared to (2/3, 1) in our previous work. Secondly, the solutions constructed here possess compact support and in particular preserve linear and angular momenta.