<p>In [<CitationRef CitationID="CR4">4</CitationRef>], Bourgain and Li establish strong ill-posedness of the 2D Euler equations for initial velocity in the critical Sobolev space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^2(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </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 this work, we extend the results of [<CitationRef CitationID="CR4">4</CitationRef>] by demonstrating strong ill-posedness in logarithmically regularized spaces which are strictly contained in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$H^2(\mathbb {R}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </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> and which contain <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> for all <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$s>2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. These spaces are constructed via application of a fractional logarithmic derivative to the critical Sobolev norm. We show that if the power <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> of the logarithmic derivative satisfies <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha \le 1/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, then the 2D Euler equations are strongly ill-posed.</p>

Ill-Posedness of the 2D Euler Equations in a Logarithmically Refined Critical Sobolev Space

期刊论文
发表日期：

2026年7月31日

查看全文

Elaine Cozzi,
Nicholas Harrison,
Zachary Radke

摘要

In [4], Bourgain and Li establish strong ill-posedness of the 2D Euler equations for initial velocity in the critical Sobolev space $H^2(\mathbb {R}^2)$ $H^{2} (R^{2})$ . In this work, we extend the results of [4] by demonstrating strong ill-posedness in logarithmically regularized spaces which are strictly contained in $H^2(\mathbb {R}^2)$ $H^{2} (R^{2})$ and which contain $H^s(\mathbb {R}^2)$ $H^{s} (R^{2})$ for all $s>2$ $s > 2$ . These spaces are constructed via application of a fractional logarithmic derivative to the critical Sobolev norm. We show that if the power $\alpha $ $α$ of the logarithmic derivative satisfies $\alpha \le 1/2$ $α \leq 1 / 2$ , then the 2D Euler equations are strongly ill-posed.