Ill-Posedness of the 2D Euler Equations in a Logarithmically Refined Critical Sobolev Space
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912666638876672 |
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| author | Cozzi, Elaine Harrison, Nicholas Radke, Zachary |
| author_facet | Cozzi, Elaine Harrison, Nicholas Radke, Zachary |
| contents | In their seminal work, Bourgain and Li establish strong ill-posedness of the 2D Euler equations for initial velocity in the critical Sobolev space $H^2(\mathbb{R}^2)$. In this work, we extend those results by demonstrating strong ill-posedness in logarithmically regularized spaces which are strictly contained in $H^2(\mathbb{R}^2)$ and which contain $H^s(\mathbb{R}^2)$ for all $s>2$. These spaces are constructed via application of a fractional logarithmic derivative to the critical Sobolev norm. We show that if the power $α$ of the logarithmic derivative satisfies $α\leq 1/2$, then the 2D Euler equations are strongly ill-posed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_20773 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ill-Posedness of the 2D Euler Equations in a Logarithmically Refined Critical Sobolev Space Cozzi, Elaine Harrison, Nicholas Radke, Zachary Analysis of PDEs In their seminal work, Bourgain and Li establish strong ill-posedness of the 2D Euler equations for initial velocity in the critical Sobolev space $H^2(\mathbb{R}^2)$. In this work, we extend those results by demonstrating strong ill-posedness in logarithmically regularized spaces which are strictly contained in $H^2(\mathbb{R}^2)$ and which contain $H^s(\mathbb{R}^2)$ for all $s>2$. These spaces are constructed via application of a fractional logarithmic derivative to the critical Sobolev norm. We show that if the power $α$ of the logarithmic derivative satisfies $α\leq 1/2$, then the 2D Euler equations are strongly ill-posed. |
| title | Ill-Posedness of the 2D Euler Equations in a Logarithmically Refined Critical Sobolev Space |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.20773 |