Breakdown of smooth solutions to the subcritical EPDiff equation
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915213193773056 |
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| author | Bauer, Martin Preston, Stephen C. Valletta, Justin |
| author_facet | Bauer, Martin Preston, Stephen C. Valletta, Justin |
| contents | We consider the EPDiff equation on $\mathbb{R}^n$ with the integer-order homogeneous Sobolev inertia operator $A=(-Δ)^k$. We prove that for arbitrary radial initial data and a sign condition on the initial momentum, the corresponding radial velocity solution has $C^1$ norm that blows up in finite time whenever $0\le k<n/2+1.$ Our approach is to use Lagrangian coordinates to formulate EPDiff as an ODE on a Banach space, enabling us to use a comparison estimate with the Liouville equation. Along the way we derive the Green function in terms of hypergeometric functions and discuss their properties. This is a step toward proving the general conjecture that the EPDiff equation is globally well-posed for any Sobolev inertia operator of any real order $k$ if and only if $k\ge n/2+1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_19780 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Breakdown of smooth solutions to the subcritical EPDiff equation Bauer, Martin Preston, Stephen C. Valletta, Justin Analysis of PDEs Classical Analysis and ODEs We consider the EPDiff equation on $\mathbb{R}^n$ with the integer-order homogeneous Sobolev inertia operator $A=(-Δ)^k$. We prove that for arbitrary radial initial data and a sign condition on the initial momentum, the corresponding radial velocity solution has $C^1$ norm that blows up in finite time whenever $0\le k<n/2+1.$ Our approach is to use Lagrangian coordinates to formulate EPDiff as an ODE on a Banach space, enabling us to use a comparison estimate with the Liouville equation. Along the way we derive the Green function in terms of hypergeometric functions and discuss their properties. This is a step toward proving the general conjecture that the EPDiff equation is globally well-posed for any Sobolev inertia operator of any real order $k$ if and only if $k\ge n/2+1$. |
| title | Breakdown of smooth solutions to the subcritical EPDiff equation |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2503.19780 |