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Breakdown of smooth solutions to the subcritical EPDiff equation

Martin Bauer, Stephen C. Preston, Justin Valletta

TL;DR

This work analyzes the breakdown of smooth radial solutions to the n-dimensional EPDiff equation with homogeneous Sobolev inertia $A=(-\Delta)^k$, proving finite-time blow-up of the $C^1$ norm for all integers $0\le k<n/2+1$ under radial data. The authors recast EPDiff as an ODE on a Banach space via Lagrangian flow, enabling a Liouville-comparison argument, and derive a Green function for $(-\Delta)^k$ using hypergeometric functions. A key contribution is a refined radial breakdown criterion and a dimension-reduction result that reduces higher-dimensional breakdown to the smallest dimension where it occurs, with immediate corollaries for higher-dimensional Burgers’, Hunter–Saxton, and Camassa–Holm-type equations. The results supply additional evidence for the conjecture that global well-posedness for Sobolev inertia hinges on the critical threshold $k\ge n/2+1$, and outline avenues for extending the analysis to non-homogeneous and non-integer orders.

Abstract

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 $k\ge n/2+1$.

Breakdown of smooth solutions to the subcritical EPDiff equation

TL;DR

This work analyzes the breakdown of smooth radial solutions to the n-dimensional EPDiff equation with homogeneous Sobolev inertia , proving finite-time blow-up of the norm for all integers under radial data. The authors recast EPDiff as an ODE on a Banach space via Lagrangian flow, enabling a Liouville-comparison argument, and derive a Green function for using hypergeometric functions. A key contribution is a refined radial breakdown criterion and a dimension-reduction result that reduces higher-dimensional breakdown to the smallest dimension where it occurs, with immediate corollaries for higher-dimensional Burgers’, Hunter–Saxton, and Camassa–Holm-type equations. The results supply additional evidence for the conjecture that global well-posedness for Sobolev inertia hinges on the critical threshold , and outline avenues for extending the analysis to non-homogeneous and non-integer orders.

Abstract

We consider the EPDiff equation on with the integer-order homogeneous Sobolev inertia operator . We prove that for arbitrary radial initial data and a sign condition on the initial momentum, the corresponding radial velocity solution has norm that blows up in finite time whenever 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 if and only if .

Paper Structure

This paper contains 17 sections, 16 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. The EPDiff equation:
  3. Known results on well-posedness and blowup of solutions:
  4. Main Contributions:
  5. Future Directions:
  6. Acknowledgements and Data availability statement:
  7. Background
  8. The EPDiff equation as an Euler-Arnold equation
  9. Momentum transport formulation
  10. Two breakdown criteria for the EPDiff equation
  11. Breakdown criteria for radial solutions
  12. Breakdown criteria by reducing the dimension
  13. Breakdown of smooth solutions for the EPDiff with homogeneous Sobolev inertia operator
  14. Hypergeometric functions
  15. Green function for Laplace operators acting on radial vector fields
  16. ...and 2 more sections

Key Result

Theorem 1

Let $k$ be an integer with $0\le k<n/2+1$. Then there is radial initial data $u_0\in H^\infty(\mathbb{R}^n,\mathbb{R}^n)$ such that the corresponding radial solution to the $n$-dimensional EPDiff equation eq:EPDiff with the homogeneous Sobolev inertia operator $A=(-\Delta)^k$ has $C^1$ norm that blo

Theorems & Definitions (30)

  • Conjecture 1
  • Theorem : Theorem \ref{['theorem:maintheorem']} in Section \ref{['breakdown']}
  • Lemma 2: Radial solutions
  • proof
  • Remark 3
  • Proposition 4: Conservation law; Lemma 3.6 in bauer2024liouville
  • Proposition 5: Proposition 3.8 in bauer2024liouville
  • Proposition 6: Lemma 3.7 from bauer2024liouville
  • Theorem 7
  • proof
  • ...and 20 more