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Local well-posedness of the higher-dimensional $b$-equation

Justin Valletta

TL;DR

The paper addresses local well-posedness for the $n$-dimensional $b$-equation by casting it as the geodesic flow of a right-invariant affine connection on the diffeomorphism group, using a Fourier multiplier inertia operator. The authors extend the Ebin–Marsden no-loss no-gain framework to a non-metric, affine-connection setting, proving a smooth geodesic spray on the Hilbert manifold $T ext{D}^s(R^n)$ and obtaining local existence and uniqueness via Picard–Lindelöf for $s>n/2+1$ with $s\ge r$ and inertia operator class $oxed{ ext{E}^r}$ ($r\ge 1$). They also establish a Kelvin-Noether circulation theorem and show no loss of spatial regularity, allowing the $H^s$-well-posedness to lift to the smooth category, yielding local well-posedness in $H^ ty(R^n,R^n)$. The results rigorously connect multi-dimensional shallow-water models to geometric hydrodynamics and provide a solid analytical foundation for the higher-dimensional $b$-equation with general inertia operators. This advances understanding of well-posedness for non-metric Euler-Arnold flows and their associated conservation structures in fluid dynamics.

Abstract

The higher-dimensional $b$-equation is a family of PDEs, introduced by Holm and Staley (2003), that describe the motion of shallow water waves in $n$-dimensions. It expresses the invariance of the Lie-transport of the momentum one-form density associated with the fluid in $b$-dimensions. The constant $b$ can also be viewed as a balance parameter between fluid convection and fluid stretching/expansion. In this article, we interpret this family of PDEs as the geodesic equation of a right-invariant affine connection on the diffeomorphism group of $\mathbb{R}^n$. Using this framework and the methods of Ebin and Marsden (1970), we show local well-posedness of the $b$-equation with a Fourier multiplier as the inertia operator. This is achieved by formulating the $b$-equation as a smooth ODE on a Hilbert manifold, applying Picard-Lindelöf, and transferring back to the smooth category by showing that there is no loss of spatial regularity during the time evolution.

Local well-posedness of the higher-dimensional $b$-equation

TL;DR

The paper addresses local well-posedness for the -dimensional -equation by casting it as the geodesic flow of a right-invariant affine connection on the diffeomorphism group, using a Fourier multiplier inertia operator. The authors extend the Ebin–Marsden no-loss no-gain framework to a non-metric, affine-connection setting, proving a smooth geodesic spray on the Hilbert manifold and obtaining local existence and uniqueness via Picard–Lindelöf for with and inertia operator class (). They also establish a Kelvin-Noether circulation theorem and show no loss of spatial regularity, allowing the -well-posedness to lift to the smooth category, yielding local well-posedness in . The results rigorously connect multi-dimensional shallow-water models to geometric hydrodynamics and provide a solid analytical foundation for the higher-dimensional -equation with general inertia operators. This advances understanding of well-posedness for non-metric Euler-Arnold flows and their associated conservation structures in fluid dynamics.

Abstract

The higher-dimensional -equation is a family of PDEs, introduced by Holm and Staley (2003), that describe the motion of shallow water waves in -dimensions. It expresses the invariance of the Lie-transport of the momentum one-form density associated with the fluid in -dimensions. The constant can also be viewed as a balance parameter between fluid convection and fluid stretching/expansion. In this article, we interpret this family of PDEs as the geodesic equation of a right-invariant affine connection on the diffeomorphism group of . Using this framework and the methods of Ebin and Marsden (1970), we show local well-posedness of the -equation with a Fourier multiplier as the inertia operator. This is achieved by formulating the -equation as a smooth ODE on a Hilbert manifold, applying Picard-Lindelöf, and transferring back to the smooth category by showing that there is no loss of spatial regularity during the time evolution.
Paper Structure (12 sections, 14 theorems, 84 equations)

This paper contains 12 sections, 14 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our contributions
  4. Acknowledgments
  5. Background material
  6. The Sobolev space $H^s(\mathbb{R}^n,\mathbb{R}^n)$
  7. The group of smooth Sobolev diffeomorphisms
  8. Fourier multipliers
  9. Geometric interpretations of the b-equation and a Kelvin-Noether circulation theorem
  10. Non-metric Euler-Arnold equation
  11. Lie-transport and a Kelvin-Noether circulation theorem
  12. Local well-posedness in Hs and Hinf

Key Result

Theorem 1

The n-dimensional $b$-equation b-equation with a Fourier multiplier of class $\mathcal{E}^r$ with $r\geq1$ as the inertia operator has for any initial data $u_0\in H^\infty(\mathbb{R}^n,\mathbb{R}^n)$, a unique non-extendable smooth solution $u\in C^\infty(J,H^\infty(\mathbb{R}^n,\mathbb{R}^n))$.

Theorems & Definitions (37)

  • Theorem : Corollary \ref{['mainresult']} in Section \ref{['wellposednesssection']}
  • Proposition 2.1: Inci, Kappeler, Topalov inci2012regularity; Section 2
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 1: Euler-Arnold equation arnold1966
  • ...and 27 more