[Paper Review] On Euler's equation and `EPDiff'

David Mumford, Peter W. Michor|arXiv (Cornell University)|Sep 28, 2012

Nonlinear Waves and Solitons6 references3 citations

TL;DR

This paper establishes that solutions to Euler's equation for incompressible, non-viscous fluid flow are limits of solutions to the EPDiff (Euler-Poincaré equation) family of geodesic equations on diffeomorphism groups, using a regularizing operator $ L_{ ho,\eta} $ with parameters $ \varepsilon, \eta \geq 0 $. The key result is a local-in-time convergence bound between EPDiff solutions and Euler’s equation, with momentum conservation under flow ensuring stability and compact support propagation.

ABSTRACT

We study a family of approximations to Euler's equation depending on two parameters $\varepsilon,η\ge 0$. When $\varepsilon=η=0$ we have Euler's equation and when both are positive we have instances of the class of integro-differential equations called EPDiff in imaging science. These are all geodesic equations on either the full diffeomorphism group $\operatorname{Diff}_{H^\infty}(\mathbb R^n)$ or, if $\varepsilon = 0$, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by $$ \|v\|_{\varepsilon,η} = \int_{\mathbb R^n} dx $$ where $L_{\varepsilon,η} = (I- frac{η^2}{p} riangle)^p \circ (I- frac1{\varepsilon^2} abla \circ ÷)$. All geodesic equations are locally well-posed, and the $L_{\varepsilon,η}$-equation admits solutions for all time if $η>0$ and $p\ge (n+3)/2$. We tie together solutions of all these equations by estimates which, however, are only local in time. This approach leads to a new notion of momentum which is transported by the flow and serves as a generalization of vorticity. We also discuss how delta distribution momenta lead to "vortex-solitons", also called "landmarks" in imaging science, and to new numeric approximations to fluids.

Motivation & Objective

• To establish a rigorous connection between Euler’s equation and the EPDiff family of geodesic equations on diffeomorphism groups.
• To analyze the regularizing effect of the operator $ L_{\varepsilon,\eta} = (I - \frac{\eta^2}{p}\Delta)^p \circ (I - \frac{1}{\varepsilon^2}\nabla\circ\text{div}) $ on fluid flow equations.
• To demonstrate that solutions of EPDiff with $ \eta > 0 $ and $ p \geq (n+3)/2 $ are globally well-posed, enabling approximation of Euler’s equation.
• To introduce a generalized momentum concept that is transported by the flow, extending the notion of vorticity.
• To explore the emergence of vortex-solitons (landmarks) from delta-distribution momenta and their role in numerical fluid approximation.

Proposed method

• Use of a right-invariant metric on $ \text{Diff}_{H^\infty}(\mathbb{R}^n) $ defined by the norm $ \|v\|_{\varepsilon,\eta}^2 = \int \langle L_{\varepsilon,\eta}v, v \rangle \, dx $, with $ L_{\varepsilon,\eta} $ combining smoothing and divergence correction.
• Derivation of the EPDiff equation $ \partial_t m = -(v\cdot\nabla)m - \text{div}(v)m - m\cdot(Dv)^T $, where $ m = L_{\varepsilon,\eta}v $, as the geodesic equation on the diffeomorphism group.
• Proof that the momentum 1-form $ \widetilde{m} $ is invariant under the flow $ \varphi $, i.e., $ \widetilde{m}(\cdot,t) = \varphi(\cdot,t)_* \widetilde{m}(\cdot,0) $, implying conservation of support and decay properties.
• Establishment of local-in-time estimates linking solutions of EPDiff with $ \varepsilon, \eta > 0 $ to solutions of Euler’s equation as $ \varepsilon, \eta \to 0 $.
• Use of the Lagrangian formulation $ \partial_t \varphi = K^{\varphi} \ast (\varphi_* \widetilde{m}(\cdot,0)) $, where $ K $ is the Green’s function of $ L_{\varepsilon,\eta} $, to express the flow in terms of initial momentum.
• Numerical exploration of landmark geodesics (vortex-solitons) via $ \delta $-function momenta, showing repulsion and attraction depending on angular momentum and energy.

Experimental results

Research questions

• RQ1 How do solutions of the EPDiff equation with regularizing parameters $ \varepsilon, \eta > 0 $ converge to solutions of Euler’s equation as $ \varepsilon, \eta \to 0 $?
• RQ2 What is the role of the generalized momentum $ m = L_{\varepsilon,\eta}v $ in preserving structure under fluid flow, and how does it generalize vorticity?
• RQ3 Under what conditions does the EPDiff equation admit global solutions, and how does this relate to the well-posedness of Euler’s equation?
• RQ4 How do delta-distribution momenta give rise to vortex-solitons (landmarks), and what dynamical behaviors do they exhibit?
• RQ5 Can the EPDiff framework serve as a stable numerical approximation to Euler’s equation, especially in the limit $ \eta \to 0 $?

Key findings

• Solutions of Euler’s equation are the limit of solutions to the EPDiff equation with $ L_{\varepsilon,\eta} $ as $ \varepsilon, \eta \to 0 $, with a local-in-time convergence bound established.
• For $ \eta > 0 $ and $ p \geq (n+3)/2 $, the EPDiff equation is globally well-posed with unique solutions for all time, ensuring regularized approximation.
• The momentum 1-form $ \widetilde{m} $ is invariant under the flow $ \varphi $, so compact support and rapid decay are preserved, implying structural stability.
• Vortex-solitons (landmarks) emerge from $ \delta $-distribution momenta and exhibit scattering or capturing dynamics depending on energy and angular momentum, with a critical energy threshold at $ E = \frac{8}{5}|\omega|^2 $.
• The vector field derived from the derivative of the kernel $ K_{0,1} $ near the origin shows spiral flowlines in the $ (x_1,x_2) $-plane and outflow along the $ x_3 $-axis, indicating localized vortex-ring-like behavior.
• Numerical integration of landmark geodesics shows that varying the background momentum $ \overline{m} $ leads to both repulsive and attractive interactions, suggesting utility in modeling fluid instabilities.