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Information geometric regularization of unidimensional pressureless Euler equations yields global strong solutions

Ruijia Cao, Florian Schäfer

TL;DR

This work addresses shock formation in the one-dimensional, pressureless Euler equations and proposes an information geometric regularization (IGR) that replaces the standard geodesic on the diffeomorphism group with dual geodesics to achieve geodesic completeness.The authors develop a variational framework and show that IGR admits global smooth solutions for $\alpha>0$, and that these solutions converge to entropy solutions of the nominal system as $\alpha\to0$ via $\Gamma$-convergence.They prove $\Gamma$-convergence of the IGR functional to the entropy-functional one, and establish regularity of the minimizers, including absolute continuity and lower bounds on the derivative, leading to time-dependent Lagrangian and Eulerian solutions.The results imply geodesic completeness of unidimensional diffeomorphism manifolds under the dual connection, supporting IGR as a rigorous inviscid regularization that preserves shock speeds while smoothing solutions.

Abstract

Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the deformation map to leave the manifold of diffeomorphisms. Information geometric regularization addresses this issue by changing the manifold geometry to make it geodesically complete. Empirical evidence suggests that this results in smooth solutions without adding artificial viscosity. This work makes a first step towards understanding this phenomenon rigorously, in the setting of the unidimensional pressureless Euler equations. It shows that their information geometric regularization has smooth global solutions. By establishing $Γ$-convergence of its variational description, it proves convergence of these solutions to entropy solutions of the nominal problem, in the limit of vanishing regularization parameter. A consequence of these results is that manifolds of unidimensional diffeomorphisms with information geometric regularization are geodesically complete.

Information geometric regularization of unidimensional pressureless Euler equations yields global strong solutions

TL;DR

This work addresses shock formation in the one-dimensional, pressureless Euler equations and proposes an information geometric regularization (IGR) that replaces the standard geodesic on the diffeomorphism group with dual geodesics to achieve geodesic completeness.The authors develop a variational framework and show that IGR admits global smooth solutions for $\alpha>0$, and that these solutions converge to entropy solutions of the nominal system as $\alpha\to0$ via $\Gamma$-convergence.They prove $\Gamma$-convergence of the IGR functional to the entropy-functional one, and establish regularity of the minimizers, including absolute continuity and lower bounds on the derivative, leading to time-dependent Lagrangian and Eulerian solutions.The results imply geodesic completeness of unidimensional diffeomorphism manifolds under the dual connection, supporting IGR as a rigorous inviscid regularization that preserves shock speeds while smoothing solutions.

Abstract

Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the deformation map to leave the manifold of diffeomorphisms. Information geometric regularization addresses this issue by changing the manifold geometry to make it geodesically complete. Empirical evidence suggests that this results in smooth solutions without adding artificial viscosity. This work makes a first step towards understanding this phenomenon rigorously, in the setting of the unidimensional pressureless Euler equations. It shows that their information geometric regularization has smooth global solutions. By establishing -convergence of its variational description, it proves convergence of these solutions to entropy solutions of the nominal problem, in the limit of vanishing regularization parameter. A consequence of these results is that manifolds of unidimensional diffeomorphisms with information geometric regularization are geodesically complete.

Paper Structure

This paper contains 21 sections, 28 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background: Information Geometric Regularization
  3. This work: Rigorous theory for unidimensional pressureless IGR
  4. Existence of variational solutions
  5. IGR solutions as dual geodesics
  6. Variational solutions
  7. Existence of minimizers
  8. Stability of minimizers, existence of variational solutions
  9. Convergence of variational solutions
  10. The nominal problem
  11. Gamma-convergence of IGR and convergence to entropy solutions
  12. Spatial regularity of minimizers
  13. Lower bound on the pushforward density
  14. Absolute continuity of the minimizer
  15. Lower bound on the derivative of the minimizer
  16. ...and 6 more sections

Key Result

Lemma 2.3

With this definition, $u(\cdot, t)$ is in $L^2(\rho_t)$, for all $t \ge 0$ outside a set of measure zero. The function $u(\cdot, t)$ does not depend on the choice of $\dot{\Phi}_t$ in the equivalence class up to $\mu$-a.e. equality.

Figures (1)

  • Figure 1: The variation $\Phi(\cdot, \varepsilon)$ in \ref{['eqn:variation']} consists of two steps. For $\delta = \varepsilon > 0$, it simply replaces $\Phi$ on the interval $[\hat{x} - \varepsilon, \hat{x} + \varepsilon]$ by a linear interpolation. For $\delta > \varepsilon$ it stretches the resulting function values on $[\hat{x} - \varepsilon, \hat{x} + \varepsilon]$ to $[\hat{x} - \delta, \hat{x} + \delta]$ and shrinks the remaining regions of the input space accordingly. Note that the first step replaces parts of the output of $\Phi$ while the second only rearranges outputs.

Theorems & Definitions (62)

  • Remark 2.1: Dual geodesics are not metric
  • Definition 2.2: Variational solution
  • Lemma 2.3
  • Proof 1
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • Proof 2
  • Corollary 2.7
  • Proof 3
  • ...and 52 more