On the global well-posedness of the inviscid generalized Proudman–Johnson equation using flow map arguments

Kogelbauer, Florian

doi:10.1016/j.jde.2019.08.040

Cited by 5 publications

(3 citation statements)

References 13 publications

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“…The results in the one-dimensional situation are in correspondence with the analysis of [27,36], where a similar p-root transform was used to study the generalized Proudman-Johnson equation. In these articles it was used as an ad-hoc simplification of some auxiliary equations; here we expose the geometry behind it, which also simplifies some of the authors' calculations.…”

Section: Introductionsupporting

confidence: 57%

“…as was first shown in [28]. See [36,27] and the references therein for analysis of this equation, also beyond the range α ∈ (−1, 1). • For M = S 1 , the L p -Fisher-Rao metric on Prob(S 1 ) can be considered as a Finsler metric on Diff(S 1 )/ Rot(S 1 ).…”

Section: 3mentioning

confidence: 73%

“…In Figure 2 we show the three types of geodesics obtained for different values of p (p = 2, 3, 5, 10 from top to bottom), and the corresponding values of α = 1−2/p. The constrained minimization of (27) was performed in Python using the Sequential Least Squares Programming (SLSQP) method provided by the Scipy minimization solver, with a discretization of 30 time points and 100 sampling points, in a straightforward implementation that was not aimed for computational efficiency. As expected, the L p -Fisher-Rao metric and the α-connection yield different geodesics on Prob(M ), except for the special case p = 2 corresponding to the Fisher-Rao metric and its Levi-Civita connection.…”

mentioning

confidence: 99%

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Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation

Bauer

Charon

Klassen

et al. 2023

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

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We introduce a family of Finsler metrics, called the L p -Fisher-Rao metrics Fp, for p ∈ (1, ∞), which generalizes the classical Fisher-Rao metric F2, both on the space of densities Dens+(M ) and probability densities Prob(M ). We then study their relations to the Amari-Cencov α-connections ∇ (α) from information geometry: on Dens+(M ), the geodesic equations of Fp and ∇ (α) coincide, for p = 2/(1 − α). Both are pullbacks of canonical constructions on L p (M ), in which geodesics are simply straight lines. In particular, this gives a new interpretation of α-geodesics as being energy minimizing curves. On Prob(M ), the Fp and ∇ (α) geodesics can still be thought as pullbacks of natural operations on the unit sphere in L p (M ), but in this case they no longer coincide unless p = 2. On this space we show that geodesics of the α-connections are pullbacks of projections of straight lines onto the unit sphere, and they cease to exists when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman-Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of Fp, and study their relation to ∇ (α) .

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Section: Introductionsupporting

confidence: 57%

Section: 3mentioning

confidence: 73%

mentioning

confidence: 99%

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Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation

Bauer

Charon

Klassen

et al. 2023

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

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A Geometric View on the Generalized Proudman–Johnson and r-Hunter–Saxton Equations

Bauer

Maor

2021

J Nonlinear Sci

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The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections

Bauer,

Le Brigant,

et al. 2024

Calc. Var.

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We introduce a family of Finsler metrics, called the $$L^p$$ L p -Fisher–Rao metrics $$F_p$$ F p , for $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) , which generalizes the classical Fisher–Rao metric $$F_2$$ F 2 , both on the space of densities $${\text {Dens}}_+(M)$$ Dens + ( M ) and probability densities $${\text {Prob}}(M)$$ Prob ( M ) . We then study their relations to the Amari–C̆encov $$\alpha $$ α -connections $$\nabla ^{(\alpha )}$$ ∇ ( α ) from information geometry: on $${\text {Dens}}_+(M)$$ Dens + ( M ) , the geodesic equations of $$F_p$$ F p and $$\nabla ^{(\alpha )}$$ ∇ ( α ) coincide, for $$p = 2/(1-\alpha )$$ p = 2 / ( 1 - α ) . Both are pullbacks of canonical constructions on $$L^p(M)$$ L p ( M ) , in which geodesics are simply straight lines. In particular, this gives a new variational interpretation of $$\alpha $$ α -geodesics as being energy minimizing curves. On $${\text {Prob}}(M)$$ Prob ( M ) , the $$F_p$$ F p and $$\nabla ^{(\alpha )}$$ ∇ ( α ) geodesics can still be thought as pullbacks of natural operations on the unit sphere in $$L^p(M)$$ L p ( M ) , but in this case they no longer coincide unless $$p=2$$ p = 2 . Using this transformation, we solve the geodesic equation of the $$\alpha $$ α -connection by showing that the geodesic are pullbacks of projections of straight lines onto the unit sphere, and they always cease to exists after finite time when they leave the positive part of the sphere. This unveils the geometric structure of solutions to the generalized Proudman–Johnson equations, and generalizes them to higher dimensions. In addition, we calculate the associate tensors of $$F_p$$ F p , and study their relation to $$\nabla ^{(\alpha )}$$ ∇ ( α ) .

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On the global well-posedness of the inviscid generalized Proudman–Johnson equation using flow map arguments

Cited by 5 publications

References 13 publications

Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation

Intrinsic Riemannian Metrics on Spaces of Curves: Theory and Computation

A Geometric View on the Generalized Proudman–Johnson and r-Hunter–Saxton Equations

The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections

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