[Paper Review] Optimal Besov differentiability for entropy solutions of the Eikonal equation

Francesco Ghiraldin, Xavier Lamy|arXiv (Cornell University)|Feb 5, 2018

Navier-Stokes equation solutions44 references21 citations

TL;DR

This paper establishes the optimal Besov differentiability of entropy solutions to the Eikonal equation in two dimensions by proving the equivalence between optimal Besov regularity, finite entropy production, and the validity of a kinetic formulation. Using velocity averaging and div-curl estimates, the authors show that such solutions belong to the endpoint Besov space $ B^{1/4}_{4,∞} $, which is sharp up to the exponent.

ABSTRACT

In this paper we study the Eikonal equation in a bounded planar domain. We prove the equivalence among optimal Besov regularity, the finiteness of every entropy production and the validity of a kinetic formulation.

Motivation & Objective

• To establish the optimal fractional differentiability of entropy solutions to the Eikonal equation in planar domains.
• To prove the equivalence between optimal Besov regularity, finiteness of all entropy productions, and the validity of a kinetic formulation.
• To improve upon prior Sobolev regularity results by identifying the sharp Besov space for solutions arising from singular perturbations of the Aviles-Giga energy.
• To investigate whether the $ 1/4 $-H"older exponent in the Besov space $ B^{1/4}_{4,∞} $ is optimal.

Proposed method

• Utilizes the connection between the Eikonal equation and scalar conservation laws, particularly Burgers' equation, to leverage entropy solution theory.
• Applies velocity averaging techniques to derive fractional differentiability estimates from the structure of entropy production measures.
• Employs a div-curl lemma argument on the renormalized fluxes $ \Sigma_{e_1,e_2}(m) $ and $ \Sigma_{\varepsilon_1,\varepsilon_2}(m) $, which are shown to be locally finite measures.
• Establishes a pointwise lower bound on the difference of these fluxes via a key inequality $ \det(X-Y) \gtrsim |X-Y|^4 $, linking it to the increment $ |D^h_e m| $.
• Uses Hodge decomposition and potential theory to estimate the $ W^{-1,p'} $ norm of the divergence of difference operators.
• Applies a logarithmic scaling $ p = -\log|h| $ to control the growth of the $ L^p $ and dual norms, leading to the sharp $ |h|\log(1/|h|) $ modulus of continuity.

Experimental results

Research questions

• RQ1 What is the optimal Besov space regularity for entropy solutions of the Eikonal equation in two dimensions?
• RQ2 How are the finiteness of all entropy productions and the validity of a kinetic formulation related to the differentiability of solutions?
• RQ3 Can the previously known $ W^{1/3-}_{\rm loc} $ regularity be improved to a sharp Besov space endpoint?
• RQ4 Is the exponent $ 1/4 $ in the Besov space $ B^{1/4}_{4,\infty} $ optimal for the differentiability of such solutions?
• RQ5 To what extent do the geometric structure of the flux maps $ \Sigma $ and the div-curl lemma contribute to regularity estimates?

Key findings

• Entropy solutions of the Eikonal equation in $ \Omega \subset \mathbb{R}^2 $ with finite entropy production for all entropies $ \Phi \in ENT $ belong to the Besov space $ B^{1/4}_{4,\infty;\rm loc}(\Omega) $.
• The regularity result is sharp in the sense that the exponent $ 1/4 $ cannot be improved, though the optimality of the integrability $ p=4 $ remains open.
• The equivalence between finite entropy production, kinetic formulation, and optimal Besov regularity is rigorously established for the first time.
• The modulus of continuity of the $ L^4 $-norm of increments $ D^h_e m $ is bounded by $ |h|\log(1/|h|) $, which implies the $ B^{1/4}_{4,\infty} $ regularity.
• The method relies on a novel application of the div-curl lemma to the difference operators of the flux maps $ \Sigma $, exploiting their measure-valued divergence.
• The result improves upon prior $ W^{1/5-}_{\rm loc} $ and $ W^{1/3-}_{\rm loc} $ regularity results by Jabin and Perthame, achieving the sharp endpoint.