Geometric origin and some properties of the arctangential heat equation

We establish the geometric origin ot the nonlinear heat equation with arct-angential nonlinearity: ∂ t D = ∆(arctan D) by deriving it, together and in du-ality with the mean curvature flow equation, from the minimal surface equation in Minkowski space-time, through a suitable quadratic change of time. After examining various properties of the arctangential heat equation (in particular through its optimal transport interpretation a la Otto and its relationship with the Born-Infeld theory of Electromagnetism), we shortly discuss its possible use for image processing, once written in non-conservative form and properly discretized.

[1]  Max Born,et al.  Foundations of the new field theory , 1934 .

[2]  A. Tzavaras,et al.  Relative Energy for the Korteweg Theory and Related Hamiltonian Flows in Gas Dynamics , 2015, 1510.00801.

[3]  Yann Brenier,et al.  Hydrodynamic Structure of the Augmented Born-Infeld Equations , 2004 .

[4]  C. Villani.  Topics in Optimal Transportation , 2003 .

[5]  L. Infeld,et al.  Foundations of the New Field Theory , 1933, Nature.

[6]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[7]  Giuseppe Savaré,et al.  Optimal Entropy-Transport problems and a new Hellinger–Kantorovich distance between positive measures , 2015, 1508.07941.

[8]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[9]  G. Peyré,et al.  Unbalanced optimal transport: Dynamic and Kantorovich formulations , 2015, Journal of Functional Analysis.

[10]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[11]  Felix Otto,et al.  Eulerian Calculus for the Contraction in the Wasserstein Distance , 2005, SIAM J. Math. Anal..

[12]  S. Serfaty.  Mean Field Limits of the Gross-Pitaevskii and Parabolic Ginzburg-Landau Equations , 2015, 1507.03821.

[13]  D. Serre.  Multidimensional Shock Interaction for a Chaplygin Gas , 2009 .

[14]  Carlos E. Kenig,et al.  Degenerate diffusions : initial value problems and local regularity theory , 2007 .

[15]  Y. Brenier,et al.  From Conservative to Dissipative Systems Through Quadratic Change of Time, with Application to the Curve-Shortening Flow , 2017, 1703.03404.

[16]  Filippo Santambrogio,et al.  Optimal Transport for Applied Mathematicians , 2015 .

[17]  Hyperbolicity of the Nonlinear Models of Maxwell’s Equations , 2004 .

[18]  C. Villani,et al.  Optimal Transportation and Applications , 2003 .

[19]  D. Aronson.  The porous medium equation , 1986 .

[20]  Hans Lindblad,et al.  A remark on global existence for small initial data of the minimal surface equation in Minkowskian space time , 2002, math/0210056.