2021/08/02 by Farhan Abedin, Abedin, Farhan, Jun Kitagawa +1 · 1 citation
Computer Science · Mathematics · #35B20 #35K96 #49Q22 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2108.01253
openalex publication_date 2021/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix a pair of smooth source and target densities \ρ and \ρ^* of equal\nmass, supported on bounded domains \Ω, \Ω^* \⊂ \ℝn.\nAlso fix a cost function c0 \∈ C4,\α(\\Ω \×\n\\Ω^*) satisfying the weak regularity criterion of Ma,\nTrudinger, and Wang, and assume \Ω and \Ω^* are uniformly c0-\nand c0^*-convex with respect to each other. We consider a parabolic version\nof the optimal transport problem between (\Ω,\ρ) and\n(\Ω^*,\ρ^*) when the cost function c is a sufficiently small C4\nperturbation of c0, and where the size of the perturbation depends on the\ngiven data. Our main result establishes global-in-time existence of a solution\nu \∈ C2xC1t( Ω \× [0, \∞)) of this parabolic\nproblem, and convergence of u(\⋅,t) as t \→ \∞ to a Kantorovich\npotential for the optimal transport map between (\Ω,\ρ) and\n(\Ω^*,\ρ^*) with cost function c. A noteworthy aspect of our work is\nthat c does \not necessarily satisfy the weak Ma-Trudinger-Wang\ncondition.\n