Title: A perturbative approach to the parabolic optimal transport problem for non-MTW costs Show Chinese title

Title: 一种针对非MTW成本的抛物型最优传输问题的摄动方法

Abstract: Fix a pair of smooth source and target densities $\rho$ and $\rho^*$ of equal mass, supported on bounded domains $\Omega, \Omega^* \subset \mathbb{R}^n$. Also fix a cost function $c_0 \in C^{4,\alpha}(\overline{\Omega} \times \overline{\Omega^*})$ satisfying the weak regularity criterion of Ma, Trudinger, and Wang, and assume $\Omega$ and $\Omega^*$ are uniformly $c_0$- and $c_0^*$-convex with respect to each other. We consider a parabolic version of the optimal transport problem between $(\Omega,\rho)$ and $(\Omega^*,\rho^*)$ when the cost function $c$ is a sufficiently small $C^4$ perturbation of $c_0$, and where the size of the perturbation depends on the given data. Our main result establishes global-in-time existence of a solution $u \in C^2_xC^1_t(\overline\Omega \times [0, \infty))$ of this parabolic problem, and convergence of $u(\cdot,t)$ as $t \to \infty$ to a Kantorovich potential for the optimal transport map between $(\Omega,\rho)$ and $(\Omega^*,\rho^*)$ with cost function $c$. A noteworthy aspect of our work is that $c$ does \emph{not} necessarily satisfy the weak Ma-Trudinger-Wang condition. Show Chinese abstract

Abstract: 固定一对光滑的源密度和目标密度$\rho$和$\rho^*$，它们的质量相等，支持在有界域$\Omega, \Omega^* \subset \mathbb{R}^n$上。 同时固定一个满足Ma、Trudinger和Wang的弱正则性条件的成本函数$c_0 \in C^{4,\alpha}(\overline{\Omega} \times \overline{\Omega^*})$，并假设$\Omega$和$\Omega^*$相对于彼此是均匀的$c_0$-和$c_0^*$-凸的。 我们考虑在成本函数$c$是$c_0$的足够小的$C^4$扰动时，$(\Omega,\rho)$和$(\Omega^*,\rho^*)$之间的抛物型最优传输问题，其中扰动的大小取决于给定的数据。 我们主要结果建立了该抛物问题解$u \in C^2_xC^1_t(\overline\Omega \times [0, \infty))$的全局存在性，并且当$t \to \infty$时，$u(\cdot,t)$收敛于在成本函数$c$下，$(\Omega,\rho)$与$(\Omega^*,\rho^*)$之间的最优传输映射的Kantorovich势。 A noteworthy aspect of our work is that $c$ does \emph{不} necessarily satisfy the weak Ma-Trudinger-Wang condition.