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The function that maps a family of probability measures to the solution ofthe dual entropic optimal transport problem is known as the Schr\"odinger map.We prove that when the cost function is $\mathcal{C}^{k+1}$ with $k\in\mathbb{N}^*$ then this map is Lipschitz continuous from the $L^2$-Wassersteinspace to the space of $\mathcal{C}^k$ functions. Our result holds on compactdomains and covers the multi-marginal case. As applications, we provedisplacement smoothness of the entropic optimal transport cost and thewell-posedness of certain Wasserstein gradient flows involving this functional,including the Sinkhorn divergence and a multi-species system.

Lipschitz Continuity of the Schr\"odinger Map in Entropic Optimal Transport