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Gradient flows

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Optimal Transport for Applied Mathematicians

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 87))

Abstract

In this chapter we present one of the most spectacular applications of optimal transport and Wasserstein distances to PDEs. We will see that several evolution PDEs can be interpreted as a steepest descent movement in the space \(\mathbb{W}_{2}\). This includes the Heat equation, the Fokker-Planck equation, and many others. We will present the main ideas, provide a rigorous analysis of the Fokker-Planck case, and, possible extension in the discussion section. The discussion also presents complementary topics about the theoretical framework of gradient flows in metric spaces, and about other models in evolutionary PDEs which are connected to optimal transport but are not gradient flows.

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Notes

  1. 1.

    We avoid here the distinction between minimizing movements and generalized minimizing movements, which is not crucial in our analysis.

  2. 2.

    Unfortunately, some knowledge of French is required (even if not forbidden, English is unusual in the Bourbaki seminar, since “Nicolas Bourbaki a une préférence pour le français”).

  3. 3.

    The original ideas which led to this theory come from [198] and [246] through what is known as “Otto’s formal calculus”, and have later been reconsidered in [15].

  4. 4.

    Note that a similar argument, based on doubling the variables, is also often performed for the differentiation of W 2 2 along curves in \(\mathbb{W}_{2}\), which we did differently in Section 5.3.5.

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  1. Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, Orsay, France

    Filippo Santambrogio

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Santambrogio, F. (2015). Gradient flows. In: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, vol 87. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20828-2_8

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