Gradient Flows pp 279-306 | Cite as

Gradient Flows and Curves of Maximal Slope in Open image in new windowp(X)

Part of the Lectures in Mathematics ETH Zürich book series (LM)


In this chapter we state some of the main results of the paper, concerning existence, uniqueness, approximation, and qualitative properties of gradient flows μ t generated by a proper, l.s.c. functional φ in Open image in new window p , X being a separable Hilbert space. Taking into account the first part of this book and the (sub)differential theory developed in the previous chapter, there are at least four possible approaches to gradient flows which can be adapted to the framework of Wasserstein spaces:
  1. 1.

    The “Minimizing Movement” approximation. We can simply consider any limit curve of the variational approximation scheme we introduced at the beginning of Chapter 2 (see Definition 2.0.6), i.e. a “Generalized minimizing movement” GMM(Φ; μ0) in the terminology suggested by E. De Giorgi. In the context of Open image in new window 2(ℝ d ) this procedure has been first used in [94, 121, 122, 120, 123] and subsequently it has been applied in many different contexts, e.g. by [93, 115, 124, 84, 85, 89, 78, 45, 46, 2, 86, 76, 15, 19].

  2. 2.
    Curves of Maximal Slope. We can look for absolutely continuous curves \( \mu _t \in AC_{loc}^p \)((0,+∞); Open image in new window p (X)) which satisfy the differential form of the Energy inequality
    $$ \frac{d} {{dt}}\varphi \left( {\mu _t } \right) \leqslant - \frac{1} {p}\left| {\mu '} \right|^p \left( t \right) - \frac{1} {q}\left| {\partial \varphi } \right|^q \left( {\mu _t } \right) \leqslant - \left| {\partial \varphi } \right|\left( {\mu _t } \right) \cdot \left| {\mu '} \right|\left( t \right) $$
    for 1-a.e. t ∈ (a, b). Notice that in the present case of Open image in new window p (X), we established in Chapter 8 a precise description of absolutely continuous curve (in terms of the continuity equation) and of the metric velocity (in terms of the Lp(μ t ;X)-norm of the related velocity vector field); moreover, in Chapter 10 we have shown an equivalent differential characterization of the slope |∂φ| in terms of the Lq(μ t ;X)-norm of the Fréchet subdifferential of φ
  3. 3.
    The pointwise differential formulation. Since we have at our disposal a notion of tangent space and the related concepts of velocity vector field v t and (sub)differential ∂φ(μ t ), we can reproduce the simple definition of gradient flow modeled on smooth Riemannian manifold, i.e.
    $$ \upsilon _t \in - \partial \varphi \left( {\mu _t } \right), $$
    trying to adapt it to the case p ≠ 2 and to extended plan subdifferentials.
  4. 4.
    Systems of Evolution Variational Inequalities (E.V.I.). When p = 2, in the case of λ-convex functionals along geodesics in Open image in new window 2(X), we can try to find solutions of the family of “metric” variational inequalities
    $$ \frac{1} {2}\frac{d} {{dt}}W_2^2 \left( {\mu _t ,\nu } \right) - \varphi \left( \nu \right) - \varphi \left( {\mu _t } \right) - \frac{\lambda } {2}W_2^2 \left( {\mu _t ,\nu } \right)\forall \nu \in D\left( \varphi \right). $$
    This formulation provides the best kind of solutions, for which in particular one can prove not only uniqueness, but also error estimates. On the other hand it imposes severe restrictions on the space (p = 2) and on the functional (λ-convexity along generalized geodesics).


Fisher Information Separable Hilbert Space Maximal Slope Limit Curve Discrete Solution 
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© Birkhäuser Verlag AG 2008

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