Gradient Flows pp 167-200 | Cite as

# Absolutely Continuous Curves in Open image in new window_{p}(*X*) and the Continuity Equation

Chapter

## Abstract

In this chapter we endow Open image in new window More precisely, given an absolutely continuous curve

_{ p }(*X*), when*X*is a separable Hilbert space, with a kind of differential structure, consistent with the metric structure introduced in the previous chapter. Our starting point is the analysis of absolutely continuous curves*μ*_{ t }: (*a, b*) → Open image in new window_{ p }(*X*) and of their metric derivative |μ′|(*t*): recall that these concepts depend only on the metric structure of Open image in new window (*X*), by Definition 1.1.1 and (1.1.3). We show in Theorem 8.3.1 that for*p*> 1 this class of curves coincides with (distributional, in the duality with smooth cylindrical test functions) solutions of the continuity equation$$
\frac{\partial }
{{\partial t}}\mu _t + \nabla .\left( {\upsilon _t \mu _t } \right) = 0inX \times \left( {a,b} \right).
$$

*μ*_{ t }, one can find a Borel time-dependent velocity field*v*_{ t }:*X → X*such that \( \left\| {\upsilon _t } \right\|_{L^p \left( {\mu _t } \right)} \leqslant \left| {\mu '} \right|\left( t \right) \) for*ℒ*^{1}-a.e.*t*∈ (*a, b*) and the continuity equation holds. Conversely, if*μt*solve the continuity equation for some Borel velocity field*v*_{ t }with \( \smallint _a^b \left\| {\upsilon _t } \right\|_{L^p \left( {\mu _t } \right)} dt < + \infty \), then*μ*_{ t }is an absolutely continuous curve and \( \left\| {\upsilon _t } \right\|_{L^p \left( {\mu _t } \right)} \geqslant \left| {\mu '} \right|\left( t \right) \) for*ℒ*^{1}-a.e.*t*∈ (*a, b*).## Keywords

Continuity Equation Tangent Space Tangent Vector Tangent Bundle Continuous Curve
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© Birkhäuser Verlag AG 2008