Gradient Flows pp 167-200 | Cite as

Absolutely Continuous Curves in Open image in new windowp(X) and the Continuity Equation

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


In this chapter we endow Open image in new window 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
$$ \frac{\partial } {{\partial t}}\mu _t + \nabla .\left( {\upsilon _t \mu _t } \right) = 0inX \times \left( {a,b} \right). $$
More precisely, given an absolutely continuous curve 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 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 1-a.e. t ∈ (a, b).


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

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