Absolutely Continuous Curves in _p (X) and the Continuity Equation

Abstract

In this chapter we endow _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) → _p (X) and of their metric derivative |μ′|(t): recall that these concepts depend only on the metric structure of (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 ℒ¹-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 ℒ¹-a.e. t ∈ (a, b).