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

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

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

).