2008 | OriginalPaper | Chapter
Absolutely Continuous Curves in p (X) and the Continuity Equation
Published in: Gradient Flows
Publisher: Birkhäuser Basel
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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
).