Optimal transport, Euler equations, Mather and DiPerna-Lions theories

4.3. THE INFINITE DIMENSIONAL CASE 47
(ii)
T
0
and divγbt ∈ L 1 (0, T ); L q (γ) ;
(iii) exp(c[divγbt] − ) ∈ L ∞ (0, T ); L 1 (γ) for some c > 0.
E
(∇bt) sym (x) q
HS dγ(x)
1/q dt < ∞, (4.3.4)
If r := max{p ′ , q ′ } and c ≥ rT , then the L r -regular flow exists and is unique in the following
sense: any two L r -regular flows X and ˜ X satisfy
X(·, x) = ˜ X(·, x) in [0, T ], for γ-a.e. x ∈ E.
Furthermore, X is L s -regular for all s ∈ [1, c
T ] and the density ut of the law of X(t, ·) under γ
satisfies
(ut) s
dγ ≤
E
exp T s[divγbt] −
dγ
L ∞ (0,T )
for all s ∈ [1, c
T ].
In particular, if exp(c[divγbt] − ) ∈ L ∞ (0, T ); L 1 (γ) for all c > 0, then the L r -regular flow exists
globally in time, and is L s -regular for all s ∈ [1, ∞).
We remark that, in the previous results in this setting by Cruzeiro [46, 47, 48], Peters [69],
and Bogachev and Wolf [34], the assumptions on the vector field were
bH ∈
L p (γ),
p∈[1,∞)
exp(c∇b L(H,H)) ∈ L 1 (γ) for all c > 0,
exp(c|divγb|) ∈ L 1 (γ) for some c > 0.
Therefore the main difference between these results and our is that we replaced exponential
integrability of b and the operator norm of ∇b by p-integrability of b and q-integrability of the
Hilbert-Schmidt norm of (the symmetric part of) ∇bt. These hypotheses are in some sense closer
to the ones in the finite dimensional case, and so our result can really be seen as an extension
of the finite dimensional theory to an infinite dimensional setting.
A natural problem, on which I would like to work in the future, is to try to understand how
much this result is optimal, and whether it can be applied to prove “a.e. well-posedness” for
PDEs, looking at them as infinite dimensional ODEs.