Nicola Gigli
email: nicola.gigli@unice.fr

Sunra J N Mosconi
email: mosconi@cibs.sns.it

A variational approach to the Navier-Stokes equations

We propose a time discretization of the Navier-Stokes equations inspired by the theory of gradient flows. This discretization produces Leray/Hopf solutions in any dimension and suitable solutions in dimension 3. We also show that in dimension 3 and for initial datum in H 1 , the scheme converges to strong solutions in some interval [0, T ) and, if the datum satisfies the classical smallness condition, it produces the smooth solution in [0, ∞).

Introduction

We consider the Navier-Stokes equations on the d-dimensional flat torus

T d = R d /Z d :    ∂ t u t + (u t • ∇)u t + ∇p t = ∆u t , in [0, ∞) × T d , ∇ • u t = 0, in T d ∀t, u 0 = u, in T d , (1)
where the initial datum u is a given divergence free vector field, say smooth. The purpose of this paper is to present a time-discretization argument, inspired by the theory of gradient flows, which allows to quickly reproduce several known results about [START_REF] Blunck | Maximal regularity of discrete and continuous time evolution equations[END_REF]. The idea is the following. Fix a parameter τ > 0, which we think as time step and, given u, define its flow map R × T d (t, x) → X u t (x) ∈ T d as the only solution of

∂ t X u t = u • X u t , X u 0 = Id. Now minimize v → 1 2 T d |∇v| 2 dL d + v • X u τ -u 2 L 2
2τ , among all L 2 and divergence free vector fields v. It is not hard to check (see Proposition 3.1) that the unique minimum u τ satisfies

u τ -u • X u -τ τ + ∇p τ = ∆u τ , (2)
where p τ is identified, up to additive constants, by

∆p τ = ∇ • u • X u -τ τ .
We claim that ( 2) is a time discretization of [START_REF] Blunck | Maximal regularity of discrete and continuous time evolution equations[END_REF]. Indeed, the term

u τ -u • X u -τ τ = u τ • X u τ -u τ • X u -τ ,
is the time discretization of the convective derivative

∂ t u t + (u t • ∇)u t = ∂ t (u t • T t ) • T -1 t ,
where here [0, ∞) × T d (t, x) → T t (x) ∈ T d is the flow map (or particle-trajectory map) associated to (u t ), i.e.:

∂ t T t = u t • T t , T 0 = Id.
and the pressure term satisfies

∆p τ = ∇ • u • X u -τ τ = ∇ • u • X u -τ -u τ ,
which is a time discretization of

∆p t = ∇ • (u t • ∇)u t = ∇ • ∂ t u t + (u t • ∇)u t ,
the latter being the formula identifying the pressure in [START_REF] Blunck | Maximal regularity of discrete and continuous time evolution equations[END_REF]. The idea is then to repeat the minimization procedure with u τ in place of u, then with the new minimizer in place of u τ and so on. This argument produces a discrete solution (u τ t ) and our goal is to show that letting τ ↓ 0 the discrete solutions converge, in a sense to be specified, to certain solutions of (1).

We remark that a time discretization based on (2) is not entirely new in this setting: O. Pirroneau [START_REF] Pironneau | On the transport-diffusion algorithm and its applications to the Navier-Stokes equations[END_REF] used the same equation (without pointing out its variational structure) in the setting of numerical analysis to investigate the rate of convergence of the discrete solutions under the assumption that a smooth solution of (1) exists on some interval [0, T ].

The authors wishes to thank L. Ambrosio, Y. Brenier and C. De Lellis for fruitful conversations.

Notation and preliminaries

With T d we denote the d-dimensional flat torus R d /Z d . A time dependent vector field [0, ∞) (t, x) → u t (x) ∈ R d will be typically denoted by (u t ), while we write u t for the static vector field x → u t (x). The subscript t will never stand for time derivative, which will be usually denoted by ∂ t . When not specified, the integral symbol without further specification on the domain will stand for integration over T d (resp. [0, +∞)) when performed w.r.t. the measure dL d (resp. dt). We will also shorten L p (T d , R d ) with L p when the space is clear from the context. Given a smooth vector vector field

u ∈ C ∞ (T d , R d ), the flow map R × T d (t, x) → X u t (x) ∈ T d is the unique solution of ∂ t X u t (x) = u • X u t (x), X u 0 (x) = x.
The classic Cauchy-Lipschitz theory ensures that X u is C ∞ in space and time as soon as u is C ∞ .

Recall that for a Borel map T : T d → T d and a non negative Borel measure µ on T d , the Borel measure Classical elliptic regularity theory ensures that p, w are C ∞ as soon as u is.

T # µ on T d is defined by T # µ(E) := µ(T -1 (E)) for all Borel sets E ⊂ T d . We will frequently use the fact that ∇ • u = 0 implies (X u t ) # L d = L d for any t ∈ R. Given a vector field u ∈ C ∞ (T d , R d ),
Given

u ∈ C ∞ (T d , R d )
, with |∇u| we will always mean the Hilbert-Smith norm of its gradient, given by

|∇u| 2 (x) := i,j |∂ i u j (x)| 2 .
We can, and will, suppose that at any time t ≥ 0, the velocity field has zero mean value. Indeed, integrating over T d equation ( 1) and integrating by parts gives for any solution of (1) it holds

d dt u t dL d = 0 ∀t ≥ 0.
Thus, if v := udL d , one can look for solutions (w t ) of ( 1) with initial data w = u -v, thus having zero mean velocity for any t ≥ 0. Letting then u t (x) = w t (x -vt) + v, it is easily checked that (u t ) is a solution of the original problem. The additional condition T d udL d = 0 implies the first of the two frequently used estimates

u L 2 * (T d ) ≤ C ∇u L 2 (T d ) , ∇u L 2 * (T d ) ≤ C ∆u L 2 (T d ) , (3)
where 2 * = 2d d-2 , while the second one follows from standard elliptic estimates, since ∇u has zero mean on T d by periodicity.

To show the convergence of the discretization scheme, we will use the Aubin-Lions lemma:

Lemma 2.1 (Aubin-Lions) Let X ⊂ Y ⊂ Z be three Banach spaces such that: X and Z are reflexive, the embedding of X in Y is compact and the one of Y into Z is continuous. Then for any p, q ∈ (1, ∞) the space

u ∈ L p ([0, T ], X) : ∂ t u ∈ L q ([0, T ], Z) , is compactly embedded in L p ([0, T ], Y ).
For a proof see for instance [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF] Chapter 3, Theorem 2.1.

Discrete solutions

Let u ∈ C ∞ (T d , R d ) be a smooth vector field and τ > 0. The functional F (v; u, τ ) is defined as

F (v; u, τ ) := T d |∇v| 2 dL d + 1 τ T d |v • X u τ -u| 2 dL d . ( 4
) Proposition 3.1 Let u ∈ C ∞ (T d , R d ) be a smooth vector field such that ∇ • u = 0 and τ > 0.
Then there exists a unique minimizer u τ of v → F (v; u, τ ) in the class of L 2 vector fields such that ∇ • v = 0. The minimum u τ is C ∞ and satisfies:

u τ -u • X u -τ τ + ∇p τ = ∆u τ , (5)
for some p τ ∈ C ∞ with p τ dL d = 0.

Proof. Existence follows by standard weak compactness-lower semicontinuity arguments. For uniqueness, observe that the map v → |∇v| 2 is convex and

v → |v • X u τ -u| 2 dL d = |v -u • X u -τ | 2 dL d ,
is strictly convex.

To write the Euler equation, notice that for ξ ∈ C ∞ (T d , R d ) with ∇ • ξ = 0, the standard perturbation argument gives

u τ -u • X u -τ τ • ξdL d = -∇u τ ∇ξdL d ,
thus u τ , p τ is a weak solution of the Stokes problem [START_REF] Giga | The Stokes operator in L r spaces[END_REF]. Standard regularity theory for the Stokes operator guarantees that both u τ and p τ are C ∞ and are classical solutions of ( 5). Now we use this minimization problem to build a time-discretized solution of the Navier-Stokes equation: Definition 3.2 (Discrete solutions) Let u ∈ C ∞ (T d , R d ) be a smooth vector field with ∇•u = 0 and τ > 0. Define the vector field (u τ t ) recursively by:

u τ 0 := u, u τ (n+1)τ := argmin ∇•v=0 F (v; u τ nτ , τ ), ∀n ∈ N u τ t := u τ τ [ t τ ] ,
∀t ≥ 0.

The discrete pressure field (p τ t ) is defined, for every t ≥ 0, by

∆p τ t = ∇ • u τ t • X u τ t -τ τ , p τ t dL d = 0.
Notice that since the smoothness of a vector field implies the smoothness of the corresponding minimizer for any τ > 0, the flow maps are always well defined and so are the discrete solutions. Also u τ t , p τ t are smooth for any t ≥ 0.

4 The results

Weak solutions

Here we prove that discrete solutions produce, when τ ↓ 0, weak solutions of the Navier-Stokes equations in any dimension.

Definition 4.1 (Hopf Solutions) We say that (u t ) is a Hopf solution of the Navier-Stokes equation starting from u provided it satisfies • Discrete energy inequality.

(0,∞)×T d -u t , ∂ t ξ t -u t , ∇ξ t • u t + ∇u t , ∇ξ t dt dL d = T d u, ξ 0 dL d , (6)
for any ξ ∈ C ∞ c ([0, ∞) × T d , R d ) with ∇ • ξ t = 0 for any t ≥ 0, 1 2 u s 2 L 2 + s 0 ∇u r 2 L 2 dr ≤ 1 2 u 2 L 2 , ∀s ≥ 0, 1 2 u s 2 L 2 + s t ∇u r 2 L 2 dr ≤ 1 2 u t 2 L 2 ,
1 2 u τ 2 L 2 + τ ∇u τ 2 L 2 ≤ 1 2 u 2 L 2 . (7)
• Discrete distributional solution.

For every ξ ∈ C ∞ (T d , R d ) it holds u τ -u τ , ξ L 2 -u, ξ • X u τ -ξ τ L 2 + p τ , ∇ • ξ L 2 = u τ , ∆ξ L 2 = -∇u τ , ∇ξ L 2 (8)
• Discrete uniform weak continuity.

For any ξ ∈ C ∞ (T d , R d ) with ∇ • ξ = 0 it holds | u τ -u, ξ L 2 | ≤ τ C(ξ) u L 2 + u 2 L 2 , (9)
where C(ξ) := max{Lip(ξ), ∆ξ L 2 }

• Rough estimate on the discrete time derivative.

u τ -u τ H -m Df ≤ C( u L 2 + u 2 L 2 ) ( 10
)
for any m > n 2 + 2, for some constant C independent on u, τ , where H -m Df is the dual space of the space of divergence free vector fields in H m (T d , R d ) with 0 mean, endowed with the norm u 2

H m := α ∂ α u 2 L 2
, where α varies over all the multiindexes of length m. Proof. To get (7) multiply ( 5) by τ u τ and integrate to get

u τ 2 L 2 -u τ , u • X u -τ L 2 = τ u τ , ∆u τ L 2 ,
and conclude noticing that

u τ , ∆u τ L 2 = -∇u τ 2 L 2 , u τ , u • X u -τ L 2 ≤ 1 2 u τ 2 L 2 + 1 2 u • X u -τ 2 L 2 = 1 2 u τ 2 L 2 + 1 2 u 2 L 2 .
To get (8) we sum and subtract u/τ , multiply (5) by ξ, and integrate by parts the terms involving the pressure and ∆u τ . For the discrete convective term we use the fact that (X u t ) # L d = L d for any t and thus, by the semigroup property of (X u t ),

u • X u -τ , ξ L 2 = u, ξ • X u τ L 2 .
For ( 9) we observe that

ξ • X u τ -ξ 2 dL d = τ 0 ∂ t (ξ • X u t )dt 2 dL d = τ 0 ∇ξ • X u t • u • X u t dt 2 dL d ≤ τ τ 0 |∇ξ • X u t • u • X u t | 2 dtdL d = τ τ 0 |∇ξ • X u t • u • X u t | 2 dL d dt = τ τ 0 |∇ξ • u| 2 dL d dt ≤ τ 2 Lip 2 (ξ) u 2 L 2 , (11)
yields ξ • X u τ -ξ L 2 ≤ τ Lip(ξ) u L 2 , (12)
and conclude using [START_REF] Sohr | On the regularity of the pressure of weak solutions of Navier-Stokes equations[END_REF]. It remains to prove [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF]. Start recalling that for any

ξ ∈ H m (T d , R d ), it holds ∆ξ L 2 + Lip(ξ) ≤ C ξ H m ,
for some constant C. Therefore from (9) we get

| u τ -u, ξ L 2 | ≤ Cτ ξ H m u L 2 + u 2 L 2 , ∀ξ ∈ C ∞ (T d , R d ) s.t. ∇ • ξ = 0,
which implies [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF].

Theorem 4.3 (Hopf solutions) For any sequence τ n ↓ 0 there exists a subsequence, not relabeled, such that u τn t weakly converges (in L 2 (T d )) to some u t as n → ∞ for any t ≥ 0, the convergence is strong for a.e. t and (u τn t ) converges strongly in

L 2 loc ([0, +∞), L 2 (T d , R d ))
to (u t ). Any limit vector field (u t ) found in this way is a Hopf solution of the Navier-Stokes equations.

Proof.

Compactness. From (7) we immediately get

u τ t L 2 ≤ u L 2 , ∀t, τ > 0. ( 13
)
Thus with a diagonalization argument, for each sequence τ n ↓ 0 we can find a subsequence, not relabeled, such that for each rational t, the sequence n → u τn t weakly converges to some

u t ∈ L 2 (T d , R d ). From (9) we easily get that for every ξ ∈ C ∞ (T d , R d ) such that ∇ • ξ = 0 it holds, u τ t -u τ s , ξ L 2 ≤ (t -s + τ )C(ξ) u L 2 + u 2 L 2 , ∀τ > 0, ∀t, s ∈ [0, ∞), (14)
which is enough to conclude that there is weak convergence for every t ≥ 0 as τ n ↓ 0.

To get the strong convergence we use the Aubin-Lions lemma. In order to do so it is better to introduce the piecewise affine interpolation of the {u τ nτ } n∈N in place of the piecewise constant one:

w τ t := 1 - t τ + t τ u τ τ [ t τ ] + t τ - t τ u τ τ ([ t τ ]+1) . (15)
It is immediate to verify that the compactness of

{(w τ t )} τ implies that of {(u τ t )} τ . To get the compactness of {(w τ t )} τ in L 2 loc ([0, ∞), L 2
) we apply the Aubin-Lions lemma to the spaces

X := u ∈ H 1 (T d , R d ) : ∇ • u = 0, T d u L d = 0 , Y := u ∈ L 2 (T d , R d ) : ∇ • u = 0, T d u L d = 0 , Z := H -m Df ,
where X, Y are endowed with the H 1 and the L 2 norms respectively. Then from [START_REF] Temam | Navier-Stokes equations. Theory and numerical analysis[END_REF] and the definition of (w τ t ) we get that

∂ t w τ t H -m Df = u τ t+τ -u τ t τ H -m Df ≤ C( u L 2 + u 2 L 2 ), a.e. t,
which is sufficient to conclude. Now that we have compactness in L 2 loc ([0, +∞), L 2 (T d , R d )), we know that for any sequence τ n ↓ 0 there exists a subsequence τ n k such that u τn k t converges strongly to u t for a.e. t > 0. Any limit is a Hopf solution. Let τ n ↓ 0 be a sequence for which we have weak convergence for all times and strong convergence for a.e. times and let (u t ) be the limit vector field. We have the uniform bound

u t L 2 ≤ lim n→∞ u τn t L 2 ≤ u L 2 , ∀t ≥ 0,
and passing to the limit in ( 14) we obtain

u t -u s , ξ L 2 ≤ (t -s)C(ξ) u L 2 + u 2 L 2 , ∀t, s ∈ [0, ∞),
which is enough to get the weak continuity of t → u t . Now let A ⊂ [0, ∞) the set of t's such that u τn t converges strongly to u t as n → ∞ and notice that certainly 0 ∈ A and L 1 ([0, ∞) \ A) = 0. Choose t ∈ A and s > t and observe that from [START_REF] Pironneau | On the transport-diffusion algorithm and its applications to the Navier-Stokes equations[END_REF] we get

1 2 u τ s 2 L 2 + τ [ s τ ]+τ τ [ t τ ]+τ ∇u τ r 2 L 2 dr ≤ 1 2 u τ t 2 L 2 , ∀τ > 0. ( 16
)
The choice of t ensures that u τn t L 2 → u t L 2 , so that from

lim n→∞     1 2 u τn s 2 L 2 + τn[ s τ n ]+τn τn[ t τn ]+τn ∇u τn r 2 L 2 dr     ≥ 1 2 u s 2 L 2 + s t lim n→∞ ∇u τn r 2 L 2 dr ≥ 1 2 u s 2 L 2 + s t ∇u r 2 L 2 dr,
we get the energy inequality.

To conclude we need to show that (u t ) satisfies [START_REF] Kunstmann | Maximal L p regularity fo Parabolic equations, Fourier Multiplier Theorems and H ∞ -functional calculus[END_REF].

Fix ξ ∈ C ∞ c ([0, ∞) × T d , R d
) such that ∇ • ξ t = 0 for every t ≥ 0, and for any k = 0, 1 . . . we consider (8) for u = u τ kτ tested with ξ kτ . Adding up one gets

- +∞ τ u τ t • ξ τ [ t τ ] -ξ τ ([ t τ ]-1) τ + u τ t • ξ τ [ t τ ] • X u τ t τ -ξ τ [ t τ ] τ + u τ t • ∆ξ τ [ t τ ] dt dL d = u • ξ 0 dL d . From the smoothness of ξ we know that ξ τ [ t τ ] -ξ τ ([ t τ ]-1) τ → ∂ t ξ t , ∆ξ τ [ t τ ] → ∆ξ t ,
uniformly in [0, ∞) × T d as τ ↓ 0. Thus from the strong convergence of (u τn t ) to (u t ) in

L 2 loc ([0, ∞), L 2 ) we get +∞ τn u τn t • ξ τn[ t τn ] -ξ τn([ t τn ]-1) τ n dt dL d → +∞ 0 u t • ∂ t ξ t dt dL d , +∞ τn u τn t • ∆ξ τn[ t τn ] dt dL d → +∞ 0 u t • ∆ξ t dt dL d = - +∞ 0 ∇u t • ∇ξ t dt dL d ,
as n → ∞. Thus to conclude it is sufficient to check that 12) and (13) it is easy to deduce that

ξ τn[ t τn ] • X u τn t τ -ξ τn[ t τn ] τ n → ∇ξ t • u t , weakly in L 2 ([0, T ], L 2 ) as n → ∞, where T is such that supp(ξ) ⊂ [0, T ] × T d . From (
1 τ n ξ τn[ t τn ] • X u τn t τ -ξ τn[ t τn ] L 2 ([0,∞),L 2 ) ≤ T u L 2 sup t Lip(ξ t ).
Thus to prove the desired weak convergence it is sufficient to prove that for every ξ

∈ C ∞ c ([0, ∞)× T d , R d ) it holds 1 τ n ξt , ξ τn[ t τn ] • X u τn t τn -ξ τn[ t τn ] dt dL d → ξt , (u t • ∇)ξ t dt dL d , as n → ∞.
To prove this, recall that

ξ τn[ t τn ] • X u τn t τn -ξ τn[ t τn ] τ n = 1 τ n τn 0 (u τn t • ∇)ξ τn[ t τn ] • X u τn t s ds,
and therefore

1 τ n ξt , ξ τn[ t τn ] • X u τn t τn -ξ τn[ t τn ] dt dL d = 1 τ n τn 0 ξt , (u τn t • ∇)ξ τn[ t τn ] • X u τn t s ds dt dL d = 1 τ n τn 0 ξt , (u τn t • ∇)ξ τn[ t τn ] • X u τn t s dL d dt ds = 1 τ n τn 0 ξt • X u τn t -s , (u τn t • ∇)ξ τn[ t τn ] dL d dt ds = 1 τ n τn 0 ξt • X u τn t -s ds, (u τn t • ∇)ξ τn[ t τn ] dL d dt. (17)
To conclude notice that the strong convergence of (u τn t ) to (u t ) and the smoothness of ξ ensure that (u τn t • ∇)ξ τn[ t τn ] strongly converges to (u t • ∇)ξ t in L 2 ([0, ∞), L 2 ). Also, from (12) with τ = s one gets

1 τ n τn 0 ξt • X u τn t -s ds -ξt 2 dL d dt = 1 τ n τn 0 ξt -ξt • X u τn t s 2 L 2 dtds ≤ 1 τ n τn 0 T 0 s 2 Lip( ξ) 2 u τn t 2 L 2 dtds ≤ τ 2 n 3 u 2 L 2 T Lip( ξ) 2 , ( 18
)
where

T is such that supp( ξ) ⊆ [0, T ]×T d . Therefore 1 τn τn 0 ξt •X u τn t -s ds-ξt L 2 ([0,∞),L 2 ) = O(τ n ), which completes the proof.

Suitable solutions in dimension 3

Here we show that in dimension 3 the discrete solutions converge to suitable solutions of the Navier-Stokes equations.

Definition 4.4 (Suitable solution)

We say that the pair (u t , p t ) is a suitable solution of the Navier-Stokes equations starting from u provided it is a distributional solution of the Navier-Stokes equation, (u t ) is a Hopf solution starting from u, and in addition

(u t ) ∈ L 3 loc ([0, ∞), L 3 (T 3 , R 3 )), (p t ) ∈ L 3/2 loc ([0, ∞), L 3/2 (T 3
)) and it holds

∂ t |u t | 2 2 + ∇ • u t |u t | 2 2 + p t + |∇u t | 2 ≤ ∆ |u t | 2 2 , ( 19
)
in the sense of distributions, that is

- |u t | 2 2 ∂ t ϕ -∇ϕ • u t |u t | 2 2 + p t + ϕ|∇u t | 2 dt dL 3 ≤ ∆ϕ |u t | 2 2 dt dL 3 , for any non negative ϕ ∈ C ∞ c ((0, ∞) × T 3 ).
We recall that the importance of suitable solutions is due to the work [START_REF] Caffarelli | Partial regularity of suitable weak solutions of the Navier-Stokes equations[END_REF] where important partial regularity results for these solutions have been achieved.

1 2 (|u τ | 2 -|u| 2 )ϕ - 1 2 |u| 2 (ϕ • X u τ -ϕ) -∇ϕ • u τ p τ + |∇u τ | 2 ϕdL d ≤ τ |u τ | 2 2 ∆ϕdL d ,
for any nonnegative ϕ ∈ C ∞ (T d ).

Proof. Multiply (5) by u τ , by ϕ ∈ C ∞ (T d ) with ϕ ≥ 0 and integrate to get

|u τ | 2 ϕ -u τ • u • X u -τ ϕ + ϕu τ • ∇p τ dL d = τ u τ • ∆u τ ϕdL d .
It holds

u τ • ∆u τ ϕdL d = -|∇u τ | 2 ϕ - i,j u i ∂ j u i ∂ j ϕ dL d = -|∇u τ | 2 ϕdL d + |u τ | 2 2 ∆ϕdL d , ϕu τ • ∇p τ dL d = -∇ϕ • u τ p τ dL d , u τ • u • X u -τ ϕdL d ≤ 1 2 |u τ | 2 ϕdL d + 1 2 |u • X u -τ | 2 ϕdL d = 1 2 |u τ | 2 ϕdL d + 1 2 |u| 2 ϕ • X u τ dL d ,
(notice that in the last inequality we used the fact that ϕ ≥ 0). The conclusion follows.

Lemma 4.6 (Estimate on the pressure) For any divergence free vector field u ∈

L 3 (T d , R d ) ∩ C ∞ (T d , R d
) and τ > 0 define p τ u as the only solution of

   ∆p τ u = ∇ • u • X u -τ τ , p τ u dL d = 0. ( 20
)
Then for some C > 0 independent on u it holds

p τ u L 3/2 ≤ C u 2 L 3 . (21)
Proof. For any ϕ ∈ C ∞ (T d ) we have

p τ u ∆ϕ dL d = u • X u -τ -u τ • ∇ϕ dL d = u • ∇ϕ • X u τ -∇ϕ τ dL d = u • 1 τ τ 0 ∇ 2 ϕ • u • X u s ds dL d ≤ u L 3 1 τ τ 0 ∇ 2 ϕ • u • X u s ds L 3/2 ≤ u L 3 1 τ τ 0 ∇ 2 ϕ • u • X u s L 3/2 ds = u L 3 ∇ 2 ϕ • u L 3/2 ≤ u 2 L 3 ∇ 2 ϕ L 3 . (22)
Standard elliptic regularity results ensure that ∇ 2 ϕ L 3 ≤ C ∆ϕ L 3 for some C > 0. Thus we get

p τ u ∆ϕdL d ≤ C u 2 L 3 ∆ϕ L 3 .
Again by standard arguments, we know that the equation ∆ϕ = ψ has a smooth solution for every smooth ψ such that ψdL d = 0. Thus the above bound yields

p τ u ψ dL d ≤ C u 2 L 3 ψ L 3 , ∀ψ ∈ C ∞ (T d ) such that ψ dL d = 0.
Since p τ u dL d = 0, from the last inequality we conclude that (21) is true.

Theorem 4.7 (Suitable solutions in dimension 3) Let (u τ t ) and (p τ t ) be defined by 3.2.

Then {(u τ t )} τ is compact in L 3 loc ([0, ∞), L 3 (T 3 , R 3 )) and {(p τ t )} τ is weakly compact in L 3/2 loc ([0, ∞), L 3/2 (T 3 )).
For any sequence τ n ↓ 0 such that (u τn t ) strongly converges to some

(u t ) in L 3 loc ([0, ∞), L 3 (T 3 , R 3
)) and (p τn t ) weakly converges in L 3/2 loc ([0, ∞), L 3/2 (T 3 )) to some (p t ), the couple (u t ), (p t ) is a suitable solution of the Navier-Stokes equation.

Proof.

Compactness. Arguing as in the proof of Theorem 4.3 and using the compact embedding of

H 1 into L 4 , we deduce that {(u τ t )} τ is relatively compact in L 2 loc ([0, ∞), L 4 ). For any (u t ) ∈ L 2 loc [0, ∞), L 4 ∩ L ∞ [0, ∞), L 2 it holds |u t | 3 dL 3 dt ≤ u t L 2 u t 2 L 4 dt = u L ∞ ([0,∞),L 2 ) u 2 L 2 ([0,+∞),L 4 ) ,
and therefore the uniform bound (13) ensures the desired strong relative compactness of {(u τ t )} τ in L 3 ([0, +∞), L 3 ). To get the weak compactness of {(p τ t )} τ , notice that p τ t is the unique solution of (20) for u = u τ t , and from the uniform bound of (u τ t ) in L 3 ([0, +∞), L 3 ), together with (21), we get weak compactness of (p τ t ) in L 3/2 ([0, +∞), L 3/2 ). Limits are suitable solutions. To simplify the notation, we assume that (

u τ t ) → (u t ) strongly in L 3 ([0, ∞), L 3 ) (p τ t ) → (p t ) weakly in L 3/2 ([0, ∞), L 3/2 ) (i.
e., we are not considering a sequence τ n ↓ 0). Theorem 4.3 guarantees that u is a Hopf solution of the Navier-Stokes system, and clearly (u

t ) ∈ L 3 ([0, ∞), L 3 ), (p t ) ∈ L 3/2 ([0, ∞), L 3/2 ).
To prove that (u t ), (p t ) is a distributional solution, fix ξ ∈ C ∞ c ((0, ∞) × T 3 , R 3 ), and for any k = 0, 1 . . . consider (8) for u = u τ kτ , tested with ξ kτ . Adding up one gets

+∞ τ u τ t • ξ τ [ t τ ] -ξ τ ([ t τ ]-1) τ +u τ t • ξ τ [ t τ ] • X u τ t τ -ξ τ [ t τ ] τ dt dL 3 = +∞ τ u τ t •∆ξ τ [ t τ ] +p τ t ∇•ξ τ [ t τ ] dt dL 3 . Now since ∇ • ξ τ [ t τ ] → ∇ • ξ t uniformly on [0, +∞) × T 3
, the weak convergence of (p τ t ) to (p t ) yields

p τ t ∇ • ξ τ [ t τ ] dt dL 3 → p t ∇ • ξ t dt dL 3 ,
while all the other terms are treated as in the proof of theorem 4.3.

To prove the generalized energy inequality, fix a non negative ϕ ∈ C ∞ c ((0, ∞) × T 3 ). We want to show that

- |u t | 2 2 ∂ t ϕ + u t • ∇ϕ |u t | 2 2 + p t + |∇u t | 2 ϕdtdL 3 ≤ |u t | 2 2 ∆ϕdt dL 3
Suppose τ is sufficiently small, such that supp(ϕ) ⊂ [τ, +∞). From Lemma 4.5 and the definition of u τ , p τ we immediately get, with the usual argument

- |u τ t | 2 2 ϕ τ [ t τ ] -ϕ τ ([ t τ ]-1) τ - |u τ t | 2 2 ϕ τ [ t τ ] • X u τ t τ -ϕ τ [ t τ ] τ -∇ϕ τ [ t τ ] • u τ t p τ t + |∇u τ t | 2 ϕ τ [ t τ ] dtdL 3 ≤ |u τ t | 2 2 ∆ϕ τ [ t τ ] dtdL 3 .
The convergence of u τ ensures that |u τ | 2 converges to |u| 2 in L 3/2 ([0, ∞), L 3/2 ), and from the smoothness of ϕ and the weak convergence of p τ it is immediate to verify that as τ ↓ 0 it holds

|u τ t | 2 2 ϕ τ [ t τ ] -ϕ τ ([ t τ ]-1) τ dtdL 3 → |u t | 2 2 ∂ t ϕ dtdL 3 , ∇ϕ τ [ t τ ] • u τ t p τ t dtdL 3 → ∇ϕ t • u t p t dtdL 3 , |u τ t | 2 2 ∆ϕ τ [ t τ ] dtdL 3 → |u t | 2 2 ∆ϕ t dtdL 3 .
Also, the non negativity of ϕ easily implies

lim inf τ ↓0 |∇u τ t | 2 ϕ τ [ t τ ] dtdL 3 ≥ |∇u t | 2 ϕ t dtdL 3 .
Thus to conclude it is sufficient to show that

|u τ t | 2 2 ϕ τ [ t τ ] • X u τ t τ -ϕ τ [ t τ ] τ dtdL 3 → |u τ t | 2 2 ∇ϕ t • u t dtdL 3 , (23)
as τ ↓ 0. Since

|u τ t | 2 → |u t | 2 in L 3/2 ([0, ∞), L 3/2 ), to prove (23) it is sufficient to check that ϕ τ [ t τ ] • X u τ t τ -ϕ τ [ t τ ] τ → ∇ϕ t • u t weakly in L 3 ([0, ∞), L 3 ). From ϕ τ [ t τ ] • X u τ t τ -ϕ τ [ t τ ] τ = 1 τ τ 0 ∇ϕ τ [ t τ ] • X u τ t s • u τ t • X u τ t s ds,
we immediately get that it holds

ϕ τ [ t τ ] • X u τ t τ -ϕ τ [ t τ ] τ L 3 ([0,∞),L 3 ) ≤ sup t Lip(ϕ t ) u L 3 ([0,∞),L 3 ) ,
which gives weak convergence for some subsequence. To conclude fix ψ ∈ C ∞ c ((0, ∞) × T 3 ) and notice that

ψ t ϕ τ [ t τ ] • X u τ t τ -ϕ τ [ t τ ] τ dtdL 3 = 1 τ τ 0 ψ t ∇ϕ τ [ t τ ] • X u τ t s • u τ t • X u τ t s dsdtdL 3 = 1 τ τ 0 ψ t ∇ϕ τ [ t τ ] • X u τ t s • u τ t • X u τ t s dL 3 dtds = 1 τ τ 0 ψ t • X u τ t -s ∇ϕ τ [ t τ ] • u τ t dL 3 dtds = 1 τ τ 0 ψ t • X u τ t -s ds ∇ϕ τ [ t τ ] • u τ t dL 3 dt. Since ∇ϕ τ [ t τ ] • u τ t → ∇ϕ t • u t strongly in L 3 ([0, ∞), L 3
), and, as in (18), we get 23) is proved.

1 τ τ 0 ψ t • X u τ t -s ds → ψ t , in L 2 ([0, +∞), L 2 ), (
Remark 4.8 We remark that one can actually prove that if τ n ↓ 0 is such that (u τn t ) strongly converges to some (u t ) in

L 3 loc ([0, ∞), L 3 ), then (p τn t ) weakly converges in L 3/2 loc ([0, ∞), L 3/2 ) to the distributional solution p of ∆p t = ∇ • (u t • ∇)u t , p t dL 3 = 0,
and thus to the pressure term of the limit equation. Indeed, as in the proof of lemma 4.6, it suffices to check that, for a.e. t, one has

p τn t ∆ϕ dL 3 → p t ∆ϕ dL 3 , ∀ϕ ∈ C ∞ (T 3 ),
as n → ∞. To prove this, start from

p τn t ∆ϕ dL 3 = u τn t • 1 τ n τn 0 ∇ 2 ϕ • u τn t • X u τn t s ds dL 3 ,
(which follows as in ( 22)), and notice that from the strong convergence of u τn t to u t in L 3 it is sufficient to check that it holds

1 τ n τn 0 ∇ 2 ϕ • u τn t • X u τn t s ds → ∇ 2 ϕ • u t ,
weakly in L 3/2 . We already know that the norms are uniformly bounded, thus the conclusion follows along the same lines of the last part of the proof of Theorem 4.3 (see in particular (17) and the conclusion thereafter -in the current situation there is no integral in t), we omit the details.

Strong solutions

Here we show how from the discretization scheme discussed, one can prove two classical results concerning smooth solutions of the Navier-Stokes equations: existence for a time of order ∇u -4 L 2 , and existence for all times if u satisfies the classical smallness condition. Notice that the calculations that we do here are classical: what we want to show is that the standard approach has a natural 'discrete analogous' in our setting. We recall that the two estimates (3) hold if we seek for solutions u t of (1) such that u t dL 3 ≡ 0 for any t ≥ 0, and we can certainly do so, by the discussion preceding the latter formula.

1 2 ∇u τ 2 L 2 - 1 2 ∇u 2 L 2 + τ 2 ∆u τ 2 L 2 ≤ Cτ ∇u 3 L 2 ∆u L 2 (24)
and

1 2 ∇u τ 2 L 2 - 1 2 ∇u 2 L 2 + τ 2 ∆u τ 2 L 2 ≤ Cτ ∇u 2 L 2 ∆u 2 L 2 . (25)
Proof. Multiplying (5) by -∆u τ and integrating we get

-u τ • ∆u τ + u • ∆u τ + (u • X u -τ -u) • ∆u τ dL 3 = -τ ∆u τ 2 L 2 ,
hence after integration by parts and by Young inequality we have

1 2 ∇u τ 2 L 2 - 1 2 ∇u 2 L 2 + τ ∆u τ 2 L 2 ≤ ∆u τ L 2 u • X u -τ -u L 2 ≤ τ 2 ∆u τ 2 L 2 + 1 2τ u • X u -τ -u 2 L 2 . (26)
With the same calculations we did for (11) we get

u • X u -τ -u 2 L 2 ≤ τ 2 |∇u| 2 |u| 2 dL 3 . (27)
To get (24) we bound the right hand side as

|∇u| 2 |u| 2 dL 3 ≤ ∇u L 6 ∇u L 2 u 2 L 6 ≤ C ∆u L 2 ∇u 3 L 2 ,
and plugging this into (26) we get (24) To get (25), we bound the right hand side of (27) as

|∇u| 2 |u| 2 dL 3 ≤ ∇u 2 L 6 u L 6 u L 2 ≤ C ∆u 2 L 2 ∇u 2 L 2 ,
which gives the statement when inserted into (26). Also, there is another constant C such that if ∇u L 2 ≤ C , then the scheme converges to the smooth solution of the Navier-Stokes equations in the whole [0, ∞).

Proof. It is well known (see e.g. [START_REF] Galdi | An Introduction to the Navier-Stokes Initial Boundary Value Problem[END_REF], Theoreom 5.2 for the case of bounded, smooth open subsets of R 3 ) that if a weak solution (u t ) of Navier-Stokes belongs to

L ∞ ([0, T ], H 1 ) ∩ L 2 ([0, T ], H 2 ),
then it is smooth on [0, T ]. Hence to prove the statement it is sufficient to check that for any ε > 0, the discrete solutions are uniformly bounded both in L ∞ ([0, T ], H 1 ) and in L 2 ([0, T ], H 2 ).

Let us fix τ for the moment, and consider (24) and (25) for u = u τ iτ , for some nonnegative integer i. Adding up the inequalities (24) from i = 0 . . . n -1 we get

∇u τ nτ + τ n i=1 ∆u τ iτ 2 L 2 ≤ ∇u 2 L 2 + 2Cτ n-1 i=0 ∇u τ iτ 3 L 2 ∆u τ iτ L 2 = ∇u 2 L 2 + 2C nτ 0 ∇u τ t 3 L 2 ∆u τ t L 2 dt ≤ ∇u 2 L 2 + 2C sup t<nτ ∇u τ t 3 L 2 nτ 0 ∆u τ t L 2 dt ≤ ∇u 2 L 2 + 2C sup t<nτ ∇u τ t 3 L 2 √ nτ nτ 0 ∆u τ t 2 L 2 dt 1 2
, and therefore sup t<(n+1)τ

∇u τ t 2 + (n+1)τ 0 ∆u τ t 2 L 2 dt ≤ ∇u 2 L 2 + τ ∆u 2 L 2 + 2C sup t<nτ ∇u τ t 3 L 2 √ nτ nτ 0 ∆u τ t 2 L 2 dt 1 2 . ( 28
)
We can proceed in a similar manner for (25), obtaining sup t<(n+1)τ

∇u τ t 2 + (n+1)τ 0 ∆u τ t 2 L 2 dt ≤ ∇u 2 L 2 + τ ∆u 2 L 2 + 2C sup t<nτ ∇u τ t 2 L 2 nτ 0 ∆u τ t 2 L 2 dt. (29)
Let us fix T > 0 and suppose 0 ≤ n ≤ T τ . We define

δ τ (u) := ∇u 2 L 2 + τ
√ T }δ τ (u) ≤ 1 4C , (30)
and let λ be the smallest one:

λ = 2δ τ (u) 1 + 1 -4C min{1, √ T }δ τ (u) ≤ 2δ τ (u).
It is easily proved by induction that a n ≤ λ for any n, since a 0 ≤ δ τ (u) ≤ λ. Therefore the family

{u τ t } is bounded both in L ∞ ([0, T -ε], H 1 ) and in L 2 ([0, T -ε], H 2 ) by 2δ τ (u). For τ ≤ ∇u 2 L 2 / ∆u 2 L 2 we get δ τ (u) ≤ 2 ∇u 2 L 2
, and condition (30) reads min{1,

√ T } ∇u 2 L 2 ≤ 1 8C ,
which gives the claims.

The case of bounded and smooth open sets

All the conclusions of this paper are still true if we work on a bounded smooth open subset Ω of R 3 (weak solutions are produced in any dimension) provided we add the Dirichlet boundary condition u t ≡ 0 on ∂Ω. In this case the minimization problem of proposition 3.1 has to be solved in the set H 1 (Ω) := {v ∈ H 1 0 (Ω) : ∇ • v = 0}, and all the results follow along the same line. The only difference from the periodic case is in the proof of the L 3/2 loc ([0, +∞), L 3/2 (Ω)) bound on the pressure p τ , needed in passing to the limit when one wants to prove suitability of the solution. To this end one has to proceed in a different way, using the mixed L p,s estimates of the evolutionary Stokes problem. The space L

s,r ([0, ∞) × Ω)) := L s ([0, ∞); L r (Ω)) is equipped with the norm v L s,r := ∞ 0 v t s L r dt 1/s Lemma 5.1 Let u τ
t be a discrete solution of the Navier-Stokes equation, as in definition 3.2, for some smooth initial data u with ∇ • u = 0 and u| ∂Ω = 0. Then

u τ -u τ • X u τ -τ L s,r ≤ τ C(u) (31)
for 3/2 ≥ r ≥ 1 and s ≥ 1 such that

4 = 3 r + 2 s (32)
Proof. Denoting by r the conjugate exponent of r, we follow the proof of (11) with exponent r instead of 2, obtaining, for any smooth u

Ω |u -u • X u -τ | r dL 3 = Ω τ 0 d dt u • X u -t dt r dL 3 ≤ τ r/r τ 0 Ω |∇u • X u -t • u • X u -t | r dL 3 dt ≤ τ r Ω |∇u • u| r dL 3 . Now, T τ is a powerbounded operator in P L r (Ω), since σ(T τ ) = 1 1 + τ λ n : λ n ∈ σ(S) ⊂ (-1, 0),
and it is also analytic, i.e.

{n(T τ -1)T n τ } is bounded. ( 35
)
To prove analyticity is suffice to observe that

T τ -1 = τ S(1 -τ S) -1 := τ S τ (
v τ n = T n τ u, noticing that v τ n+1 -v τ n L r (Ω) ≤ 1 n n(T τ -1)T n τ u L r (Ω) . (37)
We therefore seek for (uniform in τ ) optimal L s N, P L r (Ω) -regularity of the discrete time parabolic equation for w

n := u n -v n w τ n+1 = T τ w τ n + f τ n , w τ 0 = 0, in the sense that +∞ n=1 w τ n+1 -w τ n s L r (Ω) ≤ C s,r ∞ n=1 f τ n s L r (Ω) . (38)
By [1, Theorem 1.1], the discrete optimal L s N, P L r (Ω) -regularity is equivalent to the optimal L s [0, +∞), L r (Ω) regularity of the analytic semigroup generated by the operator S τ given in (36). Moreover, the constant in the optimal regularity estimate depends only on the R-bound of {itR(it, S τ ) :

t ∈ R \ {0}}, (39)
and thus it suffice to prove an R-bound for this set of operators which is indipendent of τ . To this end, a short computation shows that

itR(it, S τ ) = 1 it λ 2 τ (t)R(λ τ (t), S) + τ λ τ (t)Id where λ τ (t) = it 1 + τ it .
Notice that λ(R\{0}) is the circle through the origin, with center on the real axis and radius 1/τ , minus the origin; therefore it is contained in Σ π/2 . By the subadditivity and submultiplicativity of the R-bounds, we get that the set in (39) is R-bounded by R(S) + 1, since {λ τ (t)R(λ τ (t), S) :

t ∈ R \ {0}} is R-bounded by R(S) and |λ τ (t)/t| ≤ 1, τ |λ τ (t)| ≤ 1, ∀t ∈ R.
This shows that the constant in (38) (and thus, a fortiori, the one in ( 35)) is bounded indipendently of τ . We are now ready to prove an estimate of the pressure, which allows to prove the suitability of the limit solutions, in the case of Ω bounded and smooth. Since Ω p τ dL 3 = 0, the Sobolev-Poincaré inequality p L 5/3 (Ω) ≤ C ∇p L 15/14 (Ω) applied to the latter estimate gives the claim.

Remark 5.3 (On the smoothness of the initial datum) We point out that the restriction to smooth initial data has been made to simplify the discussion about the existence of flow maps. Actually, this is unnecessary: as showed by DiPerna-Lions in [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF], as soon as a vector field u has Sobolev regularity and is divergence free, one has existence and uniqueness of a one parameter group of maps X u t such that (X u t ) # L d = L d , X 0 = Id and ∂ t X u t = u • X u t in the sense of distributions. Thus one can use these maps as flow maps and directly get a weak/suitable solution for any initial datum in H 1 . Notice, however, that in order to get the discrete estimates needed for the existence of strong solutions, it is required that the initial datum is in H 2 , although the H 2 norm actually does not appear in the final statement.

  its Helmholtz decomposition is given by u = ∇p + w, where ∇ • w = 0 and pdL d = 0. It is not hard to see that the Helmholtz decomposition is unique and is an orthogonal decomposition of L 2 . For the existence, just solve ∆p = ∇ • u, by also ensuring pdL d = 0, and define w := u -∇p.

Proposition 4 . 2 (

 42 One step estimates) With the same notation and assumptions of Proposition 3.1 it holds:

Lemma 4 . 5 (

 45 A time step) With the same notation and assumptions of Proposition 3.1 it holds

Lemma 4 . 9 (

 49 One step estimates) Let d = 3. With the same notation and assumptions of Proposition 3.1 it holds

Proposition 4 . 10 (

 410 Strong Solutions) With the same notation and assumptions of Propositon 3.1 and Definition 3.2, there exists a constant C > 0 such that for T := C ∇u 4 the discrete solutions converge to a strong solution of the Navier-Stokes equations in [0, T ).

Theorem 5 . 2 3 L

 523 Let Ω ⊂ R 3 be a smooth and bounded open set. Let u τ t , p τ t be a discrete solution of the Navier-Stokes equation, as in definition 3.2, for some smooth initial data u with ∇ • u = 0 and u| ∂Ω = 0. For any ε > 0, there exists a constant C = C(ε, u) such that for any sufficiently small τ > 0p τ L 5/3 (Ω×[ε,∞)) ≤ C(ε, u).Proof. Let r and s satisfy (32), with s > 1, and let m = [ε/τ ]. With the same notation of the preceding discussion we first of all prove that for some constant C independent of τ , it holds the L r (Ω) resolvent estimate for the Stokes operator, it holds (1 -τ S) -1 ≤ C, and thus f τ n L r (Ω) ≤ Cτ g τ n L r (Ω) . We split u τ n = w τ n + v τ n and use this estimate and (38) for w n and (41) for v n , obtaining (40). Notice that the difference equation (33) readily gives an analogous estimate for Su τ n+1 L r (Ω) ; finally we can take the L r (Ω) norm in the original difference equation∇p τ n = ∆u τ n -coercive estimate u W 2,r (Ω) ≤ C Su L r (Ω), obtain the full regularity estimate r (Ω) ,where C ε = C/ε s-1 . We now choose s = 5/3 and r = 15/14, and use lemma 5.1 in this estimate to get ∇p τ 5/3 L 5/3 ([ε,∞);L 15/14 (Ω)) ≤ τ 15/14 (Ω) ≤ C(ε, u).

By Holder inequality with exponent 2/r on ∇u, and by the interpolation inequality

for q = 2r/(2 -r), one gets

We apply this estimate for u = u τ t , integrate in time over [0, ∞), use ( 16) and (32) to get

Finally, the condition 2

To obtain a bound for the pressure p τ one can then proceed as in [START_REF] Sohr | On the regularity of the pressure of weak solutions of Navier-Stokes equations[END_REF]: one lets

where P is the projection on the closure in L r (Ω) of {u ∈ C ∞ c : ∇ • u = 0} (the dependance on r will be implicit) and u τ n = u τ (nτ ). Then u τ n solves the difference equation in P L r (Ω)

where S = P ∆ is the Stokes operator in P L r (Ω). We recall here that, (see [START_REF] Solonnikov | Estimates for solutions of nonstationary Navier-Stokes equations[END_REF], [START_REF] Giga | The Stokes operator in L r spaces[END_REF]), S generates an analytic semigroup with optimal regularity in P L r (Ω). Setting Σ θ := {λ ∈ C \ {0} : |Arg(λ)| ≤ θ}, this is equivalent, by e.g. [START_REF] Kunstmann | Maximal L p regularity fo Parabolic equations, Fourier Multiplier Theorems and H ∞ -functional calculus[END_REF]Theorem 1.11] to {λR(λ, S) : λ ∈ Σ θ } R-bounded for some θ ≥ π/2, (and thus bounded). We will henceforth fix such a θ ≥ π/2 and let R(S) be the corresponding R-bound.

We rewrite the difference equation in the standard form

where


Nicola Gigli

Kazumasa Kuwada

Heat flow on Alexandrov spaces

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L 2 -space produces the same evolution as the gradient flow of the relative entropy in the L 2 -Wasserstein space. This means that the heat flow is well defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry-Émery gradient estimates and the Γ 2 -condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift.

Introduction

The heat equation is one of the most important evolutionary PDEs. It is a well known fact in modern analysis that such an equation, say in R n , can be seen as the gradient flow of the Dirichlet energy

(1.1) 1 2 R n |∇ f | 2 dx
in the space L 2 (R n , L n ). This viewpoint has been extended to the concept of Dirichlet form (see e.g. [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF][START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF]) and it has grown up into a huge research field in potential analysis and probability theory. More recently, Jordan, Kinderlehrer and Otto [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF] understood that the same equation can be seen as the gradient flow of the relative entropy (1.2)

R n ρ log ρ dx in the L 2 -Wasserstein space (P 2 (R n ),W 2 ), where P 2 (R n ) stands for the space of Borel probability measures on R n with finite second moment. This intuition, with the further studies of Otto [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF], has been one of the fundamental ingredients that drove the research in the field of gradient flows in relation with optimal transportation problems in the past decade (see, e.g., [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Villani | Optimal Transport, old and new[END_REF]). The aim of this paper is to carry on the study of the heat flow as gradient flow of the two very different functionals, (1.1) and (1.2) in the two metric spaces L 2 (R n , L n ) and (P 2 (R n ),W 2 ), in a non-smooth setting. The point is the following. On the one hand, it is known that these two gradient flows produce the same evolution 'in all the smooth settings', i.e., it has been proved that in R n , in Riemannian manifolds with Ricci curvature bounded below and in compact Finsler manifolds (for the latter see the work of Sturm and the third author [START_REF] Ohta | Heat flow on Finsler manifolds[END_REF]), the gradient flow of the Dirichlet energy with respect to the L 2 -distance coincides with the gradient flow of the relative entropy with respect to the Wasserstein distance W 2 (see also [START_REF] Fang | Wasserstein space over the Wiener space[END_REF][START_REF] Juillet | Diffusion by optimal transport in Heisenberg groups[END_REF] for related work on different kinds of spaces). On the other hand, these flows are studied well also in some non-smooth settings. These raise the natural question: do these two notions always coincide?

We remark that a natural abstract setting where one could try to give an answer to this question is the one of metric measure spaces with Ricci curvature bounded below (defined by Sturm [START_REF] Sturm | On the geometry of metric measure spaces[END_REF] and Lott and Villani [START_REF] Lott | Ricci curvature for metric-measure spaces via optimal transport[END_REF], see Definition 2.4). Indeed, in order to define either one of the two gradient flows, one usually needs both a metric and a measure on the space considered. Furthermore, it is known that in the Riemannian setting the fact that the heat flow does not lose mass is strictly related to the bound from below on the Ricci curvature, so that one has to assume some kind of lower Ricci curvature bound also in the non-smooth setting.

In this paper, we restrict our attention to the case of finite dimensional, compact Alexandrov spaces (X, d, H n ) of curvature bounded below without boundary, equipped with the Hausdorff measure H n . An Alexandrov space of curvature bounded from below by k with k ∈ R is a metric space of sectional curvature bounded from below by k in the sense of the triangle comparison property (see Definition 2.1). As naturally expected and recently shown in [34,[START_REF] Zhang | Ricci curvature on Alexandrov spaces and rigidity theorems[END_REF], such spaces actually have Ricci curvature bounded below in the sense of Lott-Sturm-Villani (see Remark 2.5). Our following main result is the first one establishing the equivalence of the two gradient flows in a genuinely non-smooth setting (see Theorem 3.1 for the slightly more general statement): Theorem 1.1. Let (X, d, H n ) be a finite-dimensional compact Alexandrov space without boundary. Then for any f 0 ∈ L 2 (X, H n ) with f 0 H n ∈ P(X), the gradient flow ( f t ) t∈[0,∞) of the Dirichlet energy with respect to the L 2 -distance gives the gradient flow ( f t H n ) t∈[0,∞) of the relative entropy with respect to the L 2 -Wasserstein distance W 2 , and vice versa.

On Alexandrov spaces, the structure of the heat flow as gradient flow has already been studied in [START_REF] Kuwae | Sobolev spaces, Laplacian and heat kernel on Alexandrov spaces[END_REF] for the Dirichlet energy approach and in [START_REF] Ohta | Gradient flows on Wasserstein spaces over compact Alexandrov spaces[END_REF] by the third author (see also [START_REF] Gigli | First variation formula in Wasserstein spaces over compact Alexandrov spaces[END_REF][START_REF] Savaré | Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds[END_REF]) for the relative entropy approach. However, up to now it was not clear that these two notions coincide.

We observe that the idea behind our proof of the theorem is completely different from the ones used in the aforementioned smooth settings. Indeed, all the proofs available in the smooth setting had the following structure:

• one studies the gradient flow of the Dirichlet energy with respect to L 2 and writes down the equation that it satisfies; • one studies the gradient flow of the relative entropy with respect to W 2 and writes down the equation that it satisfies; • one realizes that the two equations are actually the same and calls into play the uniqueness of the solution of the equation to conclude that the two gradient flows coincide.

Our approach, instead, is completely different and in particular does not pass from the study of the heat equation in the Alexandrov setting. Read back in R n our proof gives a new and purely metric way to prove the coincidence of these two gradient flows.

After having proved the identification, we can combine them: their interaction gives fruitful applications. Among them, the Lipschitz continuity (Theorem 4.4) of the heat kernel is derived as a corollary of a Bakry-Émery type gradient estimate (Theorem 4.3), which follows with the aid of a result of the second author [START_REF] Kuwada | Duality on gradient estimates and Wasserstein controls[END_REF]. The Lipschitz regularity improves the Hölder regularity established in [START_REF] Kuwae | Sobolev spaces, Laplacian and heat kernel on Alexandrov spaces[END_REF]. We believe that it is curious and worth to be underlined that such Lipschitz property immediately follows when one knows the equivalence of the two gradient flows, but is not at all trivial if one sticks to either one of the two approaches alone. Our Lipschitz regularity enables us to deduce the Γ 2 -condition (Theorem 4.6) from a Lott-Sturm-Villani Ricci curvature bound. The Γ 2 -condition as well as the Wasserstein contraction of heat flow and the Bakry-Émery gradient estimate for heat semigroup are known to be analytic characterizations of a lower Ricci curvature bound (see [START_REF] Von Renesse | Transport inequalities, gradient estimates, entropy and Ricci curvature[END_REF][START_REF] Bakry | On Sobolev and logarithmic Sobolev inequalities for Markov semigroups[END_REF][START_REF] Ledoux | The geometry of Markov diffusion generators[END_REF]). We show the equivalence of those three analytic conditions even on Alexandrov spaces (with sharp constants, Theorem 4.8).

It should be remarked that our approach easily generalizes to the case of heat flow with drift, where the drift is given by the gradient of a semiconvex potential V . In other words, the current approach can be used to study the heat flow on weighted Alexandrov spaces of the kind (X, d, e -V H n ) (see the end of Section 4).

After having completed the work on this paper, we got aware of a paper [START_REF] Zhang | On a new definition of Ricci curvature on Alexandrov spaces[END_REF] by Zhang and Zhu where the Lipschitz continuity of the heat kernel has been studied, via a completely different argument. In particular, their proof relies on the Lipschitz continuity of harmonic functions due to Petrunin [START_REF] Petrunin | Harmonic functions on Alexandrov spaces and their applications[END_REF]. They even proceeded their study to [START_REF] Zhang | s gradient estimate on Alexandrov spaces[END_REF][START_REF] Qian | Sharp spectral gap and Li-Yau's estimate on Alexandrov spaces[END_REF], where they showed a Bochner type formula and Li-Yau estimates under their own notion of lower Ricci curvature bound which is stronger than the one of Lott-Sturm-Villani. It is not discussed in [START_REF] Zhang | On a new definition of Ricci curvature on Alexandrov spaces[END_REF][START_REF] Zhang | s gradient estimate on Alexandrov spaces[END_REF][START_REF] Qian | Sharp spectral gap and Li-Yau's estimate on Alexandrov spaces[END_REF] whether their approach generalizes to the case of weighted Alexandrov spaces.

The organization of the article is as follows. Section 2 is devoted to preliminaries for Alexandrov geometry and known results on the gradient flow of the Dirichlet energy as well as the gradient flow of the relative entropy on compact Alexandrov spaces. We prove our main theorem in Section 3, and discuss its applications in Section 4.

Preliminaries and notations

Alexandrov spaces

We first review the basics of Alexandrov geometry, see [START_REF] Burago | Alexandrov spaces with curvatures bounded below (Russian)[END_REF][START_REF] Otsu | The Riemannian structure of Alexandrov spaces[END_REF][START_REF] Burago | A course in metric geometry[END_REF] for details. Let (X, d) be a metric space. A rectifiable curve γ : [0, l] → X is called a geodesic if it is locally length minimizing and parametrized with constant speed. (Precisely, for any

t 0 ∈ [0, l], there is ε > 0 such that d(γ(s), γ(t))/|s -t| is constant for all s,t ∈ [t 0 -ε,t 0 + ε] ∩ [0, l].
) If γ is minimizing between endpoints, then we call it a minimal geodesic. We say that (X, d) is a geodesic space if any pair of points in X are connected by a minimal geodesic.

For k ∈ R, we denote by M 2 (k) the simply-connected, two-dimensional space form of constant sectional curvature k. Given three points x, y, z ∈ X, with d(x, y)

+ d(y, z) + d(z, x) < 2π/ √ k if k > 0,
we can take a comparison triangle x ỹz ⊂ M 2 (k) such that d( x, ỹ) = d(x, y), d( ỹ, z) = d(y, z) and d(z, x) = d(z, x). Such a triangle is unique up to a difference of isometry. Definition 2.1 (Alexandrov spaces). For k ∈ R, a complete geodesic space (X, d) is called an Alexandrov space of curvature bounded from below by k if, for any three points x, y, z ∈ X (with d(x, y)+d(y, z)+d(z, x) < 2π/ √ k if k > 0) and any minimal geodesic γ : [0, 1] → X from y to z, we have d(x, γ(t)) ≥ d( x, γ(t)) for all t ∈ [0, 1], where x ỹz ⊂ M 2 (k) is a comparison triangle of xyz and γ : [0, 1] → M 2 (k) is the unique minimal geodesic from ỹ to z.

Example 2.2. (a) A complete Riemannian manifold is an Alexandrov space of curvature bounded from below by k if and only if its sectional curvature is greater than or equal to k everywhere.

(b) If (X, d) is an Alexandrov space of curvature bounded from below by k, then the scaled metric space (X, c • d) with c > 0 is an Alexandrov space of curvature bounded from below by k/c 2 .

(c) For a convex domain D in the Euclidean space R n , the boundary ∂ D equipped with the length distance is an Alexandrov space of nonnegative curvature.

(d) Let (M, g) be a Riemannian manifold of nonnegative sectional curvature and G be a compact group acting on M by isometries. Then the quotient space M/G equipped with the quotient metric is an Alexandrov space of nonnegative curvature.

(e) If a sequence of Alexandrov spaces of curvature bounded from below by k is convergent with respect to the Gromov-Hausdorff distance, then its limit space is again an Alexandrov space of curvature bounded from below by k.

Fix x ∈ X and let Σx be the set of all unit speed geodesics γ : [0, l] → X with γ(0) = x. For γ, η ∈ Σx , thanks to the curvature bound, the joint limit

∠ x (γ, η) := arccos lim s,t→0 s 2 + t 2 -d(γ(s), η(t)) 2 2st
exists and is a distance on Σx /∼ where γ ∼ η holds if ∠ x (γ, η) = 0. We define the space of directions (Σ x , ∠ x ) at x as the completion of Σx /∼ with respect to ∠ x . The tangent cone (K x , d) is the Euclidean cone over (Σ x , ∠ x ), i.e.,

K x := Σ x × [0, ∞)/Σ x × {0}, d (γ, s), (η,t) := s 2 + t 2 -2st cos ∠ x (γ, η).
The inner product on K x is defined by (γ, s), (η,t) x := st cos ∠ x (γ, η). In Riemannian manifolds, spaces of directions and tangent cones correspond to unit tangent spheres and tangent spaces, respectively. The Hausdorff dimension of X is an integer or infinity. From here on, we consider a compact n-dimensional Alexandrov space of curvature bounded from below by k without boundary equipped with the n-dimensional Hausdorff measure H n (see [START_REF] Burago | Alexandrov spaces with curvatures bounded below (Russian)[END_REF] for the definition of boundary of Alexandrov spaces). We say that x ∈ X is a singular point if K x is not isometric to the Euclidean space R n , and denote the set of singular points by S X . We remark that H n (S X ) = 0 holds whereas S X can be dense in X (see [START_REF] Burago | Alexandrov spaces with curvatures bounded below (Russian)[END_REF][START_REF] Otsu | The Riemannian structure of Alexandrov spaces[END_REF]).

Dirichlet energy and the associated gradient flow

We introduce the Sobolev space and the Dirichlet energy following [START_REF] Kuwae | Sobolev spaces, Laplacian and heat kernel on Alexandrov spaces[END_REF], and will see that it coincides with other notions of Sobolev spaces. We begin by discussing a C 1 -differentiable structure of the set of regular points X \ S X established in [START_REF] Burago | Alexandrov spaces with curvatures bounded below (Russian)[END_REF][START_REF] Otsu | The Riemannian structure of Alexandrov spaces[END_REF]. We remark that Perelman extends this to DC 1 -structure (via 'difference of concave functions', see [START_REF] Perelman | DC structure on Alexandrov space with curvature bounded below[END_REF]), but the C 1 -structure is enough for considering the Sobolev space. There is a weak

C 1 -atlas {(U φ ,V φ , φ )} φ ∈Φ in the sense that U φ ⊂ X is an open set, φ : U φ → R n is a bi-Lipschitz embedding, V φ ⊂ U φ with φ ∈Φ V φ ⊃ X \ S X , and that the coordinate change φ 2 • φ -1 1 is C 1 on φ 1 (V φ 1 ∩ V φ 2 ∩ (X \ S X )) if V φ 1 ∩ V φ 2 = / 0 ([30, Theorem 4.2(1)]
). Such charts are constructed through the distance function. Precisely, φ is introduced as φ (x) := (d(x, p 1 ), d(x, p 2 ), . . . , d(x, p n )) for suitable p 1 , p 2 , . . . , p n ∈ X, and then V φ is chosen as the set of regular points x such that a minimal geodesic between x and p i is unique for all i ([30, Section 3]). It is worth mentioning that, for any ε > 0, φ can be (1 + ε)-bi-Lipschitz by taking smaller U φ (cf. [START_REF] Burago | A course in metric geometry[END_REF]Theorem 10.9.16]). We also remark that it is possible to modify φ by taking an average so as to satisfy V φ = U φ ([30, Section 5]), but it is unnecessary for our discussion (just like [5, Remark 2.9]).

We say that a function f on X is differentiable at a regular point x ∈ X \ S X if x ∈ V φ and f • φ -1 is differentiable at φ (x) for some φ ∈ Φ. Then we can define the gradient vector ∇ f (x) ∈ K x by identifying K x and R n through φ . (To be precise, by virtue of the first variation formula ([30, Theorem 3.5]), each d i := d(•, p i ) is differentiable at x with ∇d i (x) = -v i , where v i ∈ K x is the tangent vector of the unique, minimal, unit speed geodesic from x to p i . Then K x is linearly identified with R n as ∑ i a i v i = (-a i ).) Moreover, again due to the first variation formula, we obtain the Taylor expansion

(2.1) f γ(t) = f (x) + t ∇ f (x), γ(0) x + o x (t),
where γ : [0, δ ] → X is a minimal geodesic emanating from x and o x (t) is independent of the choice of γ (see [START_REF] Bertrand | Existence and uniqueness of optimal maps on Alexandrov spaces[END_REF]Lemma 3.4], and note that the remainder term in the first variation formula for d(•, p i ) at x indeed depends only on x and p i ).

Another important fact we will use is the Rademacher theorem, namely a Lipschitz function f on X is differentiable H n -a.e.. This easily follows from the usual Rademacher theorem for f

• φ -1 (see [5, Corollary 2.14]). It follows from (2.1) that ∇ f (x), ∇ f (x) coincides with the local Lipschitz constant |∇ L f |(x) given by |∇ L f |(x) := lim y→x | f (x) -f (y)| d(x, y) .
Based on the notion of gradient vector, we define the Sobolev space and the Dirichlet energy as follows (see [START_REF] Kuwae | Sobolev spaces, Laplacian and heat kernel on Alexandrov spaces[END_REF][START_REF] Kuwae | Sobolev spaces and Dirichlet spaces over maps between metric spaces[END_REF] for details). For a function f :

X → R such that f • φ -1 ∈ W 1,2 (φ (U φ )
) for all φ ∈ Φ, we introduce the weak gradient vector ∇ f (x) ∈ K x for a.e. x ∈ V φ as the element corresponding to the weak gradient vector ∇( f • φ -1 )(φ (x)). We define the Sobolev space W 1,2 (X) and the Dirichlet energy

E by W 1,2 (X) := f ∈ L 2 (X, H n ) X ∇ f , ∇ f dH n < ∞ , E ( f , g) := X ∇ f , ∇g dH n for f , g ∈ W 1,2 (X).
We do not divide E by 2 for notational simplicity. Note that E coincides with the energy functional introduced by Korevaar and Schoen [START_REF] Korevaar | Sobolev spaces and harmonic maps for metric space targets[END_REF] (We can reduce the argument to the Euclidean case by using a (1 + ε)-bi-Lipschitz chart; see [22, Theorem 6.2]). We also remark that the set of Lipschitz functions

C Lip (X) is dense in W 1,2 (X) with respect to the Sobolev norm f 2 W 1,2 = f 2 L 2 + E ( f , f ) ([20, Theo- rem 1.1]).
Furthermore, if f is a Lipschitz function, then the weak gradient vector ∇ f (x) coincides with the gradient vector as in (2.1) a.e. x, and hence

∇ f (x), ∇ f (x) = |∇ L f |(x) holds a.e.
x. Therefore E also coincides with Cheeger's energy functional ( [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF]), because the local Lipschitz constant is the minimal generalized upper gradient ([9, Theorem 6.1]) and Lipschitz functions are dense in both Sobolev spaces (thanks to the weak Poincaré inequality for upper gradients and the volume doubling condition, [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF]Theorem 4.24]). Indeed, in our framework, the volume doubling condition directly follows from the Bishop-Gromov volume comparison theorem and the Poincaré inequality is a consequence of [START_REF] Kuwae | On generalized measure contraction property and energy functionals over Lipschitz maps[END_REF][START_REF] Ranjbar-Motlagh | Poincaré inequality for abstract spaces[END_REF], for instance.

By following the general theory of bilinear forms, there exists a nonpositive selfadjoint operator (∆, D(∆)) on L 2 (X, H n ) associated with (E ,W 1,2 (X)). It is characterized by the following identity:

(2.2) E (g, f ) = - X g∆ f dH n , f ∈ D(∆), g ∈ W 1,2 (X).
We call ∆ the Laplacian as in the classical case. Based on a general theory of functional analysis, the one-parameter semigroup of contractive symmetric linear operators

T t = e t∆ on L 2 (X, H n ) is defined associated with ∆. For any f ∈ L 2 (X, H n ), T t f solves the (linear) heat equation ∂ t u = ∆u with u(0, •) = f in the sense that T t f ∈ D(∆) for t > 0 and lim t↓0 T t f -f t = ∆ f in L 2 (X, H n ) for f ∈ D(∆), lim t↓0 T t f = f in L 2 (X, H n ) for f ∈ L 2 (X, H n ).
Note that T t is Markovian in the sense that 0 ≤ T t f ≤ 1 holds whenever 0 ≤ f ≤ 1. As shown in [20, Theorem 1.5], there exists a continuous function (t, x, y) → p t (x, y) on (0, ∞) × X × X satisfying the following properties:

(i) For any f ∈ L 2 (X, H n ), t > 0 and H n -a.e. x ∈ X,

(2.3) T t f (x) = X p t (x, y) f (y) H n (dy).
(ii) For any s,t > 0 and x, y ∈ X, p t (x, y) = p t (y, x), (2.4)

p s+t (x, y) = X p s (x, z)p t (z, y) H n (dz), (2.5) X p t (x, z) H n (dz) = 1, (2.6) p t (x, y) > 0.
See [20, Theorems 1.4, 1.5(3)] for the continuity of p t (x, y). The equality (2.6) follows from the fact that 1 ∈ W 1,2 (X) and E (1, 1) = 0, because X is assumed to be compact. As in the classical case, we call p t (x, y) the heat kernel. The existence and these properties of p t (x, y) are deduced from the Poincaré inequality for E and the volume doubling condition, together with results in [START_REF] Sturm | Analysis on local Dirichlet spaces.III. The parabolic Harnack inequality[END_REF].

Remark 2.3. The bilinear form (E ,W 1,2 (X)) is a symmetric strongly local regular Dirichlet form (see [START_REF] Kuwae | Sobolev spaces, Laplacian and heat kernel on Alexandrov spaces[END_REF] for it and further details; see [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF][START_REF] Fukushima | Dirichlet forms and symmetric Markov processes[END_REF] for basics on Dirichlet forms). Moreover, T t enjoys the strong Feller property. As a result, there exists a diffusion process ((X t ) t≥0 , (P x ) x∈X ) on the whole space X associated with

(E ,W 1,2 (X)) in the sense that E x [ f (X t )] = T t f (x) for f ∈ C(X), x ∈ X and t > 0.
Denote by P(X) the set of all Borel probability measures on X. Let us define a positive Borel measure T t ν for ν ∈ P(X) and t ≥ 0 by

T t ν(dy) :=    X p t (x, y) ν(dx) H n (dy) t > 0, ν(dy) t = 0.
Thanks to (2.6), T t ν ∈ P(X) holds. By definition, T t ν is absolutely continuous with respect to H n for t > 0. When dν = f dH n , it holds dT t ν = T t f dH n . In this paper, we call the evolution (t, ν) → T t ν the gradient flow of the Dirichlet energy (since the Dirichlet energy is a functional on the L 2 -space of functions, this terminology should be interpreted in an extended sense). Indeed, it is easy to see from

1 2 E ( f + εg, f + εg) = 1 2 E ( f , f ) -ε X g∆ f dH n + O(ε 2 )
that the Radon-Nikodym derivative dT t ν/dH n is the gradient flow of E /2 with respect to the L 2 -norm. Before closing this subsection, we review the derivation property of W 1,2 (X). It is formulated as follows: For

f 1 , . . . , f k , g ∈ W 1,2 (X) ∩ L ∞ (X, H n ) and Φ : R k → R which is C 1 on the range of ( f 1 , . . . , f k ), Φ( f 1 , . . . , f k ) belongs to W 1,2 (X) and (2.7) ∇Φ( f 1 , . . . , f k ), ∇g = k ∑ j=1 ∂ Φ ∂ x j ( f 1 , . . . , f k ) ∇ f j , ∇g H n -a.e..
This identity directly follows from the definition of W 1,2 (X) or from the strong locality of the Dirichlet form (E ,W

Gradient flows in the Wasserstein space

We next introduce the Wasserstein space and a purely metric notion of gradient flows in it. We refer to [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Villani | Optimal Transport, old and new[END_REF] for the basic theory as well as the recent diverse developments.

Given µ, ν ∈ P(X), a probability measure

π ∈ P(X × X) is called a coupling of µ and ν if π(A × X) = µ(A) and π(X × A) = ν(A) hold for all Borel sets A ⊂ X.
Then, for 1 ≤ p < ∞, we define the L p -Wasserstein distance as

W p (µ, ν) := inf π X×X d(x, y) p π(dxdy) 1/p
, where π runs over all couplings of µ and ν. In most parts, we work in the quadratic case p = 2. The L 2 -Wasserstein space (P(X),W 2 ) becomes a metric space and inherits the compactness from (X, d). Moreover, (P(X),W 2 ) is a geodesic space. If µ is absolutely continuous with respect to H n , then a minimal geodesic (µ t ) t∈[0,1] from µ to any ν is unique and µ t is also absolutely continuous for all t ∈ (0, 1) (see [START_REF] Bertrand | Existence and uniqueness of optimal maps on Alexandrov spaces[END_REF] for a more detailed characterization of µ t , and [START_REF] Figalli | Absolute continuity of Wasserstein geodesics in the Heisenberg group[END_REF] for the absolute continuity). For µ ∈ P(X), we define the relative entropy by Ent(µ) := X ρ log ρ dH n when µ = ρH n with ρ ∈ L 1 (X, H n ), and Ent(µ) := ∞ otherwise. Set P * (X) := {µ ∈ P(X) | Ent(µ) < ∞}. Note that Ent is lower semi-continuous with respect to W 2 and satisfies Ent(µ) ≥log H n (X) by Jensen's inequality.

Definition 2.4 (The curvature-dimension condition). For K ∈ R, we say that (X, d, H n ) satisfies the curvature-dimension condition CD(K, ∞) if Ent is K-geodesically convex in the sense that any pair µ, ν ∈ P(X) admits a minimal geodesic

(µ t ) t∈[0,1] from µ to ν such that Ent(µ t ) ≤ (1 -t) Ent(µ) + t Ent(ν) - K 2 (1 -t)tW 2 (µ, ν) 2
holds for all t ∈ [0, 1].

We remark that the above inequality is obvious if µ ∈ P * (X) or ν ∈ P * (X). Therefore it is sufficient to consider µ, ν ∈ P * (X), and then a minimal geodesic between them is unique.

Remark 2.5. (i) The curvature-dimension condition CD(K, ∞) for general metric measure spaces is introduced and studied independently in [START_REF] Sturm | On the geometry of metric measure spaces[END_REF][START_REF] Lott | Ricci curvature for metric-measure spaces via optimal transport[END_REF], and known to be equivalent to the lower Ricci curvature bound Ric ≥ K for complete Riemannian manifolds equipped with the Riemannian distance and the volume measure ( [START_REF] Von Renesse | Transport inequalities, gradient estimates, entropy and Ricci curvature[END_REF]).

(ii) The condition CD(K, N) for N ∈ (1, ∞) is also introduced in [START_REF] Sturm | On the geometry of metric measure spaces[END_REF][START_REF] Lott | Weak curvature conditions and functional inequalities[END_REF]. In general, CD(K, N) implies CD(K, ∞). In the Riemannian case, CD(K, N) is equivalent to Ric ≥ K and dim ≤ N.

(iii) It is recently demonstrated in [34,[START_REF] Zhang | Ricci curvature on Alexandrov spaces and rigidity theorems[END_REF] that n-dimensional Alexandrov spaces of curvature bounded from below by k satisfy CD((n -1)k, n) (and hence CD((n -1)k, ∞)), as is naturally expected from the relation between the sectional and the Ricci curvatures.

There is a well established theory on the gradient flow of geodesically convex functions as comprehensively discussed in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]. For later convenience, we recall a couple of notions in a general form. We say that a curve (µ t ) t∈I ⊂ P(X) on an interval Definition 2.6 (Gradient flows). Let E : P(X) → R ∪ {+∞} be a K-geodesically convex functional. We say that an absolutely continuous curve

I ⊂ R is absolutely continuous if there is f ∈ L 1 (I) such that (2.8) W 2 (µ t , µ s ) ≤ s t f (
(µ t ) t∈[0,∞) in P(X) is a gradient flow of E provided E(µ t ) < ∞ for t ≥ 0 and
(2.12)

E(µ t ) = E(µ s ) + 1 2 s t | μr | 2 dr + 1 2 s t |∇ -E| 2 (µ r ) dr
for all 0 ≤ t < s. The equation (2.12) is called the energy dissipation identity.

The existence of such a gradient flow comes from the general theory presented in [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Corollary 2.4.11]. Furthermore, as shown in [27, Theorem 6.2] (see also [START_REF] Gigli | First variation formula in Wasserstein spaces over compact Alexandrov spaces[END_REF]Theorem 4.2]), the gradient flow produces a contraction semigroup in the sense that for any µ, ν ∈ P(X) with E(µ) < ∞ and E(ν) < ∞, the gradient flows

(µ t ) t∈[0,∞) , (ν t ) t∈[0,∞) starting from µ, ν satisfy (2.13) W 2 (µ t , ν t ) ≤ e -Kt W 2 (µ, ν) for all t ≥ 0,
where K is the modulus of convexity of E. In particular, the uniqueness follows from considering µ = ν. (Though the strategy of the construction in [START_REF] Ohta | Gradient flows on Wasserstein spaces over compact Alexandrov spaces[END_REF] is different from [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], the resulting flow is the same by uniqueness, see [ (ii) The first author [START_REF] Gigli | On the heat flow on metric measure spaces: existence, uniqueness and stability[END_REF]Theorem 15] proved the uniqueness of the gradient flow of the relative entropy on general metric measure spaces satisfying CD(K, ∞), without relying on the contractivity. In this generality, the contractivity fails. More precisely, the heat flow on a finite-dimensional (Minkowski) normed space is not contractive (except for inner product spaces; see [START_REF] Ohta | Non-contraction of heat flow on Minkowski spaces[END_REF]).

In the sequel, we mainly study the gradient flow of the relative entropy. Then (2.13) allows us to continuously and uniquely extend the gradient flow semigroup to the full P(X) (since the closure of P * (X) is P(X)). Such an extension also satisfies (2.13).

Identification of the two gradient flows

This section contains our main result: Theorem 3.1 (Identification of the two gradient flows). Let (X, d) be a compact n-dimensional Alexandrov space without boundary. For any ν ∈ P(X), the gradient flow of the Dirichlet energy starting from ν is the gradient flow of the relative entropy, and vice versa.

Recall that CD(K, ∞) holds with K = (n -1)k, and we use this, e.g., in the proof of Proposition 3.2 below. The main technical obstacle in the proof of this theorem is to let the L 2 and the W 2 structures 'interact'. Our strategy consists in picking a gradient flow of the Dirichlet energy and in proving that it obeys the energy dissipation identity (2.12) for the relative entropy. We start with a bound on the local slope (2.9) by the Fisher information.

Proposition 3.2 (Bound on the local slope)

. Let µ = f H n ∈ P * (X) with f ∈ W 1,2 (X). Then it holds |∇ -Ent | 2 (µ) ≤ X ∇ f , ∇ f f dH n .
Proof. We first assume that f is Lipschitz and bounded away from 0. Then we know that ∇ f , ∇ f = |∇ L f | 2 H n -a.e., so that the conclusion follows from [42, Theorem 20.1] together with CD(K, ∞). Thus all we need to do is to proceed by approximation. Suppose that 0 < c ≤ f H n -a.e. for some c ∈ R. Since C Lip (X) is dense in W 1,2 (X) and √ f ∈ W 1,2 (X), we can find a sequence {g i } i∈N of Lipschitz functions such that g i converges to √ f as i → ∞ with respect to the Sobolev norm. Substituting C i max{g i , √ c} for some C i > 0 if necessary, we can assume that √ c i ≤ g i for some c i > 0 H n -a.e. as well as g i L 2 = 1 for all i. Set f i := g 2 i . As f i is Lipschitz and bounded away from 0, we have

|∇ -Ent | 2 ( f i H n ) ≤ X ∇ f i , ∇ f i f i dH n .
On the one hand, the right-hand side is equal to 4E (g i , g i ) and converges to 4E

( √ f , √ f ) = X ∇ f , ∇ f / f
|∇ -Ent |( f H n ) ≤ lim i→∞ |∇ -Ent |( f i H n ).
Combining these, we prove the claim for f . It remains to remove the assumption that f is bounded away from 0. To do this we just consider f i := (1 + i -1 ) -1 ( f + i -1 ) and apply (3.1) again.

Let us turn to considering the gradient flow of the Dirichlet energy (T t ν) t∈[0,∞) . For simplicity of notations, we write f t = dT t ν/dH n in the remainder of the section. By (2.3), f t ∈ C(X) ⊂ L 2 (X, H n ) holds for t > 0. Then (2.5), (2.4) and (2.3) imply f t ∈ D(∆). For any t > 0, there is ε t > 0 such that f t ≥ ε t holds since p t (x, y) is positive and continuous. As a well-known fact, the following bound also holds.

Lemma 3.3 (Maximum principle). Let f 0 ∈ L 2 (X, H n ). If f 0 ≥ c a.
e., then T t f 0 ≥ c a.e. for every t ≥ 0. The same holds for bounds from above.

Proof. Take A ⊂ X measurable. Since T t is Markovian, T t 1 A ≥ 0 H n -a.e.. Thus

A (T t f 0 -c) dH n = A T t ( f 0 -c) dH n = X T t 1 A • ( f 0 -c) dH n ≥ 0.
Since A is arbitrary, the assertion holds. Bounds from above follow by applying the same argument tof 0 .

By virtue of Lemma 3.3 with the remark before it, for any ε > 0, there are c,C > 0 such that

(3.2) c ≤ f t ≤ C for all t ≥ ε.
Proposition 3.4 (Entropy dissipation). Let ν ∈ P(X) and f t = dT t ν/dH n . Then the function t → Ent( f t H n ) is locally Lipschitz in (0, ∞) and, moreover, it holds

d dt Ent( f t H n ) = - X ∇ f t ,
X ∆ f t dH n = 0, we obtain d dt Ent( f t H n ) = X e ( f t )∆ f t dH n = X log( f t ) + 1 ∆ f t dH n = - X ∇ log( f t ), ∇ f t dH n = - X ∇ f t , ∇ f t f t dH n .
For the next argument, we briefly recall some properties of the Hamilton-Jacobi semigroup in our context. For f ∈ C Lip (X) and t > 0, we define

Q t f : X → R by (3.3) Q t f (x) := inf y∈X f (y) + d 2 (x, y) 2t .
Also, set Q 0 f := f . The following is shown in [START_REF] Balogh | Functional inequalities and Hamilton-Jacobi equations in geodesic spaces[END_REF]Theorem 2.5(iv)] and [25, Theorem 2.5(viii)] in the framework of general metric measure spaces supporting the volume doubling condition and the Poincaré inequality for upper gradients.

Proposition 3.5 (Hamilton-Jacobi semigroup). It holds Q t f ∈ C Lip (X) for every t ≥ 0, the map [0, ∞) t → Q t f ∈ C(X)
is Lipschitz in the uniform norm, and

(3.4) d dt Q t f (x) + |∇ L Q t f | 2 (x) 2 = 0,
for almost every t, x.

Remark 3.6. The equation (3.3) has been called the Hopf-Lax formula or the Moreau-Yosida approximation also in the literature. The former name is mainly used in the PDE context to a special solution to the Hamilton-Jacobi equation (3.4). The latter one is typically used for an approximation of an (unbounded) operator in functional analysis.

Proposition 3.7 (Absolute continuity with respect to W 2 ). For ν ∈ P(X), the curve t → T t ν is absolutely continuous in the space (P(X),W 2 ) and its metric speed | Ṫ t ν| satisfies

(3.5) | Ṫ t ν| 2 ≤ X ∇ f t , ∇ f t f t dH n , for a.e. t.
Proof. Fix t, s > 0. By the Kantorovich duality (cf., e.g., [1, Theorem 6.1.1] and [42, Theorem 5.10]) together with (3.3), we obtain

(3.6) 1 2 W 2 2 T t ν, T t+s ν = sup ϕ∈C Lip (X) X (Q 1 ϕ) f t+s dH n - X ϕ f t dH n .
By Proposition 3.5, the map r → Q r ϕ from [0, 1] to L 2 (X, H n ) is Lipschitz. Moreover, it is differentiable in L 2 (X, H n ) and the derivative is determined by (3.4).

The curve

[0, 1] r → f t+rs ∈ L 2 (X, H n ) is Lipschitz as well. Thus [0, 1] r → (Q r ϕ) f t+rs ∈ L 1 (X, H n )
is Lipschitz and its derivative can be calculated with the Leibniz rule. Thus we have

X (Q 1 ϕ) f t+s dH n - X ϕ f t dH n (3.7) = 1 0 d dr X (Q r ϕ) f t+rs dH n dr = 1 0 X - |∇ L Q r ϕ| 2 2 f t+rs + s(Q r ϕ)∆ f t+rs dH n dr.
Using formulas (2.2), (3.2) and the trivial inequality

-∇g, ∇ g ≤ 1 2s ∇g, ∇g + s 2 ∇ g, ∇ g , H n -a.e. for s > 0,
we have

X (Q r ϕ)∆ f t+rs dH n = - X ∇Q r ϕ, ∇ f t+rs dH n = - X ∇Q r ϕ, ∇ f t+rs f t+rs f t+rs dH n ≤ 1 2s X ∇Q r ϕ, ∇Q r ϕ f t+rs dH n + s 2 X ∇ f t+rs , ∇ f t+rs f t+rs dH n .
Plugging this inequality in (3.7) and recalling that ∇Q r ϕ,

∇Q r ϕ = |∇ L Q r ϕ| 2 H n -a.e. (since Q r ϕ is Lipschitz), we obtain X (Q 1 ϕ) f t+s dH n - X ϕ f t dH n ≤ s 2 2 1 0 X ∇ f t+rs , ∇ f t+rs f t+rs dH n dr.
This bound does not depend on ϕ, thus from (3.6) we deduce

(3.8) W 2 2 T t ν, T t+s ν ≤ s 2 1 0 X ∇ f t+rs , ∇ f t+rs f t+rs dH n dr.
Since we have (3.2) and the Dirichlet energy decreases along the flow t → f t , we obtain

W 2 2 T t ν, T t+s ν ≤ s 2 c 1 0 X ∇ f t+rs , ∇ f t+rs dH n dr ≤ s 2 c X ∇ f t , ∇ f t dH n ,
which gives that the map t → T t ν ∈ P(X) is locally Lipschitz. The bound (3.5) follows directly from (3.8) (by recalling the definition of the absolutely continuous curves (2.8)).

Now we are ready to prove Theorem 3.1.

Proof. of Theorem 3.1. By Propositions 3.7, 3.2 and 3.4, we have

1 2 | ḟt H n | 2 + 1 2 |∇ -Ent | 2 ( f t H n ) ≤ X ∇ f t , ∇ f t f t dH n = - d dt Ent( f t H n )
a.e. t. As the reverse inequality (2.11) is always true, equality (2.12) holds for all 0 < t ≤ s. Since ( f t H n ) t∈[0,∞) is continuous also at t = 0, it is the gradient flow of the relative entropy. The converse immediately follows from the uniqueness of the gradient flow of the relative entropy.

Remark 3.8. (i) When ν = f 0 H n with f 0 ∈ L 2 (X, H n ),
we can give a proof of Theorem 3.1 without relying on the positivity improving property (3.2). Indeed, by virtue of the contraction property of both flows, it is possible to use Propositions 3.4, 3.7 only for 0 < c ≤ f 0 ≤ C and prove the theorem via approximation. It suggests that our argument possibly works in a more general framework where the existence of the density p t (x, y) does not follow from the theory of Dirichlet forms.

In such a case, we could 'construct' p t (x, •) as the gradient flow starting from the Dirac measure δ x ∈ P(X) (via the contraction property).

(ii) In the proof of Theorem 3.1, we heavily rely on the fact

∇ f , ∇ f = |∇ L f | 2 for f ∈ C Lip (X)

Applications

In this section, we assume that (X, d) is a compact Alexandrov space without boundary satisfying CD(K, ∞), and prove some applications of Theorem 3.1. It should be stressed that the 'sectional curvature bound' k in the sense of drov appears nowhere in the sequel, and the 'Ricci curvature bound' K is essential instead (recall Remark 2.5). Indeed, those results involving K are natural extensions of the corresponding ones on a Riemannian manifold with Ric ≥ K. In this sense, the emergence of K instead of k is natural and gives sharper estimates.

Since the gradient flow of the Dirichlet energy T t ν is obviously linear and symmetric, we immediately obtain the following: Theorem 4.1 (Linearity and symmetry). For ν ∈ P(X), let (µ ν t ) t≥0 be the gradient flow of the relative entropy on P(X) with µ ν 0 = ν. Then the following hold true.

(i) For ν 0 , ν 1 ∈ P(X), λ ∈ [0, 1] and t ≥ 0, µ (1-λ )ν 0 +λ ν 1 t = (1 -λ )µ ν 0 t + λ µ ν 1 t . (ii) For f , g ∈ L 1 (X, H n ) with f , g ≥ 0 and f L 1 = g L 1 = 1, X f dµ gH n t = X g dµ f H n t .
We remark that the linearity, but not symmetry, also follows from the gradient flow approach under the local angle condition ( [START_REF] Savaré | Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds[END_REF]Theorem 8]). In general, these properties are completely nontrivial, and the linearity indeed fails in the Finsler setting ( [START_REF] Ohta | Heat flow on Finsler manifolds[END_REF]).

A new property for (T t ν) t≥0,ν∈P(X) coming from our identification with the gradient flow of the relative entropy is the L 2 -Wasserstein contraction (2.13). Together with [START_REF] Kuwada | Duality on gradient estimates and Wasserstein controls[END_REF]Corollary 3.4], we obtain the following:

Theorem 4.2 (Contraction for the heat flow). For 1 ≤ p ≤ 2, (4.1) W p (T t ν 0 , T t ν 1 ) ≤ e -Kt W p (ν 0 , ν 1 )
holds for every ν 0 , ν 1 ∈ P(X).

Furthermore, by the duality result [19, Theorem 2.2], Theorem 4.2 yields the following Bakry-Émery type L 2 -gradient estimate

(4.2) |∇ L T t f |(x) ≤ e -Kt T t (|∇ L f | 2 )(x) 1/2
for any f ∈ C Lip (X) and x ∈ X. By combining (4.2) with the regularity of the heat kernel, we can extend it to f ∈ W 1,2 (X) as follows:

Theorem 4.3 (Gradient estimate). Let f ∈ W 1,2 (X) and t > 0. Then T t f ∈ C Lip (X) and
(4.3) |∇ L T t f |(x) ≤ e -Kt T t (|∇ f | 2 )(x) 1/2
holds for all x ∈ X. In particular,

(4.4) |∇T t f |(x) ≤ e -Kt T t (|∇ f | 2 )(x) 1/2 for a.e. x
and

|∇ L T t f | ≤ e -Kt T t L 1 →L ∞ E ( f , f ) hold. Proof. Take { f i } i∈N ⊂ C Lip (X) such that f i → f in W 1,2 (X) as i → ∞. Then (4.2) yields T t f i ∈ C Lip (X).
Let y ∈ X and γ : [0, l] → X a unit speed minimal geodesic from x to y. Since |∇ L T t f i | is an upper gradient of T t f i (see [START_REF] Cheeger | Differentiability of Lipschitz functions on metric measure spaces[END_REF]Proposition 1.11] for instance), we have

|T t f i (y) -T t f i (x)| ≤ l 0 |∇ L T t f i |(γ(s)) ds (4.5) ≤ e -Kt l 0 T t (|∇ L f i | 2 )(γ(s)) 1/2 ds,
where the second inequality follows from (4.2). Thanks to (2.3) and the boundedness of p t , T t f i converges pointwisely to

T t f as i → ∞. Since T t (|∇ L f i | 2 )(z) = T t (|∇ f i | 2 )(z) for z ∈ X, T t (|∇ L f i | 2 ) converges pointwisely to T t (|∇ f | 2
) in a similar manner. Thus, by letting i → ∞ in (4.5), we obtain

(4.6) |T t f (x) -T t f (y)| ≤ e -Kt l 0 T t (|∇ f | 2 )(γ(s)) 1/2 ds.
The boundedness of p t yields that there is C > 0 satisfying (4.7)

T t g(z) ≤ C g L 1 (X,H n )
for all g ∈ L 1 (X, H n ) and z ∈ X. Since l = d(x, y), the estimate (4.7) for g = |∇ f | 2 together with (4.6) implies T t f ∈ C Lip (X). In order to show (4.3), choose a sequence {y i } i∈N in X so that y i → x as i → ∞ and

lim i→∞ |T t f (x) -T t f (y i )| d(x, y i ) = |∇ L T t f |(x).
By the continuity of p t , T t (|∇ f | 2 ) ∈ C(X) holds. Thus, applying (4.6) for y = y i , dividing both sides of it by d(x, y i ) and letting i → ∞ yield (4.3).

As an easy but important consequence of Theorem 4.3, we obtain the Lipschitz continuity of the heat kernel p t (x, y) as well as that of eigenfunctions. Recall that -∆ has discrete spectrum consisting of nonnegative eigenvalues with finite multiplicity ([20, Corollary 1.1]). (i) For ν ∈ P(X) and t > 0, let f t = dT t ν/dH n . Then f t ∈ C Lip (X) for t > 0. In particular, we have p t (x, •) ∈ C Lip (X) and T t f ∈ C Lip (X) for all x ∈ X and f ∈ L 1 (X, H n ). (ii) Let f be an L 2 -eigenfunction of ∆ corresponding to an eigenvalue -λ < 0.

Then f ∈ C Lip (X). Moreover,

|∇ L f | ≤ e (λ -K)t λ T t L 1 →L ∞ f L 2 (X,H n )
holds for each t > 0.

Proof. Since f t ∈ W 1,2 (X), Theorem 4.3 yields that f t = T t/2 f t/2 ∈ C Lip (X). For the second assertion, note that f = e λt T t f . Then the first assertion and E ( f , f ) = λ f 2 L 2 (X,H n ) yield the conclusion. Remark 4.5. (i) It has been known that the heat kernel p t (x, y) is (locally) Hölder continuous of some fractional exponent, that follows from the parabolic Harnack inequality shown in [START_REF] Kuwae | Sobolev spaces, Laplacian and heat kernel on Alexandrov spaces[END_REF].

(ii) The existence and the continuity of p t are used in the proof of Theorems 4.3, 4.4 in an essential way (cf. Remark 3.8).

(iii) To obtain a useful estimate of |∇ L T t f | along our argument, we need a nice bound for T t L 1 →L ∞ . For instance, the parabolic Harnack inequality, or the Nash inequality, implies (4.8)

p t (x, y) ≤ C H n (B √ t (x)
) with some constant C > 0 being independent of t, x, y, for small t. It gives a bound for T t L 1 →L ∞ . By a general argument, (4.8) follows from the local Poincaré inequality and the volume doubling condition, both of which depend only on the dimension n of X and a lower curvature bound (see [START_REF] Kuwae | Sobolev spaces, Laplacian and heat kernel on Alexandrov spaces[END_REF][START_REF] Kuwae | On generalized measure contraction property and energy functionals over Lipschitz maps[END_REF][START_REF] Ranjbar-Motlagh | Poincaré inequality for abstract spaces[END_REF][START_REF] Sturm | Analysis on local Dirichlet spaces.III. The parabolic Harnack inequality[END_REF] for instance). However, we should be careful if we want to know whether C in (4.8) depends on the diameter and/or the volume of X. Indeed, estimates of type (4.8) are mainly studied on non-compact state spaces and hence they did not seem to pay so much attentions on such a dependency in the literature.

In what follows, we consider two additional applications of the Bakry-Émery gradient estimate (4.4) by employing the Lipschitz continuity of T t f . The first one is the following inequality:

Theorem 4.6 (Γ 2 -condition). Let f ∈ D(∆) with ∆ f ∈ W 1,2 (X). Then, for g ∈ D(∆) ∩ L ∞ (X, H n ) with g ≥ 0 and ∆g ∈ L ∞ (X, H n ), we have (4.9) 1 2 X ∆g ∇ f , ∇ f dH n - X g ∇∆ f , ∇ f dH n ≥ K X g ∇ f , ∇ f dH n .
Remark 4.7. (i) By virtue of the analyticity of T t and Theorem 4.4, we have T t f ∈ D(∆ m+1 ) and ∆ m T t f ∈ C Lip (X) for any f ∈ L 2 (X, H n ), t > 0 and m ≥ 0. Thus there are fairly many f and g's satisfying the condition in Theorem 4.6.

(ii) The inequality (4.9) is nothing but a weak form of the Γ 2 -condition

(4.10) Γ 2 ( f , f ) := 1 2 ∆( ∇ f , ∇ f ) -2 ∇ f , ∇∆ f ≥ K ∇ f , ∇ f .
This inequality is known to be equivalent to (4.4) in an abstract framework (see [START_REF] Bakry | On Sobolev and logarithmic Sobolev inequalities for Markov semigroups[END_REF][START_REF] Ledoux | The geometry of Markov diffusion generators[END_REF] and references therein, for instance). However, the assumption involves the existence of a nice core A ⊂ D(∆) and it seems hopeless to verify it on Alexandrov spaces. When X is a complete Riemannian manifold, the inequality (4.10) is equivalent to Ric ≥ K. Indeed,

Γ 2 ( f , f ) = Ric(∇ f , ∇ f ) + | Hess f | 2 holds by the Bochner identity.
Proof. We first show the claim for f ∈ D(∆)∩C Lip (X) with ∆ f ∈ D(∆)∩L ∞ (X, H n ). By (4.4), we obtain (4.11)

X g ∇T t f , ∇T t f dH n ≤ e -2Kt X gT t ( ∇ f , ∇ f ) dH n .
The derivation property (2.7) yields that, for t ≥ 0,

X g ∇T t f , ∇T t f dH n (4.12) = X ∇(gT t f ), ∇T t f dH n - X T t f ∇g, ∇T t f dH n = - X gT t f ∆T t f dH n - 1 2 X ∇g, ∇(T t f ) 2 dH n = - X gT t f T t ∆ f dH n + 1 2 X ∆g(T t f ) 2 dH n . Hence we obtain d dt X g ∇T t f , ∇T t f dH n t=0 (4.13) = - X g(∆ f ) 2 dH n - X g f ∆ 2 f dH n + X ∆g f ∆ f dH n = X ∇(g∆ f ), ∇ f dH n + X ∇(g f ), ∇∆ f dH n - X ∇g, ∇( f ∆ f ) dH n = 2 X g ∇∆ f , ∇ f dH n by using the derivation property again. Since f ∈ C Lip (X), ∇ f , ∇ f = |∇ L f | 2 ∈ L ∞ (X, H n ) holds. Hence we have d dt e -2Kt X gT t ( ∇ f , ∇ f ) dH n t=0 (4.14) = d dt e -2Kt X (T t g) ∇ f , ∇ f dH n t=0 = X ∆g ∇ f , ∇ f dH n -2K X g ∇ f , ∇ f dH n . Since (4.11) implies (4.15) d dt X g ∇T t f , ∇T t f dH n t=0 ≤ d dt e -2Kt X gT t ( ∇ f , ∇ f ) dH n t=0 ,
we obtain (4.9) by combining (4.15) with (4.13) and (4.14).

Next we consider the case that f ∈ D(∆) with ∆ f ∈ W 1,2 (X). Then, by the above discussion, T δ f and g satisfy (4.9) for δ > 0 (cf. Remark 4.7(i)). Since g, ∆g ∈ L ∞ (X, H n ) and ∆T δ f = T δ ∆ f , it suffices to show the claim that lim δ →0 ∇T δ h, ∇T δ h = ∇h, ∇h weakly in L 1 (X, H n ) for h, h ∈ W 1,2 (X). By polarization, we may assume h = h . The spectral decomposition yields (4.16) lim

δ →0 E (T δ h -h, T δ h -h) = 0 (see [12, Lemma 1.3.3], for instance). Let ψ ∈ L ∞ (X, H n ). Then the Schwarz inequality yields X ψ ∇T δ h, ∇T δ h dH n - X ψ ∇h, ∇h dH n (4.17) ≤ X ψ 2 |∇T δ h| 2 dH n 1/2 + X ψ 2 |∇h| 2 dH n 1/2 × E (T δ h -h, T δ h -h) 1/2 ≤ 2 ψ L ∞ E (h, h) 1/2 E (T δ h -h, T δ h -h) 1/2 .
Hence the desired claim follows from (4.16) and (4.17).

While we proved the implication from (4.1) with p = 2 to (4.4) and (4.9), these conditions are equivalent to each other on complete Riemannian manifolds (see [START_REF] Von Renesse | Transport inequalities, gradient estimates, entropy and Ricci curvature[END_REF]). Such an equivalence still holds in our framework with a sharp constant, which can be different from K in our hypothesis CD(K, ∞): Theorem 4.8 (Equivalence of "Ricci curvature bound" inequalities). Given K 0 ∈ R, the following are equivalent.

(i) (4.1) holds for ν 0 , ν 1 ∈ P(X) and t ≥ 0, with p = 2 and K = K 0 . (ii) (4.4) holds for f ∈ W 1,2 (X) and t ≥ 0 with K = K 0 . (iii) (4.9) holds with K

= K 0 for f ∈ D(∆) with ∆ f ∈ W 1,2 (X) and g ∈ D(∆) ∩ L ∞ (X, H n ) with g ≥ 0 and ∆g ∈ L ∞ (X, H n ).
Proof. "(i) ⇒ (ii) ⇒ (iii)" is already shown in Theorem 4.3 and Theorem 4.6.

For "(iii) ⇒ (ii)", it follows from a standard argument of the so-called Γ 2calculus (see [START_REF] Bakry | On Sobolev and logarithmic Sobolev inequalities for Markov semigroups[END_REF][START_REF] Ledoux | The geometry of Markov diffusion generators[END_REF] for instance). For completeness, we give a sketch of the proof. Take g 0 ∈ C(X) with g 0 ≥ 0 arbitrary and let g = T δ g 0 . Then g

∈ D(∆)∩L ∞ (X, H n ) with g ≥ 0 and ∆g ∈ L ∞ (X, H n ). Let us define Ψ : [0,t] → R by Ψ(s) := X gT s (|∇T t-s f | 2 ) dH n .
By a similar calculation as in (4.12), we can easily prove that Ψ is continuous on [0,t) and C 1 on (0,t). A similar argument as in (4.17) yields that Ψ is continuous at t. Here we use the ultracontractivity T t L 1 →L ∞ < ∞. A similar calculation as in the proof of Theorem 4.3 together with the assumption in (iii) leads to the inequality Ψ (s) ≥ 2K 0 Ψ(s). Hence (4.4) follows by integrating it.

For "(ii) ⇒ (i)", we claim that (4.4) implies (4.3) for every x ∈ X. Indeed, by using a bi-Lipschitz chart, we can bring the problem locally on an open set in a Euclidean space. Then, by applying [2, Lemma 3.2.1] and by using the continuity of T t (|∇ f | 2 ), the claim follows. Then we can apply [START_REF] Kuwada | Duality on gradient estimates and Wasserstein controls[END_REF]Theorem 2.2] to conclude (i) from (4.3). Remark 4.9. By the same argument as in "(ii) ⇒ (i)" of the last proof, we can give a proof of T t f ∈ C Lip (X) for f ∈ W 1,2 (X) under the condition (4.4). In other word, a priori regularity T t f ∈ C Lip (X) by Theorem 4.3 is not used in the last proof.

As the second application of (4.4), we mention that (4.4) together with Theorem 4.4 implies some functional inequalities by means of [START_REF] Kawabi | The Littlewood-Paley-Stein inequality for diffusion processes on general metric spaces[END_REF]Theorem 1.3]. Since T t is Markovian, we can restrict T t to a contraction on L ∞ (X, H n ). Then we can further extend T t to a contraction on L p (X, H n ) for any 1 ≤ p ≤ ∞ by interpolation and the symmetry of T t . Let us denote the infinitesimal generator of T t in L p (X, H n ) by ∆ p . Let us define R (q) α f := |∇((α -∆ p ) -q/2 f )|. Corollary 4.10. Let 2 ≤ p < ∞, q > 1 and α > max{(-K), 0}. Then we have the following:

(i) There exists a constant C R > 0 which depends only on p, q and max{(α + K), α} such that R (q) α f L p ≤ C R f L p for f ∈ L p (X, H n ). (ii) Suppose q < 2. Then there exists C p,q > 0 such that |∇T t f | L p ≤ C p,q R (q) α L p →L p α q/2 + t -q/2 f L p for t > 0 and f ∈ L p (X, H n ).

Proof. It is sufficient to verify the assumption of [START_REF] Kawabi | The Littlewood-Paley-Stein inequality for diffusion processes on general metric spaces[END_REF]Theorem 1.3]. Set A := C Lip (X) ∩ D(∆ 2 ). Since we already know (4.4), we only need to show the following claim: A is dense in W 1,2 (X) and f 2 ∈ D(∆ 1 ) holds for any f ∈ A . By Theorem 4.4, T t f ∈ A holds. Thus A is dense in W 1,2 (X) since E (T t ff , T t ff ) → 0 as t → 0 (cf. (4.16)). By [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF]Proposition I.2.4.3], it is enough to prove f 2 ∈ D(∆ 2 ) for f ∈ A . Take g ∈ W 1,2 (X) arbitrary. Recall that

|∇ f | 2 = |∇ L f | 2 ∈ L ∞ (X, H n ).
The derivation property yields

|E ( f 2 , g)| = 2 X f ∇ f , ∇g dH n = 2 X ( ∇ f , ∇(g f ) -g ∇ f , ∇ f )dH n = 2 X ( f ∆ f -∇ f , ∇ f )g dH n ≤ 2 f L ∞ ∆ f L 2 + |∇ f | 2 L 2 g L 2 .
This estimate means f 2 ∈ D(∆ 2 ) and hence the proof is completed.

Finally, we observe that all of our results are generalized to the heat equation with drift, in other words, the Fokker-Planck equation. Given a potential function V ∈ C Lip (X), we modify the Dirichlet energy and the relative entropy into

E V ( f , g) = X ∇ f , ∇g e -V dH n f , g ∈ W 1,2 (X),
Ent V (µ) = Ent(µ) + X V dµ µ ∈ P(X).

We regard E V as a bilinear form on L 2 (X, e -V H n ). Observe that Ent V is nothing but the relative entropy with respect to e -V H n . Note that the semigroup T V t on L 2 (X, e -V H n ) associated with E V solves the following diffusion equation Since e -V is bounded and away from 0, e -V H n is equivalent to H n . Hence (X, d, e -V H n ) satisfies the volume doubling condition as well as the Poincaré inequality for upper gradients. Moreover, as E and E V are equivalent, the Poincaré inequality for E V is also valid. Therefore a continuous density p V t for T V t exists. Under the assumption that Ent V is K-geodesically convex, we can apply the general theory of the gradient flow on (P(X),W 2 ) to obtain the gradient flow µ t of Ent V . Furthermore, every argument in Sections 3, 4 works verbatim and gives similar results for the equation (4.18). Note that, under CD(K, ∞) for (X, d, H n ), the K -convexity of V implies the (K + K )-convexity of Ent V .

Bibliography

  for which we used the local structure of Alexandrov spaces. Actually, we use |∇ L f | in Propositions 3.2, 3.7 and |∇ f | in Propositions 3.4, 3.7.

Theorem 4 . 4 (

 44 Lipschitz continuity).

  ∆u -∇V, ∇u .

  r) dr for all t, s ∈ I with t ≤ s. Note that absolutely continuous curves are continuous. For an absolutely continuous curve (µ t ) t∈I , the metric derivative| μt | := lim h→0 W 2 (µ t , µ t+h ) |h| is well-defined for a.e. t ∈ I ([1, Theorem 1.1.2]). Moreover, | μt | belongs to L 1 (I)and is the minimal function for which (2.8) holds. Given a functional E : P(X) → R ∪ {+∞}, we consider a gradient flow of E solving " μt = -∇E(µ t )" in the following sense. For µ ∈ P(X) with E(µ) < ∞, we define the local slope as E|(µ r ) dr for all t, s ∈ I with t < s along any absolutely continuous curve (µ t ) t∈I with values in P(X) with E(µ t ) < ∞. As a consequence, it holds

	(2.9)	|∇ -E|(µ) := lim ν→µ	max{E(µ) -E(ν), 0} W 2 (µ, ν)	.
	If E is K-geodesically convex, then we have
			s
	(2.10) | μr | • |∇ -(2.11) |E(µ t ) -E(µ s )| ≤ t E(µ t ) ≤ E(µ s ) + 1 2 s t 1 | μr | 2 dr + 2

s t |∇ -E| 2 (µ r ) dr for all t < s. From (2.10) and (2.11), it is natural to give the following definition.

  dH n by construction. On the other hand, we deduce from[1,

	Corollary 2.4.10] that
	(3.1)

Partially supported by SFB 611 Partially supported by the JSPS fellowship for research abroad and the Grant-in-Aid for Young Scientists (B) 22740083 Partially supported by the Grant-in-Aid for Young Scientists (B) 20740036 Communications on Pure and Applied Mathematics, Vol. 000, 0001-0023 (2000)