Michael Goldman
email: goldman@math.univ-paris-diderot.fr

Marc Josien
email: marc.josien@polytechnique.edu

Felix Otto
email: otto@mis.mpg.de

New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations

published or not. The documents may come

New bounds for the inhomogenous Burgers and the

Kuramoto-Sivashinsky equations

Introduction

The Kuramoto-Sivashinsky equation

We consider the one-dimensional Kuramoto-Sivashinsky equation:

∂ t u + u∂ x u + ∂ 2 x u + ∂ 4 x u = 0. (K-S)
This equation appears in many physical contexts, in particular in the modeling of surface evolutions. Sivashinsky used it to describe flame fronts [START_REF] Sivashinsky | On flame propagation under conditions of stoichiometry[END_REF], wavy flow of viscous liquids on inclined planes [START_REF] Sivashinsky | On irregular wavy flow of a liquid film down a vertical plane[END_REF] and crystal growth [START_REF] Golovin | Convective Cahn-Hilliard models for kinetically controlled crystal growth[END_REF]. Although the solutions of (K-S) are smooth and even analytic [START_REF] Jolly | Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations[END_REF], they display a chaotic behavior for sufficiently large systems size L (see [START_REF] Hyman | Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces[END_REF] and Figure 1.1). The structure of the Kuramoto-Sivashinsky equation has some similarities with the Navier-Stokes equation. Therefore, it is sometimes possible to apply similar techniques to study both equations (see [START_REF] Otto | Universal bounds for the Littlewood-Paley first-order moments of the 3D Navier-Stokes equations[END_REF]).

Figure 1: Chaotic behavior of u For a given system size L > 0, we will consider L-periodic solutions of (K-S). Since the spatial average L 0 u(t , x)d x is constant over time, and since the equation is invariant under the Galilean transformation:

t = t , x = x +U t , u = u +U ,
it is not restrictive to assume that L 0 u(t , x)d x = 0 for all t ≥ 0. We can artificially cut the equation in two parts and consider separately the two mechanisms involved in (K-S):

∂ t u + ∂ 2 x u + ∂ 4 x u = 0, (1)
∂ t u + u∂ x u = 0. ( 2
)
The first equation ( 1) is linear and can be seen in Fourier space as:

∂ t F u = (ξ 2 -ξ 4 )F u. ( 3
)
The fourth partial derivative term ∂ 4

x u decreases the short wavelength part of the energy spectrum whereas the second derivative term ∂ 2

x u amplifies the long wavelength part. The second equation (2) corresponds to Burgers equation. It is nonlinear and develops shocks in finite time for non-trivial initial data. Nevertheless, as we will see later, this term has some mild regularizing effect. It is worth mentioning that for (2), the energy L 0 u 2 d x, is conserved. Therefore, one can intuitively say that in (K-S) the linear terms transport the energy from long wavelengths to short ones. Numerical simulations suggest (see the article of Wittenberg and Holmes [START_REF] Wittenberg | Scale and space localization in the Kuramoto-Sivashinsky equation[END_REF]) that the time-averaged power spectrum lim

T →∞ (LT ) -1 T 0 |F (u)(t , ξ)| 2 d t is independent of L for L 1.
Moreover, this quantity is independent of |ξ| and L 1 in the long wavelength regime L -1 | ξ | 1 and decays exponentially in the short wavelength regime |ξ| 1. In line with this, numerical simulation suggests that for all α ≥ 0:

lim sup T →∞ (LT ) -1 T 0 L 0 |∂ x | α u 2 d xd t = O(1).
This conjecture is supported by a universal bound on all stationary periodic solutions of (K-S) with mean 0, due to Michelson [START_REF] Michelson | Steady solutions of the Kuramoto-Sivashinsky equation[END_REF].

Known bounds

A first energy bound was obtained in the 80's by Nicolaenko, Scheurer and Temam [START_REF] Nicolaenko | Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attractors[END_REF], who established by the "background flow method" that for every odd (in space) solution u of (K-S):

lim sup t →∞ 1 L L 0 u 2 d x 1/2 = O(L p ),
with p = 2. This has been later generalized by Goodman [START_REF] Goodman | Stability of the Kuramoto-Sivashinsky and related systems[END_REF], and Bronski and Gambill [START_REF] Bronski | Uncertainty estimates and L 2 bounds for the Kuramoto-Sivashinsky equation[END_REF] and improved to p = 1. Using an entropy method, Giacomelli and the third author [START_REF] Giacomelli | New bounds for the Kuramoto-Sivashinsky equation[END_REF] improved this result by showing that:

lim sup t →∞ 1 L L 0 u 2 d x 1/2 = o(L).
The proof is based on the fact that the dispersion relation ξ 2 -ξ 4 in (3) vanishes for ξ → 0 and it implies that for every α ∈ [0, 2], we have:

lim sup T →∞ 1 T L T 0 L 0 |∂ x | α u 2 d xd t 1/2
= o(L), by using the energy identity,

∂ t L 0 (u(t , x)) 2 d x = L 0 (∂ x u(t , x)) 2 d x - L 0 ∂ 2 x u(t , x) d x.
In a more recent paper [START_REF] Otto | Optimal bounds on the Kuramoto-Sivashinsky equation[END_REF], the third author proved that, for all α ∈ (1/3, 2), lim sup

T →∞ 1 T L T 0 L 0 |∂ x | α u 2 d xd t 1/2 = O ln 5/3+ (L) , 1 (4)
by using two ingredients: an a priori estimate for the capillary Burgers equation

∂ t u + u∂ x u + ∂ 4 x u = |∂
x |g and an a priori estimate for the inhomogeneous Burgers equation, that is ∂ t u + u∂ x u = |∂ x |g . More precisely, the result of [START_REF] Otto | Optimal bounds on the Kuramoto-Sivashinsky equation[END_REF] states that, for every solution u of (K-S),

u B 1/3 3,3 = O ln 5/3+ (L) , (5)
where • B s p,r denotes a Besov norm (see the appendix).

Main result

In this paper, we improve and simplify the result of the third author by showing that:

Theorem 1.1. Let L > 2.
For u a smooth L-periodic solution with zero average of the equation

∂ t u + u∂ x u + ∂ 2 x u + ∂ 4 x u = 0, there holds lim sup T →∞ sup h>0 1 LT T 0 L 0 |u(t , x + h) -u(t , x)| 3 h d xd t 1/3 = O(ln 1/2+ (L)). ( 6
)
This result is indeed slightly stronger than the previous one, since by [START_REF] Wittenberg | Scale and space localization in the Kuramoto-Sivashinsky equation[END_REF], it implies an improvement of the exponent in (5) from 5/3+ to 5/6+. However, this is not the main contribution of this paper. It is rather a simplified proof of the a priori estimate for inhomogeneous Burgers equation, which was one of the main tool for proving [START_REF] Giacomelli | New bounds for the Kuramoto-Sivashinsky equation[END_REF]. For this purpose, we derive a modified Kármán-Howarth-Monin formula (see [START_REF] Otto | Optimal bounds on the Kuramoto-Sivashinsky equation[END_REF]). We also show how the proof of Golse and Perthame [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF] (based on the kinetic formulation of Burgers equation) of a similar estimate for the homogeneous Burgers equation can be reinterpreted in this light. Since we work with slightly different Besov norms compared to [START_REF] Otto | Optimal bounds on the Kuramoto-Sivashinsky equation[END_REF], we need also to adapt most of the other steps to get [START_REF] Golovin | Convective Cahn-Hilliard models for kinetically controlled crystal growth[END_REF]. Besides Proposition 2.5, which we borrow directly from [START_REF] Otto | Optimal bounds on the Kuramoto-Sivashinsky equation[END_REF], we give here self-contained proofs.

The structure of the paper is the following: In Section 2, we enunciate the main theorem and give the structure of the proof. It has several ingredients: a Besov estimate for the inhomogeneous inviscid equation (Proposition 2.3), a regularity estimate for the capillary Burgers equation (Proposition 2.5) and an inverse estimate for Besov norms on solutions of (K-S) (Proposition 2.7). The following sections (i.e. Section 3, 4 and 5) are devoted to the proofs. In the appendix, we recall definitions and a few classical results regarding Besov spaces.

General notations

We denote by D h the finite-difference operator

D h : u → u(x + h) -u(x), by L p the space L p ([0, T ] × [0, L]) and for k ∈ N, L > 0, by C k L = f ∈ C k (R), f is L -periodic .
For an L-periodic function u, the spatial Fourier transform is defined by:

F (u)(ξ) = L -1 L 0 exp(-i ξ x)u(x)d x
and for a Schwartz function φ:

F (φ)(ξ) = R exp(-i ξ x)φ(x)d x.
For v ∈ R, we let v + = max(v, 0) (and similarly, v -= max(-v, 0)).

Main theorem and structure of the proof

In this section, we state the main theorem and the results on which it is based (see the appendix for the definition and main properties of Besov spaces).

Main theorem

Theorem 2.1. Let L > 2. For a smooth L-periodic solution u with zero average of (K-S), there holds u

B 1/3 3,∞ + u B 2 2,2 = O(ln 1/2+ (L)). ( 7
)
From this theorem, we derive by interpolation (57) the following corollary:

Corollary 2.2. Let L > 2.
For a smooth L-periodic solution u with zero average of (K-S) and for indices p, s and r related by

p ∈ [1/3, 2], s = 10/p -3, 1/r = 3(1/p -1/3),
we have u B s p,r = O(ln 1/2+ (L)).

Structure of the proof

The proof of Theorem 2.1 uses four important ingredients: a regularity result for Burgers equation (Proposition 2.3), a higher regularity estimate for the capillary Burgers equation (Proposition 2.5), an energy estimate (Lemma 2.6), and a result which allows us to "increase" the r index of Besov spaces (Proposition 2.7). Let us now sketch the proof, discarding lower-order terms (in particular all the terms contain-

ing g = -|∂ x | -1 ∂ 2 x u
) and taking borderline exponents in the estimates 2 . The strategy is graphically represented in Figure 2. The starting point is Proposition 2.3, which for s = 1, p = 5/2, r = 5/2 and ξ = -|∂ x | -1 ∂ 4

x u (recall also (59)), roughly says that . It is at this last stage that we lose a logarithm since [START_REF] Wittenberg | Scale and space localization in the Kuramoto-Sivashinsky equation[END_REF] gives

u 3 B 1/3 3,∞ u B 1 5/2,
• • 2/5 • 1/2 • 3/5 • 4/5 1/2 • • • • • Legend: • Interpolation • Energy estimate • Higher regularity • Increase of the index r • Estimate of the inhomogeneous Burgers equation • • • • • • 1/3 • • 2/5 • 1/2 • 3/5 • 4/5 1/
u B 1/3 3,3 ln 1/3 (L) u B 1/3 3,∞ .
Putting all these estimates together, we find

u B 1/3 3,∞ ln 1/2 (L),
which is [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF].

As mentioned, the first ingredient is an estimate for the inhomogeneous Burgers equation. A similar estimate was obtained in [START_REF] Otto | Optimal bounds on the Kuramoto-Sivashinsky equation[END_REF]Prop. 1]. A related inequality for the homogeneous Burgers equation has been recently derived in [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF].

Let us consider the following inhomogeneous Burgers equation:

∂ t u + u∂ x u = |∂ x | g + |∂ x | ξ . ( 10
)
Proposition 2.3 (Besov estimate for the inhomogeneous Burgers equation). Let ξ, g be smooth L-periodic functions. Then, for any smooth L-periodic solution u of [START_REF] Jolly | Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations[END_REF], there holds: For s ∈]0, 1[, r, r , p, p ∈ [1, +∞] verifying 1 r + 1 r = 1 and 1 p + 1 p = 1, there exists a constant c > 0 just depending on s, r, p such that:

u 3 B 1/3 3,∞ ≤c u B 1/3 3,∞ g B 2/3 3/2,1 + u B s p,r ξ B 1-s p ,r + u(0, •) 2 L 2 [0,L] . ( 11
)
Therefore, taking the time-space average, it holds:

u 3 B 1/3 3,∞ ≤c u B 1/3 3,∞ g B 2/3 3/2,1 + u B s p,r ξ B 1-s p ,r . ( 12
)
The proof of ( 11) is based on a modified Kármán-Howarth-Monin identity:

Lemma 2.4. Let η be a smooth L-periodic function and let u be a smooth L-periodic solution with zero average of

∂ t u + u∂ x u = η, ( 13
)
then for h ∈ R, ∂ t 1 2 L 0 |D h u|D h ud x + ∂ h 1 6 L 0 |D h u| 3 d x = L 0 D h η|D h u|d x. ( 14
)
The usual Kármán-Howarth-Monin identity [START_REF] Frisch | The legacy of A. N. Kolmogorov[END_REF] states that

∂ t 1 2 L 0 D h u 2 d x + ∂ h 1 6 L 0 D h u 3 d x = L 0 D h ηD h ud x. ( 15
)
This formula can be easily checked by using equation ( 13) and the periodicity. The main difference between ( 15) and ( 14) is that in the latter, the coercive term

L 0 |D h u| 3 d x replaces the non-coercive term L 0 D h u 3 d x.
We will give two proofs of ( 14). The first is by a direct computation and the second uses the kinetic formulation of Burgers equation following ideas of [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF]. Therefore, this second proof gives a new, and hopefully interesting, interpretation of the arguments of [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF]. The second ingredient is a higher regularity result for the capillary Burgers equation (see [14, Prop. 2, p. 14]).

Proposition 2.5 (Higher regularity). Let p, q

∈ [1, +∞[, α ∈ R satisfying: p + 1 ≤ q ≤ 2p, and α = (6 + α)p/q -3 ∈]0, 1[.
Then, there exists c > 0 such that, if u, g are smooth, L-periodic in x and satisfy

∂ t u + u∂ x u + ∂ 4 x u = |∂ x | g , the following estimate holds: u B 3+α p,p ≤ c u q/p B α q,q + g B α p,p . (16)
Proposition 2.5 allows to jump from higher derivatives to smaller ones in Besov spaces. The proof, which we will not provide, is based on a narrow-band Littlewood-Paley decomposition.

The third ingredient is an elementary energy estimate, which directly bounds the

L 2 [0, T ], H 2 ([0, L]) ∼ = B 2
2,2 norm of a solution u of the inhomogeneous capillary Burgers equation.

Lemma 2.6 (Energy estimate). Let u be a smooth solution of:

∂ t u + u∂ x u + ∂ 4 x u = |∂ x | g . ( 17
)
Then, the following estimate holds:

u B 2 2,2 ≤ c u 1/2 B 1/3 3,∞ g 1/2 B 2/3 3/2,1 . ( 18
)
Proof. Since the proof is straightforward, we give it now. Integrating over the equation ( 17) over [0, T ] × [0, L], we get:

L 0 u(T, x) 2 d x - L 0 u(0, x) 2 d x = T 0 L 0 u |∂ x | g d xd t - T 0 L 0 (∂ 2 x u) 2 d xd t .
Therefore by (58)

T 0 L 0 ∂ 2 x u 2 d xd t ≤ L 0 u(0, x) 2 d x + u B 1/3 3,∞ g B 2/3 3/2,1
, taking then the time-space average yields the result.

As already mentioned, these three estimates will not be sufficient to conclude. We will also need an estimate relating Besov norms with different exponents r . One can easily see that if

r 1 > r 2 , then • B s p,r 1 ≤ • B s p,r 2
(it is a consequence of convexity inequality). In fact, it is possible to reverse the inequality for solutions of (K-S), but this comes with a price: a logarithm of the spatial period L appears. Proposition 2.7 (Increasing the index r ). There exists c > 0 such that, for all L ≥ 2, u solution of (K-S), the following estimate holds:

u B 1/3 3,3 ≤ c ln 1/3 (L) u B 1/3 3,∞ . ( 19
)
3 Proof of Theorem 2.1

In this section, we derive the main theorem from the above propositions. We now consider the rescaled Besov norm B s p,r as a point (s, 1/p, 1/r ) in the space R3 . All the norms involved in our problem lie in the rectangle P of R 3 defined by 3 :

     s = 10/p -3, 1 p ∈ [0, 1], 1 r ∈ [0, 1].
Proof of Theorem 2.1. Let u be a solution of (K-S). It is convenient to introduce the abbreviation:

D(α) = u B 10α-3 α -1 ,α -1 D * (1/3) = u B 1/3 3,∞ . Notice that D (1/2) = u B 1/2 2,2
and

D (1/3) = u B 1/3 3,3
. With this notation, interpolation inequality (57) takes the form

D(α) ≤ D θ (α 1 )D 1-θ (α 2 ) (20) for α = θα 1 + (1 -θ)α 2 and θ ∈ [0, 1].
Letting s = 10α -3, p = α -1 and r = α -1 in (12) and using (59

) for ξ = -|∂ x | -1 ∂ 4 x u, ( 12
) can be rewritten as

D 3 * (1/3) ≤ c D(α)D(1 -α) + D * (1/3) g B 2/3 3/2,1 , (21)
for α ∈]3/10, 2/5[. In turn, ( 16) with p -1 = β, q -1 = γ and α = 10β -6 (which implies α = 10γ -3) gives

D(β) ≤ c D β/γ (γ) + g B 10β-6 β -1 ,β -1 , (22)
for γ ∈]3/10, 2/5[ and γ 1-γ ≤ β ≤ 2γ. Finally, ( 18) is equivalent to

D (1/2) ≤ cD 1/2 * (1/3) g 1/2 B 2/3 3/2,1 , (23)
and [START_REF] Wittenberg | Scale and space localization in the Kuramoto-Sivashinsky equation[END_REF] to

D (1/3) ≤ c ln 1/3 (L)D * (1/3) . ( 24
)
Our first goal is to argue that for g = -|∂ x | -1 ∂ 2 x u, we can replace in the above estimates all the Besov norms involving g by D * (1/3). By ( 59) and (57),

g B 10β-6 β -1 ,β -1 ≤ c u B 10β-5 β -1 ,β -1 ≤ c u 1/2 B 10β-3 β -1 ,β -1 u 1/2 B 10β-7 β -1 ,β -1 = cD 1/2 (β) u 1/2 B 10β-7 β -1 ,β -1 .
Hence, in view of (23), Young's inequality and since by (59), g

B 2/3 3/2,1 ≤ c u B 5/3 3/2,1
, it will be enough to prove that

u B 10β-7 β -1 ,β -1 + u B 5/3 3/2,1 ≤ c(D * (1/3) + D (1/2)) (25)
for β ∈]11/15, 4/5[ (which reduces the use of ( 22) to γ ∈]1/3, 2/5[). We can indeed prove more generally that for 1/3 < s < 2, p ≤ 2 and any q ≥ 1, there holds

u B s p,q ≤ c u B 2 2,2 + u B 1/3 3,∞ . (26)
Thanks to Jensen's inequality, we have u

B s p,q ≤ u B s 2,q and u B 1/3 3,∞ ≥ u B 1/3 2,∞
. By monotonicity of the Besov norms with respect to the last index, there also holds

u B s 2,q ≤ u B s 2,1 and u B 2 2,2 ≥ u B 2 2,∞
. Therefore, we are left with proving that

u B s 2,1 ≤ c u B 1/3 2,∞ + u B 2 2,∞
.

By definition of the Besov norms, for 1/3 < s < 2,

u B s 2,1 = k≥0 2 k u k L 2 + k<0 2 k u k L 2 ≤ sup k 2 2k u k L 2 k≥0 2 -k + sup k 2 1 3 k u k L 2 k≤0 2 2 3 k ≤ c u B 1/3 2,∞ + u B 2 2,∞
, which after taking the average over time and space, finishes the proof of (26).

To sum up, we now have that (21), ( 22) and (23) together with (25) imply

D 3 * (1/3) ≤ c D(α)D(1 -α) + D 2 * (1/3) (27) for α ∈]3/10, 2/5[, D(β) ≤ c D β/γ (γ) + D * (1/3) (28) for γ ∈]1/3, 2/5[ and γ 1-γ ≤ β ≤ 2γ and D (1/2) ≤ cD * (1/3) . ( 29
)
We now gather the above estimates in order to bound D * (1/3). Passing to the logarithm in the above inequalities, we see that optimizing the parameters to get the best power of ln(L) is equivalent to a linear programming problem. Its solution thus lie at the boundaries of the admissible domain. It is not hard to see that in particular, we want to take

β 2 = γ = α with α as close as possible to 2/5. Let θ, η ∈]0, 1[ be such that (1 -α) = θβ + (1 -θ) 1 2 and α = η 1 2 + (1 -η) 1 3 ,
so that θ is close to 1/3 and η is close to 2/5. Thanks to (20),

D(1 -α) ≤ D θ (β)D 1-θ (1/2) and D(α) ≤ D η (1/2)D 1-η (1/3). ( 30
)
Since we can assume that D * (1/3) ≥ 1, we get from ( 27), ( 30) and (28),

D 3 * (1/3) ≤ cD(α) D 2θ (α) + D θ * (1/3) D 1-θ (1/2) ≤ c D 1+2θ (α)D 1-θ * (1/3) + D(α)D * (1/3) ,
where in the last inequality, we used (29). From (30), ( 29) and (24), we deduce

D 3 * (1/3) ≤ c D 2+θ * (1/3) ln 1 3 (1+2θ)(1-η) (L) + D 2 * (1/3) ln 1-η 3 (L) .
Dividing by D 2 * (1/3) this inequality and noticing that for η close to 2/5,

1-η 3 is close to 1/5, we obtain that if D * (1/3) ≥ ln 1-η 3 (L), then D * (1/3) ≤ cD θ * (1/3) ln 1 3 (1+2θ)(1-η) (L),
which gives finally

D * (1/3) ≤ c ln 1 3 (1+2θ)(1-η) 1-θ (L)
and thus the result since lim θ↑1/3,η↑2/5

1 3 (1+2θ)(1-η) 1-θ = 1/2.

Proof of Proposition 2.3

For the reader's convenience, let us recall the statement of Proposition 2.3. Let u be a smooth solution of the following inhomogeneous Burgers equation:

∂ t u + u∂ x u = |∂ x | g + |∂ x | ξ . ( 31
)
Proposition. Let ξ, g be smooth L-periodic functions. Then, for any smooth L-periodic solution u of (31), there holds: For s ∈]0, 1[, r, r , p, p ∈ [1, +∞] verifying 1 r + 1 r = 1 and

1 p + 1 p = 1
, there exists a constant c ∈ R * + just depending on s, r, p such that:

u 3 B 1/3 3,∞ ≤c u B 1/3 3,∞ g B 2/3 3/2,1 + u B s p,r ξ B 1-s p ,r + u(0, •) 2 L 2 [0,L] .
Before proceeding further, let us remark that, by approximation, this applies to any (possibly non smooth) entropy solution of Burgers equation (31). Indeed, if we consider a solution u of ∂ t u -u∂ x u -ε∂ 2 x u = 0, then it is a smooth solution of (31) with g = 0 and ξ = ε |∂ x | u. A careful inspection of the proof of Proposition 2.3 shows that for p = r = 2 it extends to s = 1, yielding

u 3 B 1/3 3,∞ ≤ c u B 1 2,2 ξ B 0 2,2 + u(0, •) 2 L 2 [0,L] ,
that is u 3

B 1/3 3,∞ ≤ c ε ∂ x u 2 L 2 + u(0, •) 2 L 2 [0,L] .
Combining this with the energy inequality:

ε ∂ x u 2 L 2 ≤ c u(0, •) 2 L 2 [0,L] ,
gives

u 3 B 1/3 3,∞ ≤ c u(0, •) 2 L 2 [0,L] ,
which passes to the limit as ε → 0. The indices are optimal in the light of the result of De Lellis and Westdickenberg [START_REF] Lellis | On the optimality of velocity averaging lemmas[END_REF] which states that we cannot hope to have more regularity, in the sense that the Besov index s cannot be better than 1/3.

As already pointed out, the proof of the aimed estimate is based on a modified Kármán-Howarth-Monin identity: Lemma 4.1. Let η be a smooth L-periodic function and let u be a smooth L-periodic solution with zero average of

∂ t u + u∂ x u = η (32) then for h ∈ R, ∂ t 1 2 L 0 |D h u|D h ud x + ∂ h 1 6 L 0 D h u 3 d x = L 0 D h η D h u d x. ( 33
)
Proof. By periodicity, (33) will be a direct consequence of the following pointwise identity:

1 2 ∂ t |D h u|D h u + 1 6 ∂ h D h u 3 + 1 2 ∂ x u|D h u|D h u + 1 3 |D h u| 3 = D h η D h u . ( 34
)
For simplicity, let us introduce the notation

u h (x) = u(x + h) (so that D h u = u h -u).
Using (32) we get:

1 2 ∂ t |D h u|D h u + 1 6 ∂ h D h u 3 = |D h u|∂ t (D h u) + 1 2 D h u D h u∂ x u h = D h u D h η -u∂ x u + 1 2 D h u u h ∂ x u h -u∂ x u h = D h η D h u + 1 2 D h u 2u∂ x u -u∂ x u h -u h ∂ x u h .
It remains to prove that

|D h u| 2u∂ x u -u∂ x u h -u h ∂ x u h = -∂ x u|D h u|D h u + 1 3 |D h u| 3 . ( 35
)
We start with

|D h u|2u∂ x u = -u 2 ∂ x |D h u| + ∂ x |D h u|u 2
and

|D h u|u∂ x u h = -∂ x |D h u|u u h + ∂ x |D h u|uu h = -uu h ∂ x |D h u| -u h |D h u|∂ x u + ∂ x |D h u|uu h , to get |D h u| 2u∂ x u -u∂ x u h -u h ∂ x u h = -u 2 ∂ x |D h u| + ∂ x |D h u|u 2 + uu h ∂ x |D h u| + u h |D h u|∂ x u -∂ x |D h u|uu h -|D h u|u h ∂ x u h = -∂ x u|D h u|D h u + uD h u∂ x |D h u| -u h |D h u|∂ x D h u. But since D h u∂ x |D h u| = |D h u|∂ x D h u, then |D h u| 2u∂ x u -u∂ x u h -u h ∂ x u h = -∂ x u|D h u|D h u -|D h u|D h u∂ x D h u = -∂ x u|D h u|D h u + 1 3 |D h u| 3 ,
which concludes the proof of (35).

Remark. Arguing along the same lines, one can prove that more generally, if a is nonnegative and if u is a smooth solution with zero average of

∂ t u + ∂ x [a(u)] = η (36) then ∂ t 1 2 L 0 |D h u|D h ud x + ∂ h L 0 |D h u|(a(u) + a(u h )) -2|A(u h ) -A(u)|d x = L 0 D h η D h u d x,
where A = a. Notice that Burgers equation corresponds to (36) with a(u) = 1 2 u 2 . If a is C 1 and monotone in the sense that there exist β ≥ 1 and C > 0, such that for v ≥ w,

a (v) -a (w) ≥ C (v -w) β ,
then one can obtain a similar estimate to (11) by using that for ū ≥ u

( ū -u)(a( ū) + a(u)) -2(A( ū) -A(u)) = ū u ū w (a (v) -a (w))d vd w ≥ C | ū -u| β+2 .
In this way, one can fully recover the results from [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF].

We now give an alternative proof of (the integrated form of ) (33) following ideas from [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF]. This proof uses the kinetic formulation of the inhomogeneous Burgers equation together with the use of an interaction identity (see (43)).

Lemma 4.2. Let η be a smooth L-periodic function and let u be a smooth L-periodic solution with zero average of

∂ t u + u∂ x u = η, ( 37
)
then for h ∈ R,

1 2 h 0 L 0 |D ∆ u|D ∆ ud xd ∆ T 0 + 1 6 T 0 L 0 D h u 3 d xd t = T 0 L 0 h 0 D ∆ η D ∆ u d ∆d xd t . ( 38
)
Proof. Before starting the proof, let us point out that since we have a direct proof of (33), we will not take care of regularity issues. Nevertheless, all passages can be rigorously justified by a suitable approximation argument.

Step 1. Without loss of generality, we can assume that h > 0. Letting:

f (t , x, v) = 1 if v ≤ u(t , x), 0 if v > u(t , x),
equation ( 37) is equivalent to the following kinetic formulation (see [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related equations[END_REF]):

∂ t f (t , x, v) + v∂ x f (t , x, v) = -∂ v f (t , x, v)η(t , x). ( 39
)
Notice that since u is bounded, D h f is integrable even though f is not. We are going to compute only integrals depending on D h f and will therefore not have to deal with integrability issues. As in [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF]Lem. 4.3], we define:

M u (v) = 1 if v ≤ u, 0 if v > u.
We first claim that for all u, ū ∈ R the following equality holds:

1 6 |u -ū| 3 = R R 1 R + (v -w) (v -w) (M u (v) -M ū (v)) (M u (w) -M ū (w)) d vd w. ( 40
)
Without loss of generality, we can suppose that u ≥ ū. Then:

M u (v) -M ū (v) = 1 ] ū,u[ (v).
Thus (40) follows from:

R R 1 R+ (v -w) (v -w) (M u (v) -M ū (v)) (M u (w) -M ū (w)) d vd w = R R 1 R+ (v -w)1 ] ū,u[ (v)1 ] ū,u[ (w) (v -w)d vd w = u ū u w (v -w)d vd w = 1 6 |u -ū| 3 .
Letting

Q(h) = T 0 L 0 R × R 1 R+ (v -w) (v -w)D h f (t , x, v)D h f (t , x, w)d vd wd xd t ,
we see that proving (38) is equivalent to

Q(h) = T 0 L 0 h 0 D ∆ η D ∆ u d ∆d xd t - 1 2 h 0 L 0 |D ∆ u|D ∆ ud xd ∆ T 0 . ( 41
)
Step 2. To cope with the quantity Q, the main tool is the following interaction identity (see [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF]), which have been introduced first by Varadhan ([18, Lem. 22.1]): Let A, B , C , D, E , F : [0, T ] × [0, L] → R be functions satisfying the following system

∂ t A + ∂ x B = C , ∂ t D + ∂ x E = F, ( 42
)
and having zero spatial average. Then the following identity holds:

T 0 L 0 (AE -B D) = T 0 L 0 A(t , x) x 0 F (t , y)d y d xd t + T 0 L 0 C (t , x) x 0 D(t , y)d y d xd t (43) - L 0 x 0 A(t , x)D(t , y)d yd x t =T t =0
. Indeed, by Taylor expansion:

T 0 L 0 A(t , x)E (t , x)d xd t = T 0 L 0 x 0 A(t , x)∂ x E (t , y)d yd xd t - T 0 L 0 A(t , x)E (t , 0)d xd t .
Since A has zero spatial average, the second term vanishes. Using equation (42) to compute the first term and integrating by parts, we get:

T 0 L 0 x 0 A(t , x)∂ x E (t , y)d yd xd t = T 0 L 0 x 0 A(t , x)F (t , y)d yd xd t - T 0 L 0 x 0 A(t , x)∂ t D(t , y)d yd xd t = T 0 L 0 x 0 A(t , x)F (t , y)d yd xd t + T 0 L 0 x 0 ∂ t A(t , x)D(t , y)d yd xd t - L 0 x 0 A(t , x)D(t , y)d yd x t =T t =0
.

Let us now compute more precisely the second term, using (42):

T 0 L 0 x 0 ∂ t A(t , x)D(t , y)d yd xd t = T 0 L 0 x 0 C (t , x)D(t , y)d yd xd t - T 0 L 0 ∂ x B (t , x) x 0 D(t , y)d yd xd t = T 0 L 0 x 0 C (t , x)D(t , y)d yd xd t + T 0 L 0 B (t , x)D(t , x)d xd t ,
which concludes the proof of (43).

Step 3. We apply the interaction identity to:

   A(t , x, v) = D h f (t , x, v), D(t , x, w) = A(t , x, w), B (t , x, v) = vD h f (t , x, v), E (t , x, w) = B (t , x, w), C (t , x, v) = -∂ v D h ( f (t , x, v)η(t , x)), F (t , x, w) = C (t , x,

w).

Note that A,B ,C ,D,E ,F implicitly depend on h. Multiplying each side of the identity by 1 R+ (vw) and integrating it, we get:

Q(h) = - R × R 1 R+ (v -w) T 0 L 0 (A(t , x, v)E (t , x, w) -B (t , x, v)D(t , x, w)) d xd t d vd w = - R × R 1 R+ (v -w) T 0 L 0 A(t , x, v) x 0 F (t , y, w)d y d xd t d vd w - R × R 1 R+ (v -w) T 0 L 0 C (t , x, v) x 0

D(t , y, w)d y d xd t d vd w

+ R × R 1 R+ (v -w) L 0 x 0 A(t , x, v)D(t , y, w)d yd x t =T t =0 d vd w. = - R × R 1 R+ (v -w) T 0 L 0 A(t , x, v) x 0 F (t , y, w)d y d xd t d vd w - R × R 1 R+ (w -v) T 0 L 0 F (t , y, w) y 0 A(t , x, v)d x d yd t d vd w + R × R 1 R+ (v -w) L 0 x 0 A(t , x, v)D(t , y, w)d yd x t =T t =0 d vd w. =Q 1 +Q 2 +Q 3 .
But, by periodicity,

y 0 A(t , x, w)d x = - L y A(t , x, w)d x, and thus L 0 F (t , y, v) y 0 A(t , x, w)d x d y = - L 0 A(t , x, w) x 0 F (t , y)d y d x.
Therefore,

Q 2 = R × R 1 R+ (w -v) T 0 L 0 A(t , x, v) x 0 F (t ,

y, w)d y d xd t d vd w.

Step 4. In the next two steps, the time variable plays no role. We will therefore consider

Q 1 = - R × R 1 R+ (v -w) L 0 A(x, v) x 0 C (y, w)d y d xd vd w and Q 2 = R × R 1 R+ (w -v) L 0 A(x, v) x 0 C (y, w)d y d xd vd w.
By definition of A and C , we have

Q 1 -Q 2 = - R × R L 0 D h f (x, v) x 0 ∂ v D h ( f (y, w)η(y))d y d xd vd w = L 0 D h u(x) x 0 D h η(y)d y d x.
The y-integral then telescopes to:

Q 1 -Q 2 = L 0 D h u(x) x+h x - h 0 η(y)d yd x.
Since by periodicity we have,

L 0 D h u(x)d x = 0, this reduces to Q 1 -Q 2 = L 0 D h u(x) x+h x η(y)d yd x = L 0 η(y) y y-h D h u(x)d xd y = L 0 h 0 η(y)(u(y + ∆) -u(y -∆))d yd ∆. ( 44
)
Step 5. Here we argue that

Q 1 +Q 2 = L 0 h 0 D ∆ η D ∆ u d ∆d x. ( 45
)
For this we prove first that

Q 1 = 1 2 h 0 L 0 D ∆ η D ∆ u + η(u(x -∆) -u(x + ∆))d xd ∆. (46)
Indeed, combined with (44), this would give,

Q 1 +Q 2 = 2Q 1 -(Q 1 -Q 2 ) = L 0 h 0 D ∆ η D ∆ u d ∆d x.
which is (45). By definition of A and C :

Q 1 = R × R 1 R+ (v -w) L 0 D h f (x, v) x 0 ∂ v D h f (y, w)η(y) d y d xd vd w = +∞ -∞ L 0 D h f (x, v) v -∞ x 0 ∂ v D h f (y, w)η(y) d yd wd xd v = +∞ -∞ L 0 D h f (x, v) x 0 D h f (y, v)η(y) -D h η(y)d yd xd v.

Arguing as in

Step 4, we find

Q 1 = L 0 η(y) h 0 +∞ -∞ D h f (y -x, v) f (y, v) -1 d xd v d y.
After changing the names of the variables to y = x and x = ∆ we obtain

Q 1 = L 0 η( x) h 0 +∞ -∞ D h f ( x -∆, v) f ( x, v) -1 d ∆d v d x
Dropping the tildas, we can rewrite the inner term using the definition of f as:

h 0 +∞ -∞ D h f (x -∆, v)( f (x, v) -1)d ∆d v = h 0 +∞ -∞ ( f (x + ∆, v) -f (x -∆, v))( f (x, v) -1)d vd ∆ = - h 0 u(x)∨u(x+∆)∨u(x-∆) u(x) f (x + ∆, v) -f (x -∆, v)d vd ∆ = - h 0 D ∆ u + -D -∆ u + d ∆.
We thus find

Q 1 = - L 0 h 0 η D ∆ u + -D -∆ u + d ∆d x. 20 Using that L 0 η D ∆ u + d x = L 0 η(x -∆) D -∆ u -d x,
and similarly

L 0 η D -∆ u + d x = L 0 η(x -∆) D ∆ u -d x,
we obtain

Q 1 = 1 2 h 0 L 0 D ∆ η D ∆ u + -D -∆ η D -∆ u + + η -∆ D -∆ u -η ∆ D ∆ ud xd ∆ = 1 2 h 0 L 0 D ∆ η D ∆ u + -D -∆ η D -∆ u + + η(u(x -∆) -u(x + ∆))d xd ∆ = 1 2 h 0 L 0 D ∆ η D ∆ u + η (u(x -∆) -u(x + ∆)) d xd ∆,
which is (46). If we now integrate (45) over time, we find

Q 1 +Q 2 = T 0 L 0 h 0 D ∆ η|D ∆ u|d ∆d xd t . ( 47
)
Step 6. We finally argue that

Q 3 = - 1 2 h 0 L 0 |D ∆ u|D ∆ ud xd ∆ T 0 . ( 48
)
Let us recall that by definition of A:

Q 3 = R × R 1 R+ (v -w) L 0 x 0 D h f (t , x, v)D h f (t , y, w)d yd xd vd w t =T t =0
.

As above:

Q 3 = - L 0 h 0 +∞ -∞ v -∞ f (t , x, v)( f (t , x + ∆, w) -f (t , x -∆, w)d wd vd ∆d x t =T t =0 and, v -∞ ( f (t , x + ∆, w) -f (t , x -∆, w)d w = (u(t , x + ∆) -v ∧ 0) -(u(t , x -∆) -v ∧ 0). Therefore +∞ -∞ v -∞ f (t , x, v)( f (t , x + ∆, w) -f (t , x -∆, w)d wd v = u(t ,x) -∞ v -∞ ( f (t , x + ∆, w) -f (t , x -∆, w)d wd v = u(t ,x) -∞ (u(t , x + ∆) -v ∧ 0) -(u(t , x -∆) -v ∧ 0)d v = u(t ,x) (u(t ,x+∆)∧u(t ,x)) u(t , x + ∆) -vd v - u(t ,x) (u(t ,x-∆)∧u(t ,x)) u(t , x -∆) -vd v which, using that for a, b ∈ R, b (a∧b) (a -v)d v = - 1 2 ((a -b ∧ 0)) 2
gives

Q 3 = 1 2 L 0 h 0 (u(t , x + ∆) -u(t , x) ∧ 0) 2 -(u(t , x -∆) -u(t , x) ∧ 0) 2 d ∆d x t =T t =0 = 1 2 L 0 h 0 D ∆ u 2 --D -∆ u 2 -d ∆d x t =T t =0 = 1 2 L 0 h 0 D ∆ u 2 --D ∆ u 2 + d ∆d x t =T t =0 = - 1 2 L 0 h 0 |D ∆ u|D ∆ ud ∆d x t =T t =0
which proves (48). Combined with (47), this yields (41).

We can now prove Proposition 2.3

Proof of Proposition 2.3. By linearity, it is enough proving the estimate for g = 0. Thanks to (38) applied to η = |∂ x | ξ, we have

T 0 L 0 D h u 3 d xd t = 6 T 0 L 0 h 0 D ∆ η D ∆ u d ∆d xd t -3 h 0 L 0 |D ∆ u|D ∆ ud xd ∆ T 0
Thanks to Theorem A.1, for s < 1 and every function v, |v| B s p,r ≤ c v B s p,r (notice that it also holds for s = 1 if p = r = 2). Using the triangle inequality and the invariance of the Besov norms with respect to translations, we obtain that for s ∈ (0, 1),

|D ∆ u| B s p,r
≤ c u B s p,r . Applying (58) we get

T 0 L 0 h 0 D ∆ η D ∆ u d ∆d xd t = T 0 L 0 h 0 |∂ x | D ∆ ξ D ∆ u d ∆d xd t ≤ 1 π h 0 D ∆ ξ B 1-s p ,r D ∆ u B s p,r ≤ ch ξ B 1-s p ,r u B s p,r .
On the other hand, since

h 0 L 0 |D ∆ u|D ∆ ud xd ∆ T 0 ≤ ch L 0 u(0, x) 2 + u(T, x) 2 d x,
and since multiplying the equation ( 31) by u and integrating gives,

L 0 1 2 u(T, x) 2 d x - L 0 1 2 u(0, x) 2 d x = T 0 L 0 ∂ t 1 2 u 2 d xd t = T 0 L 0 u |∂ x | ξ d xd t ≤c ξ B 1-s p ,r u B s p,r ,
we have

h 0 L 0 |D ∆ u|D ∆ ud xd ∆ T 0 ≤ ch ξ B 1-s p ,r u B s p,r + u(0, •) 2 L 2 .
Putting this together, we find

1 h T 0 L 0 D h u 3 d xd t ≤ c ξ B 1-s p ,r u B s p,r + u(0, •) 2 L 2
which concludes the proof.

Remark. Starting from (33), one can obtain a larger family of estimates for the inhomogeneous Burgers equation. Unfortunately, the estimate

u 3 B 1/3 3,3 ≤c u B 1/3 3,3 g B 2/3 3/2,3/2 + u B s p,r ξ B 1-s p ,r + u(0, •) 2 L 2 ,
which would allow to avoid the logarithmic correction in [START_REF] Golse | Optimal regularizing effect for scalar conservation laws[END_REF], is borderline.

Proof of Proposition 2.7

Let us remind the reader the statement we want to prove:

Proposition (Comparison between Besov norms of different index r ). There exists c > 0 such that, for all L > 2, for every u solution of (K-S) with average zero, the following estimate holds:

u B 1/3 3,3 ≤ c ln 1/3 (L) u B 1/3 3,∞ . ( 49
)
Proof. The proof is elementary and resembles [14, Prop. 4 II), Step 2 &3]. Let us first cut the term that we want to bound in three parts:

+∞ 0 D h u 3 L 3 h d h h = 0 D h u 3 L 3 h d h h + L D h u 3 L 3 h d h h + +∞ L D h u 3 L 3 h d h h =A( ) + B ( ) +C .
The large scale term C is in fact controlled by A( ) and B ( ) by periodicity. Indeed:

C = +∞ L D h u 3 L 3 h d h h = +∞ n=1 (n+1)L nL D h u 3 L 3 h d h h ≤ +∞ n=1 1 n 2 L 0 D h u 3 L 3 h d h h ≤c (A( ) + B ( )) .
The intermediate scale term B ( ) is directly handled with thanks to u B 1/3 3,∞ :

B ( ) = L D h u 3 L 3 h d h h ≤ sup h∈R * + D h u 3 L 3 h L d h h ≤ ln(L/ ) u 3 B 1/3 3,∞ .
The small scale term A( ) is the most difficult to bound. We shall prove that:

A( ) ≤ c L 0 u(0, x) 2 d x 3/2 + L 3/2 u 3 B 1/3 3,∞ . ( 50
)
Before proceeding with the proof of (50), let us show that it is sufficient to conclude. Indeed, fix now = L -3/2 . Then, we obtain:

+∞ 0 D h u 3 L 3 h d h h ≤ c ln(L) u 3 B 1/3 3,∞ + c 1 L L 0 u(0, x) 2 d x 3/2
.

Taking now the time-space average yields (49). It remains to prove (50).

First, we prove that:

T 0 L 0 D h u 3 d xd t ≤ ch 2 L 0 u(0, x) 2 d x 3/2 + T 0 L 0 u 2 d x 3/2 d t . ( 51
)
We start by noting that:

L 0 D h u(t , x) 3 d x ≤2 sup x∈[0,L] |u(t , x)| L 0 x+h x ∂ x u(t , y)d y 2 d x ≤2 sup x∈[0,L] |u(t , x)| 2 L 0 (∂ x u) 2 d x.
For the convenience of the reader, we recall the argument for

sup x∈[0,L] |u| ≤ c L 0 u 2 d x 1/4 L 0 (∂ x u) 2 d x 1/4 . (52)
In fact, starting from:

u 2 (t , x) = u 2 (t , y) + x y 1 2 u(t , z)∂ x u(t , z)d z ∀y ∈ [0, L],
and using that u has zero average and thus vanishes somewhere, Cauchy-Schwarz inequality immediately gives (52). Therefore,

L 0 D h u 3 d x ≤ch 2 L 0 u 2 d x 1/4 L 0 (∂ x u) 2 d x 5/4
, and using the following Sobolev inequality (which can easily be proved by using Fourier methods):

L 0 (∂ x u) 2 d x ≤ L 0 u 2 d x 1/2 L 0 ∂ 2 x u 2 d x 1/2
, we get:

L 0 D h u 3 d x ≤ch 2 L 0 u 2 d x 7/8 L 0 ∂ 2 x u 2 d x 5/8 . ( 53
)
Now, we have to work a bit to extract information from the energy identity. Multiplying (K-S) by u and integrating over x, we obtain:

d d t L 0 u 2 d x =2 L 0 (∂ x u) 2 d x -2 L 0 ∂ 2 x u 2 d x ≤2 L 0 u 2 d x 1/2 L 0 ∂ 2 x u 2 d x 1/2 -2 L 0 ∂ 2 x u 2 d x ≤ L 0 u 2 d x - L 0 ∂ 2 x u 2 d x.
In a first step, we get from this differential inequality that there exists c > 0 such that:

L 0 u 2 (t + s, x)d x ≤ c L 0 u 2 (t , x)d x ∀s ∈ [0, 1]. (54)
In a second step, we get:

L 0 u 2 (t + 1, x)d x - L 0 u 2 (t , x)d x ≤ t +1 t L 0 u 2 d xd s - t +1 t L 0 ∂ 2 x u 2 d xd s.
The combination of both implies:

t +1 t L 0 ∂ 2 x u 2 d xd s + sup s∈[t ,t +1] L 0 u 2 (s, x)d x ≤ c L 0 u 2 (t , x)d x. (55)
Hence, together with (53) in the form of

D h u 3 d xd s ≤ C h 2 L 0 u 2 (t , x)d x 3/2 .
Using this inequality for t = 0 and in its integrated form between zero and T , we obtain (51).

We now claim that

T 0 L 0 u 2 d x 3/2 d t ≤ L 3/2 u 3 B 1/3 3,∞ . (56)
Indeed, using Hölder's inequality, the fact that u has zero average and Jensen's inequality we obtain:

T 0 L 0 u 2 (t , x)d x 3/2 d t ≤L 1/2 T 0 L 0 u(t , x) - 1 L L 0 u(t , h)d h 3 d xd t ≤L -1/2 T 0 L 0 L 0 |u(t , x) -u(t , x + h)| 3 d hd xd t ≤L 1/2 L 0 T 0 L 0 |u(t , x + h) -u(t , x)| 3 h d xd t d h ≤L 3/2 sup h∈R + T 0 L 0 D h u(t , x) 3 h d xd t .
Conclusion Putting together (51) and (56), we get as desired

A( ) = 0 T 0 L 0 D h u 3 d xd t d h h 2 ≤c 0 d h L 0 u(0, x) 2 d x 3/2 + L 3/2 u 3 B 1/3 3,∞ ≤c L 0 u(0, x) 2 d x 3/2 + L 3/2 u 3 B 1/3

A Besov spaces

A.1 Definition of time-space Besov spaces

We recall here some basics of the theory of Besov spaces. We refer to [1, Chapter 2, p. 51-121], for the construction of a dyadic Littlewood-Paley decomposition, and most of the proofs. Definition A.1 (Dyadic Littlewood-Paley decomposition). Let φ k k∈Z be a family of Schwartz functions such that their Fourier transforms F φ k k∈Z satisfy:

F (φ 0 )(ξ) = 0 ∀ |ξ| ∉ 2 -1 , 2 , F (φ k )(ξ) = F (φ 0 ) 2 -k ξ ∀k ∈ Z, ∀ ξ ∈ R, k∈Z F (φ k )(ξ) = 1 ∀ ξ ∈ R .
Then, for a L-periodic function u, we define its Littlewood-Paley decomposition as:

u k (t , •) = φ k * u(t , •)
where * denotes the periodic convolution. This allows us to define time-space Besov space B s p,r for s ∈

[0, +∞], p ∈ [1, ∞], r ∈ [1, ∞]
by the set of functions such that:

u B s p,r = k∈Z 2 r sk u k r L p 1/r < ∞ if r < ∞, u B s p,r = sup k∈Z 2 sk u L p < ∞ if r = ∞.
We are actually interested in a rescaled homogeneous Besov norm, defined by: We derive from this identity the following Besov estimate (see also [14, Step.

u B s p,r = lim sup T →+∞ 1 (LT )
Proof. The proof is analogous to [14, Step 1 p. 17]. By definition, we have:

h k = φ k * h.
Therefore, using the properties of convolution and of the quasi-orthogonality of the dyadic partition of unity we get:

h k = φ k * k ∈[k-1;k+1] φ k * h = |∂ x | -1 ∂ m x φ k * k ∈[k-1,k+1] u k .
Then, using Young's inequality, we obtain:

h k L p ≤ R |∂ x | -1 ∂ m x φ k d x k ∈[k-1,k+1] u k L p ≤2 k(m-1) R |∂ x | -1 ∂ m x φ 0 d x k ∈[k-1,k+1] u k L p ≤c2 k(m-1) k ∈[k-1,k+1] u k L p .

1 p 1 θ r 1 + 1 -θ r 2 , 0 φL 0 D h φD h g d x d h h 2 . 0 φ

 111120020 particularly well adapted for interpolation as seen from the following theorem:Theorem A.2 (Interpolation between Besov spaces). Let (s, p, r ), (s 1 , p 1 , r 1 ), (s 2 , p 2 , r 2 ) ∈ R + ×[1, +∞] 2 , and u ∈ B s with θ ∈]0, 1[, then u ∈ B sp,r and: follows from the definition of the Besov norms and an application of Hölder's inequality. In some lemmas that we will enunciate later, we will use the partial derivative |∂ x | which is slightly different from the classical ∂ x . It is defined via Fourier series:|∂ x | : underlines a link between Besov spaces and the operator |∂ x |:Lemma A.3. For all φ, g ∈ C 1 L (R): L |∂ x | g d x = 1 π +∞ 0Proof. Let us expand φ and g in Fourier series as can explicitly compute on the one hand:L |∂ x | g d x = n∈Z 2π |n| φ n g -n ,and on the other hand: hn -1 φ n e -2i π L hn -1 g -n

4 . 1 p = 1 and 1 r + 1 r = 1 , 0 D h φD h g d xd t d h h 2 .Lemma A. 5 .

 41111025 Let φ, g ∈ C 1 L (R). Then, for all p, p , r, ∈ [1, +∞], with1 p + for all s ∈]0, 1[ the following estimate holds:Then, Hölder's inequality leads us to the result: a useful lemma relating Besov norms of derivatives.Let s > 0, p, r ∈ [1, ∞], m ∈ N and u ∈ B s+m-1 p,r . Suppose h = |∂ x | -1 ∂ m x u.Then there exists a positive constant c depending only on (s, p, r ) such that the following estimate holds: h B s p,r ≤ c u B s+m-1 p,r .

  However, since we have a relatively simple and self-contained argument for it, we preferred to include it.

3,∞ . Remark. As in [14, Prop. 4 II), Step 2 &3], we could have used an L ∞ (in time) bound on L 0 u 2 d x (proven for instance in [5, Prop. 4]) to get (50) directly.

we use the notation O(ln 5/3+ (L)) to indicate that for every κ > 5/3, there exists c > 0 such that the left-hand side is bounded by c ln κ (L).

Let us stress that we cannot reach these exponents since some of the constants (in particular the one in (8)) explode.

See Figure 2 in Section 3.2 which represents the strategy in P .

Acknowledgment

FO acknowledges many discussions with Dorian Goldman on simplifications of the proof of [14]. MG was partially funded by a Von Humboldt fellowship. Most of this work was done while MG and MJ were hosted by the MPI-MIS Leipzig whose kind hospitality is acknowledged.

Hence:

which implies the aimed inequality.


Keywords: 1991 Mathematics Subject Classification. 34M55, 34M56, 34M03 Ordinary differential equations, Moduli spaces, Connections

We give some concrete examples of moduli spaces of connections. Precisely, we explain how to explicitely construct the moduli spaces of rank 2 fuchsian systems and logarithmic connections on the Riemann sphere with 4 poles. The former ones are affine cubic surfaces; the latter ones are analytically isomorphic to affine surfaces but not algebraically: they do not carry non constant regular functions. We end with some remark on arbitrary number of poles.

INTRODUCTION TO MODULI SPACES OF CONNECTIONS: SOME EXPLICIT CONSTRUCTIONS

1. Introduction: variation on an example of Serre Given a complex smooth projective curve X, a rank 1 (flat) connection, or equivalently a rank 1 local system on X, is an element of the cohomology group H 1 (X, C * ). This is the easiest example of a moduli space of connections. From the exact sequence of sheaves 0 → C * → O * → Ω 1 → 0 (where O * → Ω 1 is given by f → df f ), we can derive the associate long exact sequence of cohomology groups and extract the following [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 \ {x 1 , • • • , x 4[END_REF] 0 → H 0 (X, Ω 1 ) → H 1 (X, C * ) → Jac(X) → 0 making H 1 (X, C * ) into a principal C g -bundle over the Jacobian of the curve. Even simpler is the case when X is an elliptic curve: we get a C-bundle over the curve X ≃ Jac(X) itself. One may compactify it by adding a section at infinity in order to get a P 1 -bundle P → X: it is one of the two undecomposable P 1 -bundles found by Atiyah in [START_REF] Atiyah | Vector bundles over an elliptic curve[END_REF]. The section at infinity σ : X → P is the unique section whose image Σ := σ(X) has zero self-intersection in the total space S := Tot(P ) of the bundle. The group H 1 (X, C * ) identifies, as a variety, with the complement S \ Σ.

Viewing X as a Riemann surface, we get the Riemann-Hilbert correspondence

RH : H 1 (X, C * ) -→ Hom(π 1 (X), C * )
which associates to a local system its monodromy representation. Once we choose a system of generators for the fundamental group, we get an identification between the space of representations and the group C * ×C * . We thus get a complex analytic group isomorphism

(2)

H 1 (X, C * ) -→ C * × C * .
However, the two underlying varieties are not isomorphic from the algebraic point of view: while C * ×C * is an affine variety, there is no non constant regular functions on S. Indeed, such a function would extend as a rational function on S with polar divisor nΣ; one easily check that the zero divisor of such function should define a complete curve not intersecting Σ. But there is no complete curve in S since it is analytically isomorphic to an affine variety. This famous example is due to Serre.

There is a natural non degenerate 2-form ω on S constructed from the exact sequence (1) by choosing non zero 1-forms on the base X and on the fiber; it is well-defined up to a scalar constant. Isomorphism [START_REF] Atiyah | Vector bundles over an elliptic curve[END_REF] sends it to the 2-form du u ∧ dv v (up to a scalar). The 2-form ω extends on S as a rational 2-form with divisor 2Σ.

Let us briefly mention two other facts. The maximal compact subgroup given by unitary connections/representation is an analytic real torus providing a real section of P → X, and turns to be real algebraic on representation-side, namely S 1 × S 1 ⊂ C * × C * . The fibration P → X is sent to the foliation defined by u∂ u + τ v∂ v where (1 : τ ) is commensurable to periods of X.

One can modify Serre's example as follows. The elliptic involution ι : X → X lifts-up as a biregular involution ϕ : S → S. This follows from the unicity of the undecomposable bundle P , or equivalently from the natural action of the involution ι on local systems. The quotient surface S fibers over X/ι ≃ P 1 . The involution ϕ has 2 fixed points over each of the 4 Weierstrass points: one of them lie on the section at infinity Σ (which is invariant) and the other one is corresponding to a 2-torsion point of H 1 (X, C * ). We thus get 8 conic singular points on the quotient that we have to blow-up. The resulting surface Ŝ → S is equipped with the quotient 2-form ω having polar divisor

2 Σ + E 1 + E 2 + E 3 + E 4 where E i are exceptional divisors of conic points along Σ (the image of Σ); ω is non degenerate on the open set M := Ŝ \ ( Σ ∪ E 1 ∪ E 2 ∪ E 3 ∪ E 4 )
. This open set is no more a group, but share all other nice properties with Serre example. For instance, the divisor 2 Σ + E 1 + E 2 + E 3 + E 4 at infinity is (from the numerical point of view) like a degenerate elliptic fiber of type I * 0 in Kodaira list, there is no non constant regular function on M and it is analytically conjugated to an affine surface, namely the Cayley cubic, quotient of

C * × C * by the involution (u, v) → ( 1 u , 1 v ).
This second example is actually a particular case of moduli space of connections arising as space of initial conditions of one of the Painlevé VI equations, namely that one found by Picard.

These nice properties are common to general moduli spaces of connections (except structure of group, particular for the rank one) and this is a good motivation to study them. On the other hand, they naturally appear as spaces of initial conditions for isomonodromy equations, but we skip this point of view from our dicussion.

Moduli space of fuchsian systems

Consider a sl 2 -fuchsian system on the Riemann sphere with 4 poles

(3) dY dx = A 1 x -t 1 + A 2 x -t 2 + A 3 x -t 3 + A 4 x -t 4 with constant matrices A i , i = 1, 2, 3, 4, satisfying (4) A i = a i b i c i -a i ∈ sl 2 (C) and A 1 + A 2 + A 3 + A 4 = 0
(here and after we assume all 4 poles t i 's in the affine part for simplicity). The group SL 2 (C) acts on the Y -variable and thus on residues A i 's by simultaneous conjugacy. Indeed, change of variable Y = M Y ′ induces change of residues A ′ i = M -1 A i M . The spectrum of each matrix A i is preserved by this action. Let us fix the spectral data [START_REF] Oblezin | Isomonodromic deformations of sl(2) Fuchsian systems on the Riemann sphere[END_REF] -

det(A i ) = a 2 i + b i c i = θ 2 i INTRODUCTION TO MODULI SPACES OF CONNECTIONS: SOME EXPLICIT CONSTRUCTIONS 3
for some θ = (θ 1 , θ 2 , θ 3 , θ 4 ) (eigenvalues are ±θ i ). We want to describe the quotient space F uchs θ (X, D) of those systems by this action:

(6) (A 1 , A 2 , A 3 , A 4 ) ∈ (sl 2 (C)) 4 ; i A i = 0, det(A i ) = -θ 2 i / SL2(C) .
A straightforward computation shows that F uchs θ (X, D) is expected to be a surface (depending on 4 parameters θ i 's). However, it is non Hausdorff (as a topological space) in general. For instance, the orbit of the triangular system (7)

A i = θ i b i 0 -θ i
under diagonal conjugacy is not closed; its closure contains the diagonal system (with all b i 's vanishing) so that they define infinitesimally closed points in the quotient. Note however that, in this case, we have

θ 1 + θ 2 + θ 3 + θ 4 = 0.
Introduce the following functions on F uchs θ (X, D):

(8) X 1 := det(A 2 + A 3 ), X 2 := det(A 1 + A 3 ), X 3 := det(A 1 + A 2 ) and Y = tr (A 1 [A 2 , A 4 ])
where [A, B] = AB -BA is the Lie bracket. These functions are clearly invariant under SL 2 (C)-action.

Proposition 1. Assume θ 4 = 0. Then, the map

(9) (X 1 , X 2 , X 3 , Y ) : F uchs θ (X, D) -→ C 4
sends the moduli space onto the affine cubic surface S θ defined by (10)

X 1 + X 2 + X 3 = θ 2 1 + θ 2 2 + θ 2 3 + θ 2 4 and Y 2 4 + X 1 X 2 X 3 + (θ 2 1 -θ 2 3 )(θ 2 2 -θ4 2 )X 1 + (θ 2 2 -θ 2 3 )(θ 2 1 -θ4 2 )X 2 = (θ 2 1 + θ 2 2 -θ 2 3 -θ4 2 )(θ 2 1 θ 2 2 -θ 2 3 θ4
2 ) The map (9) above is one-to-one over the smooth part of S θ . Singularities arise when

• θ i = 0 and A i = 0 for i = 1, 2, 3;

• ±θ 1 ± θ 2 ± θ 3 ± θ 4 = 0 and all A i 's are simultaneously triangular up to conjugacy. In particular, apart from special values of θ listed just before, the quotient F uchs θ (X, D) is Hausdorff and is isomorphic to the smooth cubic surface S θ .

Proof. Since θ 4 = 0, we can assume

A 4 = θ 4 0 0 -θ 4
. This normalization is welldefined up to diagonal conjugacy so that the monomials a i and b i c j are invariant.

Using

A 1 + A 2 + A 3 + A 4 = 0, we can now express A 3 in function of A 1 and A 2 :    a 3 = -a 1 -a 2 -θ 4 b 3 = -b 1 -b 2 c 3 = -c 1 -c 2
Spectral data [START_REF] Oblezin | Isomonodromic deformations of sl(2) Fuchsian systems on the Riemann sphere[END_REF] gives the following conditions (11)

a 2 1 + b 1 c 1 = θ 2 1 , a 2 2 + b 2 c 2 = θ 2 2 and 2a 1 a 2 + 2θ 4 (a 1 + a 2 ) + b 1 c 2 + b 2 c 1 + θ 2 1 + θ 2 2 -θ 2 3 + θ4 2 = 0.
On the other hand, we get (12)

X 1 = 2θ 4 a 1 + θ 2 4 + θ 2 1 X 2 = 2θ 4 a 2 + θ 2 4 + θ 2 2 X 3 = 2θ 4 a 3 + θ 2 4 + θ 2 3 and Y = 4θ 4 (b 1 c 2 -b 2 c 1 )
Once we know X 1 , X 2 , X 3 , Y , we get the invariants a 1 , a

Moduli space of connections

Fix D = t 1 + • • •+ t n a reduced divisor (of poles) on the Riemann sphere X := P 1 with n ≥ 4. Fix eigenvalues {θ + i , θ - i } ⊂ C for each i = 1, . . . , n with integral sum n i=1 θ + i + θ - i = -d ∈ Z. For simplicity, assume generic condition θ ± 1 + • • •+ θ ± n ∈ Z for any choice of signs ± to avoid reducible connections. Denote by θ = (θ ± 1 , . . . , θ ± n ) the spectral data.
Consider the triples (E, ∇, l) called "parabolic connections" where

• E is a rank 2 vector bundle of degree d over P 1 , • ∇ is a logarithmic connection ∇ : E → E ⊗ Ω 1 (D) with polar divisor D, having residual eigenvalues θ + i and θ - i over the pole t i , • l = (l 1 , . . . , l n ) is a parabolic structure on E such that l i ⊂ E| ti belongs to
the eigenspace generated by θ + i . When θ + i = θ - i , we note that the parabolic l i is the eigendirection of θ + i ; the parabolic structure is therefore relevant only in case of equality θ + i = θ - i . Note also that Fuchs relation says that

n i=1 θ + i + θ - i + deg(E) = 0
, which explains the constraints given by d in the above definitions.

We say that two parabolic connections (E, ∇, l) and (E ′ , ∇ ′ , l ′ ) are equivalent when there is a bundle isomorphism φ : E → E ′ conjugating connections and parabolic structures. We denote by Con θ (X, D) the moduli space of parabolic connections for this equivalence. This can be viewed as a stack (see [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 \ {x 1 , • • • , x 4[END_REF]), but it can actually be constructed by GIT method (see [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF]); under our generic assumptions on θ, we simply get:

Theorem 2 (Inaba-Iwasaki-Saito). The moduli space Con θ (X, D) is a smooth irreducible quasi-projective variety of dimension 2(n -3) equipped with a regular symplectic 2-form ω.

The construction needs a choice of weights to impose stability condition; all parabolic connections are stable under our assumptions and the resulting quotient does not actually depends on this choice. However, if we allow non generic eigenvalues, then some reducible connections become unstable, and the quotient of semistable points depends on the choice of weights. Note also that we could have avoid parabolic structure in the discussion by assuming also θ + i = θ - i , but we will use parabolics later; this is why we already introduce this notion.

Connection matrix.

Let us now be more explicit for readers that are not familiar to connections. Following Birkhoff, any rank vector bundle E on P 1 splits as a direct sum of line bundles. In particular, for the rank 2 case, we have

E = O(d 1 ) ⊕ O(d 2 ) with d 1 ≤ d 2 and d 1 + d 2 = d.
Choose sections e i of O(d i ) whose divisor (zero or pole) is supported by ∞ ∈ P 1 so that, over the affine part C = P 1 \{∞}, the vector bundle is trivial:

E| C = Ce 1 ⊕Ce 2 .
We can describe the connection ∇ as follows:

∇| C : Y → dY + Ω • Y
where Y is a section of E| C and Ω is a 2 × 2-matrix of meromorphic 1-forms:

(13) Ω = n i=1 A i x -t i dx + B(x)dx
with B holomorphic (and x the affine variable of C = P 1 \ {∞}). Let e ′ i be sections of O(d i ) defined by e i = x di e ′ i . At infinity, we must express the connection in the basis (e ′ 1 , e ′ 2 ) and, setting

Y = M Y ′ with M = x d1 0 0 x d2 we find ∇| P 1 \{0} : Y ′ → dY ′ + Ω ′ • Y ′ with Ω ′ = M -1 ΩM + M -1 dM which must be holomorphic at ∞. If we now write Ω = A(x) dx n i=1 (x-ti) , this holomorphy conditions says that A(x) = - d 1 0 0 d 2 x n-1 + a(x) b(x) c(x) d(x) with    b polynomial of degree ≤ n -2 -(d 2 -d 1 ) a, d polynomials of degree ≤ n -2 c polynomial of degree ≤ n -2 + (d 2 -d 1 )
Fuchsian systems are just logarithmic connections on the trivial bundle

d 1 = d 2 = 0. Example 3 (Case n = 4). When E = O ⊕ O(1)
, then holomorphy at infinity gives the following constraints

A 1 + A 2 + A 3 + A 4 = 0 0 ⋆ -1 and B(x) ≡ 0.
When E = O(-1) ⊕ O(1), then we get constraints

A 1 + A 2 + A 3 + A 4 = 1 0 ⋆ -1 , B(x) ≡ 0 0 ⋆ 0 (constant matrices) and i A i x -t i = ⋆ b i (x-ti) ⋆ ⋆ .
Finally, when d 2 -d 1 > 2, the [1, 2]-coefficient of all A i 's and B(x) must be zero so that we are in the reducible case. In other words, under our generic assumption on θ, we have d 2 -d 1 = 0, 1, 2. For degree d = 0 or 1, we are led to the following possibilities

• d = 0 and (d 1 , d 2 ) = (0, 0) or (-1, 1), • d = 1 and (d 1 , d 2 ) = (0, 1).
We now impose spectral data by

tr(A i ) = θ + 1 + θ - i and det(A i ) = θ + 1 • θ - i .
The parabolic structure is therefore given by

l i = C • b i θ + i -a i or C • θ + i -d i c i where A i = a i b i c i d i
except when θ + i = θ - i and A i is a scalar matrix, in which case l i may be arbitrarily choosen: it is an extra data in this case.

3.2. Bundle automorphisms. So far, we have described the space of parabolic connections up to Birkhoff normalization of the bundle. Then, we would like to quotient by the action of bundle automorphisms on connections. They can be described as follows.

• When d 1 = d 2 , then E is just the twist of the trivial bundle by O(d 1 ) and automorphisms are the same: GL 2 (C) acts by conjugacy on the connection matrix (13), i.e. on matrices A i 's (B is scalar and the action is trivial on it).

• When d 1 < d 2 , then the automorphism group is given in the trivialization chart E| C = Ce 1 ⊕ Ce 2 by M = λ 1 0 f (x) λ 2 with f (x) polynomial of degree ≤ d 2 -d 1 ,
and it is acting on connection matrix (13) by

Ω → Ω ′ = M -1 ΩM + M -1 dM.
The action on parabolics is obvious.

Example 4. Assume n = 4 and (d 1 , d 2 ) = (-1, 1); for simplicity, assume also

(θ + i , θ - i ) = (θ i , -θ i ). The matrix connection writes Ω = A(x) dx i (x-ti)
where

A(x) = x 3 + a(x) b c(x) -x 3 -a(x) with    b ∈ C a polynomial of degree ≤ 2 c polynomial of degree ≤ 4
Note that b = 0 otherwise the connection would be reducible. Conjugating by a diagonal matrix M , we can assume b = 1. The action of unipotent isomorphisms is described by

M = 1 0 f (x) 1 : x 3 + a 1 c -x 3 -a → x 3 + a + f 1 c -2(x 3 + a)f -f 2 + f ′ -x 3 -a -f
where f is any degree 2 polynomial. Choosing f = -a, we derive a unique normal form

Ω = x 3 1 c(x) -x 3 dx i (x -t i ) where c(x) i (x -t i ) = c 0 + i τ i θ 2 i x -t i with c 0 ∈ C
and constants τ i are defined by ( 14)

1 i (x -t i ) = i 1 τ i (x -t i ) , i.e. τ i = j =i (t i -t j ).
It follows that the moduli space of connections on the fixed bundle E = O(-1)⊕ O(1) is the affine line A 1 ∋ c 0 (extra parameter in the above normal form). To get the full moduli space Con θ (P 1 , {t 1 , t 2 , t 3 , t 4 }) of degree 0 bundles with the same eigenvalues (satisfying

(θ + i , θ - i ) = (θ i , -θ i ) and ±θ 1 ± θ 2 ± θ 3 ± θ 4 ∈ Z),
we have to patch together this affine line A 1 with the affine cubic surface S θ of Fuchsian systems given by Proposition 1 (see remark ending example 3). Following Theorem 2, one should obtain a smooth irreducible quasi-projective surface.

Isomorphisms between moduli spaces.

There are many isomorphisms between moduli spaces of connections Con θ (X, D) modifying spectral data. First, one can twist a parabolic connection (E, ∇, l) by a rang 1 logarithmic connection (L, ζ):

• L is a line bundle of degree k, L = O(k), • ζ : L → L ⊗ Ω 1 (D)
is determined by its eigenvalues λ i satisfying

λ 1 + • • • + λ n + k = 0.

INTRODUCTION TO MODULI SPACES OF CONNECTIONS: SOME EXPLICIT CONSTRUCTIONS 7

The action on connection matrix (13) is given by

Ω → Ω + i λ i 0 0 λ i dx x -t i
and this defines an isomorphism (15)

Con θ (X, D)

⊗(L,ζ) -→ Con θ ′ (X, D) where (θ + i ) ′ = θ + i + λ i (θ - i ) ′ = θ - i + λ i The degree is changed by d → d ′ = d + 2k.
In case of even degree, we can assume up to such an isomorphism that d = 0 and moreover θ + i + θ - i = 0 for all i (we get a sl 2 -connection). In the odd case, we can assume d = 1; then, in case n = 4, necessarily E = O ⊕ O(1) (see remark ending example 3).

The other family of isomorphisms comes from elementary transformations of parabolic bundles. Given a parabolic bundle (E, l) over (X, D), then we define (E ′ , l ′ ) := Elm - ti (E, l) by • E ′ is the vector bundle defined by the subsheaf E ′ ⊂ E generated by those sections directed by l i , i.e. defined by the exact sequence of morphisms of sheaves

0 → E ′ → E → E/ li → 0
where l i is viewed as a sky-scrapper sheaf.

• the new parabolic direction l ′ i ⊂ E ′ | ti is defined by the kernel of the inclusion morphism E ′ → E. A connection ∇ on E induces a connection ∇ ′ on the subsheaf E ′ ⊂ E; if (E, ∇, l)
is parabolic, ∇ preserves the direction l i and ∇ ′ is still parabolic with respect to (E ′ , l ′ ). Over t i , the new eigenvalues are (θ + i , θ - i ) ′ := (θ - i + 1, θ + i ) (l ′ i is now in the eigenspace corresponding to θ - i + 1). Over other t j 's, the morphism E ′ → E is a local bundle isomorphism and the eigenvalues remain unchanged. This operation induces a map (16)

Con θ (X, D)

Elm - t i -→ Con θ ′ (X, D) where    (θ + i ) ′ = θ - i + 1 (θ - i ) ′ = θ + i all other (θ ± j ) ′ = θ ± j
In local trivialization of the bundle E around t i , with basis (e 1 , e 2 ) such that e 1 generates the parabolic direction l i at t i , we get a parabolic connection matrix of the form

Ω = θ + i ⋆ 0 θ - i dx x -t i
+ holomorphic matrix, and

l i = C 1 0 .
The new vector bundle E ′ is generated near t i by (e 1 , (x -t i )e 2 ) so that the connection matrix of ∇ ′ is given by

Ω ′ = M -1 ΩM + M -1 dM where M = 1 0 0 x -t i
which gives

Ω ′ = θ + i 0 ⋆ θ - i + 1 dx x -t i
+ holomorphic matrix, and l i = C 0 1 .

One easily check that applying twice Elm - ti gives the twist by the unique logarithmic connection on O(-1) having a single pole (with residue +1) at t i :

Elm - ti • Elm - ti (E, ∇, l) = (O(-1), dx x -t i ) ⊗ (E, ∇, l).
Note that Elm - ti decreases the degree of the vector bundle by -1 (whence the sign); one can also define Elm + ti (E, l) := (O(-1), dx x-ti ) ⊗ Elm - ti (E, l) increasing the degree of the vector bundle by +1, it is actually the inverse map

Elm + ti • Elm - ti (E, ∇, l) ≃ (E, ∇, l).
In fact, all these isomorphisms commute together.

One can also define an isomorphism permuting θ + i and θ - i , modifying the parabolic structure (we switch to the other eigenspace) and get a non abelian group. We omit this from our discussion. Finally, up to isomorphism, we can always assume degree d = 0 and θ + i + θ - i = 0 for all i. However, this may be not the best choice as we shall see.

The Painlevé case n = 4

In order to get an explicit irreducible moduli space, it is obviously better to choose d = 1 instead of d = 0 since then, we can work on the single vector bundle E = O ⊕ O(1) (see remark ending example 3). 4.1. An explicit construction. We start proceeding like in example 4. The connection matrix writes Ω = A(x) dx i (x-ti) where

A(x) = a(x) b(x) c(x) -x 3 + d(x) with    b polynomial of degree ≤ 1 a, d polynomials of degree ≤ 2 c polynomial of degree ≤ 3
Again, b(x) ≡ 0 otherwise the connection is reducible. Assume for the moment that b(x) is not constant so that we can normalize it to b(x) = x -q. The action by unipotent matrices is not enough to kill a(x) however (the freedom f (x) like in example 4 is now of degree 1). The idea is to apply an elementary transformation Elm + q at the parabolic direction given by O(1). Concretely, we apply the bundle birational isomorphism given by

M = 1 0 0 1 x-q
and obtain the new connection matrix 2), we have more automorphisms and we are able to kill the polynomial coefficient a(x) by means of unipotent automorphisms. We now get the normal form ( 17)

M -1 ΩM + M -1 dM = a(x) i (x-ti) 1 i (x-ti) (x-q)c(x) i (x-ti) -x 3 +d(x) i (x-ti) -1 x-q dx On the new bundle E ′ = O ⊕ O(
Ω ′ = 0 1 i (x-ti) c(x) (x-q) i (x-ti) -x 3 + d(x) i (x-ti) -1
x-q dx with c(x) polynomial of degree 5. Taking into account the spectral data, we get

Ω = i 0 1 τi -τ i θ + i θ - i θ + i + θ - i dx x -t i + 0 0 p -1 dx x -q + 0 0 c 0 0 dx for constants p, c 0 ∈ C.
Here, we have also assumed q ∈ C \ {t 1 , t 2 , t 3 , t 4 }. Indeed, when q = t i , we apply an elementary transformation over the pole which might change the eigenvalues. Finally, note that x = q must be an apparent singular point, which means, by Fuchs' local theory, that around x = q the kernel of the residual part is also an eigenvector for the constant part of the connection matrix:

Ω • 1 p = ⋆ 1 p + o(x -q);
this fixes the value of c 0 . Finally, the initial connection is determined by the parameters (p, q) which have the following geometrical meaning:

• q stands for the position of the apparent singular point of the new connection, • p stands for the kernel of the residue of ∇ over x = q.

Note also that, over x = t i , the two eigendirections of ∇ are given by respectively correspond to the zero section, the section dx i (x-ti) and the section at infinity. If S denotes the associated ruled surface (total space of the projective bundle), then the total space of Ω 1 (D) identifies with S \Σ where Σ is the section at infinity. On fibers F i : {x = t i }, a natural affine coordinate if given by the residue: the section dx i (x-ti) has residue 1 τi at x = t i . Therefore, the two eigendirections over x = t i correspond to the points s + i , s - i ∈ F i satisfying x = t i and Res ti (s ± i ) = θ ± i . Over x = q, the kernel of the residual matrix also defines a point s:

1 τ i θ +
s ± i : 1 τ i θ ± i and s : 1 p i (q -t i )
.

We have just defined a birational map

Con θ (X, D) S ; (E, ∇, l) → s.
Now consider the blow-up of the surface π : Ŝ → S at all 8 points s ± i : we denote by E ± i the exceptional divisors and Σ, Fi the strict transforms. When θ + i = θ - i , then first blow-up the point s i := s + i = s - i , and then blow-up the intersection point between the exceptional divisor E i and the strict transform of the fiber F i ; denote by E ′ i the last exceptional divisor. Then the birational map above induces an isomorphism:

Con θ (X, D) ∼ -→ M θ := Ŝ \ Σ ∪ F1 ∪ F2 ∪ F3 ∪ F4 .
For a general point s ∈ M θ , formula (17) gives a parabolic connection with an extra apparent singular point; after elementary transformation over x = q, we get a parabolic connection in Con θ (X, D). When q → ∞, there is a limit point in Con θ (X, D) provided that s tends to an affine point of the fiber. When q → t i , one can find a limit point in Con θ (X, D) if, and only if, s tends to one of the exceptional divisors, not to Fi ; over the point s - i , we get all parabolic connections in Con θ (X, D) such that the parabolic l i lies in the destabilizing subbundle O(1).

With our notations, the 2-form dp ∧ dq has polar divisor 2 Σ + F1 + F2 + F3 + F4 and is non zero at any other point: it defines a non degenerate holomorphic volume form on the moduli space that should better viewed as a symplectic structure. 4.2. Description and properties of moduli spaces. Fix (X, D) = (P 1 , {t 1 , t 2 , t 3 , t 4 }) as before and consider the Hirzebruch surface S given by the total space of the projective bundle P(O ⊕ Ω 1 (D)). This surface may be obtained from the total space of the projective bundle P(O ⊕ Ω 1 ) by applying 4 elementary transformations directed by the section at infinity O; from this point of view, the Liouville form on the total space of Ω 1 induces a rational 2-form ω on P(O ⊕ Ω 1 (D)) with polar divisor 2Σ + F 1 + F 2 + F 3 + F 4 : as before, Σ is the section defined by O and F i is the fiber over t i . An affine chart is defined by

(x, y) : P(O ⊕ Ω 1 ) \ Σ ∪ F ∞ → C × C
where F ∞ is the fiber over x = ∞ and y is normalized so that

• y = 0 corresponds to the section defined by 0 ∈ H 0 (X, O ⊕ Ω 1 (D)),

• y = 1 corresponds to the section defined by

dx i (x-ti) ∈ H 0 (X, O ⊕ Ω 1 (D)) (and y = ∞ is defined by O).
For each finite point y ∈ F i , we can associate the residue Res ti (y dx i (x-ti) ) = y τi (see definition ( 14)). This gives us a natural parametrization of fibers F i . Choose 2 points s ± i on each fiber F i (possibly, s + i = s - i ). Then, blow-up these 8 points and denote by Ŝ the blow-up surface, and still denote by Σ and Fi the strict transforms. Then consider the open part

M θ := Ŝ \ Σ ∪ F1 ∪ F2 ∪ F3 ∪ F4 .
The 2-form ω extends as a holomorphic symplectic form on M with polar divisor

(ω) ∞ = 2 Σ + F1 + F2 + F3 + F4 .
Consider on each fiber F i the point m i defined by the arithmetic mean of s + i and s - i . Precisely, if we denote Res ti (s ± i ) = θ ± i , then m i is defined by Res ti (m i ) =

θ + i +θ - i 2 .
Since S \ Σ is the total space of Ω 1 (D), there is a one-to-one correspondence between global holomorphic sections of Ω 1 (D) and global sections of S → P 1 not intersecting Σ. These sections form a 3-dimensional family. Assume first that there exists such a section passing through all 4 points m i ; this is equivalent to say

i (θ + i + θ - i ) = 0.
In this case, it can be checked that 2 Σ + F1 + F2 + F3 + F4 is the degenerate fiber of an elliptic fibration on Ŝ. In this case, the open part M may be viewed as a moduli space of Higgs bundles and this fibration is the well-known Hitchin fibration. The fibration is defined by those elements of the linear system |2 Σ + F1 + F2 + F3 + F4 | on S passing through the 8 points s ± i . Assume now that i (θ + i + θ - i ) = 0. Then, after applying a bundle isomorphism of the form y → cy, we may assume i (θ

+ i + θ - i ) = 1. If θ ± 1 + θ ± 2 + θ ± 3 + θ ± 4 ∈
Z whatever the choice of signs, the surface M θ identifies with the moduli space of connections Con θ (X, D) discussed in the previous section. On the other hand, when

θ ± 1 + θ ± 2 + θ ± 3 + θ ± 4 ∈
Z for some (maybe several) choices of signs, then M θ can still be viewed as a moduli space of stable connections (we have to delete some reducible connection to obtain a GIT moduli space, see [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF]). By the Riemann-Hilbert correspondance (see [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF]), there is an analytic mapping

RH : M θ -→ χ ⊂ C 3
where χ is an affine cubic surface. For generic θ, this is an analytic isomorphism; for special values of θ like above, the affine surface χ is singular and the map RH is a minimal analytic resolution of singularities. It follows that the only occurence of complete curves in M θ is for special values of θ: they come from exceptional divisors in the resolution of singularities of χ: they are rational, and there are at most 4 such curves in a given M θ . In particular, the divisor 2 Σ + F1 + F2 + F3 + F4 is no more contained in an elliptic fibration.

After applying an elementary transformation

Elm - t4 : Con θ (X, D) ∼ -→ Con θ ′ (X, D)
we obtain new eigenvalues

(θ + i ) ′ = θ i (θ - i ) = -θ i for all i = 1, 2, 3, 4.
This latter moduli space splits into the disjoint union of those connection defined on the trivial bundle, namely Fuchsian systems discussed in section 2, and those defined on E = O(-1) ⊕ O(1) (see section 4). In fact, a parabolic connection

(E = O ⊕ O(1), ∇, l) of Con θ (X, D)
is sent to a connection on the non trivial bundle by the elementary transformation if, and only if, its parabolic l 4 lie on the subbundle O(1) ⊂ E. The locus of those special parabolic connections in M θ is given by the exceptional divisor E - 4 → s - 4 (or we better should say the affine part of this divisor once we have deleted the intersection point with F 4 ). Therefore, E - 4 identifies via Elm - t4 to the moduli space A 1 of those connections on O(-1) ⊕ O(1) discussed in section 4. Moreover, after deleting E - 4 from M θ , we get an isomorphism with the moduli space of Fuchsian systems computed in section 2, i.e. a cubic affine surface:

M θ \ E - 4 Elm - t 4 ∼ -→ F uchs θ ′ (X, D) ∼ -→ S θ ⊂ C 4 .
We can do in a similar way with other elementary transformations Elm ± ti (that we have to compose with a convenient twist in order to fit with sl 2 -systems discussed in section 2) and we promtly deduce that M θ becomes a cubic affine surface once we delete any one of the exceptional divisors E ± i .

The Garnier case n > 4

The computation of section 4.1 generalizes as follows. Fix D = t 1 + • • • + t n a reduced divisor one X := P 1 , n ≥ 4. Fix eigenvalues θ, θ ± i ∈ C for each i = 1, . . . , n, with n i=1 (θ + i + θ - i ) + 1 = 0. Assume θ generic for simplicity:

θ + i -θ - i ∈ Z and θ ± 1 + • • • + θ ± n ∈ Z.
Consider the moduli space Con θ (X, D) of corresponding connections. Note that we can omit parabolic structure from the discussion: under genericity of θ, the parabolic structure l is determined by the connection (E, ∇).

We would like to explicitely describe the generic connection and derive an explicit open subset of Con θ (X, D). Let us first assume E = O ⊕O(1). The special subbundle O(1) ⊂ E is stabilized by ∇ at exactly n -3 points counted with multiplicity; these are the zeroes of the composition map

L = O(1) inclusion -→ E ∇ -→ E ⊗ Ω 1 (D) quotient -→ (E/L) ⊗ Ω 1 (D).
These points are also the extra apparent singular points for the fuchsian scalar equation derived from ∇ and the cyclic vector L. Let us consider the case where these zeroes and the poles t i 's are two-by-two distinct; denote the by q 1 , . . . , q n-3 the apparent points. We also assume q j = ∞ for simplicity.

Consider now the parabolic structure l defined over each q j by the special line bundle L. Still assuming all t i 's and q j 's two-by-two distinct, apply a positive elementary transformation at each q j directed by l (i.e. by L = O(1) ⊂ E). Let (E ′ , ∇ ′ , l′ ) be the resulting parabolic connection; note that E ′ = O ⊕ O(n -2).

Proposition 5. The connection (E ′ , ∇ ′ ) is given, up to bundle automorphism, by the connection matrix (we follow notations of section 3.1)

Ω = n i=1 0 1 τi -τ i θ + i θ - i θ + i + θ - i dx x -t i + n-3 j=1 0 0 p j -1 dx x -q j + 0 0 c(x) 0 dx
where c(x) is polynomial of degree n -4, determined by the fact that all poles q j 's are non logarithmic (with scalar monodromy). The connection is determined by the new parabolic structure l′ defined by kernel of residual matrices over the q j 's.

Proof. We proceed like in section 4. Then, up to a diagonal automorphism, we have b(x) = n-3 j=1 (x -q j ). We now apply the n -3 elementary transformations and get the new connection matrix From this statement, one easily derive the following open subset of Con θ (X, D). Consider the surface S defined by the total space of P(O ⊕ Ω 1 (D)) and denote by Σ the section defined by Ω 1 (D): S \ Σ naturally identifies with the total space of Ω 1 (D). In particular, the affine part of the fiber F i over t i has the natural chart Res ti : F i \ Σ ∼ -→ C given by taking the residue of sections of Ω 1 (D); we define two points s ± i ∈ F i by Res ti (s ± i ) = θ ± i . Let π = Ŝ → S be the blow-up of S at all 2n points and denote by Σ and Fi the strict transforms. Then consider the open part

M θ := Ŝ \ ( Σ ∪ F1 ∪ • • • ∪ Fn )
and the symetric product

(M θ ) (n-3) := M θ × • • • × M θ /Perm(1, • • • , n).
Then the above normal form defines an embedding

(M θ ) (n-3) \ ∆ ֒→ Con θ (X, D)
where ∆ is the "codimension one diagonal" defined by those {q i , p i } for which q i = q j for some i = j. This is the point of view developped by Oblezin in [START_REF] Oblezin | Isomonodromic deformations of sl(2) Fuchsian systems on the Riemann sphere[END_REF] where the compactification of this picture is discussed. Another point of view has been recently studied by Saito and the author in [START_REF] Loray | Lagrangian fibration in duality on moduli space of rank two logarithmic connections over the projective line[END_REF].

2 and b 1 c 2 - 1 , b 2 c 2 and b 1 c 2 + b 2 c 1 , and therefore b 1 c 2 and b 2 c 1 .

 21211 b 2 c 1 . From relations (11), we promptly deduce b 1 c The cubic equation (10) just says that these invariants satisfy the obvious relation (b 1 c 1 )(b 2 c 2 ) = (b 1 c 2 )(b 2 c 1 ). We can thus recover (b 1 , b 2 , c 1 , c 2 ) uniquely up to the diagonal action, except when either b 1 = b 2 = 0, or c 1 = c 2 = 0; in these latter cases, there are several possible solutions (A 1 , A 2 , A 3 ) up to diagonal action, but all of them are triangular.

i and 1 τ

 1 i θ - i corresponding respectively to eigenvalues θ + i and θ - i . Let us identify O(2) = Ω 1 (D), and consider the projectivization P(O ⊕ Ω 1 (D)) of our vector bundle E as the fiber-compactification of Ω 1 (D): the line bundles in E generated by 1

1 .

 1 On the initial bundle E = O ⊕ O(1), the connection matrix writes Ω = A(x) dx i (x-ti) whereA(x) = a(x) b(x) c(x) -x n-1 + d(x) with    deg(b) ≤ n -3 deg(a), deg(d) ≤ n -2 deg(c) ≤ n -1

1 idx 1 i

 11 (x-ti) c(x) j (x-qj ) i (x-ti) --x n-1 +d(x) i (x-ti) -On the new bundle E ′ = O ⊕ O(n -2), we can use unipotent automorphisms of E ′ to kill the polynomial coefficient a(x). We now get the unique normal form (18)Ω ′ = 0 (x-ti) c(x) i (x-ti) j (x-qj ) -x n-1 + d(x)i (x-ti) -dx with c(x) and d(x) polynomials of degree 3n-7 and n-2 respectively. Now, taking into account the spectral data, we get the normal form of the statement.

The surface M θ is not affine however. In fact, there are no non-constant regular functions on M θ (see [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 \ {x 1 , • • • , x 4[END_REF]). This mainly follows from numerical properties (2 Σ + F1 + F2 + F3 + F4 ) • Σ = (2 Σ + F1 + F2 + F3 + F4 ) • Fi = 0; indeed, a rational function on Ŝ whose polar divisor is contained in the support of 2 Σ + F1 + F2 + F3 + F4 would define a fibration of complete curves inside the complement M θ , contradicting the (almost) non existence of complete curves. This generalize Serre example.

4.3.

How complete curves arise for special parameters. When θ + i = θ - i for some i, then the two points s + i and s - i coincide in S. In this case, we have to blowup twice over s + i = s - i to get the surface Ŝ. Precisely, we first blow-up this point to get an exceptional divisor E i , and then blow-up the intersection point between E i and the strict transform of F i ; we get a second exceptional divisor E ′ i and still denote by E i and Fi the strict transforms in the resulting surface Ŝ. We define similarly the open set

Then, E i is a rational curve in M θ having -2 self-intersection number. This curve corresponds to the locus of those parabolic connections (E, ∇, l) for which the residual matrix at t i is the scalar matrix θ + i • I. Similarly, when θ + i = θ - i + 1, from Fuchs' local theory, we can check that a generic element (E, ∇, l) of Con θ (X, D) will have a logarithmic singular point at t i (the local monodromy has a non trivial Jordan block); however, for some special parabolic connections, the singular point is not logarithmic anymore (it has diagonal monodromy). The locus of those special connections is a complete and smooth rational curve in M θ , namely the strict transform of the curve C ⊂ S characterized as follows:

• q : C → P 1 has degree 2 and C does not intersect Σ;

• C intersects F i at s + i and has two smooth local branches at this point; • C intersects other fibers F j at both s + j and s - j . The first condition fixes the linear system on S that contains the curve:

The two other conditions characterize C in this linear system. After blowing-up, the resulting curve Ĉ becomes smooth and does not intersect strict transforms F j 's anymore.

When θ + i -θ - i ∈ Z, the story is the same. Let us just characterize the special curve C in the case θ + i = θ - i + 2: • q : C → P 1 has degree 4 and C does not intersect Σ; • C intersects F i three times at s + i and one time at s - i ; • C intersects other fibers F j twice at both s + j and s - j . This can be checked by straightforward formal computation.

Another occurence of complete curves in M θ comes from reducible connections. When say θ + 1 + θ + 2 + θ + 3 + θ + 4 = 0, then some reducible connections arise in Con θ (X, D): each of them stabilize a trivial subbundle O ⊂ E and induces on it a logarithmic rank 1 connection with eigenvalue θ + i at t i . The locus of these reducible connections form a section Ĉ for the projection q : M θ → P 1 . The corresponding curve C ⊂ S is the section of q : C → P 1 that does not intersect Σ and intersecting each F i at s + i . 4.4. Link with Fuchsian systems. As we shall see here, if we delete one of the exceptional divisors E ± i → s ± i from M θ , then the surface becomes affine. Consider the moduli space Con θ (X, D) with eigenvalues θ satisfying