F F Golse

Clément Cyril Imbert

Clément Mouhot

A F Vasseur

Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation

Keywords: Mathematics Subject Classification. 35H10, 35B65 Hypoelliptic equations, kinetic theory, Fokker-Planck equation, Landau equation, ultraparabolic equations, Kolmogorov equation, Hölder continuity, De Giorgi method, Moser iteration, averaging lemma

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

∂ t f + v • ∇ x f = ∇ v • (A[f ]∇ v f + B[f ]f ) where        A[f ](v) = a d,γ ˆRd I - w |w| ⊗ w |w| |w| γ+2 f (v -w) dw, B[f ](v) = b d,γ ˆRd |w| γ wf (v -w) dw
with γ ∈ [-d, 0] and a d,γ > 0. We note that the main physical case is that of Coulomb interactions when γ = -d and d = 3 (giving rise to the Landau-Coulomb equation in plasma physics); the other cases are hard potentials γ ∈ (0, 1] (not covered here 1 ), Maxwellian molecules γ = 0, and soft potentials γ ∈ [-d, 0). It can be rewritten as follows

(1.2) ∂ t f + v • ∇ x f = ∇ v • (A[f ]∇ v f ) + B[f ]∇ v f + c[f ]f where c[f ](v) =    c d,γ ˆRd |w| γ f (v -w) dw if γ > -d, c d,γ f if γ = -d.
We assume that the mass, energy and entropy density of the weak solution f satisfy the following control at a given space-time point (x, t):

(1.3) C(x, t)                M 1 ≤ M (x, t) = ˆRd f (x, v, t) dv ≤ M 0 (local mass), E(x, t) = 1 2 ˆRd f (x, v, t)|v| 2 dv ≤ E 0 (local energy), H(x, t) = ˆRd f (x, v, t) ln f (x, v, t) dv ≤ H 0 (local entropy).
The weak solutions to equation (1.1) on

U x × U v × I, U x ⊂ R d open, U v ⊂ R d open, I = [a, b] with -∞ < a < b ≤ +∞, are defined as functions f ∈ L ∞ t (I, L 2 x,v (U x × U v ))) ∩ L 2 x,t (U x × I, H 1 v (U v )) such that ∂ t f +v •∇ x f ∈ L 2
x,t (U x ×I, H -1 v (U v )), f satisfies estimates (1.3) and satisfies the equation in the sense of distributions

f C α (B 1/2 ×B 1/2 ×(-1/2,0]) ≤ C f L 2 (B 1 ×B 1 ×(-1,0]) + f 1+ |γ| d L ∞ (B 1 ×B 1 ×(-1,0])
for some α and C depending on dimension, M 1 , M 0 , E 0 and H 0 .

Remark 2. After this work was completed, we heard from a nice recent preprint of Cameron, Silvestre and Snelson [START_REF] Cameron | Global a priori estimates for the inhomogeneous landau equation with moderately soft potentials[END_REF] that establishes a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials (γ ∈ [-2, 0]), with arbitrary initial data, under the assumption (1.3). When γ ∈ [-2, 0], it thus allows us to remove the L ∞ assumption on the weak solution in Theorem 1.

Under the assumptions of Theorems 1, it is known [START_REF] Desvillettes | On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications[END_REF][START_REF] Silvestre | Upper bounds for parabolic equations and the Landau equation[END_REF] that the diffusion matrix A[f ] is uniformly elliptic and B[f ] and c[f ] are essentially bounded for bounded velocities (see Lemmas 31 and 32 in Appendix). In particular, the assumption (1.7) given below, and under which our main results (Theorems 3 and 4) hold true, is satisfied. 1 Our method would apply as well in this case with no changes, we did not include it only because it requires additional the condition sup x ´v f (t, x, v)|v| 2+γ dv < ∞ on the solution and we wanted a clean statement. 2 Observe that the coefficients A[f ] and B[f ] are controlled under assumption (1.3), thanks to Lemmas 31 and 32.

1.2. The question studied and its history. We are also motivated by the study of the following nonlinear kinetic Fokker-Planck equation

(1.4) ∂ t f + v • ∇ x f = ρ[f ] ∇ v • (∇ v f + vf ) , t ≥ 0, x ∈ R d , v ∈ R d ,
(with or without periodicity conditions with respect to the space variable) where d ∈ N * , f = f (x, v, t) ≥ 0 and ρ[f ] := ´Rd f (x, v, t) dv. The construction of global smooth solutions for such a problem is one motivation of the present paper.

The linear kinetic Fokker-Planck equation

∂ t f +v •∇ x f = ∇ v •(∇ v f + vf
) is sometimes called the Kolmogorov-Fokker-Planck equation, as it was studied by Kolmogorov in the seminal paper [START_REF] Kolmogoroff | Zufällige Bewegungen (zur Theorie der Brownschen Bewegung)[END_REF]. In this note, Kolmogorov explicitely calculated the fundamental solution and deduced regularisation in both variables x and v, even though the operator ∇ v • (∇ v + v) -v • ∇ x shows ellipticity in the v variable only. It inspired Hörmander and his theory of hypoellipticity [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF], where the regularisation is recovered by more robust and more geometric commutator estimates (see also [START_REF] Preiss | Hypoelliptic differential operators and nilpotent groups[END_REF]).

Another question which has attracted a lot of attention in calculus of variations and partial differential equations along the 20th century is Hilbert's 19th problem about the analytic regularity of solutions to certain integral variational problems, when the quasilinear Euler-Lagrange equations satisfy ellipticity conditions. Several previous results had established the analyticity conditionally to some differentiability properties of the solution, but the full answer came with the landmark works of De Giorgi [START_REF] De | Sull'analiticità delle estremali degli integrali multipli[END_REF][START_REF] De | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF] and Nash [START_REF] Nash | Continuity of solutions of parabolic and elliptic equations[END_REF], where they proved that any solution to these variational problems with square integrable derivative is analytic. More precisely their key contribution is the following 3 : reformulate the quasilinear parabolic problem as

(1.5) ∂ t f = ∇ v (A(v, t)∇ v f ) , t ≥ 0, v ∈ R d
with f = f (v, t) ≥ 0 and A = A(v, t) satisfies the ellipticity condition 0 < λI ≤ A ≤ ΛI for two constants λ, Λ > 0 but is, besides that, merely measurable. Then the solution f is Hölder continuous.

The method has been extended to degenerate cases, like the p-Laplacian, first in the elliptic case by Ladyzhenskaya and Uralt'seva [START_REF] Ladyzhenskaya | Linear and quasilinear elliptic equations[END_REF], and then, degenerate parabolic cases were covered by DiBenedetto [START_REF] Dibenedetto | On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients[END_REF] (see also DiBenedetto, Gianazza and Vespri [START_REF] Dibenedetto | Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations[END_REF][START_REF] Dibenedetto | Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations[END_REF][START_REF] Dibenedetto | Harnack's inequality for degenerate and singular parabolic equations[END_REF]). More recently, the method has been extended to integral operators, such as fractional diffusion, in [START_REF] Caffarelli | Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation[END_REF][START_REF] Caffarelli | Regularity theory for parabolic nonlinear integral operators[END_REF] -see also the work of Kassmann [START_REF] Moritz Kassmann | A priori estimates for integro-differential operators with measurable kernels[END_REF] and of Kassmann and Felsinger [START_REF] Felsinger | Local regularity for parabolic nonlocal operators[END_REF]. Further application to fluid mechanics can be found in [START_REF] Caputo | Global regularity of solutions to systems of reaction-diffusion with subquadratic growth in any dimension[END_REF][START_REF] Goudon | Regularity analysis for systems of reaction-diffusion equations[END_REF][START_REF] Vasseur | Higher derivatives estimate for the 3D Navier-Stokes equation[END_REF]. 1.3. Main results. In view of the Landau equation and the nonlinear (quasilinear) equation (1.4), it is natural to ask whether a similar result as the one of De Giorgi-Nash holds for hypoelliptic equations. More precisely, we consider the following kinetic Fokker-Planck equation

(1.6) ∂ t f + v • ∇ x f = ∇ v • (A∇ v f ) + B • ∇ v f + s, t ∈ (0, T ), (x, v) ∈ Ω,
where

Ω is an open set of R 2d , f = f (x, v,
     0 < λI ≤ A ≤ ΛI |B| ≤ Λ s essentially bounded
for two constants λ, Λ. We establish the Hölder continuity of solutions to this problem. To state the result, we have to define cylinders that respect two invariant transformations of the (class of) equation(s): the scaling (x, v, t) → (r 3 x, rv, r 2 t) and the transformation

(1.8) T z 0 : z → (x 0 + x + tv 0 , v 0 + v, t 0 + t).
Given z 0 = (x 0 , v 0 , t 0 ) ∈ R 2d+1 , the cylinder Q r (z 0 ) "centered" at z 0 of "radius" r is defined as

(1.9) Q r (z 0 ) = (x, v, t) : |x -x 0 -(t -t 0 )v 0 | < r 3 , |v -v 0 | < r, t ∈ t 0 -r 2 , t 0 .
When z 0 = 0, we shall omit to specify the base point: Q r := Q r (0, 0, 0).

The weak solutions to equation (1.6) on

U x × U v × I, U x ⊂ R d open, U v ⊂ R d open, I = [a, b] with -∞ < a < b ≤ +∞, are defined as functions f ∈ L ∞ t (I, L 2 x,v (U x × U v ))) ∩ L 2 x,t (U x × I, H 1 v (U v )) such that ∂ t f + v • ∇ x f ∈ L 2 x,t (U x × I, H -1 v (U v )
) and f satisfies the equation (1.6) in the sense of distributions.

Theorem 3 (Hölder continuity). Let f be a weak solution of (1.6)

in Q ext := Q r 0 (z 0 ) and Q int := Q r 1 (z 0 ) with r 1 < r 0 . Then f is α-Hölder continuous with respect to (x, v, t) in Q int and f C α (Q int ) ≤ C f L 2 (Qext) + s L ∞ (Qext) for some α universal (i.e. α = α(d, λ, Λ)) and C = C(d, λ, Λ, Q ext , Q int ).
In order to prove such a result, we first prove that L 2 sub-solutions are locally bounded; we refer to such a result as an L 2 -L ∞ estimate. We then prove that solutions are Hölder continuous by proving a lemma which is an hypoelliptic counterpart of De Giorgi's "isoperimetric lemma".

We moreover prove a "quantitative version" of the strong maximum principle: a Harnack inequality.

Theorem 4 (Harnack inequality). If f is non-negative weak solution of (1.6) in Q 1 , then (1.10) sup Q - f ≤ C inf Q + f + s L ∞ (Q 1 )
where

Q + := Q R and Q -:= Q R (0, 0, -∆)
and C > 1 and R, ∆ ∈ (0, 1) are small (in particular Q ± ⊂ Q 1 and they are disjoint), and universal, i.e. only depend on dimension and ellipticity constants.

Remark 5. Using the transformation T z 0 (x, v, t) = (x 0 + x + tv 0 , v 0 + v, t 0 + t), we get a Harnack inequality for cylinders centered at an arbitrary point z 0 = (x 0 , v 0 , t 0 ).

1.4.

Comments and previously known results. In [START_REF] Pascucci | The Moser's iterative method for a class of ultraparabolic equations[END_REF], the authors obtain an L 2 -L ∞ estimate with completely different techniques; however they cannot reach the Hölder continuity estimate. Our techniques rely on the velocity averaging method. Velocity averaging designates a special type of smoothing effect for solutions of the free transport equation

(∂ t + v • ∇ x )f = S
observed for the first time in [START_REF] Valeri | Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation[END_REF][START_REF] Golse | Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d'un opérateur de transport[END_REF] independently, later improved and generalized in [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF][START_REF] Diperna | Global weak solutions of Vlasov-Maxwell systems[END_REF]. This smoothing effect bears on averages of f in the velocity variable v, i.e. on expressions of the form

ˆRd f (x, v, t) φ(v) dv , say for C ∞ c test functions φ.
Of course, no smoothing on f itself can be observed, since the transport operator is hyperbolic and propagates the singularities. However, when S is of the form

S = ∇ v • (A(x, v, t)∇ v f ) + s
where s is a given source term in L 2 , the smoothing effect of velocity averaging can be combined with the H 1 regularity in the v variable implied by the energy inequality in order to obtain some amount of smoothing on the solution f itself. A first observation of this type (at the level of a compactness argument) can be found in [START_REF] Lions | On Boltzmann and Landau equations[END_REF]. More recently, Bouchut [START_REF] Bouchut | Hypoelliptic regularity in kinetic equations[END_REF] has obtained more quantitative Sobolev regularity estimates. These estimates are one key ingredient in our proof.

We give two proofs of this L 2 -L ∞ estimate, one following Moser's approach, the other following De Giorgi's ideas. We emphasize that, in both approaches, the main ingredient is a local gain of integrability of non-negative sub-solutions. This latter is obtained by combining a comparison principle and a Sobolev regularity estimate. We then prove the Hölder continuity through a De Giorgi type argument on the decrease of oscillation for solutions. We also derive the Harnack inequality by combining the decrease of oscillation with a result about how the minimum of non-negative solutions deteriorates with time.

In [START_REF] Wang | The C α regularity of a class of non-homogeneous ultraparabolic equations[END_REF][START_REF] Wang | The C α regularity of weak solutions of ultraparabolic equations[END_REF], the authors get a Hölder estimate for weak solutions of so-called ultraparabolic equations, including (1.6). Their proof relies on the construction of cut-off functions and a particular form of weak Poincaré inequality satisfied by non-negative weak sub-solutions. Our paper proposes an alternate method based on velocity averaging. It illustrates the interesting connection between velocity averaging and hypoelliptic-like structures. It also provides several tools for further applications.

The C ∞ smoothing of solutions to the Landau equation has been investigated so far in two different settings: either for weak spatially homogeneous solutions (non-negative in L 1 and with finite energy) [START_REF] Aleksei | On a connection between the solution of the Boltzmann equation and the solution of the Landau-Fokker-Planck equation[END_REF][START_REF] Desvillettes | Plasma kinetic models: the Fokker-Planck-Landau equation[END_REF][START_REF] Villani | On the spatially homogeneous Landau equation for Maxwellian molecules[END_REF][START_REF] Desvillettes | On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness[END_REF] (see also the related entropy dissipation estimates in [START_REF] Desvillettes | On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications[END_REF][START_REF] Desvillettes | Entropy dissipation estimates for the Landau equation in the Coulomb case and applications[END_REF]), or for classical spatially heterogeneous solutions [START_REF] Chen | Smoothing effects for classical solutions of the full Landau equation[END_REF][START_REF] Liu | Regularizing effects for the classical solutions to the Landau equation in the whole space[END_REF]. The analytic regularisation of weak spatially homogeneous solutions was investigated in the case of Maxwellian or hard potentials in [START_REF] Chen | Analytic smoothness effect of solutions for spatially homogeneous Landau equation[END_REF]. Let us also mention that in [START_REF] Silvestre | Upper bounds for parabolic equations and the Landau equation[END_REF], Silvestre derives an L ∞ bound on the spatially homogeneous solutions for soft potentials without relying on energy methods (which implies as well the smoothing by standard parabolic techniques). Let us also mention works studying modified Landau equations [START_REF] Krieger | Global solutions to a non-local diffusion equation with quadratic nonlinearity[END_REF][START_REF] Gressman | A non-local inequality and global existence[END_REF] and the work [START_REF] Gualdani | Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential[END_REF] that shows that any weak radial solution to the Landau-Coulomb equation that belongs to L 3/2 is automatically bounded and C 2 using barrier arguments. Finally, we highlight the related results of regularisation for the Boltzmann equation without long-range interactions [START_REF] Desvillettes | About the regularizing properties of the non-cut-off Kac equation[END_REF][START_REF] Chen | Smoothing estimates for Boltzmann equation with full-range interactions: spatially homogeneous case[END_REF][START_REF] Chen | Smoothing estimates for Boltzmann equation with full-range interactions: Spatially inhomogeneous case[END_REF], and the related perturbative results for the Landau and (long-range interaction) Boltzmann equation [START_REF] Guo | The Landau equation in a periodic box[END_REF][START_REF] Gressman | Global classical solutions of the Boltzmann equation without angular cut-off[END_REF][START_REF] Alexandre | Regularizing effect and local existence for the non-cutoff Boltzmann equation[END_REF][START_REF] Alexandre | Global existence and full regularity of the Boltzmann equation without angular cutoff[END_REF][START_REF] Alexandre | A review of Boltzmann equation with singular kernels[END_REF][START_REF] Wu | Global in time estimates for the spatially homogeneous Landau equation with soft potentials[END_REF][START_REF] Alexandre | Some a priori estimates for the homogeneous Landau equation with soft potentials[END_REF]. From this review, and the best of our knowledge, the regularity of a priori non-negative locally L ∞ solutions (under our assumption (1.3)) to the spatially heterogeneous Landau equation has not investigated so far.

A part of the results of this paper were announced in [START_REF] Golse | Hölder regularity for hypoelliptic kinetic equations with rough diffusion coefficients[END_REF][START_REF] Imbert | Hölder continuity of solutions to hypoelliptic equations with bounded measurable coefficients[END_REF].

1.5. Plan of the paper. In Section 2, we prove the universal gain of integrability for non-negative sub-solutions. In Section 3, we derive from this gain of integrability a local upper bound of such non-negative sub-solutions; we give two proofs: one following de Giorgi's approach and the other one following Moser's iteration procedure. In Section 4, the Hölder estimate is derived by proving a lemma of "reduction of oscillation". In Section 5 we prove a Harnack inequality for non-negative solutions. In Section 6, we prove a local gain of regularity of sub-solutions. In Section 7, we prove that the velocity gradient of the solution is slightly better than square integrable.

1.6. Notation. We occasionally write A B in order to say that A ≤ CB for some constant C which only depends on dimension and ellipticity constants λ and Λ. Such a constant C is called universal.

The inverse transformation T -1

z 0 : z → z -1 0 • z is defined by T -1 z 0 (z) = (x -x 0 -(t -t 0 )v 0 , v -v 0 , t -t 0 ).
The notation z 0 • z and z -1 0 refers to a Lie group structure associated with the equation.

Local gain of regularity / integrability

We consider the equation (1.6) and we want to establish a local gain of integrability of solutions in order to apply De Giorgi-Moser's iteration and get a local L ∞ bound. Since we will need to perform convex changes of unknown, it is necessary to obtain this gain for all (non-negative) sub-solutions. The next theorem is stated in cylinders centered at the origin.

Theorem 6 (Gain of integrability for non-negative sub-solutions). Consider two cylinders Q int := Q r 1 and Q ext := Q r 0 with 0 < r 1 < r 0 . There exists p > 2 (only depending on dimension) such that for all non-negative sub-solution f of (1.6) in Q ext , we have

(2.1) f 2 L p (Q int ) ≤ C C 2 0,1 f 2 L 2 (Qext) + C 0,1 ˆQext |s| 2 1 f >0 with C 0,1 = 1 r 2 0 -r 2 1 + r 0 r 3 0 -r 3 1 + 1 (r 0 -r 1 ) 2 + 1 and C = C(d, λ, Λ) .
Remark 7. The exponent p is obtained by the Sobolev embedding H

1 3 (R 2d+1 ) → L p (R 2d+1
), that is to say p := 6(2d + 1)/(6d + 1). This result is a consequence of the comparison principle and the following gain of regularity.

Theorem 8 (Gain of regularity for sign-changing solutions). Consider z 0 ∈ R 2d+1 and two cylinders

Q int := Q r 1 (z 0 ) and Q ext := Q r 0 (z 0 ) with 0 < r 1 < r 0 . Then any (sign-changing) weak solution f of (1.6) in Q ext satisfies (2.2) f 2 H 1 3 x,v,t (Q int ) ≤ C f 2 L 2 (Qext) + s 2 L 2 (Qext) with C = C(d, λ, Λ, Q ext , Q int ).
Remark 9. Using Theorem 6 and De Giorgi-Moser's iteration, it is in fact possible to prove that this gain of regularity is also true for non-negative sub-solutions, as we will see in Section 6.

Global estimates and gain of regularity/integrability. Remark that our weak solutions

in f ∈ L ∞ t (I, L 2 x,v (U x ×U v )))∩L 2 x,t (U x ×I, H 1 v (U v )) are in C 0 t (I, L 2 x,v (U x ×U v )∩H 1/2 t (I, L 2 x,v (U x ×U v )
), following and adapting respectively the by-now standard arguments in [START_REF] Struwe | On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems[END_REF] and [START_REF] Giaquinta | Partial regularity for the solutions to nonlinear parabolic systems[END_REF] to the kinetic case. This justifies the calculations performed in our energy estimates in the sequel.

Lemma 10 (Global estimate). Let g be a weak solution of

(∂ t + v • ∇ x )g = ∇ v • (A∇ v g) + ∇ v • H 1 + H 0 in R 2d+1 with H 1 and H 0 in L 2 (R 2d+1 ) and g, H 0 and H 1 supported in R d × B(0, r 0 ) × R. Then (2.3) ∇ v g 2 L 2 + D 1 3 x g 2 L 2 + D 1 3 t g 2 L 2 ≤ C H 1 2 L 2 + H 0 2 L 2
where C = C(1 + r 2 0 ) and C = C(d, λ, Λ). In particular, there exists p > 2 (only depending on dimension) such that

(2.4) g 2 L p ≤ C H 1 2 L 2 + H 0 2 L 2
where

C = C(1 + r 2 0 ) and C = C(d, λ, Λ). Proof. Integrating against 2g in R 2d+1 yields 2λ ˆR2d+1 |∇ v g| 2 dx dv dt ≤ ˆR2d+1 (-2H 1 • ∇ v g + 2gH 0 ) dx dv dt ≤ λ 2 ˆR2d+1 |∇ v g| 2 dx dv dt + 2 λ ˆR2d+1 |H 1 | 2 dx dv dt + 2 ˆR2d+1 |g||H 0 | dx dv dt. Moreover 2 ˆR2d+1 |g||H 0 | dx dv dt ≤ ε ˆR2d+1 |g| 2 dx dv dt + 1 ε ˆR2d+1 |H 0 | 2 dx dv dt.
Since g is supported in B(0, r 0 ) in the velocity variable, we can use the Poincaré inequality to get

ε ˆR2d+1 |g| 2 dx dv dt ≤ C P r 2 0 ε ˆR2d+1 |∇ v g| 2 dx dv dt
and we choose ε such that C P r 2 0 ε = λ/2. This implies

(2.5) ∇ v g 2 L 2 ≤ C H 1 2 L 2 + H 0 2 L 2 . Applying [7, Theorem 1.3] with p = 2, r = 0, β = 1, m = 1, κ = 1 and Ω = 1 yields D 1 3 x g 2 L 2 + D 1 3 t g 2 L 2 g 2 L 2 + ∇ v g L 2 (1 + |v| 2 ) 1 2 H 0 L 2 + ∇ v g 4 3 L 2 (1 + |v| 2 )(H 1 + A∇ v g) 2 3 L 2 + ∇ v g L 2 (1 + |v| 2 ) 1 2 (H 1 + A∇ v g) L 2 .
Using the fact that g, H 0 and

H 1 are supported in R d × B(0, r 0 ) × R, we get D 1 3 x g 2 L 2 + D 1 3 t g 2 L 2 r 2 0 ∇ v g 2 L 2 + (1 + r 2 0 ) 1 2 ∇ v g L 2 H 0 L 2 + (1 + r 2 0 ) 2 3 ∇ v g 4 3 L 2 H 1 2 3 L 2 + ∇ v g 2 3 L 2 + (1 + r 2 0 ) 1 2 ∇ v g L 2 ( H 1 L 2 + ∇ v g L 2 ) (1 + r 2 0 ) ∇ v g 2 L 2 + H 1 2 L 2 + ∇ v g L 2 H 0 L 2 .
Combining this estimate with (2.5) yields (2.3). The proof is now complete.

2.2.

The local energy estimate. The gain of integrability with respect to v and t is classical; it derives from the natural energy estimate, after truncation. We follow here [START_REF] Moser | A Harnack inequality for parabolic differential equations[END_REF].

Lemma 11 (The local energy estimate). Under the assumptions of Theorems 6 and 8, any subsolution f satisfies

(2.6) sup t ˆQt int f 2 (•, •, t) + ˆQint |∇ v f | 2 ≤ C C 0,1 ˆQext f 2 + ˆQext |s| 2 for Q t int := {(x, v) ∈ R 2d : (x, v, t) ∈ Q int }, C = C(d, λ, Λ) and C 0,1 = 1 r 2 0 -r 2 1 + r 0 r 3 0 -r 3 1 + 1 (r 0 -r 1 ) 2 + 1 . Moreover, if the sub-solution f is non-negative, then (2.7) sup t ˆQt int f 2 (•, •, t) + ˆQint |∇ v f | 2 ≤ C C 0,1 ˆQext f 2 + ˆQext |s| 2 1 f >0 . Proof. Consider Ψ ∈ C ∞ c (R 2d × R) with 0 ≤ Ψ ≤ 1 and integrate the inequation satisfied by f against 2f Ψ 2 in R := R 2d × [t 1 , 0] with t 1 ∈ (-r 2 1 , 0] and get ˆR ∂ t (f 2 )Ψ 2 + ˆR v • ∇ x (f 2 )Ψ 2 ≤ 2 ˆR ∇ v • (A∇ v f )f Ψ 2 + 2 ˆR(B • ∇ v f )f Ψ 2 + 2 ˆR f sΨ 2 . Add ´R f 2 ∂ t (Ψ 2 ),

integrate by parts and use the upper bound on

A to get ˆR ∂ t (f 2 Ψ 2 ) + 2λ ˆR |∇ v f | 2 Ψ 2 ≤ ˆR f 2 (∂ t + v • ∇ x )(Ψ 2 ) -4 ˆR ΨA∇ v f • f ∇ v Ψ + 2 ˆR(B • ∇ v f )f Ψ 2 + 2 ˆR f sΨ 2 ≤ ˆR f 2 (∂ t + v • ∇ x )(Ψ 2 ) + 4Λ ˆR(|∇ v f |Ψ)f (Ψ + |∇ v Ψ|) + 2 ˆR f sΨ 2 ≤ ˆR f 2 (∂ t + v • ∇ x )(Ψ 2 ) + 8Λ 2 λ -1 (|∇ v Ψ| 2 + Ψ 2 ) + 2 ˆR f sΨ 2 + λ ˆR |∇ v f | 2 Ψ 2 .
We thus get (2.8)

ˆR ∂ t (f 2 Ψ 2 )+λ ˆR |∇ v f | 2 Ψ 2 ≤ C ∂ t Ψ ∞ + r 0 ∇ x Ψ ∞ + ∇ v Ψ 2 ∞ + 1 ˆR∩supp Ψ f 2 +2 ˆR f sΨ 2 with C = C(d, λ, Λ). Choose next Ψ 2 such that Ψ(t = 0) = 0 and supp Ψ ⊂ Q ext and get for t 1 ∈ R: ˆR2d f 2 (•, •, t 1 )Ψ 2 (t 1 ) dx dv + λ ˆR2d+1 |∇ v f | 2 Ψ 2 dx dv dt ≤ C ˆQext f 2 + 2 ˆQext |f | |s|. If Ψ additionally satisfies Ψ ≡ 1 in Q int , we get (2.6). Remark that (2.7
) is a simple consequence of (2.6). The proof is now complete.

Local gain: proofs.

Proof of Theorems 6 and 8. We first remark that if f is a non-negative sub-solution of (1.6), then f = f 1 f ≥0 and it is also a sub-solution of the same equation when the source term s is replaced with s1 f ≥0 .

For i = 1, 1 2 , consider f i = f χ i where χ 1 and χ 1/2 are two truncation functions such that

χ 1 ≡ 1 in Q int and χ 1 ≡ 0 outside Q mid , χ 1 2 ≡ 1 in Q mid and χ 1 2 ≡ 0 outside Q ext .
The function f 1 now satisfies

(∂ t + v • ∇ x )f 1 ≤ ∇ v • (A∇ v f 1 ) + ∇ v • H 1 + H 0 in R 2d+1
with H 1 and H 0 given by

   H 1 = (-A∇ v χ 1 )f 1 2 H 0 = (Bχ 1 -A∇ v χ 1 ) • ∇ v f 1 2 + α 1 f 1 2 + s1 {f ≥0} χ 1 with α 1 = (∂ t + v • ∇ x )χ 1 . We remark that f 1 , H 0 and H 1 are supported in Q ext .
We now consider the solution g of

(∂ t + v • ∇ x )g = ∇ v • (A∇ v g) + ∇ v • H 1 + H 0 in R 2d+1 .
We remark that g is also supported in Q ext , and since h := f 1 -g is a sub-solution of the equation

∂ t h + v • ∇ x h ≤ ∇ v (A∇ v h
) with zero initial data at t = -r 2 0 , the comparison principle implies that h ≤ 0 everywhere, and therefore 0 ≤ f 1 ≤ g. It can be proved for instance by observing that h + is also a sub-solution of the same inequation and the standard energy estimate implies that its L 2

x,v -norm is non-increasing along the time variable.

Moreover,    H 1 2 L 2 ∇ v χ 1 2 L ∞ f 2 L 2 (Qext) H 0 2 L 2 1 + ∇ v χ 1 2 L ∞ ∇ v f 2 L 2 (Q mid ) + α 1 2 L ∞ f 2 L 2 (Qext) + s1 {f ≥0} 2 L 2 (Qext)
. In view of Lemma 11, we know that

∇ v f 2 L 2 (Q mid ) C 0,1 f 2 L 2 (Qext) + s1 {f ≥0} 2 L 2 (Qext) . Hence, H 0 2 L 2 + H 1 2 L 2 (1 + ∇ v χ 1 2 L ∞ )(1 + C 0,1 ) + α 1 2 L ∞ f 2 L 2 (Qext) + 2 + ∇ v χ 1 2 L ∞ s1 {f ≥0} 2 L 1 (Qext) .
In view of the definition of C 0,1 in Lemma 11, we thus get

H 0 2 L 2 + H 1 2 L 2 C 2 0,1 f 2 L 2 (Qext) + (r 0 -r 1 ) -2 s1 {f ≥0} 2 L 1 (Qext) . Lemma 10 then yields g 2 L p (Q int ) ≤ C C 2 0,1 f 2 L 2 (Qext) + C 0,1 ˆQext |s| 2 1 f ≥0 .
We then obtain (2.1) by using the fact that 0 ≤ f 1 ≤ g. This achieves the proof of Theorem 6.

As for Theorem 8, Lemma 10 can be applied directly to f 1 and the conclusion follows along the same lines, with some simplifications.

Local upper bounds for non-negative sub-solutions

In this section, we prove that non-negative L 2 sub-solutions are in fact locally bounded.

Theorem 12 (Upper bounds for non-negative L 2 sub-solutions). Given two cylinders Q ext := Q r 0 (z 0 ) and Q ∞ := Q r∞ (z 0 ) with 0 < r ∞ < r 0 , let f be a non-negative L 2 sub-solution of (1.6) in Q ext with s ∈ L q (Q ext ) and q > (2p)/(p -1) with p only depending on dimension. There for any g >, there exists

κ = κ(d, λ, Λ, Q ext , Q ∞ , g, q) > 0 such that s L q (Qext) ≤ g f L 2 (Qext) ≤ κ ⇒ f ≤ 1 2 in Q ∞ .
Remark 13. The exponent p = 6(2d + 1)/(6d + 1) is the one given by the gain of integrability in Theorem 6 (see Remark 7).

We give two proofs of such a result. The first one sticks to the case q = +∞ with no lower order terms and use Moser's approach. The second one deals with the general case and use De Giorgi's approach.

Moser's approach.

Proof of Theorem 12 in the case without source term by Moser's iteration. Using tranformations introduced in Eq. (1.8), we reduce to the case z 0 = 0.

We first observe that, for all q > 1, the function f q satisfies

(∂ t + v∇ x )f q ≤ ∇ v • (A∇ v f q ) in Q r 0 .
We now rewrite (2.1) with s = 0 from Q rn to Q r n+1 with r n+1 < r n as follows:

(3.1)

ˆQr n+1 (0) (f q ) p 2 p ≤ CC 2 n ˆQrn(0) f 2q
where C = C(d, λ, Λ) and

(3.2) C n = 1 r 2 n -r 2 n+1 + r n r 3 n -r 3 n+1 + 1 (r n -r n+1 ) 2 + B L ∞ + 1.
Choose now q = q n = (p/2) n for n ∈ N and write a n for ( ´Qn f 2qn ) 1/(2qn) . Using that for C = C(d, λ, Λ, Q ext ) ≥ 1 large enough, we have |Q ext | ≤ C, we get from (3.1)

(3.3) a n+1 ≤ ( C) 1 2qn (C n ) 1 qn a n .
Finally we choose

r n+1 = r n - 1 a(n + 1) 2 for some a > 0 (only depending on r 0 -r ∞ ) so that (3.2) yields C n ∼ a 2 n 4 as n → +∞. In particular, we can choose C = C(d, λ, Λ, B L ∞ ) large enough so that C n ≤ C 1 2 a 2 n 4 and we get from (3.3) that a n+1 ≤ ( Ca 2 n 4 ) 1 qn a n . The convergence of the following infinite product ∞ n=0 ( Ca 2 ) 1 qn (n 4 ) 1 qn < +∞ achieves the proof. 3.2. De Giorgi's approach.
Proof of Theorem 12 by De Giorgi's approach. We again reduce to the case z 0 = 0 thanks to the transformation T -1 z 0 defined in Eq. (1.8). For n ≥ 0 integer, consider radius r n , time T n , cylinder Q n and constant C n as follows

r n = r ∞ + (r 0 -r ∞ )2 -n , T n = t 0 -r 2 n , C n = 1 2 (1 -2 -n ),
and cut-off functions Ψ n (independent of time) as follows

Ψ n ≡    1 in Q 0 rn 0 outside Q 0 r n-1 and        ∇ v Ψ n L ∞ ≤ 1 r n-1 -r n ≤ C 0,∞ 2 n ∇ x Ψ k L ∞ ≤ 1 r 3 n-1 -r 3 n ≤ C 0,∞ 2 n
where C 0,∞ = C(r 0 , r ∞ ) only depends on r 0 and r ∞ , and as before

Q τ r := {(x, v) : (x, v, τ ) ∈ Q r }.
The energy estimate. Remark that

f n = (f -C n ) + is a sub-solution of (1.6) in Q rn with s n = s1 f ≥Cn .
Then the energy estimate (2.8) obtained in the proof of Lemma 11 yields for all

T n-1 ≤ τ ≤ T n ≤ t ≤ 0, (3.4) ˆQt rn f 2 n + λ ˆQrn |∇ v f n | 2 ≤ ˆQτ rn f 2 n + r n ∇ x Ψ n ∞ + ∇ v Ψ n 2 ∞ + 1 ˆQr n-1 f 2 n + 2 ˆQr n-1 f n |s|.
Averaging both sides of the inequality in τ ∈ (T n-1 , T n ) and using the estimates on the gradients of the cut-off function yields

(3.5) U n := sup t∈(Tn,0) ˆQt rn f 2 n ≤ C4 n ˆQr n-1 f 2 n + 2 ˆQr n-1 f n |s| where C = C(r 0 , r ∞ ). Remark that, (3.6) U n ≤ U n-1 ≤ • • • ≤ U 0 ≤ κ ≤ 1 (we choose κ ≤ 1).
The non-linearization procedure. Using the (universal) exponent p > 2 given by Theorem 6, we next estimate the terms in the right hand side of (3.5) as follows

         ˆQr n-1 f 2 n ≤ ´Qr n-1 f p n 2 p |{f n ≥ 0} ∩ Q r n-1 | 1-2 p ˆQr n-1 f n |s| ≤ g ´Qr n-1 f p n 1 p |{f n ≥ 0} ∩ Q r n-1 | 1-1 p -1 q (3.7)
(we used that s L q (Qext) ≤ γ) if p and q satisfy 1 -

1 p - 1 q > 0.
We next remark that

{f n ≥ 0} = {f n-1 ≥ C n -C n-1 = 2 -k-1 } which in turn implies (3.8) |{f n ≥ 0} ∩ Q r n-1 | ≤ 2 2n+2 ˆQr n-1 f 2 n-1 ≤ C4 n U n-1 .
Combining these three estimates with (3.5) yields

(3.9) U n ≤ C2 4n   ˆQr n-1 f p n-1 2 p U 1-2 p n-1 + s L q (Qext) ˆQr n-1 f p n-1 1 p U 1-1 p -1 q n-1   (we also used that f n ≤ f n-1 ) where C = C(d, λ, Λ, r 0 , r ∞ ).
Use of the gain of integrability. In view of Theorem 6, we know that ˆQr n-1 f p n-1

2 p ≤ C 8 n ˆQr n-2 f 2 n-1 + 4 n ˆQr n-2 s 2 1 f n-1 >0
with C = C(d, λ, Λ, r 0 , r ∞ ). We next estimate the terms in the right hand side of the previous equation depending of the source term as in (3.7) but with p = 2: we use (3.8) to get

ˆQr n-2 s 2 1 f n-1 ≥0 ≤ g 2 |{f n-1 > 0} ∩ Q r n-2 | 1-2 q ≤ g 2 2 2n-4n q U 1-2 q n-2 .
Hence, we can use (3.6) and U 0 ≤ 1 again in order to write

ˆQr n-1 f p n-1 2 p ≤ C 2 3n U n-2 + 2 4n-4n q U 1-2 q n-2 ≤ C2 4n U 1-2 q n-2
with C = C(d, λ, Λ, r 0 , r ∞ , q, g). Then (3.9) and (3.6) imply

U n ≤ C2 4n 2 4n U 2-2 p -2 q n-2 + U 3 2 -1 p -2 q n-2 ≤ C2 8n U 3 2 -1 p -2 q n-2
.

Conclusion. Remark that we can assume that C ≥ 1. We rewrite it as

(3.10) V n ≤ β n V α n-1
where

V n = U 2n , β = 2 8 C and α = 3 2 -1 p -2 q . Remark that α > 1 as soon as 1 q < 1 2 1 2 - 1 p .
Applying (3.10) recursively, we get

V n ≤ β k+α(k-1)+α 2 (k-2)+•••+α k-1 V α k 0 . Remark now that n + α(n -1) + • • • + α n-1 = n(1 + α + • • • + α n-1 ) -α(1 + 2α + • • • + (n -1)α n-2 ) = n α n -1 α -1 -α d dα α n -1 α -1 = n(α n -1) α -1 -α nα n-1 (α -1) -(α n -1) (α -1) 2 = α(α n -1) -n(α -1) (α -1) 2 ≤ α (α -1) 2 α n . Hence V n ≤ β α (α-1) 2 V 0 α n .
This implies that U 2n = V n → 0 as n → +∞ as soon as

β α (α-1) 2 V 0 ≤ (2 8 C) α (α-1) 2 κ < 1 where C = C(d, λ, Λ, r 0 , r ∞ , q, g). Hence, U ∞ = ˆQr∞ f - 1 2 2 + = 0 which means that f ≤ 1/2 in Q r∞ .
This completes the proof of Theorem 12.

4. Intermediate-value lemma and Hölder continuity 4.1. A De Giorgi intermediate-value lemma. An important step in the proof of regularity in De Giorgi's method for elliptic equations is based on an inequality of isoperimetric form (see the proof of [START_REF] De | Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari[END_REF]Lemma II]). This inequality is a quantitative variant of the well-known fact that no H 1 function can have a jump discontinuity, and can also be understood as a quantitative minimum principle. More precisely, given an H 1 function u valued in [0, 1] and which takes the values 0 and 1 on sets of positive measure, De Giorgi's isoperimetric inequality provides a lower bound on the measure of the set of intermediate values {0 < u < 1}. In the present subsection, we establish an analogue of this inequality adapted to our equation and the combination of the first order transport operator and the second order elliptic operator in the velocity variable. We prove the core lemma at "unit scale". We recall that 1).

Q 2 = B 8 × B 2 × (-4, 0] and Q 1 = B 1 × B 1 × (-1, 0], Q ω = B ω 3 × B ω × (-ω 2 , 0] and we denote the shifted cube Q := Q ω (0, 0, -1) = B ω 3 × B ω × (-1 -ω 2 , -1] (see Figure

Lemma 14 (A De Giorgi intermediate-value lemma).

Let ω = 1 4 . For any (universal) constants δ 1 ∈ (0, 1), δ 2 ∈ (0, 1) there exist ν > 0 and θ ∈ (0, 1) (both universal) such that for any sub-solution f of (1.6) in Q 2 with f ≤ 1 and |s| ≤ 1 Remark 15. While De Giorgi's isoperimetric inequality is based on an explicit computation leading to a precise estimate with effective constants, the proof of Lemma 14 is obtained by an argument by contradiction, so that the values of θ and ν are not known explicitly.

and |{f ≥ 1 -θ} ∩ Q ω | ≥ δ 1 |Q ω | |{f ≤ 0} ∩ Q| ≥ δ 2 | Q| we have |{0 < f < 1 -θ} ∩ B 1 × B 1 × (-2, 0]| ≥ ν.
Remark 16. The compactness argument used in the proof is reminiscent of one used by Guo in [START_REF] Guo | The Vlasov-Poisson-Boltzmann system near Maxwellians[END_REF] and of one used by the fourth author in [START_REF] Vasseur | The De Giorgi method for elliptic and parabolic equations and some applications[END_REF].

Proof. We argue by contradiction by assuming that there exists a sequence (f k ) k≥0 of sub-solutions:

(4.1) (∂ t + v • ∇ x )f k ≤ ∇ v • (A k ∇ v f k ) + B k • ∇ v f k + s k such that f k ≤ 1 and |s k | ≤ 1 and
θ k → 0 α k → 0 as k → +∞ and |{f k ≥ 1 -θ k } ∩ Q ω | ≥ δ 1 |Q ω | |{f k ≤ 0} ∩ Q| ≥ δ 2 | Q| |{0 < f k < 1 -θ k } ∩ (Q 1 ∪ Q)| → 0 as k → +∞.
The convexity of z → z + together with |s k | ≤ 1 implies that the non-negative part f + k of f k satisfies the same inequation, and therefore

(4.2) (∂ t + v • ∇ x )f + k = ∇ v • (A k ∇ v f + k ) + B k • ∇ v f + k + 1 -µ k for some non-negative measures µ k . A priori estimates for f + k .
The natural energy estimate is obtained by multiplying the equation with f + k Ψ 2 with a smooth cut-off function Ψ supported in Q 2 and valued in [0, 1], and using the fact that f + k ≤ 1 and |s k | ≤ 1:

λ ˆR2d+1 |∇ v f + k | 2 Ψ 2 ≤ C ˆR2d+1 Ψ 2 + |∇ v Ψ| 2 + Ψ|(∂ t + v • ∇ x )Ψ| + Λ ˆR2d+1 |∇ v f + k |f + k Ψ 2 ≤ C ˆR2d+1 Ψ 2 + |∇ v Ψ| 2 + Ψ|(∂ t + v • ∇ x )Ψ| + λ 2 ˆR2d+1 |∇ v f + k | 2 Ψ 2 . Hence (4.3) λ ˆR2d+1 |∇ v f + k | 2 Ψ 2 ≤ C ˆR2d+1 Ψ 2 + |∇ v Ψ| 2 + Ψ|(∂ t + v • ∇ x )Ψ|
where C = C(d, λ, Λ). We can also multiply the equation by Ψ 2 and get

- ˆR2d+1 f + k (∂ t + v • ∇ x )(Ψ 2 ) = - ˆR2d+1 A k ∇ v f + k • ∇ v (Ψ 2 ) + ˆR2d+1 B k • ∇ v f + k Ψ 2 + ˆR2d+1 Ψ 2 - ˆR2d+1 Ψ 2 dµ k .
Combining the latter equation with (4.3), we deduce (4.4)

ˆR2d+1 Ψ 2 dµ k ≤ C ˆR2d+1 Ψ 2 + |∇ v Ψ| 2 + Ψ|(∂ t + v • ∇ x )Ψ|
where C = C(d, λ, Λ).

Passage to the limit. On the one hand, Banach-Alaoglu theorem implies that

f + k * F in L ∞ loc (Q 2 ) and (4.5) ∇ v f + k ∇ v F and    A k ∇ v f + k H 1 B k • ∇ v f + k H 0 in L 2 loc (Q 2 )
for some weak limit

F ∈ L ∞ loc (Q 2 ) ∩ (L 2 x,t H 1 v ) loc (Q 2 ). In particular, (4.3) implies that (4.6) ˆQ |∇ v F | 2 Q 1
for all Q Q 2 , with a control depending on Q. On the other hand, the bound (4.4) implies that

µ k µ in M(Q 2 ).
We thus have

(∂ t + v • ∇ x )F = ∇ v H 1 + H 0 + 1 -µ. (4.7)
By velocity averaging (see Theorem 1.8 in [START_REF] Bouchut | Kinetic equations and asymptotic theory[END_REF]), together with the bound (4.3), we deduce the strong convergence

f + k → F in L p loc (Q 2 ) for 1 ≤ p < +∞.

It implies the convergence in probability and thus the function

F satisfies |{F = 1} ∩ Q ω | ≥ δ 1 |Q ω | (4.8) |{F = 0} ∩ Q| ≥ δ 2 | Q| (4.9) |{0 < F < 1} ∩ (B 1 × B 1 × (-2, 0])| = 0 .
In view of (4.6), since indicator functions are not in H 1 unless they are constant, we have that for almost every (x, t) ∈ B 1 × (-1, 0),

either for almost every v ∈ B 1 , F (x, v, t) = 0 or for almost every v ∈ B 1 , F (x, v, t) = 1.
In other words, F (x, v, t) = 1 P (x, t) for some measurable set P ⊂ B 1 × (-1, 0). In view of (4.8) and (4.9), P satisfies (4.10)

   |P ∩ B ω 3 × (-ω 2 , 0)| > 0 |B ω 3 × (-1 -ω 2 , -1) \ P | > 0.
Propagation. We thus get from (4.7)

∂ t F + v • ∇ x F ≤ ∇ v H 1 + H 0 + 1 in B 1 × B 1 × (-2, 0). Consider a cut-off funtion ξ ∈ D(R d ) such that ˆRd ζ(z) dz = 1, ζ(z) = ζ(-z), supp ζ ⊂ B 1 2 . Given v 0 ∈ B 1 2
, since F only depends on (t, x), we can use a test-function of the form ζ(v -v 0 ), and get for all

v 0 ∈ B 1 2 , ∂ t F + v 0 • ∇ x F ≤ ˆRd |H 1 (x, v, t)∇ v ζ(v -v 0 )| + |H 0 (x, v, t)ζ(v -v 0 )| dv + 1 in (x, t) ∈ B 1 × (-2, 0). Since F is an indicator function and H 0 , H 1 ∈ L 2 loc (Q 2 ), this implies for v 0 ∈ B 1 2 , (4.11) ∂ t F + v 0 • ∇ x F ≤ 0 in B 1 × (-2, 0).
We next remark that (4.12)

   for all (x, t) ∈ B ω 3 × (-ω 2 , 0) and (x 0 , t 0 ) ∈ B ω 3 × (-1 -ω 2 , -1),
there exists v 0 ∈ B ω so that (x 0 , v 0 , t 0 ) ∈ Q and (x, t) = (x 0 + sv 0 , t 0 + s).

Indeed, the time shift s is fixed by t = t 0 + s and belongs to (1 -ω 2 , 1 + ω 2 ). Then the velocity v 0 is fixed by x = x 0 + sv 0 and satisfies

|v 0 | = |x -x 0 | t -t 0 < 2ω 3 1 -ω 2 ≤ ω since ω = 1 4 ≤ 1 √ 3 . Since |B 1 × (-1 -ω 2 , -1) \ P | > 0 (see (4. 10
)), we can use (4.11) and (4.12) and conclude that F ≡ 0 in Q ω , and contradicts (4.10). The proof is complete.

Improvement of oscillation.

It is classical that Hölder continuity is a consequence of the decrease of the oscillation of the solution "at unit scale".

Lemma 17 (Improvement of oscillation).

There exist λ 0 ∈ (0, 1), ω ∈ (0, 1/2) and β > 0 (all universal) such that any f solution of

(1.6) in Q 2 with osc Q 2 f ≤ 2 and |s| ≤ β satisfies osc Q ω 2 f ≤ 2 -λ 0 .
This lemma is a consequence of the following one.

Lemma 18 (A measure-to-pointwise estimate). Given δ 2 > 0, there exist λ 0 ∈ (0, 1), ω ∈ (0, 1/2) and β > 0 (depending on δ 2 but not on the sub-solution) such that any f sub-solution of (1.6) in

Q 2 with f ≤ 1 and |s| ≤ β such that |{f ≤ 0} ∩ Q| ≥ δ 2 | Q| satisfies (4.13) f ≤ 1 -λ 0 a.e. in Q ω 2 .
Proof of Lemma 17. Let f be a solution of (1.6) in Q 2 with osc Q 2 f ≤ 2 and |s| ≤ β. We can reduce to the case where |f | ≤ 1. Indeed, we remark that there exists a constant C such that f = f -C satisfies (1.6) in Q 2 (0) with | f | ≤ 1 and the same source term.

If |{f ≤ 0} ∩ Q| ≥ | Q|/2, then apply Lemma 18 with δ 2 = 1/2. In the other case, considering -f implies that the essential infimum of f is raised. In both cases, we get the desired improvement of the oscillation of f . This completes the proof of the lemma.

We now turn to the proof of Lemma 18.

Proof of Lemma 18. The proof proceeds in several steps.

Choice of parameters.

Theorem 12 provides us with κ correponding to the upper bound g = 1 on the source term and

Q ext = Q ω and Q ∞ = Q ω 2 .
Lemma 14 applied with δ 2 and δ 1 = √ κ/|Q ω | provides us with ν and θ universal. We choose next k 0 the smallest positive integer such that

k 0 ν > |B 1 × B 1 × (-2, 0)|.
We finally choose β such that β ≤ θ k 0 .

Iteration. We define f 0 = f and

f k+1 = 1 θ (f k -(1 -θ)) = θ -k (f -(1 -θ k )).
They satisfy f k ≤ 1 and

(∂ t + v • ∇ x )f k ≤ ∇ v • (A∇ v f k ) + B • ∇ v f k + s k with s k = θ -k s. In particular |s k | ≤ θ -k 0 β ≤ 1
|{f 0 ≤ 0} ∩ Q| ≥ δ 2 | Q| and {f k+1 ≤ 0} ⊃ {f k ≤ 0}.
Our goal is to prove that there exists at least one index k ∈ {1, . . . , k 0 } such that

|{f k ≥ 0} ∩ Q ω | ≤ δ 1 |Q ω |.
Indeed, remarking that for such an index k 1

(f k 1 ) + L 2 (Qω) ≤ {f k 1 ≥ 0} ∩ Q ω 1 2 ≤ δ 1 |Q ω | ≤ κ,
Theorem 12 then implies that

f ≤ 1 - 1 2 θ k 1 ≤ 1 - 1 2 θ k 0 in Q ω 2
which concludes the proof.

Let us prove the claim by contradiction. Assume that for all k = 1, . . . , k 0 ,

|{f k ≥ 0} ∩ Q ω | ≥ δ 1 |Q ω |. Since f k+1 = 1 θ (f k -(1 -θ)), this also implies for k = 0, . . . , k 0 -1, |{f k ≥ 1 -θ} ∩ Q ω | ≥ δ 1 |Q ω |.
But (4.14) also implies that for all k ≥ 0,

|{f k ≤ 0} ∩ Q| ≥ δ 2 | Q|.
Hence Lemma 14 implies that for k = 0, . . . , k 0 -1,

|{0 ≤ f k ≤ 1 -θ} ∩ (B 1 × B 1 × (-2, 0))| ≥ ν. Now remark that |{f k+1 ≤ 0} ∩ (B 1 × B 1 × (-2, 0))| = |{f k ≤ 0} ∩ (B 1 × B 1 × (-2, 0))| + |{0 ≤ f k ≤ 1 -θ} ∩ (B 1 × B 1 × (-2, 0))| ≥ |{f k ≤ 0} ∩ (B 1 × B 1 × (-2, 0))| + ν.
In particular

|B 1 × B 1 × (-2, 0)| ≥ |{f k 0 ≤ 0} ∩ (B 1 × B 1 × (-2, 0))| ≥ k 0 ν
which is impossible for k 0 as chosen above. The proof is now complete.

Proof of the Hölder estimate.

Proof of Theorem 3. Consider an L 2 solution f of Eq. (1.6) in a cylinder Q ext = Q r 0 (z 0 ). By Theorem 12, we know that f is locally bounded in Q ext . In particular, f is bounded in

Q mid = Q r 0 +r 1 2 (z 0 ) and f L ∞ (Q mid ) ≤ C 0 f L 2 (Qext) + s L ∞ (Qext)
for some constant

C 0 = C(d, λ, Λ, Q ext , Q mid ). If f ≡ 0 in Q ext ,
there is nothing to prove. If f is not identically 0, recalling that β is given by Lemma 17, we assume that

f L ∞ (Q mid ) ≤ 1 and s L ∞ (Qext) ≤ β by considering, if necessary, f = f C 0 f L 2 (Qext) + s L ∞ (Qext) + β -1 s L ∞ (Qext)
.

Let z 1 ∈ Q int := Q r 1 (z 0 ). We want to prove that for all r > 0 such that

Q 2r (z 1 ) ⊂ Q mid , (4.15) osc Qr(z 1 ) f ≤ Cr α
for some universal α ∈ (0, 1) and some constant C = C(d, λ, Λ, r 0 , r 1 ). Let r > 0 denote the largest r ∈ (0, 1) such that Q 2r (z 0 ) ⊂ Q mid . We remark that for r ∈ (0, r), We recall how to scale solutions. For all r ∈ (0, r), the function fr (x, v, t) = f (r 3 x, rv, r 2 t) is defined in Q 2 and satisfies (1.6) with    Br (x, v, t) = r B(r 3 x, rv, r 2 t) sr (x, v, t) = r 2 s(r 3 x, rv, r 2 t).

Q 2r (z 1 ) = T -1 z 1 (Q 2r ) where T z 1 is defined in Eq. (1.8) and f = f • T z 1 satisfies (1.
Since osc Q 2r f ≤ 2, we have osc Q 2 fr ≤ 2 and Lemma 17 implies that

osc Q ω 2 fr = osc Q ω 2 r f ≤ 2θ with θ = 1 -λ 0 /2 (
we used the fact that r ≤ 1 to ensure that sr L ∞ (Q 2 ) ≤ β). We remark that we can assume that θ ≥ 1/2 and we recall that ω ∈ (0, 1/2). We next apply Lemma 17 to θ -1 fr 1 with r1 = (ω/4)r, which rescales the L ∞ bound on the source term by a factor (ω/4) 2 θ -1 < 1 as compared to sr L ∞ (Q 2 ) ≤ β. Hence the bounds assumed are still valid and we get osc Q r2 f ≤ 2θ 2 with r2 = (ω/2)r 1 . Inductively, we deduce that osc Q rk f ≤ 2θ k with rk = (ω/2) k r/2. This yields (4.16) for r = rk with

α = ln θ ln(ω/2) and C = 2 2 r α . If now r ∈ [r k+1 , rk ], then osc Qr f ≤ osc Qr k f ≤ C rα k = C 2 ω α rα k+1 ≤ Cr α with C = C(2/ω) α .
Observe finally that the constant C and C are uniformly bounded above as z 0 varies in Q int since r ≥ r 1 -r 0 . The proof is now complete.

Harnack inequality

In this section, we derive Harnack inequality for solutions to Eq. (1.6). We use here an approach that Luis Silvestre explained to us in the stationary setting: we start with Hölder continuous solutions and we consider expanding cylinders to control the spreading of the lower bound of nonnegative solutions (see Lemma 23). The Harnack inequality is a consequence of the decrease of oscillation we proved earlier and a so-called "doubling property" that estimates how the minimum of a solution propagates with time. Let us first recall the decrease of oscillation proposition.

Proposition 19 (Decrease of oscillation).

There exist δ ∈ (0, 1) and ω ∈ (0, 1/2) (both universal) such that for any r ∈ (0, 1) and any solution f of (1.6) in some cylinder Q 2r (z) satisfies

osc Q ω 4 r (z) f ≤ (1 -δ) osc Qr(z) f + 2β -1 s L ∞ . Remark 20. The conclusion of the proposition is equivalent to osc Q ω 4 r f • T z ≤ (1 -δ) osc Qr f • T z + 2β -1 s L ∞ with T z (y, w, s) = (x + y + sv, v + w, t + s) where z = (x, v, t). Proof. By considering f = f • T z osc Q 2r (z) f /2 + s L ∞ /β ,
and a rescaling fr , we can assume that z = 0 and osc Q 2 fr ≤ 2 and s L ∞ ≤ β (we use here that r ≤ 1). We then apply Lemma 17 to fr and get the desired result with 1 -δ = 1 -λ 0 /2.

5.1.

How minima propagate with time. The goal of this subsection is to prove the following proposition. In order to state it, we introduce two cylinders which contain Q -:

Q -⊂ Q -[1] ⊂ Q -[2] ⊂ Q 1 .
See Figure 2. We recall that Q + = Q R and Q -= Q R (0, 0, -∆) and R, ∆ ∈ (0, 1) are small so that in particular Q ± ⊂ Q 1 and they are disjoint. We let

Q -[i] be equal to Q ρ i (0, 0, -∆) with R < ρ 1 < ρ 2 < 1.
In the following propositions, we introduce elongated cylinders Q el where the time is stretched longer in the past than what the scaling would induce:

Q el 1 = B (ω/4) 3 × B ω/4 × (-1, 0] Q el r (z) = T z B (ω/4) 3 r 3 × B (ω/4)r × (-r 2 , 0]
. Proposition 21 (The propagation of minima). Assume that f is a non-negative super-solution of (1.6) in Q 1 with a non-negative source term s. There exists r 0 > 0, R > 0 (universal) such that for any r ∈ (0, r 0 ) and z

∈ Q -such that Q el r (z) ⊂ Q -[2],
we have min

Q el r (z) f ≤ C pm r -q min Q + f
for some universal constants C pm and q > 0.

We first derive from Lemma 18 the following doubling property at the origin. For the two next lemmas, it is easier that 0 is the final time of the first cylinder.

+ , Q -, Q -[1] and Q -[2]
. Harnack inequality relates the supremum of a solution over Q -and its infimum over Q + . The proof consists in constructing a sequence of points lying in Q - [START_REF] Valeri | Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation[END_REF] and whose corresponding values explode. Neighborhoods of points included in Q -[2] are also considered.

Lemma 22 (The doubling property at the origin).

There exists h ∈ (0, 1) (universal) such that for any non-negative super-solution f of

(1.6) in B 8 × B 2 × (-1, 4] with s ≥ 0, we have inf Q 1 f ≥ h inf Q 0 f with Q 1 = Q el 2 (0, 0, 4) and Q 0 = Q el 1 .
Proof. We first note that since s ≥ 0, the function f is a super-solution of (1.6) with s = 0. We first prove that (5.1) inf

Q ω/2 (0,0,1) f ≥ h 0 inf Q ω/4
f for some universal constant h 0 ; see Figure 3.

Figure 3. The doubling property. On the left, the cylinders Q ω/4 and Q ω/2 (0, 0, 1). In the middle, the elongated cylinders Q 0 and Q 1 . On the right, the iterated cylinders Q 0 , . . . , Q N (Lemma 23).

If inf Q ω/4 f = 0, there is nothing to prove. If not, the function

g = f inf Q ω/4 f satisfies (1.6) in Q 2 (
up to translation in time -this is where we use that s = 0) and

|{g ≥ 1} ∩ Q ω | ≥ |Q ω/4 | = δ 2 |Q ω |
for some universal δ 2 , where Q ω plays the role of Q in Lemma 18. We then apply Lemma 18 (with time shifted by +1) to g = 1

-g ≤ 1, we get g ≥ h 0 in B (ω/2) 3 × B ω/2 × (1 -(ω/2) 2 , 1]
, that is to say, (5.1) indeed holds true.

Apply now the result to f (x, v, t) = f (x, v, t -T ) for T ∈ [0, 1 -ω 2 ] and get (5.2) inf

B (ω/2) 3 ×B ω/2 ×(0,1] f ≥ h 0 inf Q 0 f.
By applying (5.2) on time intervals (1, 2], (2, 3] and [START_REF] Alexandre | Some a priori estimates for the homogeneous Landau equation with soft potentials[END_REF][START_REF] Alexandre | Regularizing effect and local existence for the non-cutoff Boltzmann equation[END_REF], we propagate the infimum till time t = 4 and get the desired result for h = h 4 0 . Applying iteratively the previous lemma, we obtain straightforwardly the following lemma whose proof is omitted.

Lemma 23 (The iterated doubling property at the origin).

There exists h > 0 (universal) such that for any f non-negative super-solution of

(1.6) in B 2 3N × B 2 N × (-1, T N ), we have (5.3) inf Q N f ≥ h N inf Q 0 f with Q k = B R 3 k × B R k × (T k-1 , T k ] for k ≥ 1
where R k = (ω/4)2 k and T k = 4 3 (4 k -1) for k ≥ 0. Remark 24. In [START_REF] Imbert | An introduction to fully nonlinear parabolic equations[END_REF], a measure estimate is also applied iteratively to prove a Harnack inequality for fully nonlinear parabolic equations in non-divergence form.

We can now prove Proposition 21.

Proof of Proposition 21. In the following proof, we need iterated cylinders that are not centered at the origin and with arbitrary radius.

Q k r (z) := T z rQ k . The cylinder Q k is first scaled by r (this is rQ k ) and then centered around z (this is T z rQ k ). Let z ∞ ∈ Q + be such that min Q + f = f (z ∞ ).
Lemma 25. There exist R, ∆, r 0 (small, universal) such that a) for all r ∈ (0, r 0 ) and z ∈ Q -, the iterated cylinders

Q k r (z) (k ∈ N) which are included in {t ≤ 0} are in fact included in Q 1 (0); b) the union of the iterated cylinders +∞ k=1 Q k r (z) contains Q + . The proof is elementary but tedious. It is given in Appendix. Applying Lemma 23, we get inf Q el r (z) f ≤ h -N inf Q N r (z) f ≤ h -N min Q + f with N such that z ∞ ∈ Q N r (z), i.e. r -1 (z -1 • z ∞ ) ∈ Q N . In particular, r -2 (t ∞ -t) ∈ [T N -1 , T N ]. Since z ∞ ∈ Q + and z ∈ Q -, we know that 4 N -1 ≤ T N -1 ≤ t ∞ -t r 2 ≤ 1/2 + R 2 r 2 .
In particular,

h -N ≤ 1/2 + R 2 4 q 2 r -q
where q = -ln γ/ ln 2 > 0. We get the desired inequality with C pm = ((1/2 + R 2 )/4) q 2 . The proof of the proposition is thus complete. 5.2. Proof of the Harnack inequality. We can now turn to the proof of Theorem 4.

Proof of Theorem 4. We first remark that replacing f (x, v, t) with f (x, v, t) + s L ∞ t if necessary, we can assume that s ≥ 0. Dividing f by 2β -1 s L ∞ if necessary, we can assume that s L ∞ = β/2 (if s ≡ 0).

We are going to find a universal constant C = C H such that (1.10) cannot hold false. In other words, we are going to find a universal C H such that (5.4) m + 1 ≤ C H M entails a contradiction where

M := sup Q - f = f (z 0 ) and m := inf Q + f = f (z ∞ )
for some z 0 ∈ Q -and z ∞ ∈ Q + . We used here the fact that u is (Hölder) continuous.

Our goal is to construct by induction a sequence

(z k ) k≥0 in Q -[1] (we recall that Q -⊂ Q -[1] ⊂ Q -[2] ⊂ Q 1 , see Figure 2) such that (5.5) f (z k ) ≥ (1 -δ ) -k M
for some universal δ ∈ (0, 1). This implies in particular that f (z k ) → +∞ as k → +∞ which is absurd since f is bounded in Q -. Remark first that (5.5) holds true for k = 0. Let us assume that we already constructed z 0 , . . . , z k and let us construct z k+1 . Let z k = (x k , v k , t k ). We choose r k > 0 such that (5.6) f (z k ) = r -2q k m where q is given by Proposition 21. Inequality (5.4) and the induction hypothesis (5.5) imply (5.7)

r 2q k ≤ C H (1 -δ ) k .
From the decrease of oscillation (Proposition 19), we know that

1 + osc Qr k f ≥ (1 -δ) -1 osc q k f (recall 2β -1 s L ∞ = 1) with Q k = Q r k (z k ) and q k = Q ωr k /4 (z k ).
In particular,

z k ∈ q k . Let z k+1 ∈ Q k be such that max Q k f = f (z k+1 ).
Then we get

(5.8) 1 + f (z k+1 ) ≥ (1 -δ) -1 f (z k ) -min q k f .
Recall that z k ∈ Q - [START_REF] Valeri | Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation[END_REF]. Choosing C H small, we can ensure through (5.7

) that Q r k (z k ) ⊂ Q -[2]
. We also remark that

q k ⊃ Q el (ω/4) 2 r k (z k ).
We thus can apply Proposition 21 and get min

q k f ≤ min Q el (ω/4) 2 r k (z k ) f ≤ Cpm r -q k m
with Cpm = C pm (4/ω) q . The use of (5.6) in the previous inequality yields (5.9) min

q k f ≤ Cpm r q k f (z k ) ≤ Cpm C H f (z k ).
Now combining (5.8) and (5.9), we get

1 + f (z k+1 ) ≥ (1 -δ) -1 (1 -Cpm C H )f (z k ).
Use next that 1 ≤ C H M (this is a consequence of (5.4)) and the induction hypothesis and get

f (z k+1 ) ≥ (1 -δ) -1 (1 -Cpm C H )(1 -δ ) -k M -C H M ≥ j(1 -δ ) -k M. with j = (1 -δ) -1 (1 -Cpm C H ) -C H .
We thus choose δ such that (1 -δ ) -1 = j and we can choose C H small enough so that δ ∈ (0, 1). In particular we get

f (z k+1 ) ≥ (1 -δ ) -k-1 M
which is the desired inequality.

We are left with proving that the sequence {z k } stays in Q - [START_REF] Valeri | Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation[END_REF]. The fact that z k+1 lies in 2q) .

Q r k (z k ) = T z k (Q r k (0)). This implies in particular that |v k+1 -v k | ≤ r k which in turn yields |v k -v 0 | ≤ l≥0 r l ≤ C 1/(2q) H l≥0 (1 -δ ) k 2q = C 1/(2q) H 1 -(1 -δ ) 1/(
Using now that the fact that δ is explicitely given as a function of δ and C H (see above), we conclude that |v k -v 0 | can be arbitrarily small uniformly in k. We can argue in the same spirit for |x k -x 0 | and |t k -t 0 |. Since z 0 ∈ Q -, we conclude that we can indeed ensure that z k lies in Q - [START_REF] Valeri | Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation[END_REF]. The proof of the theorem is now complete.

Local gain of regularity for sub-solutions

In this section, we investigate the regularity of sub-solutions to Eq. (1.6) beyond the gain of integrability proved above. Observe that, on the one hand, Theorem 6 applies to sub-solutions but only concludes to the gain of integrability. On the other hand, Theorem 8 proves a gain of Sobolev regularity but only applies to solutions (not sub-solutions). It might seem, at first sight, that the lack of ellipticity in all directions means the gain of regularity of solutions is false, since in the elliptic and parabolic case it is entirely based on the energy estimate. However we show here that, using the local upper bound proved above by the De Giorgi-Moser iteration, and refined averaging lemmas, this result still holds in essence for our equation, even though the gain of regularity is only H s with s > 0 small. We prove the following result:

Theorem 26 (Gain of regularity for non-negative sub-solutions). Consider z 0 ∈ R 2d+1 and two cylinders Q int := Q r 1 (z 0 ) and Q ext := Q r 0 (z 0 ) with 0 < r 1 < r 0 . Then there is some s ∈ (0, 1/3) so that any weak non-negative sub-solution f of (1.6) in Q ext satisfies (6.1)

f H s x,v,t (Q int ) ≤ C f L 2 (Qext) + s L ∞ (Qext) with C = C(d, λ, Λ, Q ext , Q int ).
Proof of Theorem 26. We define Q mid in between Q int and Q ext and the same truncation functions as before. Theorem 12 implies that

f L ∞ (Q mid ) f L 2 (Qext) + s L ∞ (Qext) .
We want to apply [START_REF] Bouchut | Hypoelliptic regularity in kinetic equations[END_REF]Theorem 1.3] on f in Q mid . However since f is only a sub-solution it satisfies the equation

∂ t f + v • ∇ x f = ∇ v • (A∇ v f ) + B • ∇ v f + s -µ in Q ext
where we have included the defect non-negative measure µ ≥ 0 accounting for the inequation. We can now repeat the reasoning from the proof of Lemma 14 and reduce to the case

∂ t g + v • ∇ x g = ∇ v • (A∇ v g) + ∇ v • H 1 + H 0 -μ in R 2d+1
with g ≡ f in Q int and g, the measure μ ≥ 0, H 0 and H 1 supported in Q mid , and with g, ∇ v g, H 0 and H 1 bounded in L 2 on Q mid . Then by integrating in x, v, t we deduce that μ has bounded variation in terms of the previous bounds. Since for q > (4d + 2), the space W 1 2 ,q

x,v,t embeds into continuous bounded functions of x, v, t, we deduce that the space of measures is included in W - 1 2 ,q * x,v,t and therefore

(6.2) μ = (1 -∆ x,t ) 1 4 (1 -∆ v ) h with h ∈ L q * (Q mid )
and the bound on the L q * (Q mid ) depends on the previous bounds above, and where q * = 1/(1-1/q) is the conjugate exponent of q. Observe that q * is striclty smaller than 2 and close to one, for instance q * ∈ (1, 14/13) in dimension d = 3. We then apply [7, Theorem 1.3] with κ = 1, r = 1 2 , m = 2, β = 1, p = q * : we deduce that g belongs to W 1 8 ,p x,t L p v (observe that we use a full Laplacian derivative in v in Eq. (6.2) in order to be in the framework of [7, Theorem 1.3], even though (1 -∆ v ) 1/4 would have been enough for the purpose of having h ∈ L q * ). By interpolation with the L ∞ estimate, we obtain then that g ∈ H s

x,t L 2 v for some s ∈ (0, 1 8 ) small enough. Finally, we combine the latter estimate with the energy estimate g ∈ L 2

x,t H 1 v we conclude with g ∈ H s x,v,t . Since the truncation function is equal to one on the smaller cube Q int , it translates into f ∈ H s

x,v,t on Q int and concludes the proof.

Gain of integrability of the velocity gradient

This section is devoted to the proof of the following theorem.

Theorem 27 (Gain of integrability for ∇ v f ). Let f be a solution of (1.6) without lower order terms (B ≡ 0 and s ≡ 0) in some cylinder Q r 0 (z 0 ). There exists a universal ε > 0 such that for all

Q[i] = Q r i (z 0 ), i = 0, 1, 2 with r 2 < r 1 < r 0 , ∇ v f ∈ L 2+ε (Q 2 ) (7.1) ˆQ[2] |∇ v f | 2+ε dz ≤ C ˆQ[1] |∇ v f | 2 dz 2+ε 2 with C = C(d, λ, Λ, Q 2 , Q int , Q ext ).
The proof follows along the lines of the one of [32, Theorem 2.1]. It consists in deriving an almost reverse Hölder inequality which in turn implies the result thanks to the analogous of [START_REF] Giaquinta | On the partial regularity of weak solutions of nonlinear parabolic systems[END_REF]Proposition 1.3]. The following measure-theoretical lemma will be used as a black box in the proof of Theorem 27. It implies the use of cylinders with different shape:

Q(z 0 , r) = {z = (x, v, t) : |x i -x 0 i | < r 3 , |v i -v 0 i | < r, -r 2 < t -t 0 ≤ 0} where x = (x 1 , . . . , x d ) and v = (v 1 , . . . , v d ).
The scaling of the equation preserves this family of cylinders but not the Lie group action T z .

Lemma 28 (A Gehring lemma). Let g ≥ 0 in Q such that there exists q > 1 such that for all

z 0 ∈ Q and R such that Q 4R (z 0 ) ⊂ Q, Q R (z 0 ) g q dz ≤ b Q 4R (z 0 ) g dz q + θ Q 4R (z 0 )
g q dz for some θ > 0. There exists θ 0 = θ 0 (q, d) such that if θ < θ 0 , then g ∈ L p loc (Q) for p ∈ [q, q + ε) and

Q R g p dz 1 p ≤ c p Q 4R g q dz 1 q
, the constants ε > 0 depends only on b, q, θ and dimension, and c p further depends on p.

The proof of Lemma 28 is an easy adaptation of the one of [31, Proposition 5.1], by changing Euclidian cubes with cylinders Q R .

The proof of Theorem 27 is a consequence of some estimates involving weighted means of the solution. Given z 0 ∈ R 2d+1 , they are defined as follows

f2R (t) = 1 cR 4d ˆR2d f (t, x, v)χ 2R (x, v, t) dx dv
(for some c defined below) where χ 2R is a cut-off function such that

χ 2R (x, v, t) = d i=1 φ R 3 (x i -x 0 i )φ R (v i -v 0 i ) with φ R (a) = φ(a/R) for some φ such that √ φ ∈ C ∞ (R) and φ ≡ 1 in [-1, 1] and supp φ ⊂ [-2, 2]. We remark that χ 2R ≡ 1 in Q R and χ 2R ≡ 0 outside Q 2R . Lemma 29. Let f be a solution of (1.6) in Q 0 . Then for Q 3R (z 0 ) ⊂ Q 0 , ˆQR (z 0 ) |∇ v f | 2 dz ≤ CR -2 ˆQ2R (z 0 ) |f -f2R | 2 dz (7.2) sup t∈(t 0 -R 2 ,t 0 ] ˆQt R (z 0 ) |f (t) -fR (t)| 2 dx dv ≤ C ˆQ3R (z 0 ) |∇ v f | 2 dz (7.3) where Q t R (z 0 ) = {(x, v) : (t, x, v) ∈ Q R (z 0 )}. Remark 30. This lemma corresponds to [32, Lemmas 2.1 & 2.2].
Proof. For the sake of clarity, we put z 0 = 0 and R

= 1. Consider τ 2 ∈ C ∞ (R, R) such that 0 ≤ τ 2 ≤ 1, τ 2 ≡ 0 in (-∞, -2 2 ] and τ 2 ≡ 1 in [-1, 0]. Use 2(f -f2 )χ 2 τ 2 as a test function for (1.6) and get ˆR2d (f (0) -f2 (0)) 2 χ 2 dx dv + 2 ˆR2d+1 (A∇ v f • ∇ v f )χ 2 τ 2 dx dv dt = ˆR2d+1 (f -f2 ) 2 χ 2 (∂ t τ 2 ) dx dv dt - ˆR2d+1 v • ∇ x (f -f2 ) 2 χ 2 τ 2 dx dv dt -2 ˆR2d+1 (f -f2 )A∇ v f • ∇ v χ 2 τ 2 dx dv dt.
Remark that the definition of f2 implies that the remaining term

-2 ˆR2d+1 (∂ t f2 )(f -f2 )χ 2 τ 2 vanishes. This equality yields ˆR2d (f (0) -f2 (0)) 2 χ 2 dx dv + λ ˆR2d+1 |∇ v f | 2 χ 2 τ 2 dx dv dt ≤ ˆR2d+1 (f -f2 ) 2 χ 2 |∂ t τ 2 | + |v • ∇ x χ 2 |τ 2 + Λ 2 λ |∇ v √ χ 2 | 2 τ 2 dx dv dt
which yields (7.2). Changing the final time, we also get

sup t∈(-1,0] ˆR2d f (t) -f2 (t) 2 χ 2 (t) dx dv ≤ C ˆQ2 |f -f2 | 2 dx dv dt. Now the function F = f -f2 is such that ´F (x, v, t) dx dv = 0. In particular, we have ˆQ2 (f -f2 ) 2 dx dv dt ≤ C ˆQ2 |∇ v f | 2 + |D 1 3
x f | 2 dx dv dt.

Observe that if there are no lower order terms (B = 0 and s = 0), then we have for all q ∈ (1, 2], (7.4)

ˆQ2 |D 1 3 x f | q dx dv dt ≤ C ˆQ3 |∇ v f | q dx dv dt.
Indeed, in view of the proof of (2.2), it is enough to apply [7, Theorem 1.3] with such a q and use the Poincaré inequality (assuming the cutoff functions to have convex super-level sets).

Combining the three previous estimates yields sup t∈(-1,0] ˆQt

1 (f (t) -f2 (t)) 2 χ 2 (t) dx dv ≤ C ˆQ3 |∇ v f | 2 dx dv dt.
Finally, we write for t ∈ (-1, 0] We now turn to the proof of Theorem 27. The use of (7.4) is the main difference with [START_REF] Giaquinta | On the partial regularity of weak solutions of nonlinear parabolic systems[END_REF].

Proof of Theorem 27. Pick p > 2 and let q denote its conjugate exponent: 1 q + 1 p = 1. We follow [START_REF] Giaquinta | On the partial regularity of weak solutions of nonlinear parabolic systems[END_REF] in writing (omitting the center of cylinders z 0 ), thanks to (7.2), where (7.3) and Hölder inequality are used successively.

Q 1 |∇ v f | 2 ˆQ2 |f -f2 | 2 ≤ sup t∈(t 0 -4,t 0 ] ˆQt 2 |f -f2 | 2 1 2 ˆt0 t 0 -4 dt ˆQt 2 |f -f2 | 2 1 2 ˆQ4 |∇ v f | 2
We now use Sobolev inequalities and Hölder inequality (twice) successively to get

Q 1 |∇ v f | 2 ˆQ4 |∇ v f | 2 1 2 ×   ˆt0 t 0 -4 ˆQt 2 |∇ v f | q + |D 1/3 x f | q 1 2q dt   × ˆQt 2 |∇ v f | 2 + |D 1/3 x f | 2 1 4 ˆQ4 |∇ v f | 2 1 2 ˆQ2 |∇ v f | q + |D 1/3 x f | q 1 2q ×   ˆt0 t 0 -4 ˆQt 2 |∇ v f | 2 + |D 1/3 x f | 2 q 2(2q-1) dt   2q-1 2q ˆQ4 |∇ v f | 2 1 2 × ˆQ2 |∇ v f | q + |D 1/3 x f | q 1 2q × ˆQ2 |∇ v f | 2 + |D 1/3 x f | 2 1 4
.

We now use (7.4) and get

Q 1 |∇ v f | 2 ˆQ4 |∇ v f | 2 1 2 ˆQ2 |∇ v f | q 1 2q ˆQ2 |∇ v f | 2 1 4 ˆQ4 |∇ v f | 2 3 4 ˆQ2 |∇ v f | q 1 2q
. Now use and get for all ε > 0,

Q 1 |∇ v f | 2 Q 4 |∇ v f | 2 3 4 Q 4 |∇ v f | q 1 2q Q 4 |∇ v f | 2 3 4 Q 4 |∇ v f | q 1 2q
.

After rescaling, we get the following

Q R |∇ v f | 2 Q 4R |∇ v f | 2 3 4 Q 4R |∇ v f | q 1 2q ε Q 4R |∇ v f | 2 + c ε Q 4R |∇ v f | q 2 q
.

Apply now Proposition 28 in order to achieve the proof of Theorem 27. where c only depends on dimension, γ, M 0 , M 1 , E 0 and H 0 .

Lemma 32 (Upper bounds - [START_REF] Desvillettes | On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications[END_REF][START_REF] Silvestre | Upper bounds for parabolic equations and the Landau equation[END_REF]). Assume there exist positive constants M 1 , M 0 , E 0 and H 0 such that (1.3) holds true. Assume that f ∈ L ∞ (R d ).

Then

|A[f ]| ≤ C(1 + |v|) γ+2 if γ ∈ [-2, 0] C f |γ+2| d ∞ if γ ∈ [-d, -2) |B[f ]| ≤ C(1 + |v|) γ+1 if γ ∈ [-1, 0] C f |γ+1| d ∞ if γ ∈ [-d, -1) |c[f ]| ≤ C if γ = 0 C f |γ| d ∞ if γ ∈ [-d, 0).
where C only depends on dimension, γ, M 0 , E 0 . As far as a) is concerned, we should ensure that for all z ∈ Q -and r ∈ (0, r 0 ),

(z • rP + ) ∩ {t ≤ 0} ⊂ Q 1 (0).
If z = (x -, v -, t -) and z + = (x + , v + , t + ) ∈ rP + are such that z • z + ∈ {t ≤ 0}, we have 0 ≥ t -+ t + ≥ (-∆ -R 2 ) + 4 3 ((4/ω 2 )ρ 2 -r 2 ) ≥ -4∆ + (4 2 /3ω 2 )ρ 2

where ρ = |v + |. This implies in particular

ρ 2 ≤ 3ω 2 4 ∆.
In particular, for ∆ ∈ (0, 1),

|v -+ v + | ≤ R + ρ ≤ (1 + √ 3ω/2) √ ∆ |x -+ x + + t + v -| ≤ R 3 + ρ 3 + R ≤ (1 + ( √ 3ω/2) 3 )∆ 3/2 + ∆ ≤ (2 + ( √ 3ω/2) 3 ) √ ∆.
We thus can choose ∆ small enough (recall ω = 1/ √ 5) to ensure a). As far as b) is concerned, notice that for z + ∈ Q + and z ∈ Q -, we have

z -1 • z + = (t + -t, x + -x -(t + -t)v, v + -v).
Choosing R 2 ≤ ∆ ≤ 1 2 we have 2R ≤ (4R) (since R ≤ 1 and ∆ ≤ 1) and t + -t ≥ ∆ -R 2 . In particular

z -1 • z + ∈ rP -if ∆ -R 2 ≥ 4 3 4 2 ω 2 (4R) 1 3 -r 2 .
It is enough satisfy

∆ ≥ R 2 + 4 3 3ω 2 (4R) 1 3
.

Hence, for ∆ given, we can choose R = R(∆) small enough to get the desired inequality and in turn point b).

1 . Introduction 1 . 1 .

 111 The Landau equation. We consider the Landau equation(1.1)

2 . 1 ( 1 2

 211 Theorem Hölder continuity for the Landau equation). Assume γ ∈ [-d, 0]. Let f be an essentially bounded weak solution of (1.2) in B 1 ×B 1 ×(-1, 0]. Assume that C(x, t) (equation (1.3)) holds true for all B 1 × (-1, 0]. Then f is α-Hölder continuous with respect to (x, v, t) ∈ B

Figure 1 .

 1 Figure 1. Cylinders involved in the statement of the De Giorgi intermediate-value Lemma.

  6) in Q 2r with the source term s := s • T z 1 and the coefficients Ā := A • T z 1 and B := B • T z 1 . In particular f and s satisfy f L ∞ (Q 2r ) ≤ 1 and s L ∞ (Q 2r ) ≤ β and (4.15) is equivalent to: for all r ∈ (0, r), (4.16) osc Qr f ≤ Cr α .

Figure 2 .

 2 Figure 2. The cylinders Q+ , Q -, Q -[1] and Q -[2]. Harnack inequality relates the supremum of a solution over Q -and its infimum over Q + . The proof consists in constructing a sequence of points lying in Q -[START_REF] Valeri | Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation[END_REF] and whose corresponding values explode. Neighborhoods of points included in Q -[2] are also considered.

2 |f -f2 | q 1 2q ˆQt 2 |f -f2 | p 1

 221 2p

Appendix A .

 . Known estimates for the Landau equationLemma 31 (Lower bound -[START_REF] Desvillettes | On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications[END_REF][START_REF] Silvestre | Upper bounds for parabolic equations and the Landau equation[END_REF]). Assume there exist positive constants M 1 , M 0 , E 0 and H 0 such that (1.3) holds true. Thendet A[f ] ≥ c(1 + |v|) κ with κ = (d -1)(γ + 2) + γ if γ ∈ [-2, 0] 3γ + 2 if γ ∈ [-d, -2)

Appendix B . 3 4 2 ω 2 ρ 2 - 1 , 3 4 ω 2 ρ 2 - 1 ,

 .321321 Proof of a technical lemma Proof of Lemma 25. To justify a) and b), we remark that P -⊂ +∞ k=1 Q k ⊂ P + where P -:= {(y, w, s) : s ≥ 4 |y| ≤ ρ 3 , |w| ≤ ρ}, P + := {(y, w, s) : s ≥ 4 |y| ≤ ρ 3 , |w| ≤ ρ}, see Figure 4.

Figure 4 .

 4 Figure 4. Paraboloids containing/contained in the union of iterated cylinders.

1 3 3 |x

 13 and we get|v + -v| ≤ 2R ≤ (4R) 1 + -x -(t + -t)v| ≤ 2R 3 + (∆ + R 2 )R = 3R 3 + ∆R ≤ 4R

We give the parabolic version due to Nash here.

The authors would like to thank Luis Silvestre for fruitful comments during the preparation of this article. The work of A. F. Vasseur was partially supported by the NSF Grant DMS 1209420, and by a visiting professorship at École polytechnique. The work of C. Mouhot was partially supported by the ERC Grant MATKIT, and by a visiting professorship at Université Paris-Est Créteil.


Matthieu Caudron

Fabrice Emeriault

Richard Kastner

Marwan Al Heib

Sinkhole and soil-structure interactions: Development of an experimental model

INTRODUCTION

Subsidence is the result of collapse of natural and man-made cavities induces damage on the buildings. Nevertheless the complex soil-structure interactions occurring during a sinkhole phenomenon are not well understood. The current procedure used to evaluate potential damage to buildings standing above a future sinkhole is to determine the soil movement in greenfield conditions and then to apply these movements to the structure, neglecting the possible interactions: see for example [START_REF] Deck | Taking the soil-structure interaction into account in assessing the loading of a structure in a mining area[END_REF].

The main purpose is to develop a small-scale experimental model using Schneebeli metallic rods that will allow highlighting the soil-structure interactions during such phenomenon. This analogical material was developed several year ago but it is still used in recent works: [START_REF] Dolzhenko | Experimental modeling of urban underground works. Displacement measurement in a two-dimensionnal tunneling experiment[END_REF][START_REF] Shashin | Influence of surface loads and construction sequence on ground response due to tunneling[END_REF]. However, the results will not be quantitative but only qualitative because of the disrespect of some laws of similarities. Moreover the occurrence of sinkholes appears very suddenly. Displacements reach the surface as soon as the failure occurs. It is so difficult to define a specific state for the beginning of the failure.

A well documented sinkhole [START_REF] Vachat | Les désordres survenants dans les carrières de la région parisienne. Etude théorique et pratique de l'évolution des fontis[END_REF]) is taken as a representative example of the many sinkholes observed in Paris each year [START_REF] Igc | Inspection Générale des Carrières : 222 ans de gestion de l'après carrière[END_REF]. It has been induced by the collapse of an underground cavity presenting the usual dimensions of the Parisian quarries: 8 m in width, 2 m in height and a cover of 8 m (Figure 1). ABSTRACT: Soil-structure interactions during a sinkhole phenomenon are analyzed with a physical smallscale model developed using an analogical two-dimensional soil and a building model. A test bed allowing fully controlled test is used. The paper presents the results of a series of tests during which vertical and horizontal displacements of the Schneebeli steel rods are determined with the Digital Image Correlation technique at different steps of the creation of an underground cavity. First, repeatability tests are performed as well as a greenfield test. Then the same initial conditions are considered for a test with a model of building located at ground level and above the cavity. Several soil layers compose the cavity cover. Their individual characteristics are presented in Fine 1993 but it can be roughly considered that they define two mechanical homogeneous layers: a 6 m thick cohesionless material overlying a 2 m thick cohesive layer whose mechanical characteristics are presented in Table 1. The building model is similar to a dwelling and comprises three spans and two stories. Its dimensions are 10 m in total width and each story in 2.7 m high (Figure 2). A crawl space is also represented. The small scale model dimensions are determined in order to satisfy the laws of similarity as accurately as possible.

THE LAWS OF SIMILARITIES

Assumptions

The assumption that the test will not depend on the thermal effect allows making the problem a bit sim-pler: [START_REF] Dehousse | Les modèles réduits de structure en Génie Civil[END_REF] show that there are then nine equations controlling the scaling. Four of them are straightforward, the five remaining relations define the compatibility between the different scale factors. It appears that three scale factors can be chosen independently, the others then are deduced from the 5 compatibility equations.

The scale factors

The tests will be performed under normal gravity, the acceleration factor is thus 1. The analogical soil used has a unit weight of 65 kN/m 3 . The corresponding scale factor can then be considered as equal to 3.

The scale factor on the lengths has been chosen as the last independent scale factor and in order to have a test bed with convenient dimensions, it has been fixed to 1/40. Therefore, it appears that the conversion of all the pressure type units to the small scale model must be done with a scale factor of 3/40 in order to meet the requirements of the laws of similarity.

2.3

The small-scale model

The analogical soils and the cavity

Using the different scale factors determined previously, the analogical soils should have a Young Modulus close to 4 up to 8 MPa. Neither their friction angles nor the Poisson coefficients are affected by the scale. The cohesion for the concerned layer should be around 15 kPa. The thickness of the two layers is 50 mm for the coherent one and 150 mm for the second one.

The building model

The geometry of the building model is shown on Figure 3. The thickness of the beams and the columns is 0.8 mm. This dimension allows having a building with a stiffness comparable with the stiffness of a real structure. Consequently, the weight of the building model is too small. Additional weights are placed on each of the horizontal beams to reach a total building weight compatible with the requirements of the laws of similarity.

THE EXPERIMENTAL MODEL

The test bed

The soil model has a maximum size of 750 mm width per 500 mm high above the cavity (corresponding to 30 m and 20 m respectively at full scale).

A small apparatus, inspired from the work of Nakai et al. 1997 focusing on tunneling induced soil movements, allows an easy and repetitive creation of the cavity. Moreover, cavities of different widths can be stepwise created: from 25 mm up to 250 mm in a maximum of 10 steps of 25 mm each. The height can also range from 25 to 100 mm but it has to be defined prior to the beginning of the test. At full scale, it corresponds to an opening of 1 to 10 m wide and 1 to 4 m high.

Ten moving parts of 25mm width numbered from 1 to 10 beginning at the left compose the total width of the cavity. Each one is equipped with a small force sensor on its top end. This will allow observing the variation of the forces applied by the soil mass to the remaining moving parts.

In order to follow the movements of the particles in the soil mass, a PIV-DIC procedure is used (Particle Image Velocimetry -Digital Image Correlation). [START_REF] White | Soil deformation measurement using particle image velocimetry (PIV) and photogrammetry[END_REF] clearly showed the pros and the cons of such method against photogrammetry. SI-FASOFT, the software used to compute the displacements has been developed in LamCOS (INSA-LYON) by [START_REF] Touchal | Mise au point d'une nouvelle méthode de mesure de champs de déformations par corrélation directe d'images numériques[END_REF]. The precision of the DIC method is around 1/100 pixel in very good conditions but with the presented apparatus and the corresponding test setup and procedure, the precision is only of 1/20 pixel. This corresponds to approximately 20 µm for a single couple of photos (0.8 mm at full scale). Therefore this value is sufficiently accurate for the purpose of this study.

F=1.8 N F F F F F F F F F F F

The analogical soil

The basic analogical cohesionless soil is composed of metallic Schneebeli rods presenting three different diameters (3, 4 and 5 mm) for a length of 60 mm (see details in [START_REF] Dolzhenko | Experimental modeling of urban underground works. Displacement measurement in a two-dimensionnal tunneling experiment[END_REF].

In order to observe a sinkhole and not a subsidence trough, the underlying coherent soil layer must be represented with the same assembly of metallic rods where cohesion is introduced. This is achieved by soaking the concerned rods in an aqueous solution of glue. Then these rods are manually put on the test bed in the desired place and dried until complete dehydration. The relative density can not be determined, but the basic property of this material is to present a constant porosity. The physical and mechanical characteristics of the cohesionless and coherent materials are presented in Table 2.

The building models

The design of the building model was done assuming that it should stay in the elastic range whatever may be the foreseeable load.

The instrumentation of the structure model comprises 9 strain gauges and 4 displacement sensors (LVDT). The gauges are represented on Figure 3 by the black marks. The sensors are positioned on each foundation. They are all connected to a computer in order to log the different values at given times.

In Figure 3, the arrows represent the loading that are used during the test in order to have a convenient foundation loading.

THE RESULTS

The tests

Four different tests were performed. First, two identical tests were executed in order to analyse the repeatability of the procedure. Then a single test was conducted in greenfield conditions while the last test uses the same initial conditions but with the building model on the soil surface.

The two repeatability tests were done with the same conditions. The purpose was to test the measurement setup, to observe the mechanism of failure and several other parameters and then to compare them together. The geometrical parameters are 300 mm of total cover: with a 50 mm thick cohesive layer and a 250 mm thick granular soil layer. The cavity is 250 mm wide and 50 mm high.

For the greenfield test, only the total cover is different: 50 mm of cohesive material are still used but the granular soil is now 150 mm thick. The greenfield test is taken as a reference for the last test with the building model. The foundations are put under 15 mm of soil (60 cm in real scale, near the frost level). The left foundation of the building is located directly above the center of the cavity.

The procedure for each test comprises several steps necessary to obtain the complete failure of the cover. Digital photos, strains and displacements are logged at each step for further analysis.

The cavity is open in five steps (the parts 5 and 6 are lowered first, then 4 and 7 followed by 3 and 8 and so on), and then a stable state is reached. For the two repeatability tests, failure occurs during this last step. On the contrary, for the two other tests (Greenfield conditions and with the structure), it is necessary to weaken the cohesive layer by adding some water in the center of the cavity roof. This water is added in very small quantity. Photos and measures are logged each time when another degradation is applied.

Comparisons and interpretations

The two repeatability tests give similar results. Figure 4 shows the subsidence trough for each test when the failure of the cavity is complete. The maximum subsidence is respectively 33.2 mm and 31.6 mm (values computed by the DIC process). The distances to the point of inflection are very close: 140 mm and 150 mm. The volumes of the trough present only 7% of difference.

The empirical approach (after Peck) can be used to plot a theorical trough as shown by [START_REF] Caudron | Effondrement de cavités souterraines et interaction avec les ouvrages en surface[END_REF], using a circular equivalent cavity. The horizontal displacements of the ground surface can be determined by the DIC method. They both are displayed on Figure 6 together with the theoretical shape on the movements plotted using the formula given by [START_REF] Lake | Prediction and effects of ground movements caused by tunnelling in soft ground beneath urban areas[END_REF].

0 z x s v x x = (2)
With z 0 the depth to the cavity axis and s x the vertical displacement from the empirical approach (see Eq. 1). The maximal horizontal displacement predicted by Equation 2 is close to 14 mm while the computed ones for both tests are between 29 and 24 mm. The results are good for both tests in accordance with the theoretical shape, even though there may be some punctual differences.

During the tests, the forces on the cavity "pillars" are measured by the sensors placed on the moving pieces. Figure 7 illustrates the time-rate of the transmission of the load after the lowering of the two pieces. The phenomenon is not immediate; the transmission is gradual until reaching a state of balance with an increase close to 35% for each sensor. In the subsequent analysis, only the stable values obtained for each step are computed.

Figures 8 show the evolution of the forces acting on individual sensors during the second test. The gradual redistribution of the forces is clearly visible. Figure 9 represents the total force applied on the 10 sensors: the value is stable for the four first steps and then decreases during the two last steps. The weight of the soil mass is first totally transferred from the lowered pieces to the next ones and at the end of the creation of the cavity, part of this weight is transferred by arching effect directly to the soil mass.

Greenfield test

During the greenfield test, no significant movements on the ground surface can be observed before the weakening of the cohesive layer, even when the cavity is wide open. The maximum vertical movement measured by DIC is 0.2 mm. When water is gradually added, the occurrence of the sinkhole appears and the movements increase until the total failure of the cavity. One can notice on Figure 10 that the failure is not symmetric because of the right part of the cohesive layer still overhanging. Figure 11 shows the two main stages where noticeable subsidences reach the top of the soil mass.

The maximum settlement is approximately 44 mm (1.76 m at full scale). It is close to the height of the cavity (50 mm or 2 m at full scale). The width of the trough for the last step is 450 mm (18 m at full scale). As for the repeatability test, Peck's empirical approach can be used as presented by [START_REF] Caudron | Effondrement de cavités souterraines et interaction avec les ouvrages en surface[END_REF]. The empirical curve is plotted on Figure 11. No important differences appear but on the right due to the partial failure of the cavity. It appears that the prediction of the soil subsidence in greenfield conditions is quite satisfying in spite of the difference of failure.

In the analysis of the horizontal displacements, two main states have to be considered: the first one when the failure is still symmetric (beginning of the sinkhole phenomenon), the second one when the left part of the cohesive layer falls into the cavity. Figure 12 illustrates the difference between the two cases: for the first one, the horizontal displacements present a similar shape to those seen during the repeatability tests whereas for the second one, the movement is completely located on the left part of the model (where the second failure appears). The shape is very different and oriented mainly towards the left.

Soil-structure interactions

The second test with the building model on the soil mass allows characterizing the soil-structure interactions. Figure 13 shows the final subsidence trough, compared with the greenfield results. The maximum vertical movement is 28% smaller than with the building. On the contrary, the width of the concerned area is 26% larger. But the volume of the trough is similar for both tests: respectively 443 cm 3 in greenfield and 481 cm 3 with the structure.

The horizontal displacement can be computed and compared to the results of the greenfield test. Figure 14 shows totally different behavior of the model, in shape and amplitude. The last point, but not the less important, is the maximum slope. The difference between the two tests is important: 24.7% for the greenfield test and 17.9% with the building. Even though in both cases it would be very damageable for a building, these tests show that it could be very cost effective to consider the soil-structure interactions when studying the effect of a sinkhole or similar phenomenon on a building. If the relative settlements are more finely observed, it appears that the deformations in the structure are very different as shown in Table 3.

The approach commonly employed (the displacement in greenfield are applied directly on the building) would consider too large deformations and moreover in the wrong part of the building compared to the results given by the soil-structure interactions approach.

Improvements

Even though the results obtained in this study are interesting and convincing, some improvements need to be done in the future. The results from the small scale building instrumentation can not for now be used to understand the effect of the soil-structure interactions on the building. The dynamic effect need to be more finely observed by the use of a high-speed camera.

CONCLUSION

In conclusion, it appears that physical model presents some results in good accordance with the empirical methods. Moreover the soil-structure interactions observed when the building is used are obvious. Even though the modifications are not on the volume of the subsidence trough but on its shape which becomes very asymmetrical. The design of this experimental model allowing the study of soil-structure interaction is the first step in a larger research project. Now that the small scale model is ready and only need some improvements, some parametric studies can be considered such as the cavity geometry, the position of the building and so on. All these results will allow, in a further step, to setup a numerical model before the study of some real case.

Figure 1 .

 1 Figure 1. Geological profile of the cavity

Figure 2 .

 2 Figure 2. Dimensions of the real building.

Figure 3 .

 3 Figure 3. Dimensions and loading of the building model.

Figure 4 .

 4 Figure 4. Subsidence troughs for the two repeatability tests and resulting from the empirical approach.

Figure 5 .

 5 Figure 5. Failure mechanisms for the first test (up) and the second one (down).

  s x the vertical displacement, s max the maximum vertical displacement, x the position to the center of the cavity and i the distance to the point of inflexion. The failure mechanisms are similar as shown on Figure5. The only difference is located on the left part of the cavity, where a small void remains in the first test.

Figure 6 .

 6 Figure 6. Horizontal displacements.

Figure 7 .

 7 Figure 7. Transmission of the forces from the lowered parts to the adjacent remaining ones.

Figure 8 .

 8 Figure 8. Evolution of the forces during the creation of the cavity (second test).

Figure 9 .

 9 Figure 9. Variation of the total force measured by the force sensor.

Figure 10 .

 10 Figure 10. Difference of failure (greenfield is down, with structure is up).

Figure 11 .

 11 Figure 11. Subsidence troughs for the greenfield conditions. The two last stages are plotted where significant movements are observed.

Figure 12 .

 12 Figure 12. Horizontal displacement for the greenfield test.

Figure 13 .

 13 Figure 13. Comparison of the subsidence between the tests in greenfield and with the building model (the arrows represent the position of the four foundations).

Table 1 .

 1 Soil properties of the documented sinkhole from laboratory tests[START_REF] Vachat | Les désordres survenants dans les carrières de la région parisienne. Etude théorique et pratique de l'évolution des fontis[END_REF]

		Granular soil	Coherent soil
	E (MPa)	50-100	50-100
	ν	0.30	0.30
	ρ (kg/m 3 )	2200	2200
	φ (°)	~26	~26
	c (kPa)	0	200

Table 2 .

 2 Characteristics of the two materials Density E (MPa) φ (°) c (kPa)

	Cohesionless soil	6.5	4-8	22-24	0
	(prototype scale)
	Full scale value	2.2	50-100	22-24	0
	Coherent soil	6.5	4-8	27-30	~8
	Full scale value	2.2	50-100	27-30	~100

Table 3 .

 3 Comparison of the slopes for the three different spans named by the number of the two foundations delimiting them.

	Span n°	Slope in green field	Slope with SSI
	1-2	24.7%	5.1%
	2-3	23.5%	17.9%
	3-4	1.2%	12.3%

Figure 14. Horizontal displacement for the test with the building.

Acknowledgements

We want to thanks the French Department of Environment and Durable Development for his research project DRS02 entrusted to the INERIS.