Log-transform and the weak Harnack inequality for kinetic Fokker-Planck equations
LLog-transform and the weak Harnack inequality for kinetic Fokker-Planck equations
Jessica Guerand & Cyril ImbertFebruary 9, 2021
Abstract
This article deals with kinetic Fokker-Planck equations with essentially boundedcoefficients. A weak Harnack inequality for non-negative super-solutions is derived byconsidering their Log-transform and following S. N. Kruzhkov (1963). Such a resultrests on a new weak Poincaré inequality sharing similarities with the one introducedby W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009,2011, 2017). This functional inequality is combined with a classical covering argumentrecently adapted by L. Silvestre and the second author (2020) to kinetic equations.
This paper is concerned with local properties of solutions of linear kinetic equations ofFokker-Planck type in some cylindrical domain Q ( ∂ t + v · ∇ x ) f = ∇ v · ( A ∇ v f ) + B · ∇ v f + S (1)assuming that the diffusion matrix A is uniformly elliptic and B and S are essentiallybounded: there exist λ, Λ > such that for almost every ( t, x, v ) ∈ Q , (cid:40) eigenvalues of A ( t, x, v ) = A T ( t, x, v ) lie in [ λ, Λ] , the vector field B satisfies: | B ( t, x, v ) | ≤ Λ . (2)In particular, coefficients do not enjoy further regularity such as continuity, vanishing meanoscillation etc . For this reason, coefficients are said to be rough . We classically reduce the local study to the case where Q is at unit scale. For some reasonswe expose below, Q takes the form ( − , × B R × B R for some large constant R onlydepending on dimension and the ellipticity constants λ, Λ in (2).1 a r X i v : . [ m a t h . A P ] F e b efore stating our main result, we give the definition of (weak) super-solutions in a cylindrical open set set Ω , that is to say an open set of the form I × B x × B v . A function f : Ω → R is a weak super-solution of (1) in Ω if f ∈ L ∞ ( I, L ( B x × B v )) ∩ L ( I × B x , H ( B v )) and ( ∂ t + v · ∇ x ) f ∈ L ( I × B x , H − ( B v )) and for all non-negative ϕ ∈ D (Ω) , − (cid:90) Q f ( ∂ t + v · ∇ x ) ϕ d z ≥ − (cid:90) Q A ∇ v f · ∇ v ϕ d z + (cid:90) Q ( B · ∇ v f + S ) ϕ d z. Theorem 1.1 (weak Harnack inequality) . Let Q = ( − , × B R × B R and A, B satisfy (2) and S be essentially bounded in Q . Let f be a non-negative super-solution of (1) insome cylindrical open set Ω ⊃ Q . Then (cid:18)(cid:90) Q − f p ( z ) d z (cid:19) p ≤ C (cid:18) inf Q + f + (cid:107) S (cid:107) L ∞ ( Q + ) (cid:19) where Q + = ( − ω , × B ω × B ω and Q − = ( − , − ω ] × B ω × B ω ; the constants C , p , ω and R only depend on the dimension and the ellipticity constants λ, Λ .Remark . Combined with the fact that non-negative sub-solutions are locally bounded[32], the weak Harnack inequality implies the Harnack inequality proved in [13], see The-orem 1.3
Remark . The proof of Theorem 1.1 is constructive. As a consequence, it provides aconstructive proof of the Harnack inequality from [13].
Remark . The (weak) Harnack inequality implies Hölder regularity of weak solutions.
Remark . Such a weak Harnack inequality can be generalized to the ultraparabolic equa-tions with rough coefficients considered for instance in [32, 39, 40, 41].
Remark . This estimate can be scaled and stated in arbitrary cylinders thanks to Galileanand scaling invariances of the class of equations of the form (1). These invariances arerecalled at the end of the introduction.
Remark . As in [13, 18], the radius ω is small enough so that when “stacking cylinders”over a small initial one contained in Q − , the cylinder Q + is captured, see Lemma 4.3. Asfar as R is concerned, it is large enough so that it is possible to apply the expansion ofpositivity lemma (see Lemma 4.1) can be applied to every stacked cylinder. The weak Harnack inequality from our main theorem and the techniques we develop toestablish it are deeply rooted in the large literature about elliptic and parabolic regularity,both in divergence and non-divergence form.2 e Giorgi’s theorem and Harnack inequality.
E. De Giorgi proved that solutions ofelliptic equations in divergence form with rough coefficients are locally Hölder continuous[7, 8]. This regularity result for linear equations allowed him to solve Hilbert’s 19 th problemby proving the regularity of a non-linear elliptic equation. The case of parabolic equationswas addressed by J. Nash in [31]. Then J. Moser [29, 30] showed that a Harnack inequalitycan be derived for non-negative solutions of elliptic and parabolic equations with roughcoefficients by considering the logarithm of positive solutions. The proof of E. De Giorgiapplies not only to solutions of elliptic equations but also to functions in what is nowknown as the elliptic De Giorgi class. Parabolic De Giorgi classes were then introduced inparticular in [23]. The log-transform.
While the proof of the continuity of solutions for parabolic equa-tions by J. Nash [31] includes the study of the “entropy” of the solution, related to itslogarithm, the proof of the Harnack inequality for parabolic equations by Moser [30] reliesin an essential way on the observation that the logarithm of the solution of a parabolicequation in divergence form satisfies an equation with a dominating quadratic term inthe left hand side. This observation is then combined with a lemma that is the paraboliccounterpart of a result by F. John and L. Nirenberg about functions with bounded meanoscillation. Soon afterwards, S. N. Kruzhkov observes that the use of this lemma can beavoided thanks to a Poincaré inequality due to Sobolev, see [20, 21, Eq. (1.18)].
Weak Harnack inequality.
Moser [30] and then Trudinger [36, Theorem 1.2] proveda weak Harnack inequality for parabolic equations. Lieberman [26] makes the followingcomment: “
It should be noted [. . . ] that Trudinger was the first to recognize the significanceof the weak Harnack inequality even though it was an easy consequence of previously knownresults. [...] ” He also mentions that DiBenedetto and Trudinger [9] showed that non-negative functions in the elliptic De Giorgi class corresponding to super-solutions of ellipticequations, satisfy a weak Harnack inequality and G. L. Wang [37, 38] proves a weak Harnackinequality for functions in the corresponding parabolic De Giorgi class.
Parabolic equations in non-divergence form.
N. V. Krylov and M. V. Safonov [22]derived a Harnack inequality for equations in non-divergence form. In order to do so, theyintroduce a covering argument now known as the Ink-spots theorem, see for instance [17].Such a covering argument will be later used in the various studies of elliptic equations indivergence form, see e.g. [37, 38] or [9].
Expansion of positivity.
Ferretti and Safonov [11] establish the interior Harnack in-equality for both elliptic equations in divergence and non-divergence form by establishingwhat they call growth lemmas, allowing to control the behavior of solutions in terms ofthe measure of their super-level sets.Gianazza and Vespri introduce in [12] suitable homogeneous parabolic De Giorgi classesof order p and prove a Harnack inequality. They shed light on the fact that their main3echnical point is a expansion of positivity in the following sense: if a solution lies above (cid:96) in a ball B ρ ( x ) at time t , then it lies above µ(cid:96) in a ball B ρ ( x ) at time t + Cρ p for someuniversal constants µ and C , that is to say constants only depending on dimension andellipticity constants. They mention that G. L. Wang also used some expansion of positivityin [37].More recently, R. Schwab and L. Silvestre [35] used such ideas in order to derive a weakHarnack inequality for parabolic integro-differential equations with very irregular kernels. Hypoellipticity.
In the case where A is the identity matrix, Equation (1) was firststudied by Kolmogorov [19]. He exhibited a regularizing effect despite the fact that diffusiononly occurs in the velocity variable. This was the starting point of the hypoellipticity theorydeveloped by Hörmander [15] for equations with smooth variable coefficients (unlike A and B in (1)). Regularity theory for ultraparabolic equations.
The elliptic regularity for degener-ate Kolmogorov equations in divergence form with discontinuous coefficients, including (1)with B = 0 , started at the end of the years 1990 with contributions including [5, 28, 33, 34].As far as the rough coefficients case is concerned, A. Pascucci and S. Polidoro [32] provedthat weak (sub)solutions of (1) are locally bounded (from above). This result was laterextended in [6, 2]. Then W. Wang and L. Zhang [39, 40, 41] proved that solutions of(1) are Hölder continuous. Even if the authors do not state their result as an a prioriestimate, it is possible to derive from their proof the following result for a class of ultra-parabolic equations. Such a class containes equations of the form (1) with B = 0 . Morerecently, M. Litsgård and K. Nyström [27] established existence and uniqueness results forthe Cauchy Dirichlet problem for Kolmogorov-Fokker-Planck type equations with roughcoefficients. Theorem 1.2 (Hölder regularity – [39, 40, 41]) . There exist α ∈ (0 , only dependingon dimension, λ and Λ such that all weak solution f of (1) in some cylindrical open set Ω ⊃ Q = ( − , × B × B satisfies [ f ] C α ( Q / ) ≤ C ( (cid:107) f (cid:107) L ( Q ) + (cid:107) S (cid:107) L ∞ ( Q ) ) with Q / = ( − / , × B / × B / ; the constant C only depends on the dimension andthe ellipticity constants λ, Λ . Linear kinetic equations with rough coefficients.
Since the resolution of the 19 th Hilbert problem by E. de Giorgi [8], it is known that being able to deal with coefficientsthat are merely bounded is of interest for studying non-linear problems. There are severalmodels from the kinetic theory of gases related to equations of the form (1) with A , B and S depending on the solution itself. The most famous and important example is probablythe Landau equation [24]. 4n alternative proof of the Hölder continuity Theorem 1.2 was proposed by F. Golse,C. Mouhot, A. F. Vasseur and the second author [13] and a Harnack inequality was ob-tained. Theorem 1.3 (Harnack inequality – [13]) . Let f be a non-negative weak solution of (1) in some cylindrical open set Ω ⊃ Q := ( − , × B R × B R . Then sup Q − f ≤ C (cid:18) inf Q + f + (cid:107) S (cid:107) L ∞ ( Q ) (cid:19) where Q + = ( − ω , × B ω × B ω and Q − = ( − , − ω ] × B ω × B ω ; the constants C and ω only depend on the dimension and the ellipticity constants λ, Λ . Such a Harnack inequality implies in particular the strong maximum principle [1] relyingon a geometric construction known as Harnack chains. The Hölder regularity result of [39]was extended by Y. Zhu [42] to general transport operators ∂ t + b ( v ) ·∇ v for some non-linearfunction b .To finish with, we mention that C. Mouhot and the second author [16] initiated thestudy of a toy non-linear model and F. Anceschi and Y. Zhu continued it in [3]. Both studiesrely in an essential way on Hölder continuity of weak solutions to the linear equation (1). A functional analysis point of view.
The functional analysis framework used in [13]was clarified by S. Armstrong and J.-F. Mourrat in [4]. They show that it is sufficientto control f in L t,x H v and ( ∂ t + v · ∇ x f ) ∈ L t,x H − v (locally) to derive new Poincaréinequalities.There is an interesting bridge between the functional analysis point of view and thePDE one. Indeed, under the assumptions the authors of [4] work with, it is possible toconsider the Kolmogorov equation L K f := ( ∂ t + v · ∇ x f ) − ∆ v f = H with H ∈ L t,x H − v . In [13], the equation is rewritten under the equivalent form L K f = ∇ v · H + H . But in order to derive local properties of solutions such as their Hölder con-tinuity by elliptic regularity methods, it is necessary to be able to work with sub-solutions of the Kolmogorov equation. In this case L K f = H − µ where µ is an arbitrary Radonmeasure. Such additional terms are difficult to deal with in the functional analysis frame-work presented in [4]. With the partial differential point of view, comparison principlesare used in [13] to locally gain some integrability for non-negative sub-solutions. Kinetic equations with integral diffusions.
We would like to conclude this review ofliterature by mentioning the weak Harnack inequality derived in [18] for kinetic equations.The proof also relies on De Giorgi type arguments that are combined with a coveringargument, referred to as an Ink-spots theorem and inspired by the elliptic regularity forequations in non-divergence form (see above). The interested reader is referred to theintroduction of [18] for further details. 5 Q pos ( x, v ) t Figure 1: Expansion of positivity.
The proof of the main result of this article relies on proving that super-solutions of (1)expand positivity along times (Lemma 4.1) and to combine it with the covering argumentfrom [18] mentioned in the previous paragraph. The derivation of the weak Harnackinequality in the present article from the expansion of positivity follows very closely thereasoning in [18].In contrast with parabolic equations, it is not possible to apply the Poincaré inequalityin v for ( t, x ) fixed when studying solutions of linear Fokker-Planck equations such as(1). Instead, if a sub-solution vanishes enough, then a quantity replacing the average inthe usual Poincaré inequality is decreased in the future. See θ M in the weak Poincaréinequality in the next paragraph (Theorem 1.4).A way to circumvent this difficulty is to establish the expansion of positivity of super-solutions in the spirit of [11]. Given a small cylinder Q pos lying in the past of Q (seeFigure 1), Lemma 4.1 states that if a super-solution f lies above in “a good proportion”of Q pos , then it lies above a constant (cid:96) > in the whole cylinder Q . Roughly speaking,such a lemma transforms an information in measure about positivity in the past into apointwise positivity in the future in a (much) larger cylinder.We emphasize the fact that in the classical parabolic case, S. N. Kruzhkov does notneed to prove such an expansion of positivity thanks to an appropriate Poincaré inequalitythat can be applied at any time t > . Such an approach is inapplicable when there is theadditional variable x .Such a weak propagation of positivity was already proved in [13] thanks to a lemma ofintermediate values, in the spirit of De Giorgi’s original proof. We recall that the proof ofthis key lemma is not constructive in [13]. C. Mouhot and the first author [14] recentlymanaged to make the proof of the intermediate value lemma constructive. The proof of the expansion of positivity relies on the following weak Poincaré inequality.The geometric setting is shown in Figure 2.
Theorem 1.4 (Weak Poincaré inequality) . Let η ∈ (0 , . There exist R > and θ ∈ (0 , depending on dimension and η such that, if Q ext = ( − − η , × B R × B R and zero = ( − − η , − × B η × B η (see Figure 2), then for any non-negative function f ∈ L ( Q ext ) such that ∇ v f ∈ L ( Q ext ) , ( ∂ t + v · ∇ x ) f ∈ L (( − − η , × B R , H − ( B R )) , f ≤ M in Q and |{ f = 0 } ∩ Q zero | ≥ | Q zero | , satisfying ( ∂ t + v · ∇ x ) f ≤ H in D (cid:48) ( Q ext ) with H ∈ L t,x H − v ( Q ext ) , we have (cid:107) ( f − θ M ) + (cid:107) L ( Q ) ≤ C ( (cid:107)∇ v f (cid:107) L ( Q ext ) + (cid:107) H (cid:107) L t,x H − v ( Q ext ) ) for some constant C > only depending on dimension.Remark . In the previous statement, L t,x H − v ( Q ext ) is a short hand notation for L (( − − η , × B R , H − ( B R )) . t Q ext Q Q zero ( x, v ) Figure 2: Geometric setting of the weak Poincaré inequality. If the set where the function f vanishes in Q zero has a measure at least equal to | Q zero | , then h ≤ θ sup Q f in the redcylinder Q .A somewhat similar inequality was introduced by W. Wang and L. Zhang for sub-solutions of ultraparabolic equations, see for instance [39, Lemmas 3.3 & 3.4] and thecorresponding lemmas in [40, 41]. Even if statements and proofs look different, they sharemany similarities. The main difference between statements comes from the fact that weadopt the functional framework from [4] and forget about the equation under study. Themain difference in proofs lies on the fact that we avoid using repeatedly the exact form ofthe fundamental solution of the Kolmogorov equation and we seek for arguments closer tothe classical theory of parabolic equations presented for instance in [23] or [26]. In contrastwith [39, 40, 41], the information obtained through the log-transform is summarized inonly one weak Poincaré inequality (while it is split it several lemmas in [39, 40, 41]) andthe geometric settings of the main lemmas are as simple as possible. For instance, it isthe same for the weak Harnack inequality and for the lemma of expansion of positivity(Lemma 4.1). We also mostly use cylinders respecting the invariances of the equation (seethe defintion of Q r ( z ) in the paragraph devoted to notation), except the “large” cylinderswhere the equation is satisfied such as Q ext in Theorem 1.4. The Lie group structure.
Eq. (1) is not translation invariant in the velocity variablebecause of the free transport term. But this (class of) equation(s) comes from mathematical7hysics and it enjoys the Galilean invariance: in a frame moving with constant speed v ,the equation is the same. For z = ( t , x , v ) and z = ( t , x , v ) , we define the followingnon-commutative group product z ◦ z = ( t + t , x + x + t v , v + v ) . In particular, for z = ( t, x, v ) , the inverse element is z − = ( − t, − x + tv, − v ) . Scaling and cylinders.
Given a parameter r > , the class of equations (1) is invariantunder the scaling f r ( t, x, v ) = f ( r t, r x, rv ) . It is convenient to write S r ( z ) = ( r t, r x, rv ) if z = ( t, x, v ) . It is thus natural to considerthe following cylinders “centered” at (0 , , of radius r > : Q r = ( − r , × B r × B r = S r ( Q ) . Moreover, in view of the Galilean invariance, it is then natural to consider cylinderscentered at z ∈ R d of radius r > of the form: Q r ( z ) := z ◦ Q r which is Q r ( z ) := (cid:8) z ∈ R d : z − ◦ z ∈ Q r (0) (cid:9) := (cid:8) − r < t − t ≤ , | x − x − ( t − t ) v | < r , | v − v | < r (cid:9) . Organization of the article.
In Section 2, the definition of weak sub-solutions, super-solutions and solutions for (1) is recalled and two properties of the log-transform are given.Section 3 is devoted to the proof of the weak Poincaré inequality. In Section 4, we explainhow to derive the lemma of expansion of positivity from the weak Poincaré inequalityand how to prove the weak Harnack inequality from expansion of positivity by using acovering lemma called the Ink-spots theorem. This last result is recalled in Appendix A.In another appendix, see §B, we recall how Hölder regularity can be derived directly fromthe expansion of positivity of super-solutions. The proof of a technical lemma about stackedcylinders is given in Appendix C.
Notation.
The open ball of the Euclidian space centered at c of radius R is denoted by B R ( c ) . The measure of a Lebesgue set A of the Euclidian space is denoted by | A | . The z variable refers to ( t, x, v ) ∈ R × R d × R d = R d . For z , z ∈ R d , z ◦ z denotestheir Lie group product and z − denotes the inverse of z with respect to ◦ . For r > , S r denotes the scaling operator. A constant is said to be universal if it only depends ondimension and the ellipticity constants λ, Λ appearing in (2). The notation a (cid:46) b meansthat a ≤ Cb for some universal constant C > .For an open set Ω , D (Ω) denotes the set of C ∞ functions compactly supported in Ω while D (cid:48) (Ω) denotes the set of distributions in Ω . Acknowledgements.
The authors are indebted to C. Mouhot for the fruitful discussionsthey had together during the preparation of this paper. The first author acknowledgesfunding by the ERC grant MAFRAN 2017-2022.8
Weak solutions and Log-transform
We start with the definition of weak (super- and sub-) solutions of (1).
Definition 2.1 (Weak solutions) . Let
Ω = I × B x × B v be open. A function f : Ω → R isa weak super-solution (resp. weak sub-solution ) of (1) in Ω if f ∈ L ∞ ( I, L ( B x × B v )) ∩ L ( I × B x , H ( B v )) and ( ∂ t + v · ∇ x ) f ∈ L ( I × B x , H − ( B v )) and − (cid:90) f ( ∂ t + v · ∇ x ) ϕ d z + (cid:90) A ∇ v f · ∇ v ϕ d z − (cid:90) ( B · ∇ v f + S ) ϕ d z ≥ (resp. ≤ )for all non-negative ϕ ∈ D (Ω) . It is a weak solution of (1) in Q if it is both a weaksuper-solution and a weak sub-solution. As explained in the introduction, the local boundedness of sub-solutions has been knownsince [32]. We give below the version contained in [13].
Proposition 2.1 (Local upper bound for sub-solutions – [13]) . Consider two cylinders Q int = ( t , T ] × B r x × B r v and Q ext = ( t , T ] × B R x × B R v with t > t , r x < R x and r v < R v . Assume that f is a sub-solution of (1) in some cylindrical open set Ω ⊃ Q ext .Then ess-sup Q int f ≤ C ( (cid:107) f + (cid:107) L ( Q ext ) + (cid:107) S (cid:107) L ∞ ( Q ext ) ) where C only depends on d, λ, Λ and ( t − t , R x − r x , R v − r v ) . For technicals reasons, the positive part of the opposite of the logarithm is replaced with amore regular function G that keeps the important features of max(0 , − ln) . The function max(0 , − ln) was first considered in [20, 21]. Lemma 2.1 (A convex function) . There exists G : (0 , + ∞ ) → [0 , + ∞ ) non-increasingand C such that • G (cid:48)(cid:48) ≥ ( G (cid:48) ) and G (cid:48) ≤ in (0 , + ∞ ) , • G is supported in (0 , , • G ( t ) ∼ − ln t as t → + , • − G (cid:48) ( t ) ≤ t for t ∈ (0 , ] . Lemma 2.2 (Log-transform of solutions) . Let (cid:15) ∈ (0 , ] and f be a non-negative weaksuper-solution of (1) in a cylinder Q ext = ( t , T ] × B R x × B R v . Then g = G ( (cid:15) + f ) satisfies ( ∂ t + v · ∇ x ) g + λ |∇ v g | ≤ ∇ v · ( A ∇ v g ) + B · ∇ v g + (cid:15) − | S | in Q ext , i.e. it is a sub-solution of the corresponding equation in Q ext . roof. We first note that g ∈ L ∞ ( Q ext ) since ≤ g ≤ G ( (cid:15) ) . Moreover, ∇ v g = G (cid:48) ( (cid:15) + f ) ∇ v f with | G (cid:48) ( (cid:15) + f ) | ≤ | G (cid:48) ( (cid:15) ) | . In particular, g ∈ L (( t , T ] × B R x , H ( B R v )) . In order to obtainthe sub-equation, it is sufficient to consider the test-function G (cid:48) ( (cid:15) + f )Ψ in the definitionof super-solution for f .The following observation is key in Moser’s reasoning since the square of the L -normof ∇ v g is controlled by the mass of g . Lemma 2.3 (Time evolution of the mass of g ) . Let f be a non-negative weak sub-solutionof (1) in a cylinder Q ext = ( t , T ] × B r x × B r v and Q int = ( t , T ] × B R x × B R v with t > t , r x < R x and r v < R v . Then λ (cid:90) Q int |∇ v g | ≤ C (cid:18)(cid:90) Q ext g ( τ ) + 1 + (cid:15) − (cid:107) S (cid:107) L ∞ ( Q ext ) (cid:19) where C depends on dimension, λ , Λ , Q int and Q ext .Proof. Consider a smooth cut-off function Ψ valued in [0 , , supported in Q ext and equalto in Q int and use Ψ as a test-function for the sub-equation satisfied by g and get λ (cid:90) |∇ v g | Ψ ≤ − (cid:90) A ∇ v g · Ψ ∇ v Ψ + (cid:90) g ( ∂ t + v · ∇ x )Ψ + (cid:90) ( B · ∇ v g + (cid:15) − | S | )Ψ ≤ λ (cid:90) |∇ v g | Ψ + C (1 + (cid:90) Q ext g ) + λ (cid:90) |∇ v g | Ψ + C(cid:15) − (cid:107) S (cid:107) L ∞ ( Q ext ) . This yields the desired estimate.
In order to prove Theorem 1.4, we first derive a local Poincaré inequality (Lemma 3.1)with an error function h due to the localization. This function h satisfies (cid:40) L K h = f L K Ψ , in ( a, b ) × R d ,h = 0 , in { a } × R d , (3)where L K = ( ∂ t + v · ∇ x ) − ∆ v is the Kolmogorov operator and Ψ is a cut-off functionequal to in Q . We will estimate h in Lemma 3.2 below. Lemma 3.1 (A local estimate) . Let Q ext = ( a, × B R x × B R v be a cylinder such that Q ⊂ Q ext , and let Ψ : R d +1 → [0 , be C ∞ , supported in Q ext and Ψ = 1 in Q . Then forany function f ∈ L ( Q ext ) such that ∇ v f ∈ L ( Q ext ) and ( ∂ t + v · ∇ x ) f ≤ H in D (cid:48) ( Q ext ) with H ∈ L (( a, × B R x , H − ( B R v )) , we have (cid:107) ( f − h ) + (cid:107) L ( Q ) ≤ C ( (cid:107)∇ v f (cid:107) L ( Q ext ) + (cid:107) H (cid:107) L (( a, × B Rx ,H − ( B Rv )) where h satisfies the Cauchy problem (3) and C = c ( a )(1 + (cid:107)∇ v Ψ (cid:107) ∞ ) for some constant c ( a ) only depending on | a | . roof. Since H ∈ L (( a, × B R x , H − ( B R v )) , there exists H , H ∈ L ( Q ext ) such that H = ∇ v · H + H and such that (cid:107) H (cid:107) L ( Q ext ) + (cid:107) H (cid:107) L ( Q ext ) ≤ (cid:107) H (cid:107) L (( a, × B Rx ,H − ( B Rv )) (see for instance [10]). The function g = f Ψ satisfies L K g ≤ ∇ v · ˜ H + ˜ H + f [( ∂ t + v · ∇ x )Ψ − ∆ v Ψ] in D (cid:48) (( a, × R d ) with ˜ H = ( H − ∇ v f )Ψ and ˜ H = H Ψ − H ∇ v Ψ − ∇ v Ψ · ∇ v f . We thus get L K ( g − h ) ≤ ˜ H in D (cid:48) (( a, × R d ) with ˜ H = ∇ v · ˜ H + ˜ H . We then multiply by ( g − h ) + to get the natural energy estimatefor all T, T (cid:48) ∈ ( a, and (cid:15) > , (cid:90) ( g − h ) ( T, x, v ) d x d v + (cid:90) T (cid:48) a (cid:90) |∇ v ( g − h ) + | d t d x d v ≤ (cid:90) Ta (cid:90) | − ˜ H · ∇ v ( g − h ) + + ˜ H ( g − h ) + | d t d x d v ≤ (cid:107)∇ v ( g − h ) + (cid:107) L (( a, × R d ) + 2 (cid:107) ˜ H (cid:107) L (( a, × R d ) + 2 (cid:15) (cid:107) ( g − h ) + (cid:107) L (( a, × R d ) + 12 (cid:15) (cid:107) ˜ H (cid:107) L (( a, × R d ) . Remark that we can deal with the two terms of the left hand side separately so that wecan consider the two parameters T and T (cid:48) . Writing (cid:107) · (cid:107) L for (cid:107) · (cid:107) L (( a, × R d ) , we get afterintegrating in T from a to , and choosing T (cid:48) = 0 (cid:107) ( g − h ) + (cid:107) L ≤ − (cid:15)a (cid:107) ( g − h ) + (cid:107) L − a (cid:107) ˜ H (cid:107) L − a (cid:15) (cid:107) ˜ H (cid:107) L . Then remark that the function ( g − h ) + equals ( f − h ) + in Q and that (cid:107) ˜ H (cid:107) L (( a, × R d ) ≤ (cid:107) H (cid:107) L ( Q ext ) + (cid:107)∇ v f (cid:107) L ( Q ext ) , (cid:107) ˜ H (cid:107) L (( a, × R d ) ≤ (cid:107) H (cid:107) L ( Q ext ) + (cid:107)∇ v Ψ (cid:107) ∞ ( (cid:107) H (cid:107) L ( Q ext ) + (cid:107)∇ v f (cid:107) L ( Q ext ) ) . We get the desired inequality by combining the three previous inequalities and choosing (cid:15) = − (4 a ) − .In view of Lemma 3.1, it is sufficient to prove that if the function f satisfies |{ f = 0 } ∩ Q zero | ≥ | Q zero | , then the function h given by the Cauchy problem (3) is bounded from above by θ M forsome universal parameter θ ∈ (0 , . 11 emma 3.2 (Control of the localization term) . Let η ∈ (0 , . There exist a (large)constant R > and a (small) constant θ ∈ (0 , both depending on the dimension and η ,and a C ∞ cut-off function Ψ : R d +1 → [0 , , supported in Q ext = ( − − η , × B R × B R and equal to in Q , such that for all non-negative bounded function f : Q ext → R satisfying |{ f = 0 } ∩ Q zero | ≥ | Q zero | , (4) the solution h of the following initial value problem (cid:40) L K h = f L K Ψ in ( − − η , × R d h = 0 in {− − η } × R d satisfies: h ≤ θ (cid:107) f (cid:107) L ∞ ( Q ext ) in Q .Remark . This lemma is related to [39, Lemma 3.4] and [40, Lemma 3.3].
Remark . The conclusion of the lemma and its proof are essentially unchanged under theweaker assumption |{ f = 0 } ∩ Q zero | ≥ α | Q zero | for some α ∈ (0 , . Remark . Theorem 1.4 will be used in the proof of Lemma 4.1 about the expansion ofpositivity of super-solutions. The parameter η will be then chosen after choosing θ .The proof of Lemma 3.2 requires the following test-function whose construction iselementary. Lemma 3.3 (Cut-off function) . Given η ∈ (0 , and T ∈ (0 , η ) , there exists a smoothfunction Ψ : [ − − η , × R d × R d → [0 , , supported in [ − − η , × B × B , equal to in ( − , × B × B , such that ( ∂ t + v · ∇ x )Ψ ≥ everywhere and ( ∂ t + v · ∇ x )Ψ ≥ in ( − − η , − − T ] × B × B .Proof. Consider Ψ ( t, x, v ) = ϕ ( t ) ϕ ( x − tv ) ϕ ( v ) with• a smooth function ϕ : [ − − η , → [0 , equal to in [ − , with ϕ ( − − η ) = 0 , ϕ (cid:48) ≥ in [ − − η , and ϕ (cid:48) = 1 in [ − − η , − − T ] ;• a smooth function ϕ : R d → [0 , supported in B and equal to in B ;• a smooth function ϕ : R d → [0 , supported in B and equal to in B .It is then easy to check that the conclusion of the lemma holds true.We can now turn to the proof of Lemma 3.2. Proof of Lemma 3.2. If f = 0 in Q ext , then h = 0 . We thus can assume from now onthat f is not identically . We next reduce to the case (cid:107) f (cid:107) L ∞ ( Q ext ) = 1 by considering f / (cid:107) f (cid:107) L ∞ ( Q ext ) .We introduce a time lap T between the top of the cylinder Q zero and the bottom of thecylinder Q , see Figure 3. 12 ext Q t = − − TQ zero Figure 3: Reducing to the case with a time lap.Fix T = η / . Since | Q zero ∩ { t ≥ − − T }| = | Q zero | , then |{ f = 0 } ∩ Q zero ∩ { t ≤ − − T }| ≥ | Q zero | . (5)Let R > to be chosen later. We consider the cut-off function Ψ( t, x, v ) = Ψ ( t, x/R, v/R ) . Remark that Ψ is supported in Q ext and equal to in ( − , × B R × B R . Moreover, L K Ψ( t, x, v ) = ( ∂ t + v · ∇ x )Ψ ( t, x/R, v/R ) − R − ∆ v Ψ ( t, x/R, v/R ) , where in the last equation v · ∇ x means the scalar product of the value of third variable in Ψ (here it is vR ) with the gradient in the second variable. We then have L K ( h − Ψ) = − (1 − f )( ∂ t + v · ∇ x )Ψ ( t, x/R, v/R ) + 1 − fR ∆ v Ψ ( t, x/R, v/R ) and we can write h − Ψ = − P R + E R ( P R for positive and E R for error) with P R and E R solutions of the following Cauchyproblems in ( − − η , × R d , L K P R = (1 − f )( ∂ t + v · ∇ x )Ψ ( t, x/R, v/R ) , L K E R = 1 − fR ∆ v Ψ ( t, x/R, v/R ) , and P R = E R = 0 at time t = − − η .We claim that there exist constants C > and δ > depending on the dimension and η (in particular independent of R ) such that E R ≤ CR − and P R ≥ δ in Q . (6)As far as the estimate of E R is concerned, it is enough to remark that L K E R ≤ C R − forsome constant C = (cid:107) ∆ v Ψ (cid:107) L ∞ only depending on d, λ, Λ , η (in particular not depending13n R ). The maximum principle then yields the result for some universal constant C . Asfar as P R is concerned, we remark that L K P R ≥ I Z in ( − − η , × R d where Z = { f = 0 } ∩ Q zero ∩ { t ≤ − − T } . We use here the fact that ( ∂ t + v · ∇ x )Ψ ≥ in ( − − η , − − T ) × R d . Let P be such that L K P = I Z in ( − − η , × R d and P = 0 atthe initial time − − η . The maximum principle implies that P R ≥ P in the time interval ( − − η , , and in particular in Q . The strong maximum principle implies that P ≥ δ in Q for some constant δ > depending on the dimension and η . Indeed, one can usethe fundamental solution Γ of the Kolmogorov equation and write P ( t, x, v ) = (cid:90) Γ( z, ζ ) I Z ( ζ ) d ζ ≥ m | Q zero | = δ , with m = min Q × Q zero ∩{ t ≤− − T } Γ . The claim (6) is now proved.Inequalities from (6) imply that h ≤ − δ + CR − in Q . This yields the desired result with θ = 1 − δ / for R large enough. Note in particularthat R only depends on C and δ and consequently depends on the dimension and η . Before proving the weak Harnack inequality stated in Theorem 1.1, we investigate howEq. (1) expands positivity of super-solutions. Q ext Q Q pos Figure 4: Geometric setting of the expansion of positivity lemma. It is the same as theone of the weak Poincaré inequality, except that Q ext and Q zero are replaced with Q ext and Q pos . Here, Q pos denotes a set “in the past” where { f ≥ } occupies half of it. Lemma 4.1 (Expansion of positivity) . Let θ ∈ (0 , and Q pos = ( − − θ , − × B θ × B θ and let R be the constant given by Lemma 3.2 which depends on θ , d , λ , Λ . There exist η , (cid:96) ∈ (0 , , only depending on θ , d , λ , Λ , such that for Q ext := ( − − θ , × B R × B R and any non-negative super-solution f of (1) in some cylindrical open set Ω ⊃ Q ext and (cid:107) S (cid:107) L ∞ ( Q ext ) ≤ η and such that |{ f ≥ } ∩ Q pos | ≥ | Q pos | , we have f ≥ (cid:96) in Q . emark . The parameter θ will be chosen in such a way that the stacked cylinder Q pos m is contained in Q (see the definition of stacked cylinders in Appendix A). The cylinder Q pos m can be thought of as the union of m copies of Q pos stacked above (in time) of Q pos .Such cylinders are used in the covering argument used in the proof of the weak Harnackinequality and the parameter m ∈ N only depends on dimension. Proof of Lemma 4.1.
We consider g = G ( f + (cid:15) ) . We remark that g ≤ G ( (cid:15) ) since f isnon-negative and G is non-increasing. We also remark that | G (cid:48) ( f + (cid:15) ) | ≤ | G (cid:48) ( (cid:15) ) | ≤ (cid:15) − since f ≥ , see Lemma 2.1.We know from Lemma 2.2 that g is a non-negative sub-solution of (1) with S replacedwith SG (cid:48) ( f + (cid:15) ) . In particular, ( ∂ t + v · ∇ x ) g ≤ H with H = ∇ v · ( A ∇ v g ) + B · ∇ v g + (cid:15) − | S | .Recall that for a set Q ⊂ R d +1 , S ( r ) ( Q ) = { ( r t, r x, rv ) for z = ( t, x, v ) ∈ Q } . Weintroduce η ∈ (0 , θ ) and ι > two parameters depending on θ to chosen later in the proof.We are going to apply successively: the L − L ∞ estimate from Q to a slightly largercylinder Q ι for an accurate choice of ι ; the (scaled) weak Poincaré inequality in the bigcylinder ˜ Q ext = S (1+ ι ) ( Q ext ) with Q ext = ( − − η , × B R × B R . Then we estimatethe L -norm of ∇ v g by the square root of its mass in a cylinder larger than ˜ Q ext , namely S (1+ ι ) ( Q ext ) ⊂ Q ext . This is illustrated in Figure 5. Q ext ˜ Q ext Q S (1+ ι ) ( Q ) Q pos S (1+ ι ) ( Q zero ) Figure 5: Intermediate cylinders in the proof of the expansion of positivity. The greatparts are obtained after a scaling with a parameter ι close to . This is necessary inorder to keep f vanishing in a “good ratio” of S (1+ ι ) ( Q zero ) .The remainder of the proof is split into several steps. We explain in Step 1 how tochoose ι to ensure that cylinders are properly ordered and η so that we retain enough infor-mation from the assumption |{ f ≥ }∩ Q pos | ≥ | Q pos | . We then apply the aforementionedsuccessive estimates in Step 2 , before deriving the lower bound on f in Step 3 . Step 1.
Choose ι small enough so that Q ext ⊂ ˜ Q ext ⊂ S (1+ ι ) ( Q ext ) ⊂ Q ext . We only needto check the last inclusion. We choose ι > small enough so that (1 + ι ) (1 + η ) ≤ θ , ι ) ≤ and ι ) ≤ . Since η ∈ (0 , θ ) , to satisfy the first inequality it is enough tosatisfy (1+ ι ) (cid:16) (cid:0) θ (cid:1) (cid:17) ≤ θ . So we pick ι = min (cid:16) θ )4+ θ − , (cid:0) (cid:1) / − , (cid:0) (cid:1) / − (cid:17) . Recall that Q zero = ( − − η , − × B η × B η . In particular, S (1+ ι ) ( Q zero ) = ( − (1 + ι ) (1 + η ) , − (1 + ι ) ] × B (1+ ι ) η × B (1+ ι ) η . We next pick η ∈ (0 , small enough so that | Q pos \ S (1+ ι ) ( Q zero ) | ≥ | S (1+ ι ) ( Q zero ) | .
15t is then enough to satisfy θ ≥ (5 / / (1 + ι ) η so we pick η = (5 / − / (1 + ι ) − θ . Inparticular, the previous volume condition implies that |{ g = 0 } ∩ S (1+ ι ) ( Q zero ) | ≥ |{ f ≥ } ∩ S (1+ ι ) ( Q zero ) | ≥ | S (1+ ι ) ( Q zero ) | . Step 2.
With such an information at hand, we know that there exists θ ∈ (0 , , onlydepending on η and thus only depending on θ , such that ess-sup Q ( g − θ G ( (cid:15) )) + (cid:46) (cid:107) ( g − θ G ( (cid:15) )) + (cid:107) L ( Q ι ) + η from Proposition 2.1 (cid:46) ι (cid:107)∇ v g (cid:107) L ( ˜ Q ext ) + ( η /(cid:15) ) + η from Theorem 1.4 (cid:46) (cid:18)(cid:90) Q ext g + 1 + η /(cid:15) (cid:19) + ( η /(cid:15) ) + η from Lemma 2.3 (cid:46) ( G ( (cid:15) ) + 2) + 2 for η ≤ (cid:15) ≤ (cid:46) (cid:112) G ( (cid:15) ) for (cid:15) such that G ( (cid:15) ) ≥ . Remark that it has been necessary to scale g before applying Theorem 1.4. This generatesa constant depending on ι . This is emphasized by writing (cid:46) ι . But ι only depends ondimension, λ , Λ and θ . Step 3.
The previous computation yields g ≤ C (cid:112) G ( (cid:15) ) + θ G ( (cid:15) ) in Q for some θ ∈ (0 , depending on universal constants and θ . Since G ( (cid:15) ) → + ∞ (we canpick (cid:15) and η small enough only depending on the universal constants and θ ) , we thushave G ( f + (cid:15) ) = g ≤ θ G ( (cid:15) ) in Q . Now recall that G ( t ) ∼ − ln t as t → and − G (cid:48) ( t ) ≤ t for t ∈ (cid:0) , (cid:3) . The previousinequality thus implies that as (cid:15) → , ln( f + (cid:15) ) ≥ θ (cid:15). This yields the result with (cid:96) = (cid:15) θ − (cid:15) > .Before iterating the lemma of expansion of positivity, we state and prove a straightfor-ward consequence of it that will be used when applying the Ink-spots theorem. Lemma 4.2.
Let m ≥ be an integer and R be given by Lemma 4.1 for θ = m − / . Thereexists a constant M > only depending on m , d , λ , Λ such that for all non-negative super-solution f (1) with S = 0 in some cylindrical open set Ω ⊃ ( − , m ] × B Rm / × B Rm / ,such that |{ f ≥ M } ∩ Q | ≥ | Q | , then f ≥ in ¯ Q m = (0 , m ] × B m +2 × B . roof. Let θ = m − so that ¯ Q m ⊂ (0 , θ − ] × B θ − × B θ − . We apply Lemma 4.1 to fM with Q and ¯ Q m taking the role of Q pos and Q thanks to a rescaling argument. This yields that f ≥ (cid:96) M in (0 , θ − ] × B θ − × B θ − . We then pick M = 1 /(cid:96) and we conclude the proof.When deriving the weak Harnack inequality, we will need to estimate how the lowerbound deteriorates with time. Indeed such an information is needed in the Ink-spotstheorem: since cylinders can “leak” out of the set F , a corresponding error has to beestimated, see the term Cmr in Theorem A.1. The geometric setting is the one fromTheorem 1.1. In particular, recall that Q + = ( − ω , × B ω × B ω and Q − = ( − , − ω ] × B ω × B ω where ω is small and universal. It has to be small enough so that whenspreading positivity from a cylinder Q r ( z ) from the past, i.e. included in Q − , the unionof the stacked cylinders where positivity is expanded captures Q + . Then the radius R in the statement of weak Harnack inequality is chosen so that the expansion of positivitylemma can be applied as long as new cylinders are stacked over previous ones. These twofacts are stated precisely in the following lemma.In order to avoid the situation where the last stacked cylinder (see Q [ N + 1] in the nextlemma) leaks out of the domain where the equation is satisfied, we choose it in a way thatwe can use the information obtained in the previous cylinder Q [ N ] : the “predecessor” of Q [ N + 1] is contained in Q [ N ] (see Figure 6). Lemma 4.3 (Stacking cylinders) . Let ω < − . Given any non-empty cylinder Q r ( z ) ⊂ Q − , let T k = (cid:80) kj =1 (2 j r ) and N ≥ such that T N ≤ − t < T N +1 . Let Q [ k ] = Q k r ( z k ) for k = 1 , . . . , N,Q [ N + 1] = Q R N +1 ( z N +1 ) where z k = z ◦ ( T k , , and, letting R = | t + T N | and ρ = (4 ω ) , R N +1 = max( R, ρ ) and z N +1 = (cid:40) z N ◦ ( R , , if R ≥ ρ, (0 , , if R < ρ.
These cylinders satisfy Q + ⊂ Q [ N + 1] , N +1 (cid:91) k =1 Q [ k ] ⊂ ( − , × B × B , Q [ N ] ⊃ ˜ Q [ N ] where ˜ Q [ N ] is the “predecessor” of Q [ N + 1] : ˜ Q [ N ] = Q R N +1 / ( z N +1 ◦ ( − R N +1 , , . With such a technical lemma at hand, expansion of positivity for large times followseasily.
Lemma 4.4 (Expansion of positivity for large times) . Let R / given by Lemma 4.1 with θ = 1 / . There exist a universal constant p > such that, if f is a non-negative weak uper-solution of (1) with S = 0 in some cylindrical open set Ω ⊃ Q = ( − , × B R / × B R / such that |{ f ≥ A } ∩ Q r ( z ) | ≥ | Q r ( z ) | for some A > and for some cylinder Q r ( z ) ⊂ Q − , then f ≥ A ( r / p in Q + .Proof. We first apply Lemma 4.1 with θ = 1 / to f /A (after rescaling Q r ( z ) into Q pos )and get f /A ≥ (cid:96) in Q [1] . We then apply it to f / ( A(cid:96) ) and get f ≥ A(cid:96) in Q [2] . Byinduction, we get f ≥ A(cid:96) k in Q [ k ] for k = 1 , . . . , N .We then apply Lemma 4.1 one more time and get f ≥ A(cid:96) N +10 in Q [ N + 1] and inparticular f ≥ A(cid:96) N +10 in Q + . Since T N ≤ , we have N r ≤ . Choosing p > such that (cid:96) = (1 / p , we get f ≥ A ((1 / N +1 ) p ≥ A ( r / p . − ω t + T N ( x, v ) t Q − Q + Q [ N ] Q [ N + 1] Figure 6:
Stacking cylinders above an initial one contained in Q − . We see that the stackedcylinder obtained after N + 1 iterations by doubling the radius leaks out of the domain.This is the reason why Q [ N +1] is chosen in a way that it is contained in the domain and its“predecessor” is contained in Q [ N ] . Notice that the cylinders Q [ k ] are in fact slanted sincethey are not centered at the origin. We also mention that Q [ N + 1] is choosen centered ifthe time t + T N is too close to the final time .We finally turn to the proof of the main result of this paper, Theorem 1.1. Proof of Theorem 1.1.
We start the proof with general comments about the geometricsetting. The proof is going to use a covering argument through the application of the Ink-spots theorem. To apply this result, we will consider an arbitrary cylinder Q containedin Q − . The parameter ω used in the definition of Q − and Q + is chosen small enough( ω ≤ − ) so that the cylinder Q + is “captured” when stacking cylinders (Lemma 4.3)and propagating positivity (Lemma 4.4). We also pick the parameter R in the definition18f the cylinder Q large enough so that the stacked cylinders do not leak out of Q ; weimpose R ≥ R / where R / is given by Lemma 4.1 for θ = 1 / . We also impose R ≥ R m − / m / ω where R m − / is given by Lemma 4.1 with θ = m − / in order to bein position to apply Lemma 4.2 to cylinders contained in Q − , hence of radius smaller than ω . We first classically reduce to the case inf Q + f ≤ and S = 0 . Considering ˜ f ( t, x, v ) = f ( t, x, v ) + (cid:107) S (cid:107) L ∞ t , ˜ f is a super-solution of the same equation withno source term ( S = 0 ) and the weak Harnack inequality for ˜ f implies the one for f . Sofrom now we assume S = 0 . Considering next ˜ f = f / (inf Q + f + 1) reduces to the case inf Q + f ≤ .We then aim at proving that (cid:82) Q − f p ( z ) d z is bounded from above by a universal constantfor some universal exponent p . This amounts to prove that for all k ∈ N , |{ f > M k } ∩ Q − | ≤ C w.h.i. (1 − ˜ µ ) k for some universal parameters ˜ µ ∈ (0 , , M > and C w.h.i. > to be determined later.We can see that this property would be enough by transposing it to the continuous case( k real and above ) and by application of the layer cake formula to (cid:82) Q − f p ( z ) d z .We are going to apply Theorem A.1 with µ = 1 / . We pick m ∈ N such that m +1 m (1 − c/ ≤ − c/ . Then the constant M > is given by Lemma 4.2.We prove the result by induction. For k = 1 , we simply choose ˜ µ ≤ / and C w.h.i. suchthat | Q − | ≤ C w.h.i. . Now assume that the claim holds true for k ≥ and let us prove itfor k + 1 . We thus consider E = { f > M k +1 } ∩ Q − and F = { f > M k } ∩ Q . These two sets are bounded and measurable and such that E ⊂ F ∩ Q − . We consider acylinder Q = Q r ( t, x, v ) ⊂ Q − such that | Q ∩ E | > | Q | , that is to say |{ f > M k +1 } ∩ Q | > | Q | . We first prove that r is small, i.e. we determine a universal r which depends on k such that r < r . Lemma 4.4 (after translation in time) implies that f ≥ M k +1 ( r / p in Q + . In particular, ≥ inf Q + f ≥ M k +1 ( r / p so r p ≤ p M − ( k +1) . We thus choose r = 2 M − k +12 p .We next prove that ¯ Q m ⊂ F , i.e. ¯ Q m ⊂ { f > M k } . In order to do so, we apply Lemma 4.2 to f /M k after rescaling Q in Q where we assume ω ≤ (2 m + 3) − / to be able to rescale. 19y Theorem A.1, we conclude thanks to the induction assumption that |{ f > M k +1 } ∩ Q − | ≤ (1 − c/ (cid:18) C w.h.i. (1 − µ ) k + Cmr (cid:19) ≤ (1 − c/ (cid:18) C w.h.i. (1 − µ ) k + CmM − k +1 p (cid:19) . Then pick ˜ µ small enough so that M − /p ≤ (1 − µ ) and ˜ µ ≤ c and get, ≤ C w.h.i. (1 − c/ (cid:18) C − w.h.i. CmM − p (cid:19) (1 − ˜ µ ) k . Now pick C w.h.i. large enough (depending on c , C , m and M − /p ) and get, ≤ C w.h.i. (1 − c/ − ˜ µ ) k ≤ C w.h.i. (1 − ˜ µ ) k +1 . The proof is now complete.The full Harnack inequality is a direct consequence of the local boundedness of sub-solutions and the weak Harnack inequality.
Proof of Theorem 1.3.
Combine Proposition 2.1 and Theorem 1.1 and rescale to reach theresult. See for example [25] for more details.
A Appendix: the Ink-spots theorem
In order to state the Ink-spots theorem, we need to define stacked cylinders. Given Q = Q r ( t, x, v ) and m ∈ N , ¯ Q m denotes the cylinder { ( t, x, v ) : 0 < t − t ≤ mr , | x − x − ( t − t ) v | < ( m + 2) r , | v − v | < r } . Theorem A.1 (Ink-spots – [18]) . Let E and F be two bounded measurable sets of R × R d with E ⊂ F ∩ Q − . We assume that there exist two constants µ, r ∈ (0 , and an integer m ∈ N such that for any cylinder Q ⊂ Q − of the form Q r ( z ) such that | Q ∩ E | ≥ (1 − µ ) | Q | ,we have ¯ Q m ⊂ F and r < r . Then | E | ≤ m + 1 m (1 − cµ ) (cid:18) | F ∩ Q − | + Cmr (cid:19) where c ∈ (0 , and C > only depend on dimension d .Remark . This corresponds to [18, Corollary 10.1] with Q − instead of Q , i.e. the Ink-spots theorem with leakage, with s = 1 . Indeed, the statement in [18] is more general sincethe cylinders are of the form z ◦ Q r with Q r = ( − r s , × B r s × B r for some s ∈ (0 , .In the statement above, we only deal with s = 1 .20 Appendix : local Hölder estimate
The Hölder estimate from Theorem 1.2 is classically obtained by proving that the oscillationof the solution decays by a universal factor when zooming in. Such an improvement ofoscillation is obtained from Lemma 4.1 with θ = 1 .Of course, it is not necessary to prove this lemma in order to prove the Harnack in-equality since the Hölder estimate can be derived from it. Eventhough, we provide a proofto emphasize that it can be easily derived from Lemma 4.1. Lemma B.1 (Decrease of oscillation) . Let ¯ R > be such that Q ¯ R ⊃ ( − , × B R × B R with R universal given by Lemma 4.1 with θ = 1 . There exist (small) universal constants η , (cid:96) > such that for any solution f of (1) in some cylindrical open set Ω ⊃ Q ¯ R suchthat ≤ f ≤ in Q ¯ R and (cid:107) S (cid:107) L ∞ ( Q ¯ R ) ≤ η , then osc Q f ≤ − (cid:96) .Proof. Remark that either |{ f ≤ } ∩ ( − , − × B × B | ≤ | ( − , − × B × B | or |{ f ≤ } ∩ ( − , − × B × B | ≥ | ( − , − × B × B | . In the former case, Lemma 4.1implies that f ≥ (cid:96) in Q while in the latter, we simply consider ˜ f = 2 − f , apply Lemma 4.1to this new function and get f ≤ − (cid:96) in Q . In both cases, we get the desired reductionof oscillation: osc Q f ≤ − (cid:96) .Deriving Theorem 1.2 from Lemma B.1 is completely standard but we provide detailsfor the sake of completeness and for the reader’s convenience. Proof of Theorem 1.2.
Let f be a solution of (1) in Q . By scaling, we can reduce to thecase (cid:107) f (cid:107) L ( Q ¯ R ) ≤ and (cid:107) S (cid:107) L ∞ ( Q ¯ R ) ≤ η where η is given by Lemma B.1. We deduce fromProposition 2.1 that f is bounded in Q .In order to prove that f is Hölder continuous in Q / , it is sufficient to prove that forall z ∈ Q / and r ∈ (0 , / (9 R )) , osc Q r ( z ) f ≤ C α r α for some universal constants α ∈ (0 , and C α .We reduce to the case z = 0 by using the invariance of the equation by the transfor-mation z (cid:55)→ z ◦ z and we simply prove osc Q ¯ R − k f ≤ C (1 − δ ) k for some C and δ = (cid:96) / ∈ (0 , universal. By scaling, this amounts to prove that if osc Q ¯ R f ≤ then osc Q f ≤ − δ ) . By considering ˜ f = 1 + f (cid:107) f (cid:107) L ∞ ( Q ¯ R ) + (cid:107) S (cid:107) L ∞ ( Q ¯ R ) /η , wecan assume that ≤ f ≤ and | S | ≤ η in Q ¯ R . Remark that the L ∞ bound of S is reducedwhen zooming in. We now apply Lemma B.1 and conclude.21 Appendix: stacking cylinders
Proof of Lemma 4.3.
We first check that the sequence of cylinders is well defined for ω < / √ , say. Since r ≤ ω , we have t + T ≤ − ω + 4 r < and we know that thereexists N ≥ such that T N < − t ≤ T N +1 .We check next that Q + ⊂ Q [ N +1] . If R < ρ , we simply remark that ω ≤ ρ to conclude.In the other case, when R ≥ ρ , we have to prove that Q ω ( z − N +1 ) ⊂ Q R . In this case, wehave z − N +1 = (0 , x − t v , v ) − = (0 , − x + t v , − v ) and for z ∈ Q ω , z − N +1 ◦ z = ( t, − x + t v + x − tv , v − v ) ∈ Q R if ω ≤ R and ω + ω + ω + ω ≤ R and ω ≤ R . This is true for ω ≤ R that is to say ρ ≤ R .Let us now check that for all k ∈ { , . . . , N + 1 } , Q [ k ] ⊂ ( − , × B × B .As far as Q [ N + 1] is concerned, we use the fact that R = | t + T N | ≤ and ρ =(4 ω ) ≤ to get R N +1 ≤ . Moreover z N +1 ∈ Q and thus Q [ N + 1] ⊂ ( − , × B × B .We remark T N ≤ − t ≤ implies that (2 N r ) ≤ + r ≤ and in particular N r ≤ .If ¯ z k = ( t k , x k , v k ) ∈ Q [ k ] for k ≤ N then there exists ( t, x, v ) ∈ Q such that ¯ z k = z ◦ ( T k , , ◦ ((2 k r ) t, (2 k r ) x, k rv ) . This implies that x k = x + T k v + (2 k r ) tv + (2 k r ) x and v k = v + 2 k rv and since z ∈ Q − , | x k | ≤ ω + 2 ω + 1 ≤ and | v k | ≤ ω + 1 ≤ . In particular Q [ k ] ⊂ ( − , × B × B .We are left with proving that ˜ Q [ N ] ⊂ Q [ N ] .If R ≥ ρ , then the conclusion follows from the fact that R/ ≤ N r (since T N +1 > ).Let us deal with the case R ≤ ρ . In view of the definitions of these cylinders, this isequivalent to Q ρ/ (¯ z ) ⊂ Q N r with ¯ z = ( − T N , , ◦ z − ◦ ( − ρ , , . In order to prove this inclusion, we first estimate N r from below. Since t + T N +1 > and − t ≥ − ω , we have (4 / N +1 − r ≥ − ω and in particular N r ≥ (1 / / − / ω ) ≥ / . We conclude that N r ≥ / (2 √ . (7)With such a lower bound at hand, we now compute ¯ z = ( R − ρ , − x + ( t + ρ ) v , − v ) and get for z ∈ Q ρ/ , ¯ z ◦ z = ( R − ρ + t, − x + ( t + ρ ) v + x − tv , v − v ) ∈ Q ρ . Indeed, − ρ < R − ρ + t ≤ and | − x + ( t + ρ ) v + x − tv | ≤ ω + 3 ω + ( ρ/ ≤ ρ and | v − v | ≤ ρ . It is thus sufficient to pick ω such that ρ ≤ / (2 √ to get the desiredinclusion. This is true for ω ≤ − . 22 eferences [1] F. Anceschi, M. Eleuteri, and S. Polidoro , A geometric statement of the Har-nack inequality for a degenerate Kolmogorov equation with rough coefficients , Com-mun. Contemp. Math., 21 (2019), p. 17. Id/No 1850057.[2]
F. Anceschi, S. Polidoro, and M. A. Ragusa , Moser’s estimates for degenerateKolmogorov equations with non-negative divergence lower order coefficients , NonlinearAnal., Theory Methods Appl., Ser. A, Theory Methods, 189 (2019), p. 19. Id/No111568.[3]
F. Anceschi and Y. Zhu , On the nonlinear kinetic Fokker-Planck model: Cauchyproblem and diffusion asymptotics , 2020. In preparation.[4]
S. Armstrong and J.-C. Mourrat , Variational methods for the kinetic Fokker-Planck equation . Preprint arXiv:1902.04037, 2019.[5]
M. Bramanti, M. C. Cerutti, and M. Manfredini , L p estimates for some ultra-parabolic operators with discontinuous coefficients , J. Math. Anal. Appl., 200 (1996),pp. 332–354.[6] C. Cinti, A. Pascucci, and S. Polidoro , Pointwise estimates for a class of non-homogeneous Kolmogorov equations , Math. Ann., 340 (2008), pp. 237–264.[7]
E. De Giorgi , Sull’analiticità delle estremali degli integrali multipli , Atti Accad. Naz.Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8), 20 (1956), pp. 438–441.[8] ,
Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli rego-lari , Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3), 3 (1957), pp. 25–43.[9]
E. DiBenedetto and N. S. Trudinger , Harnack inequalities for quasi-minima ofvariational integrals , Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1 (1984), pp. 295–308.[10]
L. C. Evans , Partial differential equations , vol. 19, Providence, RI: American Math-ematical Society, 1998.[11]
E. Ferretti and M. V. Safonov , Growth theorems and Harnack inequality forsecond order parabolic equations , in Harmonic analysis and boundary value problems.Selected papers from the 25th University of Arkansas spring lecture series “Recentprogress in the study of harmonic measure from a geometric and analytic point ofview”, Fayetteville, AR, USA, March 2–4, 2000, Providence, RI: American Mathemat-ical Society (AMS), 2001, pp. 87–112.[12]
U. Gianazza and V. Vespri , Parabolic De Giorgi classes of order p and the Harnackinequality , Calc. Var. Partial Differential Equations, 26 (2006), pp. 379–399.2313] F. Golse, C. Imbert, C. Mouhot, and A. F. Vasseur , Harnack inequality forkinetic Fokker-Planck equations with rough coefficients and application to the Landauequation , Ann. Sc. Norm. Super. Pisa Cl. Sci., 19 (2019), pp. 253–295.[14]
J. Guerand and C. Mouhot , Quantitative De Giorgi methods in kinetic theory .In preparation.[15]
L. Hörmander , Hypoelliptic second order differential equations , Acta Math., 119(1967), pp. 147–171.[16]
C. Imbert and C. Mouhot , The Schauder estimate in kinetic theory with applica-tion to a toy nonlinear model , 2020.[17]
C. Imbert and L. Silvestre , An introduction to fully nonlinear parabolic equations ,in An introduction to the Kähler-Ricci flow. Selected papers based on the presentationsat several meetings of the ANR project MACK, Cham: Springer, 2013, pp. 7–88.[18] ,
The weak Harnack inequality for the Boltzmann equation without cut-off , J. Eur.Math. Soc. (JEMS), 22 (2020), pp. 507–592.[19]
A. N. Kolmogoroff , Zufällige Bewegungen (zur Theorie der Brownschen Bewe-gung) , Ann. of Math. (2), 35 (1934), pp. 116–117.[20]
S. N. Kružkov , A priori bounds for generalized solutions of second-order elliptic andparabolic equations , Dokl. Akad. Nauk SSSR, 150 (1963), pp. 748–751.[21] ,
A priori bounds and some properties of solutions of elliptic and parabolic equa-tions , Mat. Sb. (N.S.), 65 (107) (1964), pp. 522–570.[22]
N. V. Krylov and M. V. Safonov , A property of the solutions of parabolic equa-tions with measurable coefficients , Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), pp. 161–175, 239.[23]
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva , Linearand quasi-linear equations of parabolic type. Translated from the Russian by S. Smith ,vol. 23, American Mathematical Society (AMS), Providence, RI, 1968.[24]
L. D. Landau and E. M. Lifshitz , Statistical physics , Course of TheoreticalPhysics. Vol. 5. Translated from the Russian by E. Peierls and R. F. Peierls, PergamonPress Ltd., London-Paris; Addison-Wesley Publishing Company, Inc., Reading, Mass.,1958.[25]
D. Li and K. Zhang , A note on the harnack inequality for elliptic equations in diver-gence form , Proceedings of the American Mathematical Society, 145 (2017), pp. 135–137. 2426]
G. M. Lieberman , Second order parabolic differential equations , World ScientificPublishing Co., Inc., River Edge, NJ, 1996.[27]
M. Litsgård and K. Nyström , The dirichlet problem for kolmogorov-fokker-plancktype equations with rough coefficients , 2020.[28]
M. Manfredini and S. Polidoro , Interior regularity for weak solutions of ultra-parabolic equations in divergence form with discontinuous coefficients , Boll. UnioneMat. Ital., Sez. B, Artic. Ric. Mat. (8), 1 (1998), pp. 651–675.[29]
J. Moser , On Harnack’s theorem for elliptic differential equations , Commun. PureAppl. Math., 14 (1961), pp. 577–591.[30]
J. Moser , A Harnack inequality for parabolic differential equations , Comm. PureAppl. Math., 17 (1964), pp. 101–134.[31]
J. Nash , Continuity of solutions of parabolic and elliptic equations , Amer. J. Math.,80 (1958), pp. 931–954.[32]
A. Pascucci and S. Polidoro , The Moser’s iterative method for a class of ultra-parabolic equations , Commun. Contemp. Math., 6 (2004), pp. 395–417.[33]
S. Polidoro and M. A. Ragusa , Sobolev-Morrey spaces related to an ultraparabolicequation , Manuscr. Math., 96 (1998), pp. 371–392.[34] ,
Hölder regularity for solutions of ultraparabolic equations in divergence form ,Potential Anal., 14 (2001), pp. 341–350.[35]
R. W. Schwab and L. Silvestre , Regularity for parabolic integro-differential equa-tions with very irregular kernels , Anal. PDE, 9 (2016), pp. 727–772.[36]
N. S. Trudinger , Pointwise estimates and quasilinear parabolic equations , Commun.Pure Appl. Math., 21 (1968), pp. 205–226.[37]
G. L. Wang , Harnack inequalities for functions in De Giorgi parabolic class , in Partialdifferential equations (Tianjin, 1986), vol. 1306 of Lecture Notes in Math., Springer,Berlin, 1988, pp. 182–201.[38]