On a spatially inhomogeneous nonlinear Fokker-Planck equation: Cauchy problem and diffusion asymptotics
OON A SPATIALLY INHOMOGENEOUS NONLINEAR FOKKER-PLANCKEQUATION: CAUCHY PROBLEM AND DIFFUSION ASYMPTOTICS
FRANCESCA ANCESCHI AND YUZHE ZHU
Abstract.
We investigate the Cauchy problem and the diffusion asymptotics for a spatiallyinhomogeneous kinetic model associated to a nonlinear Fokker-Planck operator. Its solutiondescribes the density evolution of interacting particles whose mobility is hampered by theiraggregation. When the initial data lies below a Maxwellian, we derive the global well-posedness with instantaneous smoothness. The proof relies on hypoelliptic analogue of theclassical parabolic theory, as well as a positivity-spreading result based on the Harnackinequality and barrier function methods. Moreover, the scaled equation leads to the fastdiffusion flow under the low field limit. The relative phi-entropy method enables us to seethe connection between the overdamped dynamics of the nonlinearly coupled kinetic modeland the correlated fast diffusion. The global in time quantitative diffusion asymptotics isthen derived by combining entropic hypocoercivity, relative phi-entropy and barrier functionmethods.
Contents
1. Introduction 12. Preliminaries 63. Kolmogorov-Fokker-Planck equation 84. Well-posedness of the nonlinear model 115. Diffusion asymptotics 22Appendix A. Maximum principle 30Appendix B. Spreading of positivity 31Appendix C. Gaining regularity of spatial increment 33References 341.
Introduction
We consider the kinetic Fokker-Planck operator L FP := ∇ v · ( ∇ v + v ) and the followingspatially inhomogeneous nonlinear drift-diffusion model,(1.1) (cid:40) ( ∂ t + v · ∇ x ) f ( t, x, v ) = ρ βf ( t, x ) L FP f ( t, x, v ) ,f (0 , x, v ) = f in ( x, v ) , for an unknown f ( t, x, v ) ≥ t, x, v ) ∈ R + × T d × R d , where T d denotes the d -dimensionaltorus with unit size, the constant β ∈ [0 ,
1] and ρ f ( t, x ) := (cid:90) R d f ( t, x, v ) d v. Date : February 26, 2021. a r X i v : . [ m a t h . A P ] F e b RANCESCA ANCESCHI, YUZHE ZHU
With the constant (cid:15) ∈ (0 , ( (cid:15)∂ t + v · ∇ x ) f (cid:15) ( t, x, v ) = 1 (cid:15) ρ βf (cid:15) ( t, x ) L FP f (cid:15) ( t, x, v ) ,f (cid:15) (0 , x, v ) = f (cid:15), in ( x, v ) . Our aim is to show the global well-posedness and the trend to equilibrium with (cid:67) ∞ aprioriestimate for the equation (1.1), and the quantitative asymptotic dynamics of the equation (1.2)as (cid:15) tends to zero.1.1. Main results.
Let us recall that classical solution of an evolution equation denotes anonnegative function verifying the equation pointwise everywhere and matching the initialdata continuously. Unless otherwise specified, any solution considered below is intended inthe classical sense. For k ∈ N , (cid:67) k (Ω) is the set of functions having all derivatives of order lessthan or equal to k that are continuous in the domain Ω. For α ∈ (0 , (cid:67) α (Ω) is the classicalH¨older space on Ω with the exponent α . Besides, we write µ ( v ) := (2 π ) − d e − | v | and d µ := µ d v to denote the Gaussian function and the Gaussian measure, respectively. Theorem 1.1.
Let the constants < λ < Λ be given. (i) If f in ∈ (cid:67) (cid:0) T d × R d (cid:1) satisfies ≤ f in ≤ Λ µ in T d × R d , then there exists a solution tothe Cauchy problem (1.1) in R + × T d × R d . For any such solution f , and for any ν ∈ (0 , , k ∈ N , < T < T , there is some constant C T ,T,ν,k > depending only on d, β, λ, Λ , T , T, ν, k and the initial data such that (cid:107) µ − ν f (cid:107) (cid:67) k ( [ T ,T ] × T d × R d ) ≤ C T ,T,ν,k . Additionally, if f in is H¨older continuous and ρ f in ≥ λ in T d , then the solution is unique. (ii) If the initial data satisfies λµ ≤ f in ≤ Λ µ in T d × R d , then for any k ∈ N , there exist someconstant c > depending only on d, β, λ, Λ and some constant C k > depending additionallyon k such that for any t ≥ , (cid:13)(cid:13)(cid:13)(cid:13) f − µ (cid:82) f in d x d v √ µ (cid:13)(cid:13)(cid:13)(cid:13) (cid:67) k ( T d × R d ) ≤ C k e − ct . Remark 1.2.
The results of well-posedness and global regularity in point (i) can be extendedto the case where the spatial periodic box T d is replaced by the whole space R d , see Corol-lary 4.10, Proposition 4.11 and Remark 4.13 below for the precise statements. Besides, if theinitial data merely satisfies 0 ≤ f in ≤ Λ µ without continuity, the existence of solution stillholds in some weak sense, see Remark 4.9 below.In order to describe the diffusion asymptotics of the equation (1.2), we introduce the (Breg-man) distance characterized by the relative phi-entropy functional (cid:72) β . Definition 1.3.
Let β ∈ [0 , h ≥ h > T d × R d , the relative phi-entropy of h with respect to h is defined by (cid:72) β ( h | h ) := (cid:90) T d × R d (cid:0) ϕ β ( h ) − ϕ β ( h ) − ϕ (cid:48) β ( h )( h − h ) (cid:1) d x d µ, where ϕ β : R + → R is defined by ϕ β ( z ) := − β (cid:0) z − β − (2 − β ) z + 1 − β (cid:1) for β ∈ [0 ,
1) and ϕ ( z ) := z log z − z + 1. 2 N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION
Theorem 1.4.
Let the constants α ∈ (0 , , < λ < Λ be given and consider a sequence offunctions { f (cid:15), in } (cid:15) ∈ (0 , ⊂ (cid:67) α (cid:0) T d × R d (cid:1) satisfying ≤ f (cid:15), in ≤ Λ µ in T d × R d and ρ f (cid:15), in ≥ λ in T d . Let f (cid:15) be the solutions to (1.2) associated with these initial data. (i) If there exists some constant (cid:15) ∈ (0 , and some function ρ in ∈ (cid:67) α (cid:0) T d (cid:1) valued in [ λ, Λ] such that (cid:72) β (cid:0) µ − f (cid:15), in | ρ in (cid:1) ≤ (cid:15), then there exist some constants M, m > depending only on d, β, λ, Λ , α , (cid:107) ρ in (cid:107) (cid:67) α ( T d ) and (cid:107) f (cid:15), in (cid:107) (cid:67) α ( T d × R d ) such that for any T > , (cid:13)(cid:13) µ − f (cid:15) − ρ (cid:13)(cid:13) L ∞ ( [0 ,T ]; L ( T d × R d , d x d µ )) ≤ M e MT (cid:15) m , where ρ ( t, x ) is the solution to the following fast diffusion equation, (1.3) ∂ t ρ ( t, x ) = ∇ x · (cid:16) ρ − β ( t, x ) ∇ x ρ ( t, x ) (cid:17) in R + × T d ,ρ (0 , x ) = ρ in ( x ) in T d . (ii) If we additionally assume that f (cid:15), in ≥ λµ in T d × R d , then there exist some constants M (cid:48) , m (cid:48) > with the same dependence as M, m such that (cid:13)(cid:13) µ − f (cid:15) − ρ (cid:13)(cid:13) L ∞ ( R + ; L ( T d × R d , d x d µ )) ≤ M (cid:48) (cid:15) m (cid:48) . Strategy and background.
Cauchy problem of the nonlinear model.
The well-posedness of the nonlinear model (1.1)has been proved in [20] mixing H¨older and Sobolev spaces on the torus, and in [26] underthe regime of perturbation to the global equilibrium in the whole space. We develop it withrough initial data and without smallness or lower bound assumptions as in Theorem 1.1 andRemark 1.2.An analogue of the local well-posedness for the Landau equation with low-regularity andnon-perturbative initial data was established in [17], which is a follow-up study of their pre-vious work [16]. When the drift-diffusion coefficient ρ βf in (1.1) is proportional to the localmass of the solution, that is when β = 1, the equations (1.1), (1.2) have the same quadratichomogeneity as the Landau equation, but global bounds and conservation laws for (1.1), (1.2)are simpler. More precisely, the boundedness from above and from below by Maxwelliansof the initial data will be preserved along time for the solutions to (1.1), (1.2). The lack ofconservation of momentum and energy of (1.2) reduces its hydrodynamic limit to the fastdiffusion flow (1.3) rather than the Navier-Stokes dynamics of scaling limit of the Landauequation.1.2.2. Approach towards the well-posedness.
The breakthrough on apriori estimates to crackthe Cauchy problem of this kind of spatially inhomogeneous kinetic equations with a quasi-linear diffusive structure in velocity are the works [14] and [15, 20]. They are the kinetic(hypoelliptic) counterparts of the De Giorgi-Nash-Moser theory and the Schauder theory forclassical elliptic equations respectively, see for instance [13]. One may refer to [30] for asummary.Our first aim is to show the existence of the Cauchy problem (1.1) in H¨older spaces (seeSubsection 4.2) by means of the (Schauder) fixed point theorem, where the required compact-ness comes from the apriori estimates we mentioned above. The linear counterpart of (1.1) is3
RANCESCA ANCESCHI, YUZHE ZHU studied in Section 3. That is the Cauchy problem associated to the Kolmogorov operator,(1.4) L := ∂ t + v · ∇ x − tr (cid:0) A ( t, x, v ) D v · (cid:1) + B ( t, x, v ) · ∇ v , where the coefficients including the entries of the positive definite d × d real symmetric matrix A and the d -dimensional vector B are H¨older continuous. It serves as a preliminary to applythe fixed point theorem for the nonlinear equation. The well-posedness theory for the Cauchyproblem associated to the linear operator (1.4) was well-developed (see [27], as well as the sur-vey paper [2] and the references therein). Nevertheless, the H¨older spaces (see Definition 2.3)used in these works to study this subject is different from the one studied in [22] and [20] (seeDefinition 2.1). By contrast with [20], the (Schauder-type) apriori estimate in the previousliterature are weaker and not appropriate to bootstrap higher regularity (see Subsection 4.3).Armed with the Schauder estimate developed in [20] in kinetic H¨older spaces and the boot-strap procedure developed in [21], we are able to derive instantaneous (cid:67) ∞ regularization forthe solutions to the equation (1.1) in Subsection 4.3.1.2.3. Self-generating lower bound.
One of the obstacles for solving the nonlinear model (1.1)with the initial data only lying blow a Maxwellian is its lower bound on the ellipticity in thevelocity variable, as the drift-diffusion coefficient ρ βf depends on the local mass of the solution.A self-generating lower bound result established in Subsection 4.1 shows that the positivity ofsolutions spreads everywhere instantaneously, even if there are vacuum regions at the initialtime. Its proof is based on repeat applications of the spreading of positivity forward in time(see Lemma 4.5) and the spreading for all velocities (see Lemma 4.6), as proposed in [18]. Onthe one hand, the barrier function argument will be used in the same spirit as [18] to showthe former one. Indeed, a lower (resp. upper) barrier for to a certain equation is a subsolution(resp. supersolution) of the equation which bounds its solution from below (resp. above) onthe boundary; it then follows from the maximum principle that the barrier function performsas a lower (resp. upper) bound of the solution. On the other hand, combining the localHarnack inequality obtained in [14] with the construction of a Harnack chain yields the latterone. We remark that the idea of the Harnack chain was firstly used in [29] and an exampleof its application to Kolmogorov equations can be found in [1]. Essentially, the spreading ofpositivity can be seen as a lower bound estimate of the fundamental solution, which is thusrelated to the result in [16], where they applied a probabilistic method.A subtle point of this lower bound result lies in the possibilities of the degeneracy ofsolutions as t → + or t → ∞ , which leads to two delicate issues. First, with the samedifficulty as mentioned in [17], in order to prove the uniqueness of the Cauchy problem (1.1),the nondegeneracy of diffusion up to the initial time is required so that the apriori estimatescan be still applicable. The uniqueness is then achieved by using the scaling argument andGr¨onwall’s lemma, since the H¨older estimate around the initial time implies that the integrandin the inequality of Gr¨onwall’s type is improved to be integrable with respect to the timevariable, see the proof of Theorem 4.11 below for more details. Second, we are only able toshow the convergence to equilibrium if the drift-diffusion coefficient ρ βf decay slower than t − as t → ∞ in Proposition 5.1. Therefore, an additional lower Maxwellian bound on the initialdata is imposed in Theorem 1.1 (ii) and in Theorem 1.4 (ii). It would be expected that suchadditional lower bound assumption could be removed, especially when β is small.1.2.4. Long time behavior.
The drift-diffusion operator L FP acts only on the velocity variableand ceases to be dissipative on its unique steady state µ , which also ensures that the null spaceof L FP is spanned by µ and the conservation law of mass is satisfied. Consequently, the con-vergence to equilibrium is to be expected. With the help of the global (cid:67) ∞ regularity estimate,4 N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION we are able to pass from the exponential convergence to equilibrium in L -framework to theuniform convergence in (cid:67) ∞ in Subsection 5.1. Therein, the L -convergence is obtained by the L -hypocoercivity under a macro-micro (fluid-kinetic) decomposition scheme, which suggeststo construct some proper entropy (Lyapunov) functional that would provide an equivalent L -norm for solutions. The key ingredient is to control the macroscopic part by means of themicroscopic part in view of the decomposition. This hypocoercive theory was studied in [12],[10] and [19] from different approaches, while their ideas are essentially the same. In [12],the authors intended to develop the nonlinear energy estimate in an L - L ∞ framework. In[10] and [19], the authors studied the L -hypocoercivity theory in an abstract setting and inthe framework of pseudo-differential calculus, respectively. We remark that the L -frameworkallows us to avoid some difficulties from the nonlinearity of the operator ρ βf L FP f , in contrastwith H -entropic hypocoercivity methods (see for instance the memoir [35]).1.2.5. Diffusion asymptotics.
The diffusion approximation serves as a simplification of colli-sional kinetic equations when the mean free path is much smaller than the typical length ofobservation in a long time scale. This approximation for linear Fokker-Planck models can betraced back to [8], where the authors applied the Hilbert expansion method. One is also ableto achieve the diffusion limit for (1.2) in some weak sense by applying the strategy similar tothe one given in [11]. However, weak convergence is sometimes ineffective for application, asprecise description of the convergence is not given. The nonlinearity of the term ρ βf (cid:15) L FP f (cid:15) in(1.2) associated with non-perturbative initial data reveals some difficulties to derive a quan-titative convergence. In Subsection 5.2, we will use the phi-entropy of the solutions relativeto their limit to see the finite time asymptotics. The relative entropy method has becomean effective tool in the study of hydrodynamic limits since [36] and [4] (see also [31]). Theso-called phi-entropy (relative to the global equilibrium) was used to study the convergenceof certain kinds of Fokker-Planck equations, see for instance [3] and [9]. The method appliedto the diffusion asymptotics of the kinetic Fokker-Planck equation of the type with lineardiffusion can be found in [28]. We point out that such method relies heavily on the regularityof solutions to the target equation. Finally, combining the barrier function method with acareful treatment of the regularity estimate of the target equation enables us to deal with theasymptotic dynamics for the cases associated with general H¨older continuous initial data.1.3. Physical motivation.
The spatially inhomogeneous Fokker-Planck equation (1.1) arisesfrom modeling the evolution of some system of a large number of interacting particles fromthe statistical mechanical point of view. These models appear for instance in the study ofgranular media and biological dynamics, see [34] and [7]. Its solution can be interpreted asthe probability density of the particles lying at the position x at time t with velocity v . Thescaled model (1.2) for small (cid:15) describes the evolution of the particle density in the small meanfree path and long time regime, where the nondimensional parameter (cid:15) ∈ (0 ,
1) designatesthe ratio between the mean free path (microscopic scale) and the typical macroscopic length.The limiting equation (1.3) characterizes its macroscopic dynamics.From the perspective of a stochastic process { ( X t , V t ) : t ≥ } driven by a Brownian motion { (cid:66) t : t ≥ } , (cid:40) d X t = V t d t, d V t = ρ βf ( t, X t ) V t d t + (cid:113) ρ βf ( t, X t ) d (cid:66) t , the function µ − f is the evolving density of the law of the process { ( X t , V t ) : t ≥ } , if f is a solution to the equation (1.1), see the classical review paper [6]. The nonlinear term5 RANCESCA ANCESCHI, YUZHE ZHU ρ βf L FP f models the collisional interaction of the particles, where the mobility of these particlesis hampered by their aggregation. More precisely, the nonlinear dependence on the drift-diffusion coefficient ρ βf translates the fact that the friction effect in the interaction is positivelycorrelated to the local mass of particles occupying the position x at time t .The low field scaling t (cid:55)→ (cid:15) t , x (cid:55)→ (cid:15)x of the equation (1.1) formally implies the equa-tion (1.2). As (cid:15) tends to zero, its spatial diffusion phenomena are characterized by theequation (1.3). Regarding its physical interpretation, we point out that, in (1.2), the factormultiplying the time derivative takes into account the long time scale. The inverse of the fac-tor multiplying ρ βf (cid:15) L FP f (cid:15) stands for the scaled average distance traveled by particles betweeneach collision referred to as mean free path. In the small mean free path regime, it has beennoticed in [6] that the spatial variation occurs significantly only under the long time scale thatis consistent with the particle motion. In such an overdamped process named as low field limitor diffusion limit, the statistics of the particle motion translates into macroscopic behavior ofthe particle system. The dynamical friction contribution induced by the local mass results inthe correlated fast diffusion. This diffusion asymptotics also reveals some links between themesoscopic kinetic model and the macroscopic diffusion model.The associated phi-entropy (Definition 1.3) is also called Tsallis entropy in physics, whichgeneralizes Boltzmann–Gibbs entropy (the phi-entropy with β = 1) in nonextensive statisticalmechanics [32]. It gives some hints for the formulation of the correlated diffusion, where theindex β measures the degree of nonextensivity and nonlocality of the system, see the physicsliterature [33].1.4. Organization of the paper.
The article is organized as follows. In Section 2, we recallsome basic notions related to kinetic H¨older spaces that are adapted to the Fokker-Planckequations. Section 3 is devoted to the study of the linear Fokker-Planck equation with H¨oldercontinuous coefficients. We prove the well-posedness result Theorem 1.1 (i) in Section 4. Theasymptotic behaviors, including Theorem 1.1 (ii) and Theorem 1.4, are proved in Section 5.
Acknowledgement.
The authors are grateful to Cyril Imbert for suggesting the question,and both Fran¸cois Golse and Cyril Imbert for helpful discussions. YZ’s research hasreceived funding from the European Union’s Horizon 2020 research and innovation programmeunder the Marie Sk(cid:32)lodowska-Curie grant agreement No 754362.2.
Preliminaries
This section is devoted to the basic notations, including the invariant structure and thekinetic H¨older space for the equations we are concerned with.2.1.
The geometry associated to Kolmogorov operators.
Let z := ( t, x, v ) ∈ R × R d × R d . We define the kinetic scaling , S r ( t, x, v ) := ( r t, r x, rv ) , for r > , and the Galilean transformation,( t , x , v ) ◦ ( t, x, v ) := ( t + t, x + x + tv , v + v ) , for ( t , x , v ) ∈ R × R d × R d . With respect to the product ◦ , the inverse of z is given by z − := ( − t, − x + tv, − v ).In view of this structure of scaling and transformation, it is natural to define the cylindercentered at the origin of radius r > Q r := ( − r , × B r (0) × B r (0) . N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION
More generally, the cylinder centered at z = ( t , x , v ) with radius r is defined by Q r ( z ) := { z ◦ S r ( z ) : z ∈ Q } = (cid:8) ( t, x, v ) : t − r < t ≤ t , | x − x − ( t − t ) v | < r , | v − v | < r (cid:9) . Roughly speaking, the Kolmogorov operator L (see (1.4)) is invariant under the kineticscaling and left invariant under the Galilean transformation. It means that if f ( z ) is a solutionto the equation L f = 0 in a cylinder Q , then f ( z ◦ S r ( z )) solves an equation of the sameellipticity class in Q r ( z ).Besides, the associated quasi-norm (cid:107) · (cid:107) is defined by (cid:107) z (cid:107) := max (cid:110) | t | , | x | , | v | (cid:111) , as we notice that (cid:107) S r ( z ) (cid:107) = r (cid:107) z (cid:107) and (cid:107) z ◦ z (cid:107) ≤ (cid:107) z (cid:107) + (cid:107) z (cid:107) ). For further information onthe non-Euclidean geometry associated to Kolmogorov operators, one may refer to [22], [2]and the references therein.2.2. Kinetic H¨older spaces and differential operators.
The kinetic H¨older space andkinetic degree of basic differential operators should be adapted to the above kinetic scalingand Galilean transformation, so that they are homogeneous under these transforms. Theirdefinitions were introduced in [22] (see also [20]).Given a monomial m ( t, x, v ) = t k x k . . . x k d d v k d +1 . . . v k d d , we define its kinetic degree asdeg kin ( m ) = 2 k + 3 d (cid:88) j =1 k j + d (cid:88) j = d +1 k j . Any polynomial p ∈ R [ t, x, v ] can be uniquely written as a linear combination of monomialsand its kinetic degree deg kin ( p ) is defined by the maximal kinetic degree of the monomialsappearing in p . This definition is justified by the fact that p ( S r ( z )) = r deg kin ( p ) p ( z ). Definition 2.1.
Let the constant α > ⊂ R × R d × R d be given. Wesay a function f : Ω → R is (cid:67) αl -continuous at a point z ∈ Ω, if there exists a polynomial p ∈ R [ t, x, v ] with deg kin ( p ) < α and a constant C > z ∈ Ω with z ◦ z ∈ Ω,(2.1) | f ( z ◦ z ) − p ( z ) | ≤ C (cid:107) z (cid:107) α . If this property holds for any z , z on each compact subset of Ω, then we say f ∈ (cid:67) αl (Ω).If the constant C in (2.1) is uniformly bounded for any z , z ∈ Ω, we define the smallestvalue of C as the semi-norm [ f ] (cid:67) αl (Ω) , and the norm (cid:107) f (cid:107) (cid:67) αl (Ω) := [ f ] (cid:67) l (Ω) + [ f ] (cid:67) αl (Ω) , where weadditionally define (cid:67) l (Ω) := (cid:67) (Ω), the space of continuous functions on Ω, with the norm (cid:107) f (cid:107) (cid:67) l (Ω) := [ f ] (cid:67) l (Ω) := (cid:107) f (cid:107) (cid:67) (Ω) = (cid:107) f (cid:107) L ∞ (Ω) . Remark 2.2.
For α ∈ [0 , (cid:67) αl -continuity is equivalent to the standard definition of (cid:67) α -continuity with respect to the distance (cid:107) · (cid:107) .We also mention another kind of H¨older space suitable for the study of Kolmogorov oper-ators that was first used in [27]. Definition 2.3.
Let α ∈ [0 ,
1) and Ω ⊂ R × R d × R d be given. The space (cid:67) α kin (Ω) consistsof functions f ∈ (cid:67) l (Ω) such that D v f , ( ∂ t + v · ∇ x ) f ∈ (cid:67) αl (Ω), equipped with the norm (cid:107) f (cid:107) (cid:67) α kin (Ω) := (cid:107) f (cid:107) (cid:67) l (Ω) + (cid:107) D v f (cid:107) (cid:67) αl (Ω) + (cid:107) ( ∂ t + v · ∇ x ) f (cid:107) (cid:67) αl (Ω) . RANCESCA ANCESCHI, YUZHE ZHU
The consistence between these two definitions is given by [22, Lemma 2.7] (see also [20,Lemma 2.4]), a result that we state here as follows.
Lemma 2.4.
Let α ∈ (0 , and f ∈ (cid:67) αl ( Q ) . Then, there exists some constant C > depending only on the dimension d such that (cid:107)∇ v f (cid:107) (cid:67) αl ( Q ) ≤ C (cid:107) f (cid:107) (cid:67) αl ( Q ) and (cid:107) D v f (cid:107) (cid:67) αl ( Q ) + (cid:107) ( ∂ t + v · ∇ x ) f (cid:107) (cid:67) αl ( Q ) ≤ C (cid:107) f (cid:107) (cid:67) αl ( Q ) . Remark 2.5.
For α >
2, one can easily check that the polynomial p in (2.1) has the form p ( t, x, v ) = f ( z ) + ( ∂ t + v · ∇ x ) f ( z ) t + ∇ v f ( z ) · v + 12 D v f ( z ) v · v + . . . . In particular, if α ∈ (2 , x -variable. Remark 2.6.
A subtle difference between (cid:67) l and (cid:67) comes from the fact that for f ∈ (cid:67) l , D v f , ( ∂ t + v · ∇ x ) f are lying in L ∞ rather than (cid:67) .We will also employ the following notions of weighted H¨older norms in Section 3. Definition 2.7.
Let z = ( t, x, v ) ∈ Ω := (0 , T ] × R d × R d with T ∈ R + . For f ∈ (cid:67) αl (Ω) with α > σ ∈ R such that α + σ ≥
0, we define[ f ] ( σ )0 := sup z ∈ Ω (cid:108) σ [ f ] (cid:67) l ( Q (cid:108) ( z )) , [ f ] ( σ ) α := sup z ∈ Ω (cid:108) α + σ [ f ] (cid:67) αl ( Q (cid:108) ( z )) and (cid:107) f (cid:107) ( σ ) α := [ f ] ( σ )0 + [ f ] ( σ ) α , where (cid:108) := min (cid:8) , t (cid:9) measures the distance between z and the (parabolic) boundary of Ω.2.3. Other notations.
Throughout the article, B R denotes the Euclidean ball in R d centeredat the origin with radius R >
0. We employ the Japanese bracket defined as (cid:104)·(cid:105) := (cid:0) | · | (cid:1) .By abuse of notation, (cid:104)·(cid:105) will also denote the velocity mean in Section 5.Moreover, we assume 0 < λ < Λ. We denote by C a universal constant, that is to say aconstant depending only on β, d, λ, Λ , α, σ, α specified in context. Finally, we write X (cid:46) Y to denote that X ≤ CY for some universal constant C >
0, and X (cid:46) q Y to denote that X ≤ C q Y for C q > q .3. Kolmogorov-Fokker-Planck equation
This section is devoted to the study of the Cauchy problem associated to the operator (1.4),(3.1) L f := ( ∂ t + v · ∇ x ) f − tr( AD v f ) − B · ∇ v f = s in (0 , T ] × R d × R d ,f (0 , x, v ) = f in ( x, v ) in R d × R d , lim sup | z |→∞ | f ( z ) | ≤ Λ , where the d × d symmetric matrix A ( t, x, v ) and the d -dimensional vector B ( t, x, v ) satisfythe following condition,(3.2) (cid:40) A ξ · ξ ≥ λ | ξ | for any ξ ∈ R d , (cid:107) A (cid:107) (cid:67) αl (Ω) + (cid:107) B (cid:107) (cid:67) αl (Ω) ≤ Λ , where α ∈ (0 ,
1) and the norm (cid:107) · (cid:107) (cid:67) αl (Ω) of matrix denotes the summation of the norm ofeach entry. The boundedness condition at infinity means that the solution shall be bounded,which is intended for the validity of maximum principle, see the proof of Lemma A.1 below.The aim of this section is to solve the Cauchy problem 3.1 by virtue of the weighted H¨oldernorm (Definition 2.7) and by means of the standard continuity method alongside with the8 N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION
Schauder estimates. One may refer to [13, Subsection 6.5] for the similar treatment in classicalelliptic theory.Throughout this section we work with the domain Ω := (0 , T ] × R d × R d with T ∈ R + .We shed light on the fact that all of the results below can be restricted to (0 , T ] × T d × R d whenever required.3.1. Schauder estimates.
In order to apply the continuity method, first of all one needsto prove a global apriori estimate for solutions to (3.1) with respect to the weighted H¨oldernorm. In the kinetic setting, we have at our disposal the following interior Schauder estimatesproved in [20, Theorem 3.9].
Proposition 3.1 (Interior Schauder estimate) . Let the constant α ∈ (0 , be universal, s ∈ (cid:67) αl (Ω) and f verify the equation (1.4) in Ω with the condition (3.2) . Then, for any r ∈ (0 , and z ∈ Ω such that Q r ( z ) ⊂ Ω , we have (3.3) r α [ f ] (cid:67) αl ( Q r ( z )) (cid:46) (cid:107) f (cid:107) L ∞ ( Q r ( z )) + r α [ s ] (cid:67) αl ( Q r ( z )) ; in particular, the right hand side controls r (cid:107) ( ∂ t + v · ∇ x ) f (cid:107) L ∞ ( Q r ( z )) + r (cid:107) D v f (cid:107) L ∞ ( Q r ( z )) . First of all, we enhance this result to a global estimate for the Cauchy problem (3.1) undera vanishing condition for the initial data.
Proposition 3.2 (Global Schauder estimate) . Let the constants α ∈ (0 , , σ ∈ (0 , beuniversal, s ∈ (cid:67) αl (Ω) such that (cid:107) s (cid:107) (2 − σ ) α < ∞ and f be a solution to the Cauchy problem (3.1) with the condition (3.2) . If the initial data f in = 0 , then we have (cid:107) f (cid:107) ( − σ )2+ α (cid:46) (cid:107) s (cid:107) (2 − σ ) α . Proof.
In view of Proposition 3.1, it suffices to deal with the estimates around the initial time.Without loss of generality, we assume T ≤ z = ( t , x , v ) ∈ Ω and 2 r = t . Applying the interior Schauder estimate (3.3) yields r α [ f ] (cid:67) αl ( Q r ( z )) (cid:46) (cid:107) f (cid:107) L ∞ ( Q r ( z )) + r α [ s ] (cid:67) αl ( Q r ( z )) . It then follows from the arbitrariness of z that, for any σ < f ] ( − σ )0 < ∞ ,(3.4) [ f ] ( − σ )2+ α (cid:46) [ f ] ( − σ )0 + [ s ] (2 − σ ) α . With σ ∈ (0 , L (cid:18) σ [ s ] (2 − σ )0 t σ ± f (cid:19) ≥ σ [ s ] (2 − σ )0 t σ ± f = 0 on { t = 0 } , we have [ f ] ( − σ )0 (cid:46) [ s ] (2 − σ )0 . Combining this estimate with (3.4), we get the desired result. (cid:3)
Cauchy problem for the linear equation.
Our goal of this subsection is the well-posedness of the Cauchy problem (3.1) with H¨older continuous coefficients.
Proposition 3.3.
Let α ∈ (0 , , σ ∈ (0 , be given and the condition (3.2) hold. Then, forany s ∈ (cid:67) αl (Ω) such that (cid:107) s (cid:107) (2 − σ ) α < ∞ and f in ∈ (cid:67) (cid:0) R d × R d (cid:1) , there exists a unique solution f ∈ (cid:67) αl (Ω) ∩ (cid:67) (Ω) to the Cauchy problem (3.1) . RANCESCA ANCESCHI, YUZHE ZHU
The simplest setting of (3.1) with the condition (3.2) is recovered by choosing A = I and B = 0, which turns out to be the classical Kolmogorov operator L := ∂ t + v · ∇ x − ∆ v . Suchoperator was first studied in [25], where its fundamental solution was calculated explicitly,(3.5) Γ( t, x, v ) = (cid:16) √ πt (cid:17) d e − | x + t v | t − | v | t for t > , t ≤ . One can easily see that Γ is smooth outside of its pole (the origin). In fact, in this latter casethe following result holds true.
Lemma 3.4.
Let α ∈ (0 , . For any s ∈ (cid:67) αl (Ω) such that (cid:107) s (cid:107) (2 − σ ) α < ∞ , the function (3.6) f ( t, x, v ) = (cid:90) R d × R d Γ (cid:0) ( τ, ξ, η ) − ◦ ( t, x, v ) (cid:1) s ( τ, ξ, η ) d τ d ξ d η is the unique solution in (cid:67) αl (Ω) to (3.1) with L replaced by L and f in = 0 . Remark 3.5.
When the spatial domain is T d , one can apply the Green function G ( t, x, v ) := (cid:80) n ∈ Z d Γ( t, x + n, v ), which is well-defined due to the decay of Γ.We are now in a position to apply the standard continuity method to derive Proposition 3.3. Proof of Proposition 3.3.
We first establish the case for vanishing initial data and then dealwith general continuous initial data.
Step 1.
Assume that f in = 0. Let the constant σ ∈ (0 ,
2) and consider the Banach space (cid:89) := (cid:16) (cid:67) αl (Ω) , (cid:107) · (cid:107) ( − σ )2+ α (cid:17) . In particular, every function lying in (cid:89) vanishes at t = 0.For τ ∈ [0 , L τ := (1 − τ ) L + τ L , which can be written in theform of L τ = ∂ t + v · ∇ x − tr( A τ D v · ) − τ B · ∇ v , where its coefficients A τ := (1 − τ ) I + τ A and τ B still satisfy the condition (3.2) (with λ , Λreplaced by min { , λ } , max { , Λ } , respectively). For any w ∈ (cid:89) , we have(3.7) (cid:107) L τ w (cid:107) (2 − σ ) α (cid:46) (cid:16) (cid:107) A τ (cid:107) (2) α (cid:17) (cid:107) w (cid:107) ( − σ )2+ α + (cid:107) B (cid:107) (2) α (cid:107) w (cid:107) ( − σ )1+ α (cid:46) (cid:107) w (cid:107) ( − σ )2+ α . Let the set (cid:73) be the collection of τ ∈ [0 ,
1] such that the Cauchy problem (3.8) is solvablefor any s ∈ (cid:67) αl (Ω) with (cid:107) s (cid:107) (2 − σ ) α < ∞ : there is a unique solution f ∈ (cid:89) verifying(3.8) L τ f = s in Ω ,f (0 , x, v ) = 0 in R d × R d , lim sup | z |→∞ | f ( z ) | ≤ Λ . By Lemma 3.4, we see that 0 ∈ (cid:73) ; in particular, (cid:73) is not empty.It now suffices to show that 1 ∈ (cid:73) . Pick τ ∈ (cid:73) , then the global Schauder estimate providedby Proposition 3.2 implies that, for any s ∈ (cid:67) αl (Ω) with (cid:107) s (cid:107) (2 − σ ) α < ∞ , f = L − τ s satisfies(3.9) (cid:107) L − τ s (cid:107) ( − σ )2+ α (cid:46) (cid:107) s (cid:107) (2 − σ ) α . N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION
For any w ∈ (cid:89) , since τ ∈ (cid:73) and (3.7) holds, the following Cauchy problem is solvable for any s ∈ (cid:67) αl (Ω) with (cid:107) s (cid:107) (2 − σ ) α < ∞ , L τ f = s + ( τ − τ ) ( L − L ) w in Ω ,f (0 , x, v ) = 0 in R d × R d , lim sup | z |→∞ | f ( z ) | ≤ Λ . Thus, we can define the mapping F : (cid:89) → (cid:89) by setting F ( w ) = f . Armed with (3.9) and(3.7), there exists a universal constant C > u, w ∈ (cid:89) , (cid:107) F ( u ) − F ( w ) (cid:107) ( − σ )2+ α ≤ C | τ − τ | (cid:107) ( L − L ) ( u − w ) (cid:107) (2 − σ ) α ≤ C | τ − τ |(cid:107) u − w (cid:107) ( − σ )2+ α . Hence, F is a contraction mapping, provided that | τ − τ | ≤ δ := C . Then, F gives a uniquefixed point f ∈ (cid:89) , that is the unique solution to the Cauchy problem (3.8) in (cid:89) . Finally, bydividing the interval [0 ,
1] into subintervals of length less than δ , we conclude that 1 ∈ (cid:73) . Step 2.
For general f in ∈ (cid:67) (cid:0) R d × R d (cid:1) , we approximate f in uniformly as ε → { f ε in } on R d × R d . Thus, the function f − f ε in is a solution to (3.1), with thesource term equal to s − L f ε in , and associated with the vanishing initial data. The procedurepresented in the previous step ensures a unique solution f ε to (3.1) for each f ε in .The uniform convergence of { f ε in } and the maximum principle (Lemma A.1) implies the uni-form convergence of { f ε } . We may denote its limit by f ∈ (cid:67) (Ω), which satisfies f (0 , x, v ) = f in ( x, v ) on R d × R d . Thanks to the interior Schauder estimate (Proposition 3.1), { f ε } isprecompact in (cid:67) ( K ) for any compact subset K ⊂ Ω; then sending ε → f ε yields that the limit function f ∈ (cid:67) αl (Ω) is a solution to (3.1). Its uniquenessis again given by the maximum principle. This concludes the proof. (cid:3) Well-posedness of the nonlinear model
This section is devoted to the proof of Theorem 1.1 (i), including a self-generating lowerbound given in Subsection 4.1, the existence and uniqueness given in Subsection 4.2 and (cid:67) ∞ apriori estimate given in Subsection 4.3.First of all, the Cauchy problem (1.1) can be recast in terms of the unknown function g ( t, x, v ) := µ ( v ) − f ( t, x, v ) with g in ( x, v ) := µ ( v ) − f in ( x, v ) as follows, (cid:40) ( ∂ t + v · ∇ x ) g = (cid:82) [ g ] (cid:85) [ g ] ,g (0 , x, v ) = g in ( x, v ) , (4.1)where (cid:82) [ g ] and (cid:85) [ g ] on the right hand side are defined as follows: (cid:82) [ g ] := (cid:18)(cid:90) R d gµ d v (cid:19) β and (cid:85) [ g ] := µ − ∇ v (cid:16) µ ∇ v (cid:16) µ − g (cid:17)(cid:17) = ∆ v g + (cid:18) d − | v | (cid:19) g. The main advantage of this formulation is that it allows us to get rid of the first order termin v and the zero order term is bounded since g is bounded from above by a Maxwellian.For convenience, we are also concerned with the substitution h ( t, x, v ) := µ ( v ) − f ( t, x, v )and the Ornstein-Uhlenbeck operator L OU := ( ∇ v − v ) · ∇ v . The equation (1.1) is thenequivalent to(4.2) ( ∂ t + v · ∇ x ) h ( t, x, v ) = R h ( t, x ) L OU h ( t, x, v ) , R h ( t, x ) := (cid:18)(cid:90) R d h ( t, x, v ) d µ (cid:19) β . RANCESCA ANCESCHI, YUZHE ZHU
In contrast with the equation (1.1), the zero order term disappears. Let us begin by exhibitingthe global bounds of solutions to (4.2), which is a variant of [20, Lemma 4.1].
Lemma 4.1 (Global bounds) . Let Ω x = T d or R d and (cid:97) ( t, x ) ∈ L ∞ ((0 , T ) × Ω x ) be non-negative. Assume that h ( t, x, v ) ∈ L ∞ (cid:0) (0 , T ); H (cid:0) Ω x × R d , d x d µ (cid:1)(cid:1) verifies the equation ( ∂ t + v · ∇ x ) h = (cid:97) L OU h in (0 , T ) × Ω x × R d in the sense of distributions. If h (0 , · , · ) ≤ Λ in Ω x × R d , then h ≤ Λ in (0 , T ) × Ω x × R d ; if h (0 , · , · ) ≥ λ in Ω x × R d , then h ≥ λ in (0 , T ) × Ω x × R d .Proof. Integrating the equation ( ∂ t + v · ∇ x ) ( h − Λ) = (cid:97) L OU ( h − Λ) against ( h − Λ) + yields12 (cid:90) Ω x × R d (cid:2) ( h ( t, · , · ) − Λ) − ( h (0 , · , · ) − Λ) (cid:3) d x d µ = − (cid:90) [0 ,t ] × Ω x × R d (cid:97) |∇ v ( h − Λ) + | d x d µ ≤ , for any t ∈ [0 , T ]. This means that the upper bound is preserved along time. Similarly, thelower bound can be obtained by integrating the equation ( ∂ t + v · ∇ x ) ( λ − h ) = (cid:97) L OU ( λ − h )against ( λ − h ) + . (cid:3) In particular, the above global bounds preserving result holds for solutions to the equa-tions (4.2) and (5.1). We will also apply such result to the substitution g = µ h appearing inthe following Subsection 4.2 below. Unless otherwise specified, throughout this section we setthe domain Ω := (0 , T ] × T d × R d with T ∈ R + . Nevertheless, as it is specified in Remark 4.4,Corollary 4.10 and Proposition 4.11 below, the results of this section also hold if the spatialdomain is R d .4.1. Self-generating lower bound.
Throughout this subsection, we assume that the boundedsolution h of (1.1) lies below the universal constant Λ, which is guaranteed by Lemma 4.1 ifthe initial data lies below Λ. The aim of this subsection is to show the following positivity-spreading result. We remark that this proposition only relies on the mixing structure ofthe classical parabolic-type maximum principle and the transport operator, but not on thestructure of the mass conservation. Proposition 4.2 (Lower bound) . Let δ > , T ∈ (0 , T ) and h be a bounded solution to (4.2) in Ω satisfying (4.3) h (0 , x, v ) ≥ δ {| x − x | In particular, the functions η ( t ) , η ( t ) are positive and bounded on any com-pact subset of (0 , T ], but η might degenerate to zero and η may go to infinity as t tends tozero or infinity. Remark 4.4. If one is concerned with the problem in the whole space, that is Ω = (0 , T ] × R d × R d , we can proceed along the same lines of the proof in Appendix B to see that (4.3)implies the following lower bound(4.5) h ( t, x, v ) ≥ η ( t, x ) − e − η ( t,x ) | v | , where the functions η ( t, x ) , η ( t, x ) on (0 , T ] × R d are positive, continuous and only depend onuniversal constants, T, δ, r and v . Compared with (4.4), η ( t, x ) , η ( t, x ) lose the uniformityin x as R d is not compact (see Step 3 of the proof of the proposition in Appendix B). Besides,12 N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION the exponential tail with respect to v cannot be improved to a Gaussian type, since there isno uniform-in- x lower bound on the local mass (cid:82) h d µ so that Step 4 in Appendix B fails.Note that the proof of the proposition is composed mainly of two lemmas. On the onehand, Lemma 4.5 below extends the lower bounds forward a short time from a neighborhoodof any given point in T d × R d and at any given time. On the other hand, Lemma 4.6 below isused to spread the lower bound for all velocities. The spreading of the lower bound in space isgiven by selecting proper velocity to transport the positivity which is guaranteed by Lemma4.5. By applying these two lemmas repeatedly, as proposed in [18], we are able to controlthe solution from below for any finite time. We postpone the full proof of Proposition 4.2,obtained by combining the previously introduced lemmas, until Appendix B. Lemma 4.5 (Lower bound forward in time) . Let δ, τ, r ∈ (0 , and h be a bounded solutionto (4.2) in Ω with h (0 , x, v ) ≥ δ { | x − x | Let us consider the barrier function h ( t, x, v ) := − C t + δ (cid:0) − r − | x − x − tv | − τ r − | v − v | (cid:1) , where the constant C > (cid:81) := (cid:8) t ≤ min { T, τ } , | x − x − tv | + τ | v − v | < r (cid:9) contains (cid:80) . A direct computation yields that | L OU h | ≤ | ∆ v h | + | v · ∇ v h | (cid:46) δ (cid:10) τ r − (cid:11) (cid:104) v (cid:105) in (cid:81) . By choosing C := c δ (cid:10) τ r − (cid:11) (cid:104) v (cid:105) for some (small) universal constant c > 0, we have(4.6) ( ∂ t + v · ∇ x − R h L OU ) h ≤ − C + Λ β | L OU h | < (cid:81) . Besides, h ( t, x, v ) ≥ δ in (cid:110) t ≤ c (cid:10) τ r − (cid:11) − (cid:104) v (cid:105) − , | x − x − tv | + τ | v − v | < r (cid:111) ⊃ (cid:80) .Then, observing that h − h ≤ { t = 0 }∩ (cid:81) , (cid:8) t ≤ min { T, τ } , | x − x − tv | + τ | v − v | = r (cid:9) , and applying the classical maximum principle to h − h in (cid:81) yields the result. (cid:3) The spreading of lower bound to all velocities relies on the construction of a Harnack chainthrough iterative application of the local Harnack inequality [14, Theorem 1.6]. Althoughsome coefficients of the equation (4.2) are unbounded globally over v ∈ R d , we remark thattheir local boundedness is sufficient for us to achieve the result through a careful study onthe rescaling during the construction of the Harnack chain. Lemma 4.6 (Lower bound for all velocities) . Let δ > , T, R ∈ (0 , , T ∈ (0 , T ) and h be abounded solution to (4.2) in Ω such that, for any t ∈ [0 , T ] , (4.7) h ( t, x, v ) ≥ δ {| x − x − tv | Proof. For any z := ( t, x, v ) ∈ (cid:8) t ∈ [ T , T ] , | x − x − tv | < R , v ∈ R d (cid:9) , we will construct afinite sequence of points to reach z from the region (cid:8) t ≤ T, | x − x − tv | < R, | v − v | < R (cid:9) where the solution is positive by the assumption. In particular, x does not exit this region.The nonlocality of the coefficient R h , with the assumption (4.7), implies the nondegeneracyof the diffusion in velocity so that the positivity of the solution h propagates over v ∈ R d ina localized space region. Step 1. Iterate the Harnack inequality.For i ∈ { , , ..., N + 1 } with N ∈ N , we define z N +1 := z and z i := ( t i , x i , v i ) by the relation z i = z i +1 ◦ S r (cid:18) − τ , , − τ v − v | v − v | (cid:19) , where the parameters N, r, τ , τ > z := (˜ t, ˜ x, ˜ v ) ∈ Q , h i (˜ z ) := h ( z i ◦ S r (˜ z )) = h ( t i + r ˜ t, x i + r ˜ x + r ˜ tv i , v i + r ˜ v ) . We observe that if it is true for any ˜ z ∈ Q that t i +1 + r ˜ t ∈ [0 , T ] , N rτ ≤ | v − v | , (4.9) (cid:12)(cid:12) x i +1 + r ˜ x + r ˜ tv i +1 − x − (cid:0) t i +1 + r ˜ t (cid:1) v (cid:12)(cid:12) < R, (4.10)then the function h i +1 (˜ z ), for 1 ≤ i ≤ N , verifies the equation( ∂ ˜ t + ˜ v · ∇ ˜ x ) h i = R h ( z i ◦ S r (˜ z )) (∆ ˜ v h i − r ( v i + r ˜ v ) · ∇ ˜ v h i ) in Q , where the coefficients satisfy δ β R dβ (cid:46) R h (cid:46) | r ( v i + r ˜ v ) | ≤ r (1 + | v | + | v − v | ) ≤ , provided that r ≤ (1 + | v | + | v − v | ) − . Applying the Harnack inequality [14, Theorem 1.6]to h i implies that there exist constants c , τ ∈ (0 , 1) depending only on universal constants, δ and R such that, for any τ ∈ [0 , − τ ] and 1 ≤ i ≤ N , we have(4.11) h ( z i +1 ) = h i +1 (0 , , ≥ c h i +1 (cid:16) − τ , , − τ v − v | v − v | (cid:17) = c h ( z i ) . Hence, it remains to determine the chain { z i } ≤ i ≤ N +1 such that the conditions (4.9), (4.10)hold and the point z stays in the region (cid:8) ( t, x, v ) : t ≤ T, | x − x − tv | < R, | v − v | < R (cid:9) . Step 2. Determine the Harnack chain (including N, r, τ ) from a proper starting time t .For (cid:77) > 0, we set t := max (cid:26) T , t − R | v | + | v − v | ) − (cid:27) and r := R (cid:77) (1 + | v | + | v − v | ) − . Recalling that T, R ∈ (0 , (cid:77) ≥ T + τ − τ (cid:16) T (cid:17) , we have r ≤ T τ := rτ | v − v | t − t ≤ − τ . To determine the parameter (cid:77) > 0, we point out that there exists some constant C dependingonly on universal constants, T , δ, R and v such that (cid:77) ≤ C and N := t − t r τ ∈ N + . Thus, N rτ = | v − v | . This setting then guarantees the condition (4.9).14 N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION It also follows from the iteration relation that v = v , and for 1 ≤ i ≤ N + 1,(4.12) t i = t + ( i − r τ , v i = v + ( i − rτ v − v | v − v | , x i = x − r τ N (cid:88) j = i v j +1 . Step 3. Determine the starting point x .For any 1 ≤ i ≤ N , we estimate the departure distance from the expression (4.12), | x i +1 − x − ( t i +1 − t ) v | = i ( i + 1)2 r τ τ ≤ N r τ τ = ( t − t ) | v − v | ≤ R . Therefore, for any x ∈ B R ( x + tv ), there exists some x ∈ B R ( x + t v ) such that x N +1 = x .In this setting, for any 1 ≤ i ≤ N , we also have (cid:12)(cid:12) x i +1 + r ˜ x + r ˜ tv i +1 − x − (cid:0) t i +1 + r ˜ t (cid:1) v (cid:12)(cid:12) ≤ | x i +1 − x − ( t i +1 − t ) v | + | x − x − t v | + r | r ˜ x + ˜ tv i +1 − ˜ tv |≤ R R r (1 + | v − v | ) < R R (cid:77) < R. Thus, the condition (4.10) ensuring the inequality (4.11) is satisfied for 1 ≤ i ≤ N , whichyields h ( t, x, v ) ≥ c N h ( t , x , v ) ≥ δe − N log c . Recalling that c ∈ (0 , 1) appears in (4.11) and N ≤ T C (1+ | v | + | v − v | ) τ R , we obtain the desiredresult (4.8). (cid:3) Existence and uniqueness. Let us begin by summarizing some basic apriori estimatesfor solutions to the equation (4.1). Lemma 4.7 (H¨older estimates) . Let Ω x = T d or R d , and g be a solution to (4.1) in [0 , T ] × Ω x × R d satisfying (cid:82) [ g ] ≥ λ in [0 , T ] × Ω x and 0 ≤ g in ≤ Λ µ in Ω x × R d . (i) Let T ∈ (0 , T ) and δ ∈ (cid:0) , (cid:1) . There exists some universal constant a ∈ (0 , such that,for any Q r ( z ) ⊂ [ T , T ] × Ω x × R d , we have (4.13) (cid:107) g (cid:107) (cid:67) αl ( Q r ( z )) (cid:46) T ,δ µ δ ( v ) . (ii) If g in ∈ (cid:67) α (cid:0) Ω x × R d (cid:1) with (universal) α ∈ (0 , , then there exists some universalconstant α ∈ (0 , such that (cid:107) g (cid:107) (cid:67) αl ([0 ,T ] × Ω x × B ( v )) (cid:46) g in ] (cid:67) α ( Ω x × R d ) . We remark that, armed with Lemma 4.1, the assertions (i) and (ii) in the above lemmadirectly follow from [20, Proposition 4.4] and [37, Corollary 4.6], respectively. Proposition 4.8 (Existence) . For any g in ∈ (cid:67) (cid:0) T d × R d (cid:1) such that ≤ g in ≤ Λ µ in T d × R d , there exists a (classical) solution g to the Cauchy problem (4.1) in Ω . RANCESCA ANCESCHI, YUZHE ZHU Remark 4.9. If the initial data merely satisfies 0 ≤ g in ≤ Λ µ in T d × R d , then we also havethe existence of g ∈ (cid:67) (Ω) in the weak sense as follows. For any φ ∈ (cid:67) ∞ c (cid:0) [0 , T ) × T d × R d (cid:1) ,(4.14) (cid:90) T d × R d g in φ (cid:12)(cid:12) t =0 = (cid:90) Ω (cid:26) − g ( ∂ t + v · ∇ x ) φ + (cid:82) [ g ] ∇ v g · ∇ v φ − (cid:82) [ g ] (cid:18) d − | v | (cid:19) gφ (cid:27) . Indeed, as solutions become regular instantaneously, the difference between the weak solutionand the classical one lies only in the continuity around the initial time. Proof. We may assume that g in is not identically zero so that g in ≥ δ {| x − x | 0. By Proposition 4.2, for anysolution g to (4.1) and for any T ∈ (0 , T ), there is some λ ∗ > T , T, δ, r and v such that,(4.15) (cid:82) [ g ]( t, x ) ≥ λ ∗ in [ T , T ] × T d . Step 1. We first approximate the initial data g in by g (cid:15) in := g in ∗ (cid:37) ε + εµ , where (cid:37) ε ( x, v ) := ε d (cid:37) (cid:0) xε , vε (cid:1) with ( x, v ) ∈ T d × R d , ε ∈ (0 , 1] and (cid:37) ∈ (cid:67) ∞ c ( B × B ) is a nonnegative bumpfunction such that (cid:82) R d (cid:37) = 1. Then, we have εµ ≤ g ε in ≤ (1 + Λ) µ in T d × R d .Let us fix ε ∈ (0 , g ε in , we are going to find a fixed point of the mapping F : w (cid:55)→ g defined bysolving the Cauchy problem, (cid:40) ( ∂ t + v · ∇ x ) g = (cid:82) [ w ] (cid:85) [ g ] in Ω ,g (0 , · , · ) = g ε in in T d × R d , (4.16)on the closed convex subset (cid:75) of the Banach space (cid:67) γl (Ω), (cid:75) := (cid:110) w ∈ (cid:67) γl (Ω) : (cid:107) w (cid:107) (cid:67) γl (Ω) ≤ (cid:78) , εµ ≤ w ≤ (1 + Λ) µ in Ω (cid:111) , where the constants γ ∈ (0 , 1) and (cid:78) > εµ ≤ g ≤ (1 + Λ) µ in Ω, by Lemma 4.1 and the fact that (cid:82) [ w ] ≥ ε .In particular, the lower order term (cid:12)(cid:12) (cid:82) [ w ] (cid:0) d − | v | (cid:1) g (cid:12)(cid:12) (cid:46) w ∈ (cid:75) . Thus, the globalH¨older estimate [37, Corollary 4.6] implies that there exist some constants γ ∈ (0 , 1) and (cid:78) > ε such that (cid:107) g (cid:107) (cid:67) γl (Ω) ≤ (cid:78) . It then followsfrom Proposition 3.3 with the interior Schauder estimate (Proposition 3.1) that the mapping F : (cid:75) → (cid:75) ∩ (cid:67) γl (Ω) ∩ (cid:67) γl (Ω) is well-defined. Besides, with the help of the Arzel`a–Ascolitheorem, we know that F ( (cid:75) ) is precompact in (cid:67) γl (Ω).As far as the continuity of F is concerned, we take a sequence { w n } converging to w ∞ in (cid:67) γl (Ω). Since { F ( w n ) } is precompact in (cid:67) γl (Ω), there exists a converging subsequence whoselimit is g ∞ ∈ (cid:67) γl (Ω) which satisfies g ∞ (0 , · , · ) = g ε in in T d × R d . In view of the interior Schauderestimate (Proposition 3.1), { F ( w n ) } is precompact in (cid:67) ( K ) for any compact subset K ⊂ Ωand g ∞ ∈ (cid:67) (Ω) ∩ (cid:67) (Ω). Sending n → ∞ in (4.16) satisfied by ( w, g ) = ( w n , F ( w n )), we seethat the equation (4.16) also holds for the couple of limits ( w, g ) = ( w ∞ , g ∞ ). Then, applyingthe maximum principle (Lemma A.1) to ( ∂ t + v · ∇ x ) (cid:16) µ − ( g ∞ − F ( w ∞ )) (cid:17) = (cid:82) [ w ∞ ] L OU (cid:16) µ − ( g ∞ − F ( w ∞ )) (cid:17) in Ω , ( g ∞ − F ( w ∞ ))(0 , · , · ) = 0 in T d × R d , we arrive at g ∞ = F ( w ∞ ). 16 N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION Then for every ε ∈ (0 , g ε ∈ (cid:67) (Ω) ∩ (cid:67) (Ω) such that F ( g ε ) = g ε , which isa (classical) solution to (4.1) associated with the initial data g ε in . Step 2. Passage to the limit.Recalling the lower bound (4.15) on the coefficient and the higher order H¨older estimategiven by Lemma 4.7 (i), we point out that for any T ∈ (0 , T ), { g ε } is uniformly boundedin (cid:67) α ∗ l (cid:0) [ T , T ] × T d × R d (cid:1) , for some constant α ∗ ∈ (0 , 1) with the same dependence as λ ∗ .Hence, g ε converges uniformly to g in (cid:67) (cid:0) [ T , T ] × T d × R d (cid:1) , up to a subsequence.Write the equation satisfied by g ε in the weak formulation, that is, for any φ ∈ (cid:67) ∞ c (Ω), (cid:90) T d × R d [ g ε ( T, x, v ) φ ( T, x, v ) − g ε in ( x, v ) φ (0 , x, v )]= (cid:90) Ω (cid:26) g ε ( ∂ t + v · ∇ x ) φ − (cid:82) [ g ε ] ∇ v g ε · ∇ v φ + (cid:82) [ g ε ] (cid:18) d − | v | (cid:19) g ε φ (cid:27) . (4.17)Combining the energy estimate derived by choosing φ = g ε above with the upper bound of g ε provided by Lemma 4.1, we have (cid:90) Ω | (cid:82) [ g ε ] ∇ v g ε | (cid:46) (cid:90) Ω (cid:82) [ g ε ] |∇ v g ε | ≤ (cid:90) T d × R d | g ε in | + (cid:90) Ω (cid:82) [ g ε ] (cid:18) d − | v | (cid:19) | g ε | (cid:46) . Therefore, after passing to a subsequence, (cid:82) [ g ε ] ∇ v g ε converges weakly in L (Ω). On accountof its local uniform convergence, we know that its weak limit is (cid:82) [ g ] ∇ v g . Besides, since µ − g ε is uniformly bounded, by their local uniform convergence, we can also derive that thesequences g ε and (cid:82) [ g ε ] (cid:16) d − | v | (cid:17) g ε converge to g and (cid:82) [ g ] (cid:16) d − | v | (cid:17) g , respectively, weaklyin L (Ω), up to a subsequence. Then, for any φ ∈ (cid:67) ∞ c (cid:0) [0 , T ) × T d × R d (cid:1) , sending ε → g in is continuous, then the barrier function method showsthat the continuity around the initial time depends only on the upper bound of the solution andthe continuity of g in , see the derivation of the estimate (5.23) of a general type in Subsection 5.2below. Indeed, by (5.23) (with (cid:15) = 1, R = (cid:104) v (cid:105) , h (cid:15) = µ − g and h (cid:15), in = µ − g in ), we see thatfor any fixed δ ∈ (0 , x , v ) ∈ T d × R d and for any ( t, x, v ) ∈ (cid:2) , δ (cid:104) v (cid:105) (cid:3) × B δ ( x , v ), | g ( t, x, v ) − g in ( x , v ) | (cid:46) δ − (cid:104) v (cid:105) µ ( v ) t + δ − µ ( v ) (cid:0) | x − x − tv | + | v − v | (cid:1) + µ ( v ) sup B δ ( x ,v ) | g in ( x, v ) − g in ( x , v ) | (cid:46) δ − (cid:16) t + | x − x | + µ ( v ) | v − v | (cid:17) + sup B δ ( x ,v ) | g in ( x, v ) − g in ( x , v ) | . (4.18)It implies the continuity of the solution g around t = 0. This finishes the proof. (cid:3) One may extend the above existence result to the case where the spacial domain T d isreplaced by R d . Corollary 4.10. For any g in ∈ (cid:67) (cid:0) R d × R d (cid:1) such that ≤ g in ≤ Λ µ in R d × R d , thereexists a solution g to the Cauchy problem (4.1) in (0 , T ] × R d × R d . RANCESCA ANCESCHI, YUZHE ZHU Proof. For R > 0, we set g R in = g in for x ∈ [ − R, R ] d with periodic extension to R d . In the lightof Proposition 4.8, we take a solution g R to the equation (4.1) associated with the initial data g R in in (0 , T ] × [ − R, R ] d × R d , where [ − R, R ] d is considered as a periodic box. After extractinga subsequence, we define the function g := lim R →∞ g R in (0 , T ] × R d × R d pointwisely.Unless the initial data is identically zero, we may assume that g in ≥ δ {| x − x | 0. Consider R > | x | + r .Applying the the lower bound of the solution given by (4.5) yields that, for any compactsubset K ⊂ (0 , T ] × R d × R d , the coefficient (cid:82) [ g R ] ≥ λ ∗ , where the constant λ ∗ > δ, r, v and K . In view of the higher order H¨older estimategiven by Lemma 4.7 (i), we know that g R uniformly converges to g in (cid:67) ( K ), up to asubsequence. Besides, due to the estimate derived in (4.18), the solution g matches the initialdata g in continuously. The proof is complete. (cid:3) The following proposition concerned with the uniqueness of the Cauchy problem (4.1) isderived from a Gr¨onwall-type argument. The standard scaling technique and the H¨olderestimate up to the initial time given by Lemma 4.7 (ii) can improve the integrability withrespect to t in the energy estimate so that Gr¨onwall’s inequality becomes admissible, see (4.23)below for the precise expression. This kind of phenomena was also noticed in [17] (see theremarks in § | x | → ∞ . Towork it out, we take advantage of the idea originated from the uniformly local space that wasused in [23] and [16]. We also remark that such a technique is not necessary when workingwith the periodic box T d . Proposition 4.11 (Uniqueness) . Let Ω x = T d or R d , and g , g be two solutions to (4.1) in (0 , T ] × Ω x × R d associated with the same initial data g in ∈ (cid:67) α (cid:0) Ω x × R d (cid:1) such that (cid:90) R d g in µ d v ≥ λ in Ω x and 0 ≤ g in ≤ Λ µ in Ω x × R d . Then, g = g in [0 , T ] × Ω x × R d .Proof. In view of the lower bound given by Lemma 4.5, Proposition 4.2 and the upper boundgiven Lemma 4.1, we know that there is some constant λ ∗ ∈ (0 , 1) depending only on universalconstants, T and the initial data such that(4.19) (cid:90) R d g i µ d v ≥ λ ∗ in [0 , T ] × Ω x and 0 ≤ g i ≤ Λ µ in [0 , T ] × Ω x × R d , i = 1 , . Therefore, we may assume T = Λ − with Λ > 1. Let us set the difference ˜ g := e − | v | t ( g − g ).We have to show that ˜ g is identically zero.In view of the equation (4.1), a direct computation yields that the function ˜ g satisfies( ∂ t + v · ∇ x ) ˜ g + | v | g = e − | v | t ( (cid:82) [ g ] − (cid:82) [ g ]) (cid:85) [ g ] + (cid:82) [ g ] (cid:18) (cid:85) [˜ g ] + t v · ∇ v ˜ g + (cid:18) dt | v | t (cid:19) ˜ g (cid:19) , (4.20)with the initial condition ˜ g (0 , x, v ) = 0 in Ω x × R d .Let y ∈ R d . We introduce a cut-off function φ y ( x ) := φ ( x − y ), where φ ∈ C ∞ c (cid:0) R d (cid:1) is valuedin [0 , 1] such that φ | B ≡ φ | B c ≡ |∇ φ | (cid:46) R d . For any t ∈ (0 , T ], integrating the18 N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION equation (4.20) against φ y ˜ g in Ω x × R d and applying integration by parts yields that12 (cid:90) Ω x × R d φ y ˜ g ( t ) = (cid:90) t (cid:90) Ω x × R d (cid:26) ( v · ∇ φ y ) φ y ˜ g − | v | φ y ˜ g + e − | v | t ( (cid:82) [ g ] − (cid:82) [ g ]) (cid:85) [ g ] φ ˜ g − (cid:82) [ g ] (cid:18)(cid:12)(cid:12)(cid:12) ∇ v (cid:16) µ − ˜ g (cid:17)(cid:12)(cid:12)(cid:12) µφ y + dt φ y ˜ g − (cid:18) dt | v | t (cid:19) φ y ˜ g (cid:19) (cid:27) . Since (cid:82) [ g ] ∈ [0 , Λ], for any t ∈ (0 , T ], we have12 (cid:90) Ω x × R d φ y ˜ g ( t ) ≤ (cid:90) t (cid:90) Ω x × R d (cid:26) | v ||∇ φ y | φ y ˜ g − | v | φ y ˜ g + µ − | (cid:82) [ g ] − (cid:82) [ g ] | | (cid:85) [ g ] | φ y | ˜ g | (cid:27) . Due to the elementary inequality | z β − | ≤ | z − | ( z ∈ R + ) and the lower bound estimate in(4.19), we have | (cid:82) [ g ] − (cid:82) [ g ] | ≤ (cid:82) [ g ] β − β | (cid:82) [˜ g ] | β ≤ λ ∗ (cid:90) R d | ˜ g | ( t, x, · ) µ in [0 , × Ω x . Thus, the following estimate holds for any t ∈ (0 , T ],12 (cid:90) Ω x × R d φ y ˜ g ( t ) ≤ (cid:90) t (cid:90) Ω x × R d (cid:18) | v ||∇ φ y | φ y ˜ g − | v | φ y ˜ g (cid:19) + 1 λ ∗ (cid:90) t (cid:13)(cid:13) µ − (cid:85) [ g ] (cid:13)(cid:13) L ∞ x,v (cid:90) Ω x × R dv φ y | ˜ g | ( t, x, v ) µ (cid:90) R dξ | ˜ g | ( t, x, ξ ) µ d ξ (cid:46) λ ∗ (cid:90) t (cid:90) Ω x × R d |∇ φ y | ˜ g + (cid:90) t (cid:13)(cid:13) µ − (cid:85) [ g ] (cid:13)(cid:13) L ∞ x,v (cid:90) Ω x × R d φ y ˜ g , (4.21)where we used the Cauchy–Schwarz inequality and H¨older’s inequality in the last line. Re-calling that φ y ( x ) = φ ( x − y ) ∈ C ∞ c (cid:0) R d (cid:1) and |∇ φ | (cid:46) R d , we havesup y ∈ R d (cid:90) Ω x × R d |∇ φ y | ˜ g (cid:46) sup y ∈ R d (cid:90) Ω x × R d φ y ˜ g . By the definition of (cid:85) [ g ] and the upper bound Λ µ on g provided by Lemma 4.1, (cid:13)(cid:13) µ − (cid:85) [ g ] (cid:13)(cid:13) L ∞ x,v (cid:46) (cid:13)(cid:13) µ − ∆ v g (cid:13)(cid:13) L ∞ x,v . Hence, for any t ∈ (0 , T ], taking supremum over y ∈ R d in (4.21), we obtainsup y ∈ R d (cid:90) Ω x × R d φ y ˜ g ( t ) (cid:46) λ ∗ (cid:90) t (cid:16) (cid:13)(cid:13) µ − ∆ v g (cid:13)(cid:13) L ∞ x,v (cid:17) sup y ∈ R d (cid:90) Ω x × R d φ y ˜ g . (4.22)Now we have to consider the pointwise estimate on D v g . Let z = ( t , x , v ) ∈ (0 , T ] × Ω x × R d and 2 r = t . In view of (4.19), Lemma 4.7 (ii) implies that there exists some constant α ∗ ∈ (0 , 1) with the same dependence as λ ∗ such that (cid:107) g (cid:107) (cid:67) α ∗ l ([0 ,T ] × Ω x × B ( v )) (cid:46) λ ∗ g in ] (cid:67) α ( Ω x × R d ) . RANCESCA ANCESCHI, YUZHE ZHU Then, applying the interior Schauder estimate (Proposition 3.1) and Lemma 4.1 yields that (cid:13)(cid:13) D v g (cid:13)(cid:13) L ∞ ( Q r ( z )) (cid:46) λ ∗ r − (cid:107) g − g ( z ) (cid:107) L ∞ ( Q r ( z )) + r α ∗ (cid:20) (cid:82) [ g ] (cid:18) d − | v | (cid:19) g (cid:21) (cid:67) α ∗ l ( Q r ( z )) (cid:46) λ ∗ r − α ∗ µ ( v )[ g ] (cid:67) α ∗ l ( Q r ( z )) + µ ( v )[ g ] (cid:67) α ∗ l ( Q r ( z )) (cid:46) λ ∗ t − α ∗ µ ( v ) (cid:16) g in ] (cid:67) α ( Ω x × R d ) (cid:17) . By the arbitrariness of z , we know that for any s ∈ (0 , T ], (cid:13)(cid:13) µ − ∆ v g ( s ) (cid:13)(cid:13) L ∞ ( Ω x × R d ) (cid:46) λ ∗ (cid:16) g in ] (cid:67) α ( Ω x × R d ) (cid:17) s − α ∗ . Dragging this estimate into (4.22) yields that for any t ∈ (0 , T ],sup y ∈ R d (cid:90) Ω x × R d φ y ˜ g ( t ) ≤ C ∗ (cid:90) t d s (cid:16) s − α ∗ (cid:17) sup y ∈ R d (cid:90) Ω x × R d φ y ˜ g ( s ) , (4.23)where the constant C ∗ > (cid:3) Global regularity. The instantaneous (cid:67) ∞ apriori estimate in Theorem 1.1 (i) is madeup of the lower bound given by Proposition 4.2 and the following proposition. Proposition 4.12. Let Ω x = T d or R d , T ∈ (0 , T ) and g be a solution to the equation (4.1) in (0 , T ) × Ω x × R d such that (cid:82) [ g ] ≥ λ in [ T / , T ] × Ω x and 0 ≤ g in ≤ Λ µ in Ω x × R d . Then, for any ν ∈ (cid:0) , (cid:1) and k ∈ N , we have (cid:107) µ − ν g (cid:107) (cid:67) k ( [ T ,T ] × Ω x × R d ) ≤ C T ,ν,k , for some constant C T ,ν,k > depending only on universal constants, T , ν and k . Remark 4.13. For general solution of the Cauchy problem (4.1) in (0 , T ] × R d × R d associatedwith the initial data 0 ≤ g in ≤ Λ µ , Remark 4.4 says that the lower bound of the solutionis no longer uniform in x . Hence, the regularity estimate will not be uniform in x . Moreprecisely, it holds that for any ν ∈ (cid:0) , (cid:1) , k ∈ N and for any compact subset K ⊂ (0 , T ] × R d , (cid:107) µ − ν g (cid:107) (cid:67) k ( K × R d ) ≤ C ν,k,K , where the constant C ν,k,K > ν, k and K .In order to show the higher regularity, we will apply the bootstrap procedure developedin [21] which was intended for the non-cutoff Boltzmann equation. The classical bootstrapiteration proceeds by differentiating the equation, using apriori estimates to the new equa-tion to improve the regularity of solutions, and repeating such procedure. Nevertheless, since (cid:67) αl (cid:54)⊂ (cid:67) x for any α ∈ (0 , 1) by their definitions, the hypoelliptic structure of the equa-tion (4.1) lacks gain of enough regularity in x -variable which disables the x -differentiation ateach iteration. Indeed, the Schauder type estimate provided by Lemma 4.7 (i) only showsthat the solution to (4.1) belongs to (cid:67) α with respect to x -variable. In order to overcomeit, we have to apply estimates to increments of the solution to recover a full derivative. Fromnow on, for y ∈ R d and w ∈ R × R d × R d , we denote the spatial increment δ y g ( z ) := g ( w ◦ (0 , y, − g ( w ) . N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION Let us proceed with the proof of the regularity estimate. Proof of Proposition 4.12. We are going to show that for any multi-index k := ( k t , k x , k v ) ∈ N × N d × N d and ν ∈ (cid:0) , (cid:1) , there exists some constant α k ∈ (0 , 1) depending only on | k | suchthat for any Q r ( z ) ⊂ [ T / , T ] × Ω x × R d ,(4.24) (cid:107) ∂ k t t ∂ k x x ∂ k v v g (cid:107) (cid:67) αkl ( Q r ( z )) (cid:46) T ,ν,k µ ν ( v ) . For simplicity, we will omit the domain in estimates below, since the estimates can bealways localized around the center z . Step 0. The case of k = (0 , , 0) in (4.24) is a direct consequence of Lemma 4.7 (i). Step 1. We will establish that (4.24) holds for any differential operators of the type ∂ k x x . Itsuffices to show that for any n ∈ N , k x ∈ N d with | k x | = n , ν ∈ (cid:0) , (cid:1) and y ∈ B r ,(4.25) (cid:107) δ y ∂ k x x g (cid:107) (cid:67) αnl (cid:46) T ,ν,n | y | µ ν ( v ) . Indeed, sending y → | k x | = n , we suppose that (4.25) holds for any | k x | ≤ n − 1, whichimplies for any k x ∈ N d with | k x | ≤ n ,(4.26) (cid:107) ∂ k x x g (cid:107) (cid:67) αnl (cid:46) T ,ν,n µ ν ( v ) . We remark that the induction here begin with (4.26) for | k x | = 0, which holds due to theprevious step.Let q := δ y ∂ k x x g with | k x | = n . Lemma C.2 and (4.26) gives(4.27) (cid:107) q (cid:107) (cid:67) αnl (cid:46) (cid:107) ∂ k x x g (cid:107) (cid:67) αnl (cid:107) (0 , y, (cid:107) (cid:46) T ,ν,n | y | µ ν ( v ) . Therefore, we have to enhance the exponent on the right hand side to 1; as a sacrifice, theH¨older exponent on the left hand side will decrease.Set τ y g ( w ) := g ( w ◦ (0 , y, y ∈ R d and w ∈ R × R d × R d . A direct computation showsthat q verifies the equation,(4.28) ( ∂ t + v · ∇ x ) q = (cid:82) [ g ] (cid:85) [ q ] + (cid:88) | i |≤ ni ≤ kx δ y ˆ D i (cid:82) [ g ] (cid:85) [ τ y D i g ] + (cid:88) | i |≤ n − i ≤ kx ˆ D i (cid:82) [ g ] (cid:85) [ δ y D i g ] , where the multi-indices such that i ≤ k means each component of i is lower or equal than thecorresponding component of k and ˆ D i denotes the differential operator satisfying ∂ k t t D k x x =ˆ D i ◦ D i .In view of (4.26) (4.27) and the induction hypothesis, each term in two summations on theright hand side of (4.28) is bounded in (cid:67) α n l by C n (cid:107) (0 , y, (cid:107) µ ν (cid:48) ( v ) for any ν (cid:48) ∈ (0 , ν ). Then,by the interior Schauder estimate (Proposition 3.1),(4.29) (cid:107) q (cid:107) (cid:67) αnl (cid:46) T ,ν (cid:48) ,n (cid:107) (0 , y, (cid:107) µ ν (cid:48) ( v ) . Combining Lemma C.1 with (4.27) and (4.29), we obtain (4.25). Step 2. For (4.24) in the case of k v = 0, we proceed a bidimensional induction on ( m, n ) =( k t , | k x | ) such that for any ν ∈ (cid:0) , (cid:1) ,(4.30) (cid:107) ∂ k t t D k x x g (cid:107) (cid:67) αm,nl (cid:46) T ,ν,m,n µ ν ( v ) . Based on the previous step ( m = 0), we have to show that (4.30) holds for k t = m ≥ | k x | = n , under the induction hypothesis that (4.30) holds for any k t ≤ m − | k x | ≤ n + 1,21 RANCESCA ANCESCHI, YUZHE ZHU With k t = m > | k x | = n , set q := ∂ k t t D k x x g . Then, there holds(4.31) ( ∂ t + v · ∇ x ) q = (cid:82) [ g ] (cid:85) [ q ] + (cid:88) i ≤ ( kt,kx, i (cid:54) =( kt,kx, ˆ D i (cid:82) [ g ] (cid:85) [ D i g ] , where we use the notation ˆ D i such that ∂ k t t D k x x = ˆ D i ◦ D i .By the induction hypothesis, each term in the remainder (the summation on the right handside of (4.31)) with i (cid:54) = (0 , , 0) can be controlled in (cid:67) α m,n l . It now suffices to deal with theexceptional term ∂ k t t D k x x (cid:82) [ g ] so that the whole remainder can be controlled in (cid:67) α m,n l ; andthen (4.30) follows from the interior Schauder estimate (Proposition 3.1). To end this, usingLemma 2.4 and the induction hypothesis with the pair ( m − , n ) yields(4.32) (cid:107) ( ∂ t + v · ∇ x ) ∂ m − t D k x x g (cid:107) (cid:67) αm,nl (cid:46) T ,ν,m,n µ ν ( v ) . Due to the induction hypothesis with the pair ( m − , n + 1), for any ν (cid:48) ∈ (0 , ν ), µ − ν (cid:48) ( v ) (cid:107) ( v · ∇ x ) ∂ m − t D k x x g (cid:107) (cid:67) αm,nl (cid:46) ν,ν (cid:48) µ − ν ( v ) (cid:107) ∂ m − t ∇ x D k x x g (cid:107) (cid:67) αm,nl (cid:46) T ,ν,m,n . (4.33)Then, (4.32) and (4.33) produce the bound on µ − ν (cid:48) ( v ) (cid:107) q (cid:107) (cid:67) αm,nl . Step 3. Similarly, to show (4.24) for any differential operator ∂ k t t D k x x D k v v , we proceed abidimensional induction on ( m, n ) = ( k t + | k x | , k v ) such that for any ν ∈ (cid:0) , (cid:1) ,(4.34) (cid:107) ∂ k t t D k x x D k v v g (cid:107) (cid:67) αm,nl (cid:46) T ,ν,m,n µ ν ( v ) . The case n = 0 is treated in the previous step. By Lemma 2.4 and the induction hypothesis(4.34) with k t + | k x | = m and | k v | = n − n ≥ 1, we have (cid:107) ∂ v ∂ k t t ∂ k x x ∂ k v v g (cid:107) (cid:67) αm,nl (cid:46) (cid:107) ∂ k t t ∂ k x x ∂ k v v g (cid:107) (cid:67) αm,nl (cid:46) T ,ν,m,n µ ν ( v ) . Computing the equation satisfied by ∂ v ∂ k t t ∂ k x x ∂ n − v g and proceeding the argument as in theprevious step, we conclude the proof. (cid:3) Diffusion asymptotics This section is devoted to the study of the global in time quantitative diffusion asymptoticswhich consists of the (uniform-in- (cid:15) ) convergence towards the equilibrium over long times andof the finite time asymptotics, including the results of Theorem 1.1 (ii) and Theorem 1.4.First of all, let us introduce the required notation. For any scalar or vector valued functionΨ ∈ L ( R d , d µ ), we denote its velocity mean by (cid:104) Ψ (cid:105) := (cid:90) R d Ψ( v ) d µ. For any couple of functions (scalar, vectors or d × d -matrices) Ψ , Ψ ∈ L (cid:0) T d × R d , d x d µ (cid:1) ,we denote their L inner product with respect to the measure d x d µ by(Ψ , Ψ ) := (cid:90) T d × R d Ψ ( x, v )Ψ ( x, v ) d x d µ, where the multiplication between the couple in the integrand is replaced by scalar contractionproduct, if Ψ , Ψ is a couple of vectors or matrices.22 N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION Recalling our notation for the Ornstein-Uhlenbeck operator L OU = ( ∇ v − v ) · ∇ v , we applythe substitutions f (cid:15) = µh (cid:15) , f (cid:15), in = µh (cid:15), in to (1.2) and obtain(5.1) ( (cid:15)∂ t + v · ∇ x ) h (cid:15) ( t, x, v ) = 1 (cid:15) (cid:104) h (cid:15) (cid:105) β ( t, x ) L OU h (cid:15) ( t, x, v ) ,h (cid:15) (0 , x, v ) = h (cid:15), in ( x, v ) , In this setting, by applying integration by parts, for any h , h ∈ (cid:67) ∞ c (cid:0) T d × R d (cid:1) we get( h , L OU h ) = − ( ∇ v h , ∇ v h ) . We will use this identity repeatedly in the computation below. Then, the operator L OU isself-adjoint with respect to the inner product ( · , · ) and the bracket (cid:104)·(cid:105) is a projection on thenull space of L OU . Moreover, as the total mass is conserved, we define(5.2) M := (cid:90) T d × R d h (cid:15) d x d µ = (cid:90) T d (cid:104) h (cid:15) (cid:105) d x. Proceeding with the macro-micro (fluid-kinetic) decomposition, we define the orthogonal com-plement of the projection (cid:104)·(cid:105) of h (cid:15) as h ⊥ (cid:15) ( t, x, v ) := h (cid:15) ( t, x, v ) − (cid:104) h (cid:15) (cid:105) ( t, x ) . In this framework, the local mass (cid:104) h (cid:15) (cid:105) is the macroscopic (fluid) part and the complement h ⊥ (cid:15) is the microscopic (kinetic) part. Besides, taking the bracket (cid:104)·(cid:105) after multiplying theequation in (5.1) with 1 and v leads to the following macroscopic equations, (cid:15)∂ t (cid:104) h (cid:15) (cid:105) + ∇ x · (cid:104) vh (cid:15) (cid:105) = 0 , (5.3) (cid:15)∂ t (cid:104) vh (cid:15) (cid:105) + ∇ x · (cid:104) v ⊗ h (cid:15) (cid:105) = − (cid:15) (cid:104) h (cid:15) (cid:105) β (cid:104) vh (cid:15) (cid:105) , (5.4)where (cid:104) vh (cid:15) (cid:105) and (cid:104) v ⊗ h (cid:15) (cid:105) represent the local momentum and the stress tensor, respectively.5.1. Long time behavior. Our aim is to establish the (uniform-in- (cid:15) ) exponential decaytowards the equilibrium M for (5.1). In particular, when (cid:15) = 1, it sets up the exponentialconvergence in each order derivative based on the (cid:67) ∞ apriori estimates given in Subsection 4.3.We remark that the classical coercive method is not applicable in our case to obtain theconvergence to equilibrium due to the degeneracy of the ellipticity of the spatially inho-mogeneous equation. Indeed, the Poincar´e inequality only produces a spectral gap on theorthogonal complement of the projection (cid:104)·(cid:105) , see (5.6) below. As we mentioned in Subsec-tion 1.2, there are several ways to achieve the long time asymptotics. We will mainly followthe argument presented in [12] (see also [24]) in a simpler scenario. It would also allow us tosee some similarity among [12], [10] and [19]. Proposition 5.1. Let the function λ t : R + → [0 , Λ] with the derivative λ (cid:48) t ≤ on R + . If h isa solution to (5.1) in R + × T d × R d , associated with the initial data ≤ h (cid:15), in ≤ Λ , satisfying (cid:104) h (cid:15) (cid:105) β ( t, x ) ≥ λ t in R + × T d and (cid:90) R + (cid:0) λ t + λ (cid:48) t (cid:1) d t = ∞ , then the solution h (cid:15) converges to the state M in L (d x d µ ) as t → ∞ ; more precisely, thereexists some universal constant c > such that for any t > , we have (5.5) (cid:107) h (cid:15) ( t, · , · ) − M (cid:107) L (d x d µ ) (cid:46) (cid:107) h (cid:15), in − M (cid:107) L (d x d µ ) exp (cid:18) − c (cid:90) t (cid:0) λ s + λ (cid:48) s (cid:1) d s (cid:19) . RANCESCA ANCESCHI, YUZHE ZHU Proof. Since the velocity mean of the microscopic part vanishes, (cid:104) h ⊥ (cid:15) (cid:105) = 0, using the equa-tion (1.3) and the Poincar´e inequality yields that12 dd t (cid:107) h (cid:15) − M (cid:107) L (d x d µ ) = 1 (cid:15) (cid:16) (cid:104) h (cid:15) (cid:105) β L OU h (cid:15) , h (cid:15) − M (cid:17) = − (cid:15) (cid:16) (cid:104) h (cid:15) (cid:105) β ∇ v h ⊥ (cid:15) , ∇ v h ⊥ (cid:15) (cid:17) ≤ − λ t (cid:15) (cid:107)∇ v h ⊥ (cid:15) (cid:107) L (d x d µ ) (cid:46) − λ t (cid:15) (cid:107) h ⊥ (cid:15) (cid:107) L (d x d µ ) . (5.6)Now we have to recover a new entropy that would give some bound on the projection (cid:104) h (cid:15) (cid:105)− M .For every test function v · Ψ( t, x ) µ , where Ψ ∈ H t,x (cid:0) R + × T d , R d (cid:1) is a vector-valued func-tion, we write the weak formulation of (5.1) as followsdd t ( v · Ψ , h (cid:15) ) = 1 (cid:15) (cid:0) v ⊗ : ∇ x Ψ , h (cid:15) (cid:1) + ( v · ∂ t Ψ , h (cid:15) ) − (cid:15) (cid:16) (cid:104) h (cid:15) (cid:105) β L OU v · Ψ , h (cid:15) (cid:17) . Taking the macro-micro decomposition into account, we obtain from the above expressiondd t (cid:16) v · Ψ , h ⊥ (cid:15) (cid:17) = 1 (cid:15) (cid:0) | v | tr( ∇ x Ψ) , (cid:104) h (cid:15) (cid:105) − M (cid:1) + 1 (cid:15) (cid:16) v ⊗ : ∇ x Ψ , h ⊥ (cid:15) (cid:17) + (cid:16) v · ∂ t Ψ , h ⊥ (cid:15) (cid:17) − (cid:15) (cid:16) (cid:104) h (cid:15) (cid:105) β Lv · Ψ , h ⊥ (cid:15) (cid:17) . (5.7)In view of the condition (5.2), applying the elliptic estimate on the equation,(5.8) − ∆ x u = (cid:104) h (cid:15) (cid:105) − M in T d , yields that(5.9) (cid:107)∇ x u (cid:107) L x + (cid:107)∇ x u (cid:107) L x (cid:46) (cid:107)(cid:104) h (cid:15) (cid:105) − M (cid:107) L x . Besides, observing that (cid:104) vh (cid:15) (cid:105) = (cid:104) vh ⊥ (cid:15) (cid:105) , from (5.3), we get (cid:15)∂ t (cid:104) h (cid:15) (cid:105) + ∇ x · (cid:104) vh ⊥ (cid:15) (cid:105) = 0 . Combining this macroscopic relation with (5.8), we have (cid:90) T d |∇ x ( ∂ t u ) | = (cid:90) T d ∂ t u ∂ t (cid:104) h (cid:15) (cid:105) = − (cid:15) (cid:90) T d ∂ t u ∇ x · (cid:104) vh ⊥ (cid:15) (cid:105) = 1 (cid:15) (cid:90) T d ∇ x ( ∂ t u ) · (cid:104) vh ⊥ (cid:15) (cid:105) . It then follows from H¨older’s inequality that(5.10) (cid:107)∇ x ( ∂ t u ) (cid:107) L x ≤ (cid:15) (cid:107)(cid:104) vh ⊥ (cid:15) (cid:105)(cid:107) L x (cid:46) (cid:15) (cid:107) h ⊥ (cid:15) (cid:107) L (d x d µ ) . Choosing Φ = ∇ x u in (5.7) yields − (cid:15) (cid:0) | v | ∆ x u, (cid:104) h (cid:15) (cid:105) − M (cid:1) + dd t (cid:16) v · ∇ x u, h ⊥ (cid:15) (cid:17) (cid:46) (cid:18) (cid:15) (cid:107)∇ x u (cid:107) L x + (cid:107)∇ x ( ∂ t u ) (cid:107) L x + 1 (cid:15) (cid:107)∇ x u (cid:107) L x (cid:19) (cid:107) h ⊥ (cid:15) (cid:107) L (d x d µ ) . Applying (5.8), (5.9) and (5.10), we have1 (cid:15) (cid:107)(cid:104) h (cid:15) (cid:105) − M (cid:107) L x + dd t (cid:16) v · ∇ x u, h ⊥ (cid:15) (cid:17) (cid:46) (cid:15) (cid:107)(cid:104) h (cid:15) (cid:105) − M (cid:107) L x (cid:107) h ⊥ (cid:15) (cid:107) L (d x d µ ) + 1 (cid:15) (cid:107) h ⊥ (cid:15) (cid:107) L (d x d µ ) . By the Cauchy–Schwarz inequality, we arrive at(5.11) (cid:107)(cid:104) h (cid:15) (cid:105) − M (cid:107) L x + (cid:15) dd t (cid:16) v · ∇ x u, h ⊥ (cid:15) (cid:17) (cid:46) (cid:15) (cid:107) h ⊥ (cid:15) (cid:107) L (d x d µ ) . N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION Then, (5.11) combined with (5.6) implies thatdd t (cid:69) (cid:15) ( t ) (cid:46) − − δ(cid:15) (cid:107) h ⊥ (cid:15) (cid:107) L (d x d µ ) − δλ t (cid:107)(cid:104) h (cid:15) (cid:105) − M (cid:107) L x + δ(cid:15)λ (cid:48) t (cid:16) v · ∇ x u, h ⊥ (cid:15) (cid:17) ≤ − δλ t (cid:107) h (cid:15) − M (cid:107) L (d x d µ ) − δλ (cid:48) t (cid:12)(cid:12)(cid:12)(cid:16) v · ∇ x u, h ⊥ (cid:15) (cid:17)(cid:12)(cid:12)(cid:12) , where the constant δ ∈ (cid:0) , (cid:1) will be determined and the modified entropy (cid:69) (cid:15) is defined by (cid:69) (cid:15) ( t ) := (cid:107) h (cid:15) − M (cid:107) L (d x d µ ) + δ(cid:15)λ t (cid:16) v · ∇ x u, h ⊥ (cid:15) (cid:17) . Noticing (5.9) also implies that (cid:12)(cid:12)(cid:12)(cid:16) v · ∇ x u, h ⊥ (cid:15) (cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:107)(cid:104) h (cid:15) (cid:105) − M (cid:107) L x (cid:107) h ⊥ (cid:15) (cid:107) L (d x d µ ) ≤ (cid:107) h (cid:15) − M (cid:107) L (d x d µ ) . It means that the modified entropy (cid:69) (cid:15) is equivalent (independent of (cid:15) ) to the square of the L (d x d µ )-distance between h (cid:15) and M , when the constant δ > t (cid:69) (cid:15) ( t ) (cid:46) − (cid:0) λ t + λ (cid:48) t (cid:1) (cid:69) (cid:15) ( t ) . The conclusion (5.5) then follows from Gr¨onwall’s inequality and the equivalence between (cid:69) (cid:15) ( t ) and (cid:107) h (cid:15) ( t, · , · ) − M (cid:107) L (d x d µ ) . (cid:3) As far as the case (cid:15) = 1 is concerned, the exponential convergence to equilibrium in eachorder derivative is given. Proof of Theorem 1.1 (ii). Consider g := µ h . In view of the assumption on initial data andLemma 4.1, we know that λµ ≤ g ≤ Λ µ in R + × T d × R d . Proposition 5.1 then impliesthat there is some universal constant c > (cid:13)(cid:13)(cid:13) g ( t ) − M µ (cid:13)(cid:13)(cid:13) L ( T d × R d ) (cid:46) e − ct . Combining this estimate with the Sobolev embedding and the interpolation, we derive that,for any k ∈ N with k > d , (cid:13)(cid:13)(cid:13) g ( t ) − M µ (cid:13)(cid:13)(cid:13) (cid:67) k ( T d × R d ) (cid:46) (cid:13)(cid:13)(cid:13) g ( t ) − M µ (cid:13)(cid:13)(cid:13) H k ( T d × R d ) (cid:46) (cid:13)(cid:13)(cid:13) g ( t ) − M µ (cid:13)(cid:13)(cid:13) H k ( T d × R d ) (cid:13)(cid:13)(cid:13) g ( t ) − M µ (cid:13)(cid:13)(cid:13) L ( T d × R d ) . Since the H k -norm on the right hand side is bounded due to the global regularity estimategiven by Proposition 4.12, we arrive at the desired result. (cid:3) Finite time asymptotics. The study of macroscopic dynamics for the nonlinear kineticmodel (5.1) in this subsection relies on the regularity of the target equation (1.3). On accountof this, let us begin with mentioning some standard results for the equation (1.3) withoutproof. If the initial data satisfies λ ≤ ρ in ≤ Λ, then such bounds are preserved along times, λ ≤ ρ ≤ Λ, in the same spirit as Lemma 4.1. Combining the parabolic De Giorgi-Nash-Mosertheory with the Schauder theory, we know that the solution ρ is smooth for any positive time.We state the apriori estimate precisely as follows, where its behavior near the initial time istaken into account in view of the standard scaling technique.25 RANCESCA ANCESCHI, YUZHE ZHU Lemma 5.2. Let ρ in ∈ (cid:67) α (cid:0) T d (cid:1) valued in [ λ, Λ] with α ∈ (0 , , and ρ be the solution to (1.3) in R + × T d . Then, there is some universal constant α ∈ (0 , such that (5.12) (cid:107) ρ (cid:107) (cid:67) α ( R + × T d ) (cid:46) (cid:107) ρ in (cid:107) (cid:67) α ( T d ) . Moreover, there exists some constant C ρ > depending only on universal constants and (cid:107) ρ in (cid:107) (cid:67) α ( T d ) , such that for any t ∈ (0 , and x ∈ T d , we have (5.13) t − α |∇ x ρ ( t, x ) | + t − α | ∂ t ρ ( t, x ) | + t − α (cid:12)(cid:12) ∇ x ρ ( t, x ) (cid:12)(cid:12) + t − α | ∂ t ∇ x ρ ( t, x ) | ≤ C ρ ; and for any t ≥ , we have (5.14) (cid:107)∇ x ρ ( t, · ) (cid:107) L ∞ ( T d ) + (cid:107) ∂ t ρ ( t, · ) (cid:107) L ∞ ( T d ) + (cid:13)(cid:13) ∇ x ρ ( t, · ) (cid:13)(cid:13) L ∞ ( T d ) + (cid:107) ∂ t ∇ x ρ ( t, · ) (cid:107) L ∞ ( T d ) (cid:46) . We measure the distance between solutions to the scaled nonlinear kinetic model (1.2) andsolutions to the fast diffusion equation (1.3) by the relative phi-entropy functional (cid:72) β (seeDefinition 1.3). The following lemma shows the effectiveness of the relative phi-entropy formeasuring L -distance, by virtue of the uniform convexity of ϕ β . It can be seen as a simpleversion of the Csisz´ar-Kullback inequality on the relative entropy. We give its statement belowwith a proof taken from [9, Proposition 2.1] for the sake of completeness. Lemma 5.3. Let h and h be two functions valued in [0 , Λ] . Then, we have (5.15) (cid:72) β ( h | h ) ≥ (cid:18) − β (cid:19) Λ − β (cid:107) h − h (cid:107) L (d x d µ ) . If we additionally assume the lower bound that h , h ≥ λ , then (cid:72) β ( h | h ) ≤ (cid:18) − β (cid:19) λ − β (cid:107) h − h (cid:107) L (d x d µ ) . Proof. Since ϕ β (1) = ϕ (cid:48) β (1) = 0 and β ∈ [0 , z ∈ R + , there exists ξ z ∈ R + lyingbetween 1 and z so that ϕ β ( z ) = 12 ϕ (cid:48)(cid:48) β ( ξ z )( z − = 2 − β ξ − βz ( z − . Since min { z, } ≤ ξ z ≤ max { z, } , we have (cid:90) T d × R d max (cid:110) h − β , h − β (cid:111) | h − h | d x d µ ≤ − β (cid:72) β ( h | h ) ≤ (cid:90) T d × R d min (cid:110) h − β , h − β (cid:111) | h − h | d x d µ, which implies the desired results by using the boundedness of h , h . (cid:3) Let us consider the finite time diffusion asymptotics. Proposition 5.4. Let ρ in ∈ (cid:67) α (cid:0) T d (cid:1) valued in [ λ, Λ] with α ∈ (0 , , and the sequence offunctions { h (cid:15), in } (cid:15) ∈ (0 , ⊂ (cid:67) α (cid:0) T d × R d (cid:1) satisfying (cid:104) h (cid:15), in (cid:105) ≥ λ in T d and 0 ≤ h (cid:15), in ≤ Λ in T d × R d . Let h (cid:15) be the solutions of (5.1) associated with these initial data. Then, there exist someuniversal constants α ∈ (0 , , C > , and some constant C ρ > depending only on universal N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION constants and (cid:107) ρ in (cid:107) (cid:67) α ( T d ) , such that for any (cid:15) ∈ (0 , and for any t ∈ [ T , with T ∈ (0 , ,the following estimate holds, (cid:72) β ( h (cid:15) | ρ )( t ) ≤ C ρ (cid:72) β ( h (cid:15) | ρ )( T ) + C ρ (cid:15) (cid:16) t α − + (cid:15)t α − (cid:17) , (5.16) where ρ ( t, x ) is the solution to (1.3) associated with the initial data ρ in ; and for any t ≥ , wehave (cid:72) β ( h (cid:15) | ρ )( t ) ≤ (cid:104) (cid:72) β ( h (cid:15) | ρ )( t )(1) + C(cid:15) (cid:16) t (cid:17)(cid:105) e Ct . (5.17) Proof. For β ∈ [0 , h (cid:15) relative to ρ reads (cid:72) β ( h (cid:15) | ρ ) = (cid:72) β ( h (cid:15) | − (cid:72) β ( ρ | − − β − β (cid:16) (cid:104) h (cid:15) (cid:105) − ρ, ρ − β − (cid:17) . As far as the entropy (cid:72) β ( h (cid:15) | 1) is concerned, the entropy dissipation is derived by the equa-tion (5.1), integration by parts and using H¨older’s inequality (cid:104)∇ v h (cid:15) (cid:105) ≤ (cid:104) h (cid:15) (cid:105) β (cid:104) h − β(cid:15) |∇ v h (cid:15) | (cid:105) ,dd t (cid:72) β ( h (cid:15) | 1) = 2 − β − β (cid:16) h − β , h t (cid:17) = − − β(cid:15) (cid:16) h − β(cid:15) ∇ v h (cid:15) , (cid:104) h (cid:15) (cid:105) β ∇ v h (cid:15) (cid:17) ≤ − − β(cid:15) (cid:107)(cid:104)∇ v h (cid:15) (cid:105)(cid:107) L x = − − β(cid:15) (cid:107)(cid:104) vh (cid:15) (cid:105)(cid:107) L x . (5.18)In view of the limiting equation (1.3), we havedd t (cid:72) β ( ρ | 1) = 2 − β − β (cid:16) ρ − β , ∂ t ρ (cid:17) = − (2 − β ) (cid:16) ρ − β ∇ x ρ, ρ − β ∇ x ρ (cid:17) . (5.19)A direct computation with the macroscopic equation (5.3) and the equation (1.3) leads todd t (cid:16) (cid:104) h (cid:15) (cid:105) − ρ, ρ − β − (cid:17) = (cid:16) ρ − β , ∂ t (cid:104) h (cid:15) (cid:105) (cid:17) + (cid:16) (1 − β ) ρ − β (cid:104) h (cid:15) (cid:105) − (2 − β ) ρ − β , ∂ t ρ (cid:17) = 1 − β(cid:15) (cid:16) ρ − β ∇ x ρ, (cid:104) vh (cid:15) (cid:105) (cid:17) − (cid:16) (1 − β ) ∇ x (cid:0) ρ − β (cid:104) h (cid:15) (cid:105) (cid:1) − (2 − β ) ρ − β ∇ x ρ, ρ − β ∇ x ρ (cid:17) (5.20)The evolution of (cid:72) β ( h (cid:15) | ρ ) is then estimated by combining (5.18), (5.19) and (5.20),12 − β dd t (cid:72) β ( h (cid:15) | ρ ) ≤ − (cid:15) (cid:107)(cid:104) vh (cid:15) (cid:105)(cid:107) L x − (cid:15) (cid:16) (cid:104) vh (cid:15) (cid:105) , ρ − β ∇ x ρ (cid:17) + (cid:16) ∇ x (cid:0) ρ − β (cid:104) h (cid:15) (cid:105) − ρ − β (cid:1) , ρ − β ∇ x ρ (cid:17) = − (cid:13)(cid:13)(cid:13) (cid:15) − (cid:104) vh (cid:15) (cid:105) + ρ − β ∇ x ρ (cid:13)(cid:13)(cid:13) L x + (cid:16) (cid:15) − (cid:104) vh (cid:15) (cid:105) , ρ − β ∇ x ρ (cid:17) + (cid:16) ∇ x (cid:104) h (cid:15) (cid:105) + β (cid:0) − ρ − (cid:104) h (cid:15) (cid:105) (cid:1) ∇ x ρ, ρ − β ∇ x ρ (cid:17) We remark that the above inequality also holds for β = 1 by a similar computation. Abbre-viate Q (cid:15) := (cid:15) − (cid:104) vh (cid:15) (cid:105) + ρ − β ∇ x ρ , R (cid:15) := −∇ x · (cid:104) v ⊗ ∇ v h (cid:15) (cid:105) − (cid:15)∂ t (cid:104) vh (cid:15) (cid:105) and write the macroscopicequation (5.4) in the form of ∇ x (cid:104) h (cid:15) (cid:105) = − (cid:15) − (cid:104) h (cid:15) (cid:105) β (cid:104) vh (cid:15) (cid:105) + R (cid:15) . It then turns out that12 − β dd t (cid:72) β ( h (cid:15) | ρ ) ≤ − (cid:107) Q (cid:15) (cid:107) L x + (cid:16)(cid:0) − ρ − β (cid:104) h (cid:15) (cid:105) β (cid:1) Q (cid:15) , ρ − β ∇ x ρ (cid:17) + (cid:16) R (cid:15) , ρ − β ∇ x ρ (cid:17) + (cid:16) ρ − β (cid:104) h (cid:15) (cid:105) β − βρ − (cid:104) h (cid:15) (cid:105) − β, ρ − β |∇ x ρ | (cid:17) ≤ (cid:13)(cid:13)(cid:13) ρ − − β |∇ x ρ | (cid:13)(cid:13)(cid:13) L ∞ t,x (cid:107)(cid:104) h (cid:15) (cid:105) − ρ (cid:107) L x + (cid:16) R (cid:15) , ρ − β ∇ x ρ (cid:17) , (5.21)where for the second inequality, we used the Cauchy–Schwarz inequality2 (cid:16)(cid:0) − ρ − β (cid:104) h (cid:15) (cid:105) β (cid:1) Q (cid:15) , ρ − β ∇ x ρ (cid:17) ≤ (cid:107) Q (cid:15) (cid:107) L x + (cid:16)(cid:12)(cid:12) − ρ − β (cid:104) h (cid:15) (cid:105) β (cid:12)(cid:12) , (cid:12)(cid:12) ρ − β ∇ x ρ (cid:12)(cid:12) (cid:17) , RANCESCA ANCESCHI, YUZHE ZHU and the following two elementary inequalities, with β ∈ [0 , (cid:12)(cid:12) z β − (cid:12)(cid:12) ≤ | z − | and (cid:12)(cid:12) z β − βz − β (cid:12)(cid:12) ≤ | z − | , for any z ∈ R + . In view of H¨older’s inequality and the inequality (5.15) given in Lemma 5.3, we know that (cid:107)(cid:104) h (cid:15) (cid:105) − ρ (cid:107) L x ≤ (cid:107) h (cid:15) − ρ (cid:107) L (d x d µ ) ≤ β (cid:72) β ( h (cid:15) | ρ ) . Combining this with (5.21) and (5.13), we derive that, for any t ∈ (0 , t (cid:72) β ( h (cid:15) | ρ ) ≤ C ρ t α − (cid:72) β ( h (cid:15) | ρ ) + 2 (cid:16) R (cid:15) , ρ − β ∇ x ρ (cid:17) , (5.22)where the constants α ∈ (0 , 1) and C ρ > R (cid:15) isof order O ( (cid:15) ) due to the control of the entropy production and the regularity of the limitingequation. Indeed, (cid:90) t (cid:16) R (cid:15) , ρ − β ∇ x ρ (cid:17) = (cid:90) t (cid:16) (cid:104) v ⊗ ∇ v h (cid:15) (cid:105) , ∇ x (cid:0) ρ − β ∇ x ρ (cid:1)(cid:17) + (cid:15) (cid:90) t (cid:16) (cid:104) vh (cid:15) (cid:105) , ∂ t (cid:0) ρ − β ∇ x ρ (cid:1)(cid:17) − (cid:15) (cid:16) (cid:104) vh (cid:15) (cid:105) , ρ − β ∇ x ρ (cid:17) ( t ) + (cid:15) (cid:16) (cid:104) vh (cid:15) (cid:105) , ρ − β ∇ x ρ (cid:17) (0) (cid:46) (cid:18)(cid:13)(cid:13)(cid:13) ∇ x (cid:0) ρ − β ∇ x ρ (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t,x + (cid:15) (cid:13)(cid:13)(cid:13) ∂ t (cid:0) ρ − β ∇ x ρ (cid:1)(cid:13)(cid:13)(cid:13) L ∞ t,x (cid:19) (cid:90) t (cid:107)(cid:104) vh (cid:15) (cid:105)(cid:107) L x + (cid:15) (cid:13)(cid:13)(cid:13) ρ − β ∇ x ρ (cid:13)(cid:13)(cid:13) L ∞ t,x (cid:107)(cid:104) vh (cid:15) (cid:105)(cid:107) L ∞ ([0 ,T ]; L x ) . It then follows from (5.13), (5.18) and the global upper bound of h (cid:15) (Lemma 4.1) that, forany t ∈ (0 , (cid:90) t (cid:16) R (cid:15) , ρ − β ∇ x ρ (cid:17) ≤ C ρ (cid:16) t α − + (cid:15)t α − (cid:17) (cid:18)(cid:90) t (cid:107)(cid:104) vh (cid:15) (cid:105)(cid:107) L x (cid:19) + C ρ (cid:15)t α − ≤ C ρ (cid:16) (cid:15)t α − + (cid:15) t α − (cid:17) sup s ∈ [0 ,t ] (cid:113) (cid:72) β ( h (cid:15) | s ) + C ρ (cid:15)t α − ≤ C ρ (cid:16) (cid:15)t α − + (cid:15) t α − (cid:17) + C ρ (cid:15)t α − . Combining this estimate with (5.22), as well as Gr¨onwall’s inequality, we conclude (5.16).Besides, we arrive at (5.17), if we apply Lemma 5.2 with (5.14) instead of (5.13) in aboveargument. This completes the proof. (cid:3) We are now in a position to conclude the global in time diffusion asymptotics. Proof of Theorem 1.4. We are going to combine Proposition 5.1, 5.4 with a delicate analysison the relative entropy around the initial time to get Theorem 1.4. The analysis is based on thebarrier function method. Let us assume the constant α ∈ (0 , 1) provided by Proposition 5.4. Step 1. Pointwise estimate.Let us fix δ ∈ (0 , x , v ) ∈ T d × B R with R > 0, and consider the function h ( t, x, v ) := C t + C (cid:0) | x − x − (cid:15) − tv | + | v − v | (cid:1) where the constants C , C > t ≤ (cid:15)δ (cid:104) R (cid:105) , we have h ≥ C (cid:0) | x − x | − (cid:15) − t | x − x || v | (cid:1) ≥ C δ ∂B δ ( x , v ) , N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION where we chose C := 2 δ − Λ. For any ( x, v ) ∈ B δ ( x , v ), |(cid:104) h (cid:105) β L OU h | (cid:46) | ∆ v h | + | v · ∇ v h | (cid:46) δ − (cid:10) (cid:15) − t (cid:11) (cid:104) R (cid:105) . Therefore, for any t ≤ (cid:15)δ (cid:104) R (cid:105) and ( x, v ) ∈ B δ ( x , v ), (cid:16) ∂ t + (cid:15) − v · ∇ x − (cid:15) − (cid:104) h (cid:105) β L OU (cid:17) h ≥ C − C (cid:15) − δ − (cid:10) (cid:15) − t (cid:11) (cid:104) R (cid:105) ≥ , where the constant C > C := 2 C (cid:15) − δ − (cid:104) R (cid:105) . Then, themaximum principle implies that, for any t ≤ (cid:15)δ (cid:104) R (cid:105) and ( x, v ) ∈ B δ ( x , v ),(5.23) | h (cid:15) ( t, x, v ) − h (cid:15), in ( x , v ) | ≤ h ( t, x, v ) + sup B δ ( x ,v ) | h (cid:15), in ( x, v ) − h (cid:15), in ( x , v ) | . In particular, for any t ≤ (cid:15)δ (cid:104) R (cid:105) and ( x , v ) ∈ T d × B R ,(5.24) | h (cid:15) ( t, x , v ) − h (cid:15), in ( x , v ) | (cid:46) (cid:15) − δ − (cid:104) R (cid:105) t + (cid:107) h (cid:15), in (cid:107) (cid:67) α ( T d × R d ) δ α . As far as the solution ρ to the limiting equation (1.3) is concerned, using the H¨olderestimate (5.12) in Lemma 5.2, we derive that, for any t ∈ R + , (cid:107) ρ ( t ) − ρ in (cid:107) L ∞ ( T d ) (cid:46) (cid:16) (cid:107) ρ in (cid:107) (cid:67) α ( T d ) (cid:17) t α . (5.25) Step 2. Estimate of the relative entropy around the initial time.Let us restrict our attention on the time interval [0 , T ] with T ∈ (0 , 1) to be determined.Compute the relative entropy (cid:72) β ( h (cid:15) | ρ ) in terms of initial data as follows, (cid:72) β ( h (cid:15) | ρ ) = (cid:72) β ( h (cid:15), in | ρ in ) + (cid:90) T d × R d [( ϕ β ( h (cid:15) ) − ϕ β ( h (cid:15), in )) − ( ϕ β ( ρ ) − ϕ β ( ρ in ))] d x d µ − (cid:90) T d × R d (cid:2)(cid:0) ϕ (cid:48) β ( ρ ) − ϕ (cid:48) β ( ρ in ) (cid:1) ( h (cid:15), in − ρ in ) + ϕ (cid:48) β ( ρ )( h (cid:15) − h (cid:15), in ) − ϕ (cid:48) β ( ρ )( ρ − ρ in ) (cid:3) d x d µ. (5.26)Consider a truncation in v for the integrals on the right hand side. Since h (cid:15) , h (cid:15), in , ρ, ρ in areall bounded from above (Lemma 4.1), we have(5.27) (cid:90) T d × B cR [ | ϕ β ( h (cid:15) ) − ϕ β ( h (cid:15), in ) | + | ϕ β ( ρ ) − ϕ β ( ρ in ) | ] d x d µ (cid:46) (cid:90) B cR d µ (cid:46) R − . Observe that for any a, b ∈ (0 , Λ], there exists ξ ∈ R + lying between a and b such that ϕ β ( a ) − ϕ β ( b ) = 2 ξϕ (cid:48) β ( ξ )( a − b ). Meanwhile, for any ξ ∈ (0 , Λ], | ξϕ (cid:48) β ( ξ ) | (cid:46) 1. Thus, | ϕ β ( h (cid:15) ) − ϕ β ( h (cid:15), in ) | (cid:46) | h (cid:15) − h (cid:15), in | and | ϕ β ( ρ ) − ϕ β ( ρ in ) | (cid:46) | ρ − ρ in | . Set R := (cid:15) − η , δ := (cid:15) η and T := (cid:15) η for some constant η ∈ (0 , 1) to be determined. In thissetting, T ≤ (cid:15)δ (cid:104) R (cid:105) . It then follows from (5.24), (5.25) and (5.27) that, for any t ≤ T , (cid:90) T d × R d [ | ϕ β ( h (cid:15) ) − ϕ β ( h (cid:15), in ) | + | ϕ β ( ρ ) − ϕ β ( ρ in ) | ] d x d µ (cid:46) R − + (cid:90) T d × B R (cid:104) | h (cid:15) − h (cid:15), in | + | ρ − ρ in | (cid:105) d x d µ (cid:46) (cid:15) η + (cid:15) η + (cid:107) h (cid:15), in (cid:107) (cid:67) α ( T d × R d ) (cid:15) αη + (cid:16) (cid:107) ρ in (cid:107) (cid:67) α ( T d ) (cid:17) (cid:15) α + αη . (5.28) 29 RANCESCA ANCESCHI, YUZHE ZHU Besides, (cid:107) h (cid:15) − h (cid:15), in (cid:107) L ( T d × B cR , d x d µ ) (cid:46) R − and | ϕ (cid:48) β | (cid:46) λ, Λ]. Therefore, combining (5.24),(5.25) with the inequality (5.15) given in Lemma 5.3 yields that, for any t ≤ T , (cid:90) T d × R d (cid:2)(cid:12)(cid:12)(cid:0) ϕ (cid:48) β ( ρ ) − ϕ (cid:48) β ( ρ in ) (cid:1) ( h (cid:15), in − ρ in ) (cid:12)(cid:12) + (cid:12)(cid:12) ϕ (cid:48) β ( ρ )( h (cid:15) − h (cid:15), in ) (cid:12)(cid:12) + (cid:12)(cid:12) ϕ (cid:48) β ( ρ )( ρ − ρ in ) (cid:12)(cid:12)(cid:3) d x d µ (cid:46) (cid:107) h (cid:15), in − ρ in (cid:107) L ( T d × R d , d x d µ ) + (cid:107) h (cid:15) − h (cid:15), in (cid:107) L ( T d × R d , d x d µ ) + (cid:107) ρ − ρ in (cid:107) L ( T d ) (cid:46) (cid:72) β ( h (cid:15), in | ρ in ) + R − + (cid:107) h (cid:15) − h (cid:15), in (cid:107) L ( T d × B R , d x d µ ) + (cid:107) ρ − ρ in (cid:107) L ( T d ) (cid:46) (cid:15) + (cid:15) η + (cid:107) h (cid:15), in (cid:107) (cid:67) α ( T d × R d ) (cid:15) αη + (cid:16) (cid:107) ρ in (cid:107) (cid:67) α ( T d ) (cid:17) (cid:15) α +2 αη . (5.29)Plugging (5.28) and (5.29) into the expression (5.26), we derive that, for any t ≤ T ,(5.30) (cid:72) β ( h (cid:15) | ρ )( t ) (cid:46) (cid:16) (cid:107) h (cid:15), in (cid:107) (cid:67) α ( T d × R d ) + (cid:107) ρ in (cid:107) (cid:67) α ( T d ) (cid:17) (cid:15) αη . Step 3. Conclusion.Recall that we have chosen T = (cid:15) η . In view of (5.30) and the estimate (5.16) given inLemma 5.4, one may optimize in η to get the result. For simplicity, we pick η := α − α so that T = (cid:15) − α − α and (cid:15) (cid:16) T α − + (cid:15)T α − (cid:17) (cid:46) (cid:15) α . It turns out that for any t ∈ [0 , (cid:72) β ( h (cid:15) | ρ )( t ) ≤ C ρ (cid:72) β ( h (cid:15) | ρ )( T ) + C ρ (cid:15) (cid:16) T α − + (cid:15)T α − (cid:17) ≤ C ∗ (cid:15) αη , (5.31)where the constant C ρ > C ∗ > (cid:107) ρ in (cid:107) (cid:67) α ( T d ) and (cid:107) h (cid:15), in (cid:107) (cid:67) α ( T d × R d ). Then, using the estimate (5.17)given in Lemma 5.4 with (5.15), we arrive at point (i) of Theorem 1.4.As for point (ii) of Theorem 1.4, applying (5.17), together with (5.15) and (5.31), for any t ∈ [1 , T ], we have (cid:107) h (cid:15) ( t ) − ρ ( t ) (cid:107) L (d x d µ ) (cid:46) (cid:104) (cid:72) β ( h (cid:15) | ρ )(1) + (cid:15) (cid:16) T (cid:17)(cid:105) e CT (cid:46) (cid:104) C ∗ (cid:15) αη + (cid:15) (cid:16) − ι log (cid:15) ) (cid:17)(cid:105) (cid:15) − αη (cid:46) C ∗ (cid:15) αη . where we picked T := − ι log (cid:15) with ι := αη C . Finally, using Proposition 5.1 with the assump-tion on initial data, we know that there is some universal constant c > t ≥ T , (cid:107) h (cid:15) ( t ) − ρ ( t ) (cid:107) L (d x d µ ) ≤ (cid:107) h (cid:15) ( t ) − M (cid:107) L (d x d µ ) + (cid:107) ρ ( t ) − M (cid:107) L x (cid:46) e − cT = (cid:15) cι . The proof now is complete. (cid:3) Appendix A. Maximum principle The following maximum principle (on a not necessarily bounded domain) is repeatedlyapplied throughout the article. We state it in the more suitable fashion for the Fokker-Planckequations of our concern, whose proof is in the same spirit as [5, Lemma A.2]. Lemma A.1. Let the domain ω ⊂ R d × R d and the parabolic cylinder ω T := (0 , T ] × ω . If f ∈ (cid:67) ( ω T ) ∩ (cid:67) ( ω T ) is a bounded subsolution in the sense that (A.1) L f := ( ∂ t + v · ∇ x ) f − tr( AD v f ) − B · ∇ v f ≤ ω T , with the coefficients A ( t, x, v ) , B ( t, x, v ) ∈ (cid:67) ( ω T ) satisfying λ | ξ | ≤ A ( t, x, v ) ξ · ξ ≤ Λ | ξ | , | B ( t, x, v ) · ξ | ≤ Λ (cid:104) v (cid:105)| ξ | for any ξ ∈ R d , ( t, x, v ) ∈ ω T , N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION then sup ω T f ≤ sup ∂ p ω T f , where the parabolic boundary ∂ p ω T := [0 , T ] × ω − (0 , T ] × ω .Proof. If the domain ω is bounded, then the result is classical. For general (unbounded) ω , weconsider the auxiliary functions φ ( t, v ) := e C t (cid:104) v (cid:105) and φ ( t, x ) := e C t (cid:104) x (cid:105) with C , C > f is bounded, for any ε , ε > 0, there exists R ( ε ) , R ( ε ) > C , C )such that f − ε φ − ε φ ≤ sup ∂ p ω T f in ω T ∩ {| x | ≥ R ( ε ) or | v | ≥ R ( ε ) } .By choosing C = 3Λ, we have L φ = e C t ( C (cid:104) v (cid:105) − Av · v − B · v ) ≥ ( C − (cid:104) v (cid:105) = 0 in ω T . For any R ≥ R ( ε ), there exists C > R such that L φ = e C t (cid:0) C (cid:104) x (cid:105) + v · x (cid:1) ≥ ( C − (cid:104) x (cid:105) − | v | ≥ ω T ∩ {| v | < R } . Therefore, for any R > R ( ε ), f − ε φ − ε φ is a subsolution to (3.1) in the boundeddomain (0 , T ] × ( ω ∩ ( B R × B R )) with the data smaller than sup ∂ p ω T f on the boundaryportion contained in {| x | = R or | v | = R } . Then, applying the classical maximum principleyields f − ε φ − ε φ ≤ sup ∂ p ω T f in (0 , T ] × ( ω ∩ ( B R × B R )) . Sending R → ε → R → ε → (cid:3) Appendix B. Spreading of positivity This appendix is devoted to the proof of Proposition 4.2. The argument follows the onepresented in [18] and it is based on the combination of Lemma 4.5 and Lemma 4.6. Proof of Proposition 4.2. The proof is split into four steps. Step 1. Spreading positivity for all velocities for short times.Applying Lemma 4.5 ( τ = 1) yields that there is some universal constant c > ≤ t ≤ min (cid:8) , T, c (cid:10) r − (cid:11) − (cid:104) v (cid:105) − (cid:9) , h ( t, x, v ) ≥ δ { | x − x − tv | < r , | v − v | < r } ≥ δ { | x − x − tv | < r , | v − v | < r } . Let r := min (cid:8) , r (cid:9) and t := T . Then, Lemma 4.6 implies that there exists C > T , δ , r and v such that, for any 0 < t ≤ t ≤ T with T := min (cid:8) , T, c (cid:10) r − (cid:11) − (cid:104) v (cid:105) − , r (cid:104) v (cid:105) − (cid:9) and v ∈ R d , we have(B.1) h ( t, x, v ) ≥ C − e − C | v − v | {| x − x − tv | < r } ≥ C − e − C | v − v | {| x − x | Spreading positivity in space for short times.For any fixed t ∈ [ t, T ] and x ∈ T d , we set v := x − x t − t . In view of (B.1), by Lemma 4.5( τ = 2( t − t ), v = v ), we deduce that, if t − t ≤ c (cid:10) t − t ) r − (cid:11) − (cid:104) v (cid:105) − , in particular if(B.2) t ≤ t + t with t := c r r (cid:104) x − x (cid:105) − , then there exists δ > C such that, for any t ∈ [ t, t ], h ( t, x, v ) ≥ δ { | x − x − ( t − t ) v | < r , t − t ) | v − v | < r } ≥ δ { | x − x − ( t − t ) v | < r , | v − v | < r } . Then, Lemma 4.6 ( v = v ) implies that, for any 0 < t ≤ t ≤ t + t and v ∈ R d ,(B.3) h ( t, x, v ) ≥ C − e − C | v | { | x − x − ( t − t ) v | < r } , RANCESCA ANCESCHI, YUZHE ZHU for some constant C > T , δ, r, v and | x − x | . Inparticular, for any 0 < t ≤ t ≤ t + t and v ∈ R d ,(B.4) h ( t, x, v ) ≥ C − e − C | v | . Step 3. Spreading positivity for any finite time.We observe that the time interval above is restricted (see (B.2)), but it can be removed byapplying the lemmas again. Based on the previous step, it suffices to deal with the case that t > t . By a similar proof to (B.3), we derive h ( t , x, v ) ≥ δ { | x − x | < r , | v | < r } , for some constant δ > C . In view of this data, applyingLemma 4.5 to h ( t + · , · , · ) (with τ = 1, v = 0), we see that, for any t ∈ (cid:2) t , min (cid:8) T , t + T (cid:9)(cid:3) with T := c (cid:68) r (cid:69) − , h ( t, x, v ) ≥ δ { | x − x | < r , | v | < r } . It then follows from Lemma 4.6 that, for any t ∈ (cid:2) t + t, min (cid:8) T , t + T (cid:9)(cid:3) and v ∈ R d , h ( t, x, v ) ≥ C − e − C | v | { | x − x | < r } , for some constant C > C .Combining this with (B.4), as well as recalling that T = 2 t and the space domain T d iscompact, we know that there exists C > T , δ, r and v such that, for any ( t, x, v ) ∈ [ T , min { T , T } ] × T d × R d , h ( t, x, v ) ≥ C − e − C | v | . Since T and T depend only on universal constants, r and v , by applying the above argumentsiterately, we obtain the result for any finite time. Step 4. Improving the exponential tail.We remark that this step is not necessary for the applications of the lower bound result, butit show a more precise decay rate as | v | → ∞ .By the previous step, there is some c > T , T, δ, r and v such that h ≥ c in [ T , T ] × T d × B . Consider the barrier function h ( t, x, v ) := ce − C ( t − T ) − | v | in [ T , T ] × T d × B c , where the constant C > ∂ t + v · ∇ x ) h − R h L OU h = C R h h ( t − T ) (cid:0) R − h + 2( d − | v | )( t − T ) − C | v | (cid:1) ≤ C R h h ( t − T ) (cid:16) c − β + 2 dT − C (cid:17) in ( T , T ] × T d × B c . In particular, by choosing C sufficiently large (with the same dependence as c ), we have( ∂ t + v · ∇ x ) ( h − h ) − R h L OU ( h − h ) ≤ T , T ] × T d × B c . Besides, by its definition, h ≥ h on the boundary { t ∈ [ T , T ] , | v | = 1 } ∪ { t = 2 t, | v | ≥ } . Themaximum principle (Lemma A.1) then implies that h ≥ h in [ T , T ] × T d × B c . Therefore, weachieve the Gaussian type lower bound for any ( t, x, v ) ∈ [2 T , T ] × T d × R d . The proof nowis complete. (cid:3) N THE NONLINEAR KINETIC FOKKER-PLANCK EQUATION Appendix C. Gaining regularity of spatial increment This appendix is devoted to the proof of two technical lemmas for spatial increments in-volved in the bootstrapping of higher regularity for solutions to the equation (4.1) presentedin Subsection 4.3. For the convenience of the reader, we report a brief proof following thelines of [21, Lemma 8.1] with s = 1, α = β = 2. Lemma C.1. Let α ∈ (0 , and a bounded continuous function g defined in Q . If thereexists some constant M > such that for any y ∈ B , [ δ y g ] (cid:67) l ( Q ) ≤ M and [ δ y g ] (cid:67) αl ( Q ) ≤ M (cid:107) (0 , y, (cid:107) , then there exists some universal constant η ∈ (0 , such that for any y ∈ B , (cid:107) δ y g (cid:107) (cid:67) ηl ( Q ) (cid:46) M (cid:107) (0 , y, (cid:107) . Proof of Lemma C.1. In view of the assumption and Remark 2.5, for fixed y ∈ B , we considerthe the polynomial expansion p of δ y g at z ∈ Q with deg kin ( p ) = 2, p ( z ) = δ y g ( z ) + ( ∂ t + v · ∇ x ) δ y g ( z ) t + ∇ v δ y g ( z ) · v + 12 D v δ y g ( z ) v · v, for z := ( t, x, v ) ∈ R × R d × R d . For any z such that z ◦ z ∈ Q , we have(C.1) | δ y g ( z ◦ z ) − p ( z ) | ≤ M (cid:107) (0 , y, (cid:107) (cid:107) z (cid:107) α . In particular, p (0 , y, 0) = δ y g ( z ) so that for any y ∈ B , | δ y g ( z ) − δ y g ( z ) | = | δ y g ( z ◦ (0 , y, − δ y g ( z ) | = | δ y g ( z ◦ (0 , y, − p (0 , y, |≤ M (cid:107) (0 , y, (cid:107) α It then follows that for any z ∈ Q and for any k ∈ N such that z ◦ (0 , k y, ∈ Q , (cid:12)(cid:12) δ y g ( z ) − − k δ k y g ( z ) (cid:12)(cid:12) ≤ k (cid:88) j =1 − j (cid:12)(cid:12) δ j y g ( z ) − δ j − y g ( z ) (cid:12)(cid:12) ≤ M (cid:107) (0 , y, (cid:107) α k (cid:88) j =1 (1+ α ) j ≤ M (cid:107) (0 , y, (cid:107) α (1+ α ) k . (C.2)Picking k ∈ N such that (cid:107) k − (0 , y, (cid:107) ≤ < (cid:107) k (0 , y, (cid:107) and using the assumption yields | δ y g ( z ) | ≤ − k | δ k y g ( z ) | + 2 M (cid:107) (0 , y, (cid:107) α (1+ α ) k ≤ (cid:107) δ k y g (cid:107) (cid:67) l ( Q ) (cid:107) (0 , y, (cid:107) + 4 M (cid:107) (0 , y, (cid:107) ≤ M (cid:107) (0 , y, (cid:107) . (C.3)It remains to show that there exists some constant η > α such that(C.4) | δ y g ( z ◦ z ) − δ y g ( z ) | (cid:46) M (cid:107) (0 , y, (cid:107) (cid:107) z (cid:107) η . By (C.1) and Lemma 2.4, we know that for any z ∈ Q and z ◦ z ∈ Q , | δ y g ( z ◦ z ) − δ y g ( z ) | ≤ (cid:0) | ( ∂ t + v · ∇ x ) δ y g ( z ) | + | D v δ y g ( z ) | (cid:1) (cid:107) z (cid:107) + |∇ v δ y g ( z ) |(cid:107) z (cid:107) + M (cid:107) (0 , y, (cid:107) (cid:107) z (cid:107) α (cid:46) (cid:16) [ δ y g ] (cid:67) αl ( Q ) (cid:107) z (cid:107) + [ δ y g ] (cid:67) αl ( Q ) [ δ y g ] (cid:67) l ( Q ) + [ δ y g ] (cid:67) l ( Q ) (cid:17) (cid:107) z (cid:107) + M (cid:107) (0 , y, (cid:107) (cid:107) z (cid:107) α . RANCESCA ANCESCHI, YUZHE ZHU If (cid:107) z (cid:107) ≤ (cid:107) (0 , y, (cid:107) , then combining the above expression with the assumption and (C.3)implies (C.4) with η = . In particular, if k ∈ N such that (cid:107) z (cid:107) < (cid:107) k (0 , y, (cid:107) , then we have(C.5) 2 − k | δ k y g ( z ◦ z ) − δ k y g z ( z ) | (cid:46) − k M (cid:107) (0 , k y, (cid:107) (cid:107) z (cid:107) η = M (cid:107) (0 , y, (cid:107) (cid:107) z (cid:107) η . Now if (cid:107) z (cid:107) ≥ (cid:107) (0 , y, (cid:107) , applying (C.2) at points z and z ◦ z , with k ∈ N such that (cid:107) k − (0 , y, (cid:107) ≤ (cid:107) z (cid:107) < (cid:107) k (0 , y, (cid:107) , yields that | δ y g ( z ) − − k δ k y g ( z ) | ≤ M (cid:107) (0 , y, (cid:107) (cid:107) z (cid:107) α , (C.6) | δ y g ( z ◦ z ) − − k δ k y g ( z ◦ z ) | ≤ M (cid:107) (0 , y, (cid:107) (cid:107) z (cid:107) α . (C.7)Summing up (C.5), (C.6) and (C.7), we arrive at (C.4). (cid:3) Following the lines of the above proof and taking into account that (cid:107) g (cid:107) (cid:67) αl ( Q ) ≤ M , oneis also able to prove the following result. Lemma C.2. If g ∈ (cid:67) αl ( Q ) with α ∈ (0 , , then for any y ∈ B , we have (cid:107) δ y g (cid:107) (cid:67) αl ( Q ) (cid:46) (cid:107) g (cid:107) (cid:67) αl ( Q ) (cid:107) (0 , y, (cid:107) . References [1] Francesca Anceschi, Michela Eleuteri, and Sergio Polidoro. 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