The Landau equation as a Gradient Flow
José A. Carrillo, Matias G. Delgadino, Laurent Desvillettes, Jeremy Wu
aa r X i v : . [ m a t h . A P ] J u l The Landau equation as a Gradient Flow
Jos´e A. Carrillo ∗ , Matias G. Delgadino † , Laurent Desvillettes ‡ , Jeremy Wu § July 20, 2020
Abstract
We propose a gradient flow perspective to the spatially homogeneous Landau equa-tion for soft potentials. We construct a tailored metric on the space of probabilitymeasures based on the entropy dissipation of the Landau equation. Under this met-ric, the Landau equation can be characterized as the gradient flow of the Boltzmannentropy. In particular, we characterize the dynamics of the PDE through a func-tional inequality which is usually referred as the Energy Dissipation Inequality (EDI).Furthermore, analogous to the optimal transportation setting, we show that this in-terpretation can be used in a minimizing movement scheme to construct solutions toa regularized Landau equation.
The Landau equation is an important partial differential equation in kinetic theory. It givesa description of colliding particles in plasma physics [37], and it can be formally derived asa limit of the Boltzmann equation where grazing collisions are dominant [16, 44]. Similar tothe Boltzmann equation (see [7] for a consistency result and related derivation issues), therigorous derivation of the Landau equation from particle dynamics is still a huge challenge.For a spatially homogeneous density of particles f = f t ( v ) for t ∈ (0 , ∞ ) , v ∈ R d thehomogeneous Landau equation reads ∂ t f ( v ) = ∇ v · (cid:18) f ( v ) Z R d | v − v ∗ | γ Π[ v − v ∗ ]( ∇ v log f ( v ) − ∇ v ∗ log f ( v ∗ )) f ( v ∗ ) dv ∗ (cid:19) . (1)For notational convenience, we sometimes abbreviate f = f t ( v ) and f ∗ = f t ( v ∗ ) . We alsodenote the differentiations by ∇ = ∇ v and ∇ ∗ = ∇ v ∗ . The physically relevant parametersare usually d = 2 , γ ≥ − d − z ] = I − z ⊗ z | z | being the projection matrix onto { z } ⊥ . In this paper, for simplicity we will focus in the case d = 3 and vary the weight ∗ Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK ([email protected]) † Pontifical Catholic University of Rio de Janeiro, 38097 RJ, Brazil ([email protected]) ‡ Universit´e de Paris, Sorbonne Universit´e, CNRS, Institut de Math´ematiques de Jussieu-Paris RiveGauche ([email protected]) § Dept. Mathematics, Imperial College London, London SW7 2AZ, UK ([email protected]) γ , although most of our results are valid in arbitrary dimension. The regime0 < γ < hard potentials while γ < softpotentials with a further classification of − ≤ γ < − ≤ γ < − γ = 0 and γ = − d areknown as the Maxwellian and Coulomb cases respectively.The purpose of this work is to propose a new perspective inspired from gradient flows forweak solutions to (1), which is in analogy with the relationship of the heat equation and the2-Wasserstein metric, see [36, 3]. One of the fundamental steps is to symmetrize the righthand of (1). More specifically, if we consider a test function φ ∈ C ∞ c ( R d ) we can formallycharacterize the equation by ddt Z R d φf dv = − Z Z R d f f ∗ | v − v ∗ | γ ( ∇ φ − ∇ ∗ φ ∗ ) · Π[ v − v ∗ ]( ∇ log f − ∇ ∗ log f ∗ ) dv ∗ dv, (2)where the change of variables v ↔ v ∗ has been exploited. Building our analogy with theheat equation and the 2-Wasserstein distance, we define an appropriate gradient˜ ∇ φ := | v − v ∗ | γ Π[ v − v ∗ ]( ∇ φ − ∇ ∗ φ ∗ ) , so that equation (2) now looks like ddt Z R d φf dv = − Z Z R d f f ∗ ˜ ∇ φ · ˜ ∇ log f dv ∗ dv, noting that Π = Π. To highlight the use of this interpretation, we notice that ˜ ∇ φ = 0,when we choose as test functions φ = 1 , v i , | v | for i = 1 , . . . , d which immediately showsthat formally the equation conserves mass, momentum and energy. The action functionaldefining the Landau metric mimics the Benamou-Brenier formula [6] for the 2-Wassersteindistance, see [23, 24, 26] for other distances defined analogously for nonlinear and non-localmobilities. In fact, the Landau metric is built by considering a minimizing action principleover curves that are solutions to the appropriate continuity equation, that is d L ( f, g ) := min ∂ t µ + ˜ ∇· ( V µµ ∗ )=0 µ = f, µ = g (cid:26) Z Z Z R d | V | dµ ( v ) dµ ( v ∗ ) dt (cid:27) , (3)where the ˜ ∇· is the appropriate divergence; the formal adjoint to the appropriate gradient(see Section 2.1).Also, we notice that analogously to the heat equation, written as the continuity equation ∂ t f = ∇ · ( f ∇ log f ), the Landau equation can be formally re-written as ∂ t f = 12 ˜ ∇ · ( f f ∗ ˜ ∇ log f ) , equivalent to the continuity equation with non-local velocity field given by ∂ t f + ∇ · ( U ( f ) f ) = 0 U ( f ) := − Z R d | v − v ∗ | γ Π[ v − v ∗ ] ( ∇ log f − ∇ ∗ log f ∗ ) f ∗ dv ∗ . (4)2onsidering the evolution of Boltzmann entropy we formally obtain ddt Z R d f log f dv =: − D ( f t ) = − Z Z R d | ˜ ∇ log f | f f ∗ dv ∗ dv ≤ . (5)In physical terms this is referred to as the entropy dissipation or entropy production for itformally shows that the entropy functional H [ f ] := Z R d f log f dv is non-increasing along the dynamics of the Landau equation. Moreover, by integratingequation (5) in time one formally obtains H [ f t ] + Z t D ( f s ) ds = H [ f ] . (6)In [44], Villani introduced the notion of H-solution, which captures this formal property.Motivated by the physical considerations of certain conserved quantities and entropy dissi-pation, H-solutions provided a step towards well-posedness of the Landau equation in thesoft potential case. One advantage to this approach is that it avoids assuming that thesolutions belongs to L p ( R ) for p >
1. For moderately soft potentials, the propagation of L p norms is proven and this is enough to make sense of classical weak solutions [47]. In the verysoft potential case, there is no longer a guarantee of L p propagation due to the singularityof the weight. We refer to [17, Section 1.2] for a heuristic description of this difficulty.Similar to H-solutions our approach will also be based on the entropy dissipation (6).Following De Giorgi’s minimizing movement ideas [2, 3], we characterize the Landau equa-tion by its associated Energy Dissipation Inequality. More specifically, we show that weaksolutions to (1) with initial data f are completely determined by the following functionalinequality: H [ f t ] + 12 Z t | ˙ f | d L ( s ) ds + 12 Z t D ( f s ) ds ≤ H [ f ] for a.e. every t > | ˙ f | d L ( s ) stands for the metric derivative associated to the Landau metric defined above.Our analysis is also largely inspired by Erbar’s approach in viewing the Boltzmann equationas a gradient flow [25] and recent numerical simulations of the homogeneous Landau equationin [15] based on a regularized version of (4). In contrast with the classical 2-Wassersteinmetric, one of the main features of the Landau equation (1) and metric (3) is that theyare non-local. Hence, the convergence analysis usually relying on convexity and lower-semicontinuity needs to be adapted to deal with the non-locality of this equation. In particular,our characterization Theorem 11 is based in using (expected) a-priori estimates to deal withthe non-locality through appropriate bounds.On the other hand, the state of the art related to the uniqueness for the Landau equationdepends on the range of values γ may take. In the cases of hard potentials or Maxwellian,the uniqueness theory is very well understood due to Villani and the third author [21, 22,45]. In the soft potential case, one of the first major contributions to the general theory3f the spatially inhomogeneous Landau equation ( γ ≥ −
3) was the global existence anduniqueness result by Guo [35]. This result was achieved in a perturbative framework withhigh regularity assumptions on the initial data. Through probabilistic arguments, the nextmajor improvement to uniqueness for γ ∈ ( − ,
0) came from Fournier and Gu´erin [27]. Theirresult established uniqueness in a class of solutions that shrinks as γ decreases towards − L p and moments assumptions are needed. In their proof, uniqueness is shown byproving stability with respect to the 2-Wasserstein metric.Still lots of open questions for the soft potential case remain. In particular, a fundamen-tal question like uniqueness for the Coulomb case is unresolved. To tackle this and otherproblems an array of novel methods have been employed. Here is an incomplete sample ofthe contributions made in this direction which highlight the difficulties of the soft potentialcase [22, 21, 1, 12, 11, 47, 33, 31, 34, 32, 43, 30, 42, 29]. A brief glance at some of thesereferences illustrates the breadth of techniques that have found partial success at answeringthe open questions; probability-based arguments, kinetic and parabolic theory, and manymore.The purpose of this paper is to bring in another set of techniques to help answer some ofthese fundamental questions. The gradient flow theory applied to PDEs has flourished in thelast decades. In their seminal paper [36], Jordan, Kinderlehrer, and Otto proposed a varia-tional approach (JKO scheme) extended later on to a wide class of PDEs using the optimaltransportation distance of probability measures. These results and many more achievementsfrom their contemporaries allowed for novel approaches to questions of existence, uniqueness,convergence to equilibrium, and other aspects of a large class of PDE; we mention [3, 40] fora coherent exposition of these techniques and the relevant literature, even as more advanceshave been made since then.The advantage of our variational characterization of the Landau equation is that it unveilsnew possible routes of showing convergence results for this equation. First of all, it allows fornatural regularizations of the Landau equation by taking the steepest descent of regularizedentropy functionals instead of the Boltzmann entropy as in [14]. This idea was recentlydeveloped in [15] leading to structure preserving particle schemes with good accuracy. Wecan also consider the framework of convergence of gradient flows based on Γ-convergenceintroduced in [39, 41] to attack the convergence of these numerical methods [15]. Moreover,this approach is flexible enough to also study the rigorous convergence of the grazing col-lision limit of the Boltzmann equation to the Landau equation. In this case, we can takeadvantage of the similar developed framework of Erbar for the Boltzmann equation [25] toset this question in simple terms. Namely, the convergence of the associated metrics and thelower-semicontinuous limit of the dissipations. Being able to do this even at the regularizedlevel would be already a breakthrough in understanding the connection between these equa-tions. Finally, deriving uniqueness from the variational structure is classically done throughconvexity properties of the entropy functional with respect to the geodesics of the Landaumetric. This is another important avenue of research that our work opens.The plan of this paper is as follows. Section 2 introduces the prerequisites and containsthe statements of the main results. We first construct and analyze in Section 3 the Landaumetric based on (3). For a regularized problem, Section 4 shows the equivalence betweenweak solutions and gradient flows, while Section 5 shows the existence of gradient flowsolutions via a Minimizing Movement scheme. Finally, we show in Section 6 that a gradient4ow solution is equivalent to H-solutions of the Landau equation (1) under some integrabilityassumptions. Appendix A is devoted to some technical lemmata needed in the proof of themain theorems regarding the chain rule identity behind the definition of weak solutions forthe regularized Landau equation. We start by introducing the necessary notation and definitions together with a quick overviewof gradient flow concepts to make our main results fully self-contained.
We denote a . ... b ⇐⇒ ∃ C ( . . . ) > a ≤ C ( . . . ) b. We adopt the Japanese angle bracket notation for a smooth alternative to absolute value h v i = 1 + | v | , v ∈ R d . For ǫ >
0, we denote our regularization kernel to be an exponential distribution G ǫ ( v ) = ǫ − d G ( v/ǫ ) , G ( v ) = C d exp ( − h v i ) , C d = (cid:18)Z R d exp( − h v i ) dv (cid:19) − . Our results work for some general tailed behaviour in the kernels given by G s,ǫ ( v ) = ǫ − d G s ( v/ǫ ) , G s ( v ) = C s,d exp( − h v i s ) , C s,d = (cid:18)Z R d exp( − h v i s ) dv (cid:19) − , for s >
0; we point out some of the limitations and restrictions on s > G ,ǫ as the Maxwellian regularization. We denote the space ofprobability measures over R d by P ( R d ), endowed with the weak topology against boundedcontinuous functions. We will mostly be dealing with the Lebesgue measure on R d as ourreference measure which we denote by L . The subset P a ( R d ) ⊂ P ( R d ) denotes the set ofabsolutely continuous probability measures with respect to Lebesgue measure. For p > p -moments P p ( R d ) by P p ( R d ) := (cid:26) µ ∈ P ( R d ) (cid:12)(cid:12)(cid:12)(cid:12) m p ( µ ) := Z R d h v i p dµ ( v ) < ∞ (cid:27) . Finally, for
E >
0, we consider the subset P p,E ( R d ) ⊂ P p ( R d ) of probability measures with p -moments uniformly bounded by E ; P p,E ( R d ) := (cid:26) µ ∈ P p ( R d ) (cid:12)(cid:12)(cid:12)(cid:12) m p ( µ ) ≤ E (cid:27) . We denote by M the space of signed Radon measures on R d × R d with the standardweak* topology against the continuous and compactly supported functions of R d × R d . The5pace M d is the space of signed d -length Radon measures. For T >
0, we will add the timecontribution of the measures by denoting M T to be the space of signed Radon measures on R d × R d × [0 , T ] with the usual weak* topology. Similarly, M dT will be the space of signed d -length Radon measures on R d × R d × [0 , T ].For µ ∈ P ( R d ), we define a family of regularized entropies H ǫ [ µ ] by H ǫ [ µ ] := Z R d [ µ ∗ G ǫ ]( v ) log[ µ ∗ G ǫ ]( v ) dv, which we shall see is well-defined provided µ has a finite moment in Lemma 29. Formally,one can calculate the first variation of this functional in P as δ H ǫ δµ ( v ) = G ǫ ∗ log[ µ ∗ G ǫ ]( v ) . For a functional F : P a ( R d ) → R with first variation δ F δf , we refer to the F Landau equationas ∂ t f = ∇ · (cid:18) f Z R d f ∗ | v − v ∗ | γ Π[ v − v ∗ ] (cid:18) ∇ δ F δf − ∇ ∗ δ F ∗ δf ∗ (cid:19) dv ∗ (cid:19) . (7)To clarify the meaning of ˜ ∇· , for a given test function φ and vector-valued test function A ,we have Z Z R d [ ˜ ∇ φ ]( v, v ∗ ) · A ( v, v ∗ ) dv ∗ dv = − Z R d φ ( v )[ ˜ ∇ · A ]( v ) dv. In this way, the F Landau equation (7) can be concisely written as ∂ t f = 12 ˜ ∇ · (cid:18) f f ∗ ˜ ∇ δ F δf (cid:19) . Note, by formally testing (7) with φ = δ F δf , one obtains an analogy of Boltzmann’s H-theoremwith the functional F ; ddt F [ f t ] = − D F ( f t ) := − Z Z R d f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ F δf (cid:12)(cid:12)(cid:12)(cid:12) dvdv ∗ ≤ . We will refer to D F as the F dissipation. These notations induce our notion of weak solutionsto the F Landau equation (7).
Definition 1 (Weak F solutions) . For
T >
0, we say that a curve f ∈ C ([0 , T ]; L ( R d )) isa weak solution to the F Landau equation (7) if the following hold.1. f L is a probability measure with uniformly bounded second moment so that f t ≥ , Z R d f t ( v ) dv = 1 , ∀ t ∈ [0 , T ] , sup t ∈ [0 ,T ] Z R d h v i f t ( v ) dv < ∞ .
2. The F dissipation is time integrable Z T D F ( f t ) dt = 12 Z T Z Z R d f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ F δf (cid:12)(cid:12)(cid:12)(cid:12) dvdv ∗ dt < ∞ .
6. For every test function φ ∈ C ∞ c ((0 , T ) × R d ), equation (7) is satisfied in weak form Z T Z R d ∂ t φf t ( v ) dvdt = 12 Z T Z Z R d f f ∗ ˜ ∇ φ · ˜ ∇ δ F δf dvdv ∗ dt. For ǫ >
0, we will refer to the weak H ǫ solutions as ǫ -solutions and, recalling H is theBoltzmann entropy, we will refer to weak H solutions as just weak solutions or H-solutions . We recall the basic definitions of gradient flow theory that can be found in more generalityin [3, Chapter 1]. Throughout, (
X, d ) denotes a complete (pseudo)-metric space X with(pseudo)-metric d . Points a < b ∈ R will refer to endpoints of some interval. F : X → ( −∞ , ∞ ] will denote a proper function. Definition 2 (Absolutely continuous curve) . A function µ : t ∈ ( a, b ) µ t ∈ X is said to bean absolutely continuous curve if there exists m ∈ L ( a, b ) such that for every s ≤ t ∈ ( a, b ) d ( µ t , µ s ) ≤ Z ts m ( r ) dr. Among all possible functions m in Definition 2, one can make the following minimalselection. Definition 3 (Metric derivative) . For an absolutely continuous curve µ : ( a, b ) → X , wedefine its metric derivative at every t ∈ ( a, b ) by | ˙ µ | ( t ) := lim h → d ( µ t + h , µ t ) | h | . Further properties of the metric derivative can be found in [3, Theorem 1.1.2].
Definition 4 (Strong upper gradient) . The function g : X → [0 , ∞ ] is a strong uppergradient with respect to F if for every absolutely continuous curve µ : t ∈ ( a, b ) µ t ∈ X we have that g ◦ µ : ( a, b ) → [0 , ∞ ] is Borel and the following inequality holds | F [ µ t ] − F [ µ s ] | ≤ Z ts g ( µ r ) | ˙ µ | ( r ) dr, ∀ a < s ≤ t < b. Using Young’s inequality and moving everything to one side, the inequality in Definition 4implies F [ µ t ] − F [ µ s ] + 12 Z ts g ( µ r ) dr + 12 Z ts | ˙ µ | ( r ) dr ≥ , ∀ a < s ≤ t < b. If the reverse inequality also holds, one obtains the stronger Energy Dissipation
Equality .This leads to our notion of gradient flows. 7 efinition 5 (Curve of maximal slope) . An absolutely continuous curve µ : ( a, b ) → X is said to be a curve of maximal slope for F with respect to its strong upper gradient g : X → [0 , ∞ ] if F ◦ µ : ( a, b ) → [0 , ∞ ] is non-increasing and the following inequality holds F [ µ t ] − F [ µ s ] + 12 Z ts g ( µ r ) dr + 12 Z ts | ˙ µ | ( r ) dr ≤ , ∀ a < s ≤ t < b.F has the following natural candidates for upper gradient. Definition 6 (Slopes) . We define the local slope of F by | ∂F | ( µ ) := lim sup ν → µ ( F ( ν ) − F ( µ )) + d ( ν, µ ) . The superscript ‘+’ refers to the positive part. The relaxed slope of F is given by | ∂ − F | ( µ ) := inf { lim inf n →∞ | ∂F | ( µ n ) : µ n → µ, sup n ∈ N ( d ( µ n , µ ) , F ( µ n )) < + ∞} . In order to understand the Landau equation as a gradient flow, we need to clarify what typeof object the corresponding metric is.
Theorem 7 (Distance on P ,E ( R d )) . The (pseudo)-metric d L on P ,E ( R d ) , satisfies: • d L -convergent sequences are weakly convergent. • d L -bounded sets are weakly compact. • The map ( µ , µ ) d L ( µ , µ ) is weakly lower semicontinuous. • For any τ ∈ P ( R d ) the subset P τ ( R d ) := (cid:8) µ ∈ P ,m ( τ ) ( R d ) | d L ( µ, τ ) < ∞ (cid:9) is acomplete geodesic space. The content of this theorem is essentially that our new proposed distance actually pro-vides a meaningful topological structure on P ,E ( R d ). Furthermore, the connection to ǫ -solutions of Landau is established when considering the previous notions of slope and uppergradient with respect to d L . Theorem 8 (Epsilon equivalence) . Fix any ǫ, E > , γ ∈ [ − , . Assume that a curve µ : [0 , T ] → P ,E ( R d ) has a density µ t = f t L . Then µ is a curve of maximal slope for H ǫ with respect to its upper gradient p D H ǫ if and only if its density f is an ǫ -solution to theLandau equation. From the numerical perspective, we can also construct ǫ -solutions using the JKO scheme(see Section 5) which is the following Theorem 9 (Existence of curves of maximal slope) . For any ǫ, E > , γ ∈ [ − , , andinitial data µ ∈ P ,E ( R d ) , there exists a curve of maximal slope in P ,E ( R d ) for H ǫ withrespect to its upper gradient p D H ǫ . emark 10. The choice of an exponential convolution kernel G ǫ is perhaps unnatural com-pared to the Maxwellian regularization G ,ǫ for the regularized entropy H ǫ . We discuss inmore detail the estimates that fail using G ,ǫ in Remark 32 as it pertains to Theorem 8.With respect to Theorem 9, the general construction of some curve can be done even withthe Maxwellian regularization. However, due to the same lack of estimates, this curve mightnot be a curve of maximal slope with respect to p D H ǫ . This is discussed in Remark 36.Motivated by recent numerical experiments [15], Theorems 8 and 9 provide the theoreticalbasis to this ǫ approximated Landau equation. In the limit ǫ →
0, more assumptions arerequired.
Theorem 11 (Full equivalence) . We fix d = 3 and γ ∈ ( − , . Suppose that for some T > , a curve µ : [0 , T ] → P ( R ) has a density µ t = f t L that satisfies the following set ofassumptions( A1 ) (Moments and L p ) Assume that for some < η ≤ γ + 3 , we have h v i − γ f t ( v ) ∈ L ∞ t (0 , T ; L v ∩ L − η γ − η v ( R )) . ( A2 ) (Finite entropy) We assume that the the entropy is bounded in time H ( f t ) = Z R f t log f t ∈ L ∞ t (0 , T ) . ( A3 ) (Finite entropy-dissipation) We assume that the entropy-dissipation of f is integrablein time D ( f t ) = D H ( f t ) = 12 Z Z R f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H δf (cid:12)(cid:12)(cid:12)(cid:12) dvdv ∗ =12 Z Z R f f ∗ | v − v ∗ | γ +2 | Π[ v − v ∗ ]( ∇ log f − ∇ ∗ log f ∗ ) | dvdv ∗ ∈ L t (0 , T ) . Then µ is a curve of maximal slope for H with respect to its upper gradient √ D if and onlyif its density f is a weak solution of the Landau equation. Remark 12.
When γ ∈ [ − , L p spaces for p large enough and for a sufficient power-like weight), weak solutions of Landauequation satisfying ( A1 )–( A3 ) are known to exist (and to be strong and unique under extraconditions). We refer to [47], and Appendix B of [18] when γ > −
2, for details.When γ ∈ ( − , − A1 ) is not known to hold for global weak solutionswith large initial data. Solutions satisfying ( A1 )–( A3 ) are nevertheless known to exist forinitial data close to equilibrium (cf. [35], in a much larger spatially inhomogeneous context),or in the Coulomb case γ = − − η γ − η being replaced by ∞ ) for large initialdata, but on specific intervals of times only ([20, 4]).9t is an open problem to find the range of values γ under which we can show the existenceof curves of maximal slope for the original Landau equation (1), or equivalently, contructingsolutions of the original Landau equation passing ǫ → ǫ and the compactness of sequences with bounded in ǫ regularized entropydissipation D H ǫ . The rest of this work is devoted to show the main four theorems in thenext four sections. d L Our approach to defining the distance d L mentioned in Theorem 7 closely follows the dynamicformulation of transport distances originally due to Benamou and Brenier [6] and furtherextended by Dolbeault, Nazaret, and Savar´e [23]. We also refer the reader to Erbar [25] fora similar approach. We consider for γ ∈ [ − ,
0] the grazing continuity equation : ∂ t µ t + 12 ˜ ∇ · M t = 0 , in (0 , T ) × R d , (8)which is interpreted in the sense of distributions. For every φ ∈ C ∞ c ((0 , T ) × R d ), we have Z T Z R d ∂ t φ ( t, v ) dµ t ( v ) dt + 12 Z T Z Z R d [ ˜ ∇ φ ]( t, v, v ∗ ) dM t ( v, v ∗ ) dt = 0 . Equivalently, for ζ ∈ C ∞ c ( R d ), ddt Z R d ζ ( v ) dµ t ( v ) = 12 Z Z R d ˜ ∇ ζ ( v, v ∗ ) dM t ( v, v ∗ ) . (9)The curves ( µ t ) t ∈ [0 ,T ] , ( M t ) t ∈ [0 ,T ] are Borel families of measures belonging to M + and M d respectively. We will refer to µ from the pair as a curve and M as a grazing rate . For someregularity properties, we will also need to assume the following moment condition Z T Z Z R d (1 + | v | + | v ∗ | ) d | M t | ( v, v ∗ ) dt < ∞ . (10)We first establish some a-priori properties of solutions to the grazing continuity equation. Lemma 13 (Continuous representative) . For families ( µ t ) , ( M t ) satisfying the grazing con-tinuity equation and the finite momement condition (10) , there exists a unique weakly* con-tinuous representative curve (˜ µ t ) t ∈ [0 ,T ] such that ˜ µ t = µ t a.e. t ∈ [0 , T ] . Furthermore, forany φ ∈ C ∞ c ((0 , T ) × R d ) and any t , t ∈ [0 , T ] , we have the following formula Z R d φ t d ˜ µ t − Z R d φ t d ˜ µ t = Z t t Z R d ∂ t φdµ t dt + 12 Z t t Z Z R d ˜ ∇ φdM t dt. roof. This proof is nearly identical to [3, Lemma 8.1.2]. There, it was crucial to estimatethe distributional time derivative of t µ t . We perform the analogous estimate here tohighlight the difference in our context. Fix ζ ∈ C ∞ c ( R d ) and consider the map t ∈ (0 , T ) µ t ( ζ ) = Z R d ζ ( v ) dµ t ( v ) ∈ R . According to (9), the distributional time derivative is˙ µ t ( ζ ) = 12 Z Z R d ˜ ∇ ζ dM t ( v, v ∗ ) = 12 Z Z R d | v − v ∗ | γ Π[ v − v ∗ ]( ∇ ζ − ∇ ∗ ζ ∗ ) dM t ( v, v ∗ ) . Using the moment condition (10) and a mean-value estimate for γ ∈ [ − , − γ ∈ [ − , | ˙ µ t ( ζ ) | . (cid:26) sup w ∈ R d |∇ ζ ( w ) | RR R d (1 + | v | + | v ∗ | ) d | M t | ( v, v ∗ ) γ ∈ [ − , sup w ∈ R d | D ζ ( w ) | RR R d (1 + | v | + | v ∗ | ) d | M t | ( v, v ∗ ) γ ∈ [ − , − . The rest of the proof proceeds as in [3, Lemma 8.1.2] using the C -norm of ζ for the softpotentials γ ∈ [ − , −
2) as opposed to their C control of ζ .Define m ( µ t ) := Z R d vdµ t ( v ) , E ( µ t ) := Z R d | v | dµ t ( v ) . Lemma 14 (Conservation lemma) . Fix γ ∈ [ − , and let ( µ t ) t ∈ [0 ,T ] , ( M t ) t ∈ [0 ,T ] be Borelfamilies of measures in M + , M d respectively satisfying (8) and the moment condition (10) .Assume further that ( µ t ) t ∈ [0 ,T ] is weakly* continuous with respect to t . We have that massand momentum are conserved; µ t ( R d ) = µ ( R d ) , m ( µ t ) = m ( µ ) , ∀ t ∈ [0 , T ] . In the case γ ∈ [ − , − we have that the energy is conserved; E ( µ t ) = E ( µ ) , ∀ t ∈ [0 , T ] . Proof.
We show the proof of the conservation of energy for γ ∈ [ − , − ϕ ∈ C ∞ c ( B ) which satisfies0 ≤ ϕ ≤ ϕ ( v ) = 1 in B . We denote ϕ R ( v ) = ϕ ( v/R ) . Using the grazing continuity equation, we have, recalling w ( | v − v ∗ | ) = | v − v ∗ | γ , that Z R d | v | ϕ R ( v ) dµ t ( v ) − Z R d | v | ϕ R ( v ) dµ ( v )= Z t Z Z R d w Π (cid:18) vϕ R ( v ) + | v | ∇ ϕ ( v/R ) R − v ∗ ϕ R ( v ∗ ) − | v ∗ | ∇ ϕ ( v ∗ /R ) R (cid:19) dM s ( v, v ∗ ) ds (11)11stimating, using that Φ R ( v ) we use the cancelation from the projection Π to obtain (cid:12)(cid:12)(cid:12)(cid:12)Z t Z Z R d w Π ( vϕ R ( v ) − v ∗ ϕ R ( v ∗ )) dM s (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t Z Z ( B R × B R ) c w | vϕ R ( v ) − v ∗ ϕ R ( v ∗ ) | d | M s | . Z t Z Z ( B R × B R ) c | v | + | v ∗ | d | M s | , where we have used γ ∈ [ − , −
2] to bound w | vϕ R ( v ) − v ∗ ϕ R ( v ∗ ) | . ( | v − v ∗ | ≤ | v | + | v ∗ | | v − v ∗ | ≥ . Similarly, using that ∇ φ R is supported in B R \ B R and that (cid:12)(cid:12)(cid:12) D n | v | ∇ ϕ ( v/R ) R o(cid:12)(cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12)Z t Z Z R d w Π (cid:18) | v | ∇ ϕ ( v/R ) R − | v ∗ | ∇ ϕ ( v ∗ /R ) R (cid:19) dM s (cid:12)(cid:12)(cid:12)(cid:12) . Z Z ( B R × B R ) c | v | + | v ∗ | d | M s | , where we have controlled the difference with a mean-value type estimate. From the previousbounds, we can use hypothesis (10) to take R → ∞ in (11) and obtain the conservation ofenergy Z R d | v | ϕ R ( v ) dµ t ( v ) = Z R d | v | ϕ R ( v ) dµ ( v ) . The proofs for conservation of mass and momentum involve testing the grazing continuityequation against φ R and v i φ R respectively where v i is the i -th component of v . For thesestatements, the case γ ∈ [ − , −
2] follows the same as just presented. For γ ∈ [ − , Remark 15.
Note that as γ increases into the range ( − , w startsadding growth so the mean-value type argument in Lemma 14 no longer helps unless moremoments of M are assumed than (10). Due to the conservation of mass, the unique weakly*continuous representative (˜ µ t ) of Lemma 13 has the additional property of being weaklycontinuous in the context of P ( R d ).Based on the previous results, we propose the following definition. Definition 16 (Grazing continuity equation) . For some terminal time
T >
0, we define
GCE T to be the set of pairs of measures ( µ t , M t ) t ∈ [0 ,T ] satisfying the following:1. µ t ∈ P ( R d ) is weakly continuous with respect to t ∈ [0 , T ]. ( M t ) t ∈ [0 ,T ] is a family ofBorel measures belonging to M d .2. We have the moment bound Z T Z Z R d (1 + | v | + | v ∗ | ) d | M t | ( v, v ∗ ) dt < ∞ .
12. The grazing continuity equation (8) is satisfied in the distributional sense. That is, forevery φ ∈ C ∞ c ((0 , T ) × R d ), Z T Z R d ∂ t φdµ t dt + 12 Z T Z Z R d ˜ ∇ φdM t dt = 0 , or equivalently for every ζ ∈ C ∞ c ( R d ), ddt Z R d ζ ( v ) dµ t ( v ) = 12 Z Z R d ˜ ∇ ζ ( v, v ∗ ) dM t ( v, v ∗ ) . For fixed probability measures λ, ν , we may also specify the subset
GCE ( λ, ν ) as those pairs( µ, M ) ∈ GCE T such that µ = λ, µ T = ν . For E >
0, we will speak of curves ( µ, M ) ∈GCE ,ET such that m ( µ t ) = Z R d h v i dµ t ( v ) ≤ E, ∀ t ∈ [0 , T ] . In this section, we construct the action of a curve under the grazing continuity equation. Weintroduce the following function α : R d × R ≥ → [0 , ∞ ] by α ( u, s ) := | u | s , s = 00 , s = 0 , u = 0 ∞ , s = 0 , u = 0 . Lemma 17. α is lower semi-continuous (lsc), convex, and positively 1-homogeneous. For fixed µ ∈ P ( R d ) , M ∈ M d , we define µ ∈ P ( R d × R d ) by µ ( dv, dv ∗ ) := µ ( dv ) µ ( dv ∗ ) . Consider τ ∈ M given by τ = µ + | M | and the decompositions µ = f τ and M = N τ . Wedefine the action functional as A ( µ, M ) := Z Z R d α ( N, f ) dτ. (12)This is well-defined by the 1-homogeneity of α . The following lemma establishes a moreconcrete expression for the action functional. Lemma 18.
Let µ ∈ P ( R d ) be absolutely continuous with respect to L and µ = f L . Let M ∈ M d be given such that A ( µ, M ) < ∞ . Then, M is absolutely continuous with respectto f f ∗ dvdv ∗ given by density U : R d × R d → R d such that M = f f ∗ U dvdv ∗ = mdvdv ∗ and A ( µ, M ) = 12 Z Z R d f f ∗ | U | dvdv ∗ = 12 Z Z R d | m | f f ∗ dvdv ∗ . Proof.
The proof is identical to [25, Lemma 3.6] up to appropriate modifications.13 emma 19 (Lower semi-continuity of action functional) . The action functional A as definedin (12) is lower semi-continuous in both arguments. Specifically, if µ n ⇀ µ weakly in P ( R d ) and M n ∗ ⇀ M weakly* in M d , we have A ( µ, M ) ≤ lim inf n →∞ A ( µ n , M n ) . Proof.
This result is an application of the general lsc result in [8, Theorem 3.4.3] since α satisfies the required convexity, lsc, and homogeneity assumptions by Lemma 17.Another useful property of the action functional is the compactness provided by boundedaction. We first state Lemma 20.
Let F : R d → [0 , ∞ ] be measurable and fix any µ ∈ P ( R d ) , M ∈ M d . Wehave the following bound: Z Z R d F ( v, v ∗ ) d | M | ( v, v ∗ ) ≤ √ A ( µ, M ) (cid:18)Z Z R d F ( v, v ∗ ) dµ ( v ) dµ ( v ∗ ) (cid:19) (13) Proof.
This proof follows [25, Lemma 3.8]. We provide the simple argument by Cauchy-Schwarz for completeness. By considering τ = µ ⊗ µ + | M | , we estimate Z Z R d F d | M | ( v, v ∗ ) ≤ Z Z R d F (cid:12)(cid:12)(cid:12)(cid:12) dMdτ (cid:12)(cid:12)(cid:12)(cid:12) dτ ( v, v ∗ ) = Z Z R d F (cid:12)(cid:12)(cid:12)(cid:12) dMdτ (cid:12)(cid:12)(cid:12)(cid:12),r dµ ⊗ µdτ ! r dµ ⊗ µdτ dτ ≤ (cid:18)Z Z R d α (cid:18) dMdτ , dµ ⊗ µdτ (cid:19) dτ (cid:19) (cid:18)Z Z R d F dµ ⊗ µ (cid:19) = √ A ( µ, M ) (cid:18)Z Z R d F ( v, v ∗ ) dµ ( v ) dµ ( v ∗ ) (cid:19) . Remark 21.
Suppose we have µ t ∈ P ( R d ) such that m ( µ t ) = Z T Z R d | v | dµ t ( v ) dt < ∞ , then for M ∈ M dT the previous estimate (13) yields Z T Z Z R d | v | + | v ∗ | d | M t | ( v, v ∗ ) dt . Z T A ( µ t , M t ) (cid:18) Z R d | v | dµ t (cid:19) dt. (14)Therefore, if the integral in time of the second moment of µ is bounded, then M satisfiesthe moments conditions (10) and the energy is conserved (14). In the sequel, we will beconsidering curves that have bounded second moment which guarantee (14).14 roposition 22. Let ( µ nt , M nt ) n be a sequence in GCE T such that ( µ n ) n is tight and we havethe following uniform bounds sup n ∈ N Z T Z R d | v | dµ nt dt < ∞ and sup n ∈ N Z T A ( µ nt , M nt ) dt < ∞ . (15) Then, there exists ( µ t , M t ) ∈ GCE T such that, possibly after extracting a subsequence, wehave the following convergences µ nt ⇀ µ t weakly in P ( R d ) , ∀ t ∈ [0 , T ] M nt dt ∗ ⇀ M t dt weakly* in M dT . Furthermore, along this subsequence we have the following lower semi-continuity Z T A ( µ t , M t ) dt ≤ lim inf n →∞ Z T A ( µ nt , M nt ) dt. Sketch proof.
This result follows from a similar proof to [23, Lemma 4.5] and [25, Proposition3.11] which we sketch. The second moment bound for µ n in (15) produces a limit µ . Thebounded action in (15) and the estimate (14) produce a limit M t dt for a subsequence of M nt dt . The lower semi-continuity follows from Fatou’s lemma and Lemma 19. We define the distance, d L induced by the action functional on P ,E ( R d ). Throughout, wewill be working in the grazing continuity equation space defined earlier by GCE ,ET for T >
E >
Definition 23.
For λ, ν ∈ P ,E ( R d ) we define the (square of the) Landau distance by d L ( λ, ν ) := inf (cid:26) T Z T A ( µ t , M t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ( µ, M ) ∈ GCE ,ET ( λ, ν ) (cid:27) . (16)We have an equivalent characterization of d L which can be seen in other PDE contextssuch as [25, 23]. Lemma 24.
Given λ, ν ∈ P ,E ( R d ) , we have d L ( λ, ν ) = inf (cid:26)Z T p A ( µ t , M t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ( µ, M ) ∈ GCE ,ET ( λ, ν ) (cid:27) . (17) Proof.
This proof uses the same reparameterisation technique in [23, Theorem 5.4].
Proposition 25 (Minimizing curve) . Suppose that µ , µ ∈ P ,E ( R d ) are probability mea-sures such that d L ( µ , µ ) < ∞ . Then there exists a curve ( µ, M ) ∈ GCE ,E ( µ , µ ) attainingthe infimum of (16) (equivalently, also (17) ) and A ( µ t , M t ) = d L ( µ , µ ) for almost every t ∈ [0 , . roof. This result follows from the direct method of calculus of variations where the lowersemicontinuity comes from Proposition 22.
Proof of Theorem 7.
We prove the statements in exactly the order they are presented inthe theorem, starting with the properties of the proposed Landau distance as a metric. Thepositivity of d L follows from the positivity of α . We now check that d L satisfies the propertiesof a metric. d L distinguishes points Fix µ , µ ∈ P ,E ( R d ), we check that d L ( µ , µ ) = 0 ⇐⇒ µ = µ . Suppose that d L ( µ , µ ) = 0. By Proposition 25 we can find ( µ, M ) ∈ GCE ,E ( µ , µ ) which is a minimiz-ing curve and moreover 0 = d L ( µ , µ ) = A ( µ t , M t ) implies M = 0. The grazing continuityequation reduces to ∂ t µ t = 0 which implies µ t is constant in time.The converse statement follows similarly by pairing the constant curve µ : t µ = µ with the zero measure so that ( µ, ∈ GCE ,E ( µ , µ ). Symmetry
Symmetry follows because time can be reversed for every curve. For instance, if ( µ, M ) ∈GCE ,ET ( µ , µ ), then one can check that the pair µ r : t µ ( T − t ) , M r : t
7→ − M ( T − t )belong to GCE ,ET ( µ , µ ) with the same action. Triangle inequality
We sketch the argument using a glueing lemma as in [23, Lemma 4.4]. Let µ , µ , µ ∈ P ,E ( R d ) be such that d L ( µ , µ ) < ∞ and d L ( µ , µ ) < ∞ . If not, d L ( µ , µ ) ≤ d L ( µ , µ ) + d L ( µ , µ ) holds trivially. By Proposition 25, we can find minimizing curves connecting theseprobability measures (cid:26) ( µ → , M → ) ∈ GCE ,E ( µ , µ )( µ → , M → ) ∈ GCE ,E ( µ , µ ) (cid:27) . Their concatenation from time 0 to 1 is given by µ t := (cid:26) µ → t , ≤ t ≤ / µ → t − / , / ≤ t ≤ , M t := (cid:26) M → t , ≤ t ≤ / M → t − / , / < t ≤ . One can check that ( µ, M ) ∈ GCE ,E ( µ , µ ), so it is an admissible competitor in the com-putation of d L ( µ , µ ). By looking at the action on the different time pieces, we obtain d L ( µ , µ ) ≤ Z A ( µ t , M t ) dt = d L ( µ , µ ) + d L ( µ , µ ) .d L -convergence/boundedness implies weak convergence/compactness Fix µ n , µ ∞ ∈ P ,E for n ∈ N be such that d L ( µ ∞ , µ n ) → n → ∞ . By Proposition 25,take minimizing curves ( ν n , M n ) ∈ GCE ,E ( µ ∞ , µ n ) such that d L ( µ ∞ , µ n ) = A ( ν nt , M nt ) , a.e. t ∈ [0 , .
16y compactness in Proposition 22, there are limits ( ν, M ) ∈ GCE ,E such that ν n ⇀ ν and M n ∗ ⇀ M up to a subsequence. Moreover, the lower semicontinuity in Proposition 22 gives A ( ν t , M t ) ≤ lim inf n →∞ A ( ν nt , M nt ) = 0 , hence M = 0 so that ν is a constant in time. Since ν (0) = µ ∞ , this implies µ ∞ = ν (1) =lim n →∞ µ n which establishes the weak convergence.( P τ , d L ) is a complete geodesic space We start with the geodesic property from completely analogous arguments to Erbar [25],the remaining statement that P τ equipped with d L is a complete geodesic space follows.Fix τ ∈ P ,E ( R d ) with µ , µ ∈ P τ , the triangle inequality ensures d L ( µ , µ ) < ∞ soProposition 25 guarantees the existence of a minimizing curve ( µ, M ) ∈ GCE ,E ( µ , µ ). Oneeasily sees that this also induces a minimizing curve for intermediate times. More precisely,for every 0 ≤ r ≤ s ≤
1, we have that ( t µ t + r , t M t + r ) ∈ GCE ,Es − r ( µ r , µ s ) also minimizes d L ( µ r , µ s ).To show completeness, let ( µ n ) n ∈ N be a Cauchy sequence in P τ . The sequence is certainly d L -bounded so by Proposition 22, we can find, up to extraction of a weakly convergentsubsequence, µ ∞ ∈ P ,E ( R d ) such that µ n ⇀ µ ∞ in P ,E ( R d ). Lower semi-continuity of d L and the Cauchy property of the subsequence gives d L ( µ n , µ ∞ ) ≤ lim inf m →∞ d L ( µ n , µ m ) → , as n → ∞ . For any n ∈ N the triangle inequality gives d L ( µ ∞ , τ ) ≤ d L ( µ ∞ , µ n ) + d L ( µ n , τ ) < ∞ , So µ ∞ ∈ P τ . Proposition 26 (Metric derivative) . A curve ( µ t ) t ∈ [0 ,T ] ⊂ P ,E ( R d ) is absolutely continuouswith respect to d L if and only if there exists a Borel family ( M t ) t ∈ [0 .T ] belonging to M dT suchthat ( µ, M ) ∈ GCE ,ET with the property that Z T p A ( µ t , M t ) dt < ∞ . In this equivalence, we have a bound on the metric derivative lim h ↓ d L ( µ t + h , µ t ) h =: | ˙ µ | ( t ) ≤ A ( µ t , M t ) , a.e. t ∈ (0 , T ) . Furthermore, there exists a unique Borel family ( ˜ M t ) t ∈ [0 ,T ] belonging to M d which is charac-terized by M t = U µ t ⊗ µ t and U ∈ T µ := { ˜ ∇ φ | φ ∈ C ∞ c ( R d ) } L ( µ t ⊗ µ t ) such that ( µ, ˜ M ) ∈ GCE ET ( µ , µ T ) where we have equality: | ˙ µ | ( t ) = A ( µ t , ˜ M t ) , a.e. t ∈ (0 , T ) . Proof.
The argument follows exactly as in [23, Theorem 5.17].17
Energy dissipation equality
The goal in this section is to prove Theorem 8 which states that the notions of gradientflow solutions coincide with ǫ -solutions to the Landau equation. To fix ideas, we recall theregularized entropy functionals acting on probability measures H ǫ [ µ ] = Z R d ( µ ∗ G ǫ )( v ) log( µ ∗ G ǫ )( v ) dv, with G ǫ ( v ) given by G ǫ ( v ) = ǫ − d C d exp n − D vǫ Eo . The crucial ingredient to prove Theorem 8 is the following
Proposition 27 (Chain Rule ǫ ) . Suppose ( µ, M ) ∈ GCE ,ET and Z T A ( µ t , M t ) dt < ∞ . Then, sup t ∈ [0 ,T ] H ǫ [ µ t ] < ∞ and the ‘chain rule’ holds H ǫ [ µ r ] − H ǫ [ µ s ] = 12 Z rs Z Z R d ˜ ∇ (cid:20) δ H ǫ δµ (cid:21) · dM t dt, ∀ ≤ s ≤ r ≤ T. (18) Remark 28.
Recall the expression for the dissipation D ǫ [ µ ] = 12 Z Z R d (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H ǫ δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( v ) dµ ( v ∗ ) . Using a time integrated version of Lemma 20, we have the estimate12 Z rs Z Z R d (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H ǫ δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) · d | M t | ( v, v ∗ ) dt ≤ Z rs A ( µ t , M t ) D ǫ [ µ t ] dt. Therefore, under the hypothesis of Proposition 27, we have that |H ǫ ( µ r ) − H ǫ ( µ r ) | ≤ Z rs | ˙ µ | ( t ) D ǫ [ µ t ] dt, which implies that D ǫ [ µ t ] is a strong upper gradient of H ǫ , see Definition 4.Taking Proposition 27 for granted, we can prove Theorem 8. Proof of Theorem 8.
Throughout, µ = f L is a curve of probability measures with uniformlybounded second moment. Weak ǫ -solution = ⇒ Curve of maximal slope
Consider f an ǫ -solution to the Landau equation. Define m = − f f ∗ ˜ ∇ δ H ǫ δf so that the pairof measures ( µ = f L , M = m L ⊗ L ) therefore belong to
GCE ET . Indeed, the distributional18razing continuity equation from Definition 16 is precisely the weak ǫ Landau equation.Based on the definition of M and the finite H ǫ dissipation, we have the bound Z T A ( µ t , M t ) dt = Z T D ǫ ( f t ) dt < ∞ , which implies the weak continuity of µ . By Proposition 26, we have | ˙ µ | ( t ) = A ( µ t , M t ) = D ǫ ( f t ) < ∞ , a.e. t ∈ [0 , T ] . Using Proposition 27, we have for any 0 ≤ s ≤ r ≤ T H ǫ [ µ r ] − H ǫ [ µ s ] + 12 Z rs D ǫ ( µ t ) dt + 12 Z rs | ˙ µ | ( t ) dt ≤ . According to Definition 5, this is the curve of maximal slope property.
Curve of maximal slope = ⇒ weak ǫ -solution Assume that µ = f L is a curve of maximal slope for H ǫ with respect to the upper gradient √ D ǫ . Since µ is absolutely continuous with respect to d L , Proposition 26 guarantees existenceof a unique curve M : t ∈ [0 , T ] M t ∈ M d such that R T p A ( µ t , M t ) dt < ∞ and | ˙ µ | ( t ) = A ( µ t , M t ) a.e. t ∈ [0 , T ]. Furthermore, the pair ( µ, M ) ∈ GCE ET . According toLemma 18, let M = m L ⊗ L for some measurable function m . We apply the chain rule (18)with Cauchy-Schwarz and Young’s inequalities with minus signs in the follow computations. H ǫ [ f T ] − H ǫ [ f ] = 12 Z T Z Z R d ˜ ∇ δ H ǫ δf · mdvdv ∗ dt ≥ − Z T Z Z R d f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δf (cid:12)(cid:12)(cid:12)(cid:12) dvdv ∗ ! (cid:18)Z Z R d | m | f f ∗ dvdv ∗ (cid:19) dt ≥ − Z T Z Z R d f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δf (cid:12)(cid:12)(cid:12)(cid:12) dvdv ∗ ! dt − Z T (cid:18) Z Z R d | m | f f ∗ dvdv ∗ (cid:19) dt = − Z T D ǫ ( f t ) dt − Z T | ˙ f | ( t ) dt. All the inequalities in the calculations above are actually equalities owing to the fact that µ isa curve of maximal slope. In particular, since we have the equality in the Young’s inequality,this implies that m √ ff ∗ = −√ f f ∗ ˜ ∇ δ H ǫ δf . As in the previous direction, the weak ǫ Landauequation coincides with the grazing continuity equation when m is equal to − f f ∗ ˜ ∇ δ H ǫ δf .The rest of this section is devoted to proving Proposition 27. We need some lemmata toestablish crucial estimates. The following result is a variation of [10, Lemma 2.6]. Lemma 29 (Carlen-Carvalho [10]) . Let µ be a probability measure on R d with finite secondmoment/energy, m ( µ ) ≤ E for E > . Then, for every ǫ > , there exists a constant C = C ( ǫ, E ) > such that | log( µ ∗ G ǫ )( v ) | ≤ C D vǫ E . roof. Starting with an upper bound, we easily see µ ∗ G ǫ ( v ) = Z R d G ǫ ( v − v ′ ) dµ ( v ′ ) . ǫ . Turning to the lower bound, we cut off the integration domain to | v ′ | ≤ R , for some R > ǫ > (cid:28) v − v ′ ǫ (cid:29) = s (cid:12)(cid:12)(cid:12)(cid:12) v − v ′ ǫ (cid:12)(cid:12)(cid:12)(cid:12) ≤ s (cid:12)(cid:12)(cid:12) vǫ (cid:12)(cid:12)(cid:12) + 2 (cid:18) Rǫ (cid:19) ≤ √ (cid:18)D vǫ E + (cid:28) Rǫ (cid:29)(cid:19) . This is substituted into G ǫ ( v − v ′ ) to obtain µ ∗ G ǫ ( v ) ≥ Z | v ′ |≤ R G ǫ ( v − v ′ ) dµ ( v ′ ) & ǫ exp (cid:26) −√ (cid:18)D vǫ E + (cid:28) Rǫ (cid:29)(cid:19)(cid:27) Z | v ′ |≤ R dµ ( v ′ ) . At this point, we appeal to Chebyshev’s inequality to see Z | v ′ |≤ R dµ ( v ′ ) = 1 − Z | v ′ |≥ R dµ ( v ′ ) ≥ − R Z | v ′ |≥ R | v ′ | dµ ( v ′ ) . We can now choose, for example, large R such that 1 − ER ≥ to uniformly lower bound theintegral R | v ′ |≤ R dµ ( v ′ ) away from 0 and then conclude the result after applying logarithms. Lemma 30 (log-derivative estimates) . For fixed ǫ > we have the formula ∇ G ǫ ( v ) = − ǫ D vǫ E − G ǫ ( v ) vǫ . (19) For µ ∈ P ( R d ) , denoting ∂ i = ∂∂v i and ∂ ij = ∂ ∂v i ∂v j , we obtain |∇ log( µ ∗ G ǫ )( v ) | ≤ ǫ , (cid:12)(cid:12) ∂ ij log( µ ∗ G ǫ )( v ) (cid:12)(cid:12) ≤ ǫ . (20) Proof.
Equation (19) is a direct computation after noticing ∇ G ǫ G ǫ = ∇ log G ǫ = ∇ (cid:16) − D vǫ E + const. (cid:17) = − ǫ D vǫ E − vǫ . The first order log-derivative estimate of (20) is calculated using formula (19) to obtain |∇ ( µ ∗ G ǫ )( v ) | = | µ ∗ ∇ G ǫ ( v ) | ≤ ǫ Z R d (cid:28) v − v ′ ǫ (cid:29) − (cid:12)(cid:12)(cid:12)(cid:12) v − v ′ ǫ (cid:12)(cid:12)(cid:12)(cid:12) G ǫ ( v − v ′ ) dµ ( v ′ ) ≤ ǫ Z R d G ǫ ( v − v ′ ) dµ ( v ′ ) = 1 ǫ ( µ ∗ G ǫ )( v ) . ∂ ij µ ∗ G ǫ which can be computed with the help of (19) | ∂ ij µ ∗ G ǫ ( v ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ i − ǫ Z R d (cid:28) v − v ′ ǫ (cid:29) − v j − v ′ j ǫ G ǫ ( v − v ′ ) dµ ( v ′ ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ǫ Z R d (cid:28) v − v ′ ǫ (cid:29) − v i − v ′ i ǫ v j − v ′ j ǫ + δ ij (cid:28) v − v ′ ǫ (cid:29) − − (cid:28) v − v ′ ǫ (cid:29) − v i − v ′ i ǫ v j − v ′ j ǫ ! G ǫ ( v − v ′ ) dµ ( v ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ µ ∗ G ǫ ( v ) . Combining this estimate with the previous first order one, we have (cid:12)(cid:12) ∂ ij log( µ ∗ G ǫ )( v ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∂ ij µ ∗ G ǫ µ ∗ G ǫ − ( ∂ i µ ∗ G ǫ )( ∂ j µ ∗ G ǫ )( µ ∗ G ǫ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ . Lemma 31.
Fix ǫ > and γ ∈ [ − , with µ ∈ P ,E ( R d ) for some E > . We have1. Moderately soft case γ ∈ [ − , : (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˜ ∇ [ G ǫ ∗ log( µ ∗ G ǫ )]( v, v ∗ ) (cid:12)(cid:12)(cid:12) . ǫ | v | γ + | v ∗ | γ .
2. Very soft case γ ∈ [ − , − : (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ (cid:12)(cid:12)(cid:12)(cid:12) . ǫ . In particular, it holds
Z Z R d (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ (cid:12)(cid:12)(cid:12)(cid:12) dµ ( v ) dµ ( v ∗ ) ≤ E. Proof.
We develop the expression for ˜ ∇ δ H ǫ δµ in integral form to be used throughout this proof.˜ ∇ δ H ǫ δµ = ˜ ∇ G ǫ ∗ log( µ ∗ G ǫ )( v, v ∗ )= | v − v ∗ | γ Π[ v − v ∗ ]( ∇ v G ǫ ∗ log( µ ∗ G ǫ )( v ) − ∇ v ∗ G ǫ ∗ log( µ ∗ G ǫ )( v ∗ ))= | v − v ∗ | γ Π[ v − v ∗ ] Z R d G ǫ ( v ′ ) (cid:18) ∇ µ ∗ G ǫ µ ∗ G ǫ ( v − v ′ ) − ∇ µ ∗ G ǫ µ ∗ G ǫ ( v ∗ − v ′ ) (cid:19) dv ′ . (21)1. Moderately soft case γ ∈ [ − , : We use (a concave version of) the triangle inequality(valid since 1 + γ ≥
0) and the first estimate of (20) to bound the last line of (21) (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ γ ( | v | γ + | v ∗ | γ ) 2 ǫ Z R d G ǫ ( v ′ ) dv ′ . ǫ | v | γ + | v ∗ | γ . Very soft case γ ∈ [ − , − : We perform estimates in two cases, the far field | v − v ∗ | ≥ | v − v ∗ | ≤ | v − v ∗ | ≥ : In the far field, we have | v − v ∗ | γ ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ . | v − v ∗ | ≤ : We can remove the singularity from the weight with a mean-value estimate and thesecond estimate of (20) (cid:12)(cid:12)(cid:12)(cid:12) ∇ µ ∗ G ǫ µ ∗ G ǫ ( v − v ′ ) − ∇ µ ∗ G ǫ µ ∗ G ǫ ( v ∗ − v ′ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup i,j =1 ,...,d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ i (cid:18) ∂ j µ ∗ G ǫ µ ∗ G ǫ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ | v − v ∗ | ≤ ǫ | v − v ∗ | . Inserting this into (21), we have (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ | v − v ∗ | γ Z R d G ǫ ( v ′ ) dv ′ ≤ ǫ . Remark 32.
Originally, we considered the general family of convolution kernels G s,ǫ de-scribed in Section 2.1. Besides the context of the Landau equation, Lemma 30 (excludingthe second order log-derivative estimate) can be generalized to this family of s -order tailedexponential distributions with additional moment assumptions on µ . In particular, equa-tions (19) and (20) (for s ≥
1) become ∇ G s,ǫ G s,ǫ ( v ) = − sǫ D vǫ E s − vǫ , |∇ ( µ ∗ G s,ǫ ) | µ ∗ G s,ǫ ( v ) . ǫ s h v i s − . Since Maxwellians are known to be stationary solutions for the Landau equation, we wantedto perform the regularization with s = 2. However, the analogous estimates of Lemma 30 for s = 2 are not sufficient for Lemma 31 in the P framework. For example, in the moderatelysoft potential case, the estimate reads (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ,ǫ δµ (cid:12)(cid:12)(cid:12)(cid:12) . ǫ h v i γ + h v ∗ i γ / ∈ L ( µ ⊗ µ ) . However, there is one value of γ = − G ,ǫ . A restriction to P resolves the issue mentioned above for themoderately soft potential case, but then a fourth moment propagation is needed which wedid not pursue. A similar issue is present in the very soft potential case. Proof of Proposition 27.
To prove equation (18), our strategy is to regularize the pair ( µ, M )in time with parameter δ > δ needed to take the limit δ →
0. 22 inite regularized entropy
We have the following chain of inequalities H ǫ [ µ t ] = Z R d ( µ t ∗ G ǫ )( v ) log( µ t ∗ G ǫ )( v ) dv . ǫ,E Z R d ( µ t ∗ G ǫ )( v ) h v i dv . ǫ E. The first inequality comes from Lemma 29 because log( µ t ∗ G ǫ ) has linear growth (uniformin time) while in the second inequality, one realises that µ t ∗ G ǫ has as many moments as µ t with computable constants. Time regularization with δ > µ be the weakly time continuous representative (Lemma 13)and M be the optimal grazing rate (Proposition 26) achieving the finite distance d L . We firstregularize the pair ( µ, M ) in time for a fixed parameter δ > η ∈ C ∞ ( R )with the following propertiessupp η ⊂ ( − , , η ≥ , η ( t ) = η ( − t ) , Z − η ( t ) dt = 1 . We define the following measures for t ∈ [0 , T ], by taking convex combinations µ δt := Z − η ( t ′ ) µ t − δt ′ dt ′ , M δt := Z − η ( t ′ ) M t − δt ′ dt ′ . Here, we constantly extend the measures in time. That is, if t − δt ′ ∈ [ − δ, µ t − δt ′ = µ , M t − δt ′ = 0. For the other end point, if t − δt ′ ∈ [ T, T + δ ], we set µ t − δt ′ = µ T , M t − δt ′ = 0. This transformation is stable so that ( µ δ , M δ ) ∈ GCE T and in particular, thedistributional grazing continuity equation holds ∂ t µ δt + 12 ˜ ∇ · M δt = 0 . We derive equation (18) using this regularized grazing continuity equation. Consider H ǫ [ µ δt ] = Z R d ( µ δt ∗ G ǫ )( v ) log( µ δt ∗ G ǫ )( v ) dv, which we differentiate with respect to t by appealing to the Dominated Convergence Theo-rem. Firstly, due to the time regularization, we have ∂ t (cid:8) ( µ δt ∗ G ǫ ) log( µ δt ∗ G ǫ ) (cid:9) = (cid:2) ( ∂ t µ δt ) ∗ G ǫ (cid:3) (log( µ δt ∗ G ǫ ) + 1) . The L v bound is obtained on the following difference quotient for a fixed time step h > (cid:12)(cid:12)(cid:12)(cid:12) h [( µ δt + h ∗ G ǫ ) log( µ δt + h ∗ G ǫ ) − ( µ δt ∗ G ǫ ) log( µ δt ∗ G ǫ )] (cid:12)(cid:12)(cid:12)(cid:12) ≤ h (cid:12)(cid:12) ( µ δt + h ∗ G ǫ ) − ( µ δt ∗ G ǫ ) (cid:12)(cid:12) sup s ∈ [ t,t + h ] (cid:12)(cid:12) log( µ δs ∗ G ǫ ) + 1 (cid:12)(cid:12) . (cid:12)(cid:12)(cid:12)(cid:12) h [( µ δt + h ∗ G ǫ ) log( µ δt + h ∗ G ǫ ) − ( µ δt ∗ G ǫ ) log( µ δt ∗ G ǫ )] (cid:12)(cid:12)(cid:12)(cid:12) . ǫ,E h (cid:12)(cid:12) ( µ δt + h ∗ G ǫ ) − ( µ δt ∗ G ǫ ) (cid:12)(cid:12) h v i . We apply the Mean Value Theorem on the difference quotient again to get (cid:12)(cid:12)(cid:12)(cid:12) h [( µ δt + h ∗ G ǫ ) log( µ δt + h ∗ G ǫ ) − ( µ δt ∗ G ǫ ) log( µ δt ∗ G ǫ )] (cid:12)(cid:12)(cid:12)(cid:12) . δ,ǫ || η ′ || L ∞ (cid:18) µ ∗ G ǫ + Z T µ t ∗ G ǫ dt (cid:19) h v i . Since µ has finite second order moments, this last expression belongs to L v . By the Domi-nated Convergence Theorem, ddt H ǫ [ µ δt ] = Z R d (cid:2) ( ∂ t µ δt ) ∗ G ǫ (cid:3) (log( µ δt ∗ G ǫ ) + 1) dv = Z R d ( ∂ t µ δt ) · [ G ǫ ∗ log( µ δt ∗ G ǫ )] dv The last line is achieved by the self-adjointness of convolution with G ǫ and eliminating theconstant term due to the conserved mass of µ δ . Integrating in t , we obtain H ǫ [ µ δr ] − H ǫ [ µ δs ] = Z rs Z R d ( ∂ t µ δt ) · [ G ǫ ∗ log( µ δt ∗ G ǫ )] dvdt = 12 Z rs Z Z R d [ ˜ ∇ G ǫ ∗ log( µ δt ∗ G ǫ )] · dM δt dt = 12 Z rs Z Z R d ˜ ∇ δ H ǫ δµ δt · dM δt dt. (22)We now turn to establishing estimates independent of δ > Estimates on the right-hand side of (22) : According to Lemma 31, we have the estimate (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ δ (cid:12)(cid:12)(cid:12)(cid:12) . ǫ,E | v | p + | v ∗ | p , where p ≤
1. By the first moment assumption of M t , we have Z T Z Z R d (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ δt (cid:12)(cid:12)(cid:12)(cid:12) d | M t | ( v, v ∗ ) dt . ǫ,E Z T Z Z R d | v | + | v ∗ | d | M t | ( v, v ∗ ) dt < ∞ . This estimate also extends to M δt Z T Z Z R d (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ δt (cid:12)(cid:12)(cid:12)(cid:12) d | M δt | ( v, v ∗ ) dt < ∞ . Note that these estimates are independent of δ > Convergence δ → : Using the weak in time continuity of µ , we can consider | µ δt ∗ G ǫ ( v ′ ) − µ t ∗ G ǫ ( v ′ ) | ≤ Z − η ( t ′ ) |h µ t − δt ′ , G ǫ ( v ′ − · ) i − h µ t , G ǫ ( v ′ − · ) i| dt ′ . · stands for the convoluted variable. Since t belongs to a compact set, the function t
7→ h µ t , G ǫ ( v ′ − · ) i is uniformly continuous from the weak continuity of µ . In particular,using the continuity in v ′ and the lower bound from Lemma 29 we conclude that for any R > | log( µ δt ∗ G ǫ ) − log( µ t ∗ G ǫ ) | → B R . (23)Therefore by Lemma 29, (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ δt − ˜ ∇ δ H ǫ δµ t (cid:12)(cid:12)(cid:12)(cid:12) = | ˜ ∇ G ǫ ∗ log( µ δt ∗ G ǫ )( v, v ∗ ) − ˜ ∇ G ǫ ∗ log( µ t ∗ G ǫ )( v, v ∗ ) |≤ Z R d w |∇ G ǫ ( v − v ′ ) − ∇ G ǫ ( v ∗ − v ′ ) || log( µ δt ∗ G ǫ ( v ′ )) − log( µ t ∗ G ǫ ( v ′ )) | dv ′ ≤ Z B cR w |∇ G ǫ ( v − v ′ ) − ∇ G ǫ ( v ∗ − v ′ ) | C ǫ h v ′ i dv ′ + sup B R | log( µ δt ∗ G ǫ ) − log( µ t ∗ G ǫ ) | Z B R w |∇ G ǫ ( v − v ′ ) − ∇ G ǫ ( v ∗ − v ′ ) | dv ′ . For a fixed ( v, v ∗ ), we obtain the convergence to zero by taking δ → R → ∞ in theprevious estimate. Using continuity, we obtain that for any R > (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ δt ( v, v ∗ ) − ˜ ∇ δ H ǫ δµ t ( v, v ∗ ) (cid:12)(cid:12)(cid:12)(cid:12) → , T ] × B R × B R . (24)We turn to the limit estimate for the right hand side of (22). For any R >
0, we have (cid:12)(cid:12)(cid:12)(cid:12)Z rs Z Z R d ˜ ∇ δ H ǫ δµ δt · dM δt dt − Z rs Z Z R d ˜ ∇ δ H ǫ δµ t · dM t dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z rs Z Z R d (cid:18) ˜ ∇ δ H ǫ δµ δt − ˜ ∇ δ H ǫ δµ t (cid:19) · dM δt dt (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z rs Z Z R d ˜ ∇ δ H ǫ δµ t · dM δt dt − Z rs Z Z R d ˜ ∇ δ H ǫ δµ t · dM t dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z rs Z Z B R × B R (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ δt − ˜ ∇ δ H ǫ δµ t (cid:12)(cid:12)(cid:12)(cid:12) d | M δt | dt + Z rs Z Z ( B R × B R ) C (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ δ H ǫ δµ δt − ˜ ∇ δ H ǫ δµ t (cid:12)(cid:12)(cid:12)(cid:12) d | M δt | dt + o (1) . The last term is o (1) as δ → δ → R → ∞ (the second termvanishes again due to the estimate from the previous step), we obtain the convergencelim δ → Z rs Z Z R d ˜ ∇ δ H ǫ δµ δt · dM δt dt = 12 Z rs Z Z R d ˜ ∇ δ H ǫ δµ t · dM δt dt. (25) Convergence of the left-hand side of (22)By (23), Lemma 29 and the uniform bound on the second moment, we have that |H ǫ [ µ δt ] − H ǫ [ µ t ] | ≤ Z R d | ( µ δt ∗ G ǫ ) log( µ δt ∗ G ǫ )( v ) − ( µ t ∗ G ǫ ) log( µ t ∗ G ǫ )( v ) | dv → , as δ → . Therefore, by the previous equation and (25) we can take δ → H ǫ [ µ r ] − H ǫ [ µ s ] = 12 Z rs Z Z R d ˜ ∇ δ H ǫ δµ t · dM t ( v, v ∗ ) dt, which is the desired result. 25 JKO scheme for ǫ -Landau equation This section is devoted to the proof of Theorem 9 after a series of preliminary lemmata.Our construction of curves of maximal slope in Theorem 9 uses the basic minimizing move-ment/variational approximation scheme of Jordan et al. [36]. Fix a small time step τ > µ ∈ P ,E ( R d ) and consider the recursive minimization procedure for n ∈ N ν τ := µ , ν τn ∈ argmin λ ∈ P ,E (cid:20) H ǫ ( λ ) + 12 τ d L ( ν τn − , λ ) (cid:21) . (26)Then, we concatenate these minimizers into a curve by setting µ τ := µ , µ τt := ν τn , for t ∈ (( n − τ, nτ ] . (27)The scheme given by (26) and (27) satisfies the abstract formulation in [3] giving Proposition 33 (Landau JKO scheme) . For any τ > and µ ∈ P ,E ( R d ) , there exists ν τn ∈ P ,E ( R d ) for every n ∈ N as described in (26) . Furthermore, up to a subsequence of µ τt described in (27) as τ → , there exists a locally absolutely continuous curve ( µ t ) t ≥ suchthat µ τt ⇀ µ t , ∀ t ∈ [0 , ∞ ) . Proof.
Our metric setting is ( P µ , d L ) (see Theorem 7) with the weak topology σ . Thisspace is essentially P ,E ( R d ) except we need to make sure that d L is a proper metric, hencewe remove the probability measures with infinite Landau distance. We follow the proof ofErbar [25] which consists in verifying [3, Assumptions 2.1 a,b,c]. These assumptions arelisted and verified now.1. H ǫ is sequentially σ -lsc on d L -bounded sets: Suppose µ n ∈ P ,E ( R d ) ⇀ µ ∈ P ,E ( R d ), this implies µ n ∗ G ǫ ⇀ µ ∗ G ǫ in P ( R d ). It is known that H ( µ ) = (cid:26) R R d f ( v ) log f ( v ) dv, µ = f L + ∞ , elseis σ -lsc and since H ǫ ( µ ) = H ( µ ∗ G ǫ ), we achieve the first property.2. H ǫ is lower bounded: By Carlen-Carvalho Lemma 29 for fixed ǫ >
0, log( µ ∗ G ǫ )is uniformly lower bounded by a linearly growing term. For fixed µ ∈ P ,E ( R d ), wehave, with Cauchy-Schwarz H ǫ ( µ ) & ǫ − Z R d h v i µ ∗ G ǫ ( v ) dv ≥ − (cid:18)Z R d h v i µ ∗ G ǫ ( v ) dv (cid:19) ≥ − ( O ( ǫ ) + E ) > −∞ . d L -bounded sets are relatively sequentially σ -compact: This is one of the con-sequences from Theorem 7.The existence of minimizers, ν τn , to (26) and limits, µ t , to (27) is guaranteed from [3, Corollary2.2.2] and [3, Proposition 2.2.3], respectively.26t the abstract level, the limit curve constructed in Proposition 33 has no relation to √ D ǫ . The following lemmata bridge this gap. Lemma 34.
For any µ ∈ P ( R d ) , we have p D ǫ ( µ ) ≤ | ∂ − H ǫ | ( µ ) . Proof.
For fixed ǫ, R , R > γ ∈ R , take T > µ ∈ C ([0 , T ]; P ( R d )) to (cid:26) ∂ t µ = ∇ · { µφ R R R d φ R ∗ ψ R ( v − v ∗ ) | v − v ∗ | γ +2 Π[ v − v ∗ ]( J ǫ − J ǫ ∗ ) dµ ( v ∗ ) } µ (0) = µ . The functions 0 ≤ φ R , ψ R ≤ φ R ( v ) = (cid:26) , | v | ≤ R , | v | ≥ R + 1 , ψ R ( z ) = (cid:26) , | z | ≤ /R , | z | ≥ /R . For t > J ǫt to be the gradient of the first variation of H ǫ applied to µ t , meaning J ǫt = ∇ G ǫ ∗ log[ µ t ∗ G ǫ ] ∈ C ∞ ( R d ; R d ) . For this proof alone, we define the reduced ǫ -entropy-dissipation D R ,R ǫ ( µ ) := 12 Z Z R d φ R φ R ∗ ψ R ( v − v ∗ ) | v − v ∗ | γ +2 | Π[ v − v ∗ ]( J ǫ − J ǫ ∗ ) | dµ ( v ) dµ ( v ∗ ) . On the other hand, as the ǫ -entropy dissipation comes from the negative time derivative ofentropy, we have D R ,R ǫ ( µ ) = lim t ↓ H ǫ ( µ ) − H ǫ ( µ t ) t = lim t ↓ H ǫ ( µ ) − H ǫ ( µ t ) d L ( µ , µ t ) d L ( µ , µ t ) t ≤ lim t ↓ (cid:26) H ǫ ( µ ) − H ǫ ( µ t ) d L ( µ , µ t ) × t × Z t s Z Z R d φ R φ R ∗ ψ R | v − v ∗ | γ +2 ( J ǫs − J ǫs ∗ ) · Π[ v − v ∗ ]( J ǫ − J ǫ ∗ ) dµ s ( v ) dµ s ( v ∗ ) ds !) ≤ | ∂ − H ǫ | ( µ ) q D R ,R ǫ ( µ ) . In the last inequality, we have used the Lebesgue differentiation theorem with strong-weakconvergence since µ is continuous in time as well as the fact that φ R ≤ φ R and ψ R ≤ ψ R since 0 ≤ φ R , ψ R ≤
1. We are left with the inequality q D R ,R ǫ ( µ ) ≤ | ∂ − H ǫ | ( µ ) , ∀ R , R > . As functions of R , R individually, D R ,R ǫ ( µ ) is non-decreasing. Furthermore, the integrandof D R ,R ǫ ( µ ) converges to the integrand of D ǫ ( µ ) pointwise µ -almost every v, v ∗ . Thus,an application of the monotone convergence theorem in the limit R , R → ∞ on the aboveinequality completes the proof. 27 emma 35. | ∂ − H ǫ | is a strong upper gradient for H ǫ in P µ ( R d ) where µ ∈ P ,E ( R d ) .Proof. Fix λ, ν ∈ P µ ( R d ) so that by the triangle inequality of Theorem 7, we have d L ( λ, ν ) < ∞ . Now by Proposition 25, there exists a pair of curves ( µ, M ) ∈ GCE E connecting λ, ν and A ( µ t , M t ) = d L ( λ, ν ) for almost every t ∈ [0 , |H ǫ ( λ ) − H ǫ ( ν ) | ≤ Z p D ǫ ( µ t ) | ˙ µ | ( t ) dt ≤ Z | ∂ − H ǫ | ( µ t ) | ˙ µ | ( t ) dt. We now have all the ingredients to prove Theorem 9 so that we can relate curves ofmaximal slope to weak solutions of the ǫ -Landau equation. Proof of Theorem 9.
Take a limit curve µ t constructed in Proposition 33. By the previousLemma 35, the assumptions of [3, Theorem 2.3.3] are fulfilled so the curve is a maximal slopewith respect to | ∂ − H ǫ | and satisfies the associated energy dissipation inequality H ǫ ( µ r ) − H ǫ ( µ s ) + 12 Z rs | ∂ − H ǫ ( µ t ) | dt + 12 Z rs | ˙ µ | ( t ) dt ≤ . The inequality of Lemma 34 gives H ǫ ( µ r ) − H ǫ ( µ s ) + 12 Z rs D ǫ ( µ t ) dt + 12 Z rs | ˙ µ | ( t ) dt ≤ , which is precisely the statement that the limit curve µ t is a curve of maximal slope withrespect to √ D ǫ . Remark 36.
The results of Proposition 33 and Lemma 34 can be generalized to otherregularization kernels G s,ǫ , in particular, the Maxwellian regularization. However, this is notthe case for Lemma 35 since the proof relies on Proposition 27, see Remark 32. ǫ → Theorems 8 and 9 provide the basic existence theory for the ǫ > ǫ ↓ Sketch proof of Theorem 11.
By repeating the proof of Theorem 8, we see that the crucialingredient is the chain rule (18) in Proposition 27. For now assume the following
Claim 37.
Assume ( A1 ), ( A2 ), ( A3 ) and let M be any grazing rate such that ( µ, M ) ∈GCE ET and Z T A ( µ t , M t ) dt < ∞ . Then we have the chain rule H [ µ r ] − H [ µ s ] = 12 Z rs Z Z R ˜ ∇ (cid:20) δ H δµ (cid:21) · dM t dt. (28)28y following the steps of the proof of Theorem 8 and using (28) instead of (18), onecompletes the proof of Theorem 11. We dedicate this section to proving Claim 37.Equation (28) is clearly the ǫ ↓ A2 ) and the fact that ǫ
7→ H ǫ [ µ t ] is non-increasing for every t . We refer to [25, Proof of Proposition 4.2; Step 4:part d)] for more details on a similar argument.The difficulty remains in deducing that the right-hand side of (18) converges to theright-hand side of (28) as ǫ ↓ Z T Z Z R ˜ ∇ δ H ǫ δµ · dM t dt → Z T Z Z R ˜ ∇ δ H δµ · dM t dt, ǫ ↓ A1 ), ( A2 ), ( A3 ) on f . The key result which we willuse repeatedly in this section is the following theorem which is a specific case of the resultin [38, Chapter 4, Theorem 17]. Theorem 38 (Extended Dominated Convergence Theorem (EDCT)) . Let ( H ǫ ) ǫ> and ( I ǫ ) ǫ> be sequences of measurable functions on X satisfying I ǫ ≥ and suppose there exists mea-surable functions H, I satisfying1. | H ǫ | ≤ I ǫ for every ǫ > and pointwise a.e.2. H ǫ and I ǫ converge pointwise a.e. to H and I , respectively.3. lim ǫ ↓ Z X I ǫ = Z X I < ∞ . Then, we have the convergence lim ǫ ↓ Z X H ǫ = Z X H. Setting M = m L ⊗ L (valid by Proposition 18) and using Young’s inequality on theright-hand side of (18), we obtain the majorants˜ ∇ (cid:20) δ H ǫ δµ (cid:21) · m t ≤ f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H ǫ δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) + 12 | m t | f f ∗ . Notice that the first term is precisely the integrand of D ǫ while the second term is theintegrand of the action functional A ( µ t , M t ) which has no dependence on ǫ and is henceforthignored. We can apply EDCT 38 with X = (0 , T ) × R to prove (29) once we show Z T Z Z R f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H ǫ δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dv ∗ dvdt → Z T Z Z R f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dv ∗ dvdt, ǫ ↓ . (30)The pointwise a.e. convergence hypothesis of EDCT 38 is straightforward based on theregularization of H ǫ through G ǫ . Focusing on (30), we will use a standard Dominated29onvergence Theorem (DCT) for the integration in the t variable, by proving Z Z R f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H ǫ δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dv ∗ dv → Z Z R f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dv ∗ dv, a.e. t, Z Z R f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H ǫ δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dv ∗ dv ≤ C Z Z R f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dv ∗ dv, a.e. t ∀ ǫ > , (31)where C > ǫ >
0. The estimate of (31) guarantees the L t majorisation due to the finite entropy-dissipation assumption ( A3 ).Our estimates in this section accomplish both the convergence and the estimate of (31)by nested application of EDCT 38. The significance of all three assumptions ( A1 ), ( A2 ),and ( A3 ) will be apparent in proving the convergence in (31). Remark 39.
In this section, the only properties of G ǫ we use are that it is a non-negativeradial approximate identity with sufficiently many moments. As in the construction ofminimizing movement curves in Section 5, the results of this section can be achieved withother radial approximate identities. (31) The need to apply EDCT 38 instead of the more classical Lebesgue DCT is that we are unableto prove pointwise estimates in v for the function v → f R R f ∗ (cid:12)(cid:12)(cid:12) ˜ ∇ h δ H ǫ δf i(cid:12)(cid:12)(cid:12) dv ∗ . Instead, ourestimates in this section rely on the self-adjointness of convolution against radial exponentials(SACRE) to construct a convergent majorant in ǫ . Step 1: Finding majorants and appealing to EDCT 38
We seek to find pointwise a.e. majorants in the v variable f Z R f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H ǫ δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dv ∗ ≤ I ǫ ( v ) , where I ǫ ( v ) satisfies the hypothesis for the majorant in EDCT 38. We show that I ǫ convergespointwise to some I , since I ǫ depends on ǫ only through convolutions against G ǫ , which isan approximation of the identity. Hence, we are left with showing the integral convergenceItem 3 of EDCT 38 Z R I ǫ ( v ) dvdt → Z R I ( v ) dv, ǫ → . Step 2: Use SACRE with G ǫ To show the integral convergence for I ǫ , we find functions A and B such that I ǫ ( v ) ≤ A ( v )( G ǫ ∗ B )( v )and apply EDCT 38. As in the previous step, the pointwise convergence is easily proved.Hence, we are left to show the integral convergence Z R A ( G ǫ ∗ B ) dv → Z R A B , ǫ → . Z R A ( G ǫ ∗ B ) = Z R ( G ǫ ∗ A ) B | {z } =: I ǫ . Therefore, we have reduced the problem to showing integral convergence Item 3 of EDCTfor I ǫ (as the pointwise convergence is easily proved). Step 3: Reiterate step 2
We repeat the process outlined in Step 2 by finding functions A and B such that we havethe pointwise bound I ǫ ( v ) ≤ A ( v )( G ǫ ∗ B )( v ) . Again the pointwise convergence for the majorant follows easily, hence we only need to checkthe integral convergence Item 3 of EDCT 38 given by Z R A ( G ǫ ∗ B ) → Z R A B . Using SACRE, we study instead the integral convergence of I ǫ ( v ) = ( G ǫ ∗ A ) B . Eventually, after a finite number of times of finding majorants and applying SACRE, wewill obtain a majorant I iǫ for which the estimates and the convergence as ǫ → A3 ). As mentioned in the previous section, for the final step of the proof we need a bound on theweighted Fisher information and a closely related variant in terms of the entropy-dissipationoriginally discovered by the third author in [19].
Theorem 40.
Suppose γ ∈ ( − , and let f ≥ be a probability density belong to L − γ ∩ L log L ( R ) . We have Z R f ( v ) h v i γ (cid:12)(cid:12)(cid:12)(cid:12) ∇ δ H δf (cid:12)(cid:12)(cid:12)(cid:12) dv + Z R f ( v ) h v i γ (cid:12)(cid:12)(cid:12)(cid:12) v × ∇ δ H δf (cid:12)(cid:12)(cid:12)(cid:12) dv ≤ C (1 + D w, H ( f )) , where C > is a constant depending only on the bounds of m − γ ( f ) and the Boltzmannentropy, H ( f ) , of f . The estimate in this precise form can be found in [18, Proposition 4, p. 10]. We will referto the second term on the left-hand side as a ‘cross Fisher information’.To decompose the entropy-dissipation in a manageable way that makes the cross Fisherterm more apparent, we have the following linear algebra fact.
Lemma 41.
For x, y ∈ R , we have | x | ( y · Π[ x ] y ) = | x × y | roof. Without loss of generality, we assume neither x, y = 0 or else the statement holdstrivially. Let θ be an oriented angle between x and y . We expand the definition of Π[ x ] andobserve | x | ( y · Π[ x ] y ) = y · ( | x | I − x ⊗ x ) y = | x | | y | − | x · y | = | x | | y | (1 − cos θ ) = | x | | y | sin θ = | x × y | . The following lemma shows how we use assumption ( A1 ) to control the singularity ofthe weight. Lemma 42.
Given γ ∈ ( − , , assume that f satisfies ( A1 ) for some < η ≤ γ + 3 , thenwe have for a.e. t Z R f ∗ ( t ) | v − v ∗ | γ dv ∗ ≤ C ( t ) h v i γ , Z R f ∗ ( t ) | v ∗ | | v − v ∗ | γ dv ∗ ≤ C ( t ) h v i γ , (32) where || C || L ∞ (0 ,T ) . γ,η || h·i − γ f ( t ) || L ∞ (cid:18) ,T ; L ∩ L − η γ − η ( R ) (cid:19) || C || L ∞ (0 ,T ) . γ,η || h·i − γ f ( t ) || L ∞ (cid:18) ,T ; L ∩ L − η γ − η ( R ) (cid:19) . Proof.
We will only prove the first inequality of (32) since the second inequality uses thesame procedure. We split the estimation for local | v | ≤ | v | ≥ | v | ≤ v ∗ into two regions Z R f ∗ | v − v ∗ | γ dv ∗ = Z | v − v ∗ |≥ f ∗ | v − v ∗ | γ dv ∗ + Z | v − v ∗ |≤ f ∗ | v − v ∗ | γ dv ∗ ≤ Z | v − v ∗ |≤ f ∗ | v − v ∗ | γ dv ∗ , where we have used that R R f = 1 and γ ≤
0. For the integral with the singularity, we applyYoung’s convolution inequality with conjugate exponents (cid:16) − η γ − η , − ηγ (cid:17)Z | v − v ∗ |≤ f ∗ | v − v ∗ | γ dv ∗ ≤ || f ∗ ( χ B |·| γ ) || L ∞ ≤ || f || L − η γ − η || χ B |·| γ || L − ηγ ≤ (cid:18) ω η (cid:19) − ηγ || f || L − η γ − η . Here, ω is the volume of the unit sphere in R . | v | ≥ Z R f ∗ | v − v ∗ | γ dv ∗ = Z | v ∗ |≤ | v | f ∗ | v − v ∗ | γ dv ∗ + Z | v ∗ |≥ | v | f ∗ | v − v ∗ | γ dv ∗ ≤ − γ | v | γ Z | v ∗ |≤ | v | f ∗ dv ∗ + 2 − γ | v | γ Z | v ∗ |≥ | v | f ∗ | v ∗ | − γ | v − v ∗ | γ dv ∗ . | v − v ∗ | ≥ | v | − | v ∗ | ≥ | v | , ≤ − γ | v | γ | v ∗ | − γ . We estimate the first integral using the unit mass of f , while the second integral is moredelicate but again uses the splitting of the previous step to obtain Z R f ∗ | v − v ∗ | γ dv ∗ ≤ − γ | v | γ +2 − γ | v | γ (cid:18)Z | v − v ∗ |≥ f ∗ | v ∗ | − γ | v − v ∗ | γ dv ∗ + Z | v − v ∗ |≤ f ∗ | v ∗ | − γ | v − v ∗ | γ dv ∗ (cid:19) . In the large brackets, the first integral can be estimated by m − γ ( f ). Now we use the sameYoung’s inequality argument for the remaining integral to obtain Z R f ∗ | v − v ∗ | γ dv ∗ ≤ − γ | v | γ + 2 − γ | v | γ m − γ ( f ) + (cid:18) ω η (cid:19) − ηγ ||| · | − γ f || L − η γ − η ( R ) ! . The proof is complete by combining the estimates for | v | ≤ | v | ≥ Lemma 43 (Peetre) . For any p ∈ R and x, y ∈ R d , we have h x i p h y i p ≤ | p | / h x − y i | p | . Proof.
Our proof follows [5]. Starting with the case p = 2, for fixed vectors a, b ∈ R d wehave, with the help of Young’s inequality,1 + | a − b | ≤ | a | + 2 | a || b | + | b | ≤ | a | + 2 | b | ≤ | a | + 2 | a | | b | + 2 | b | = 2(1 + | a | )(1 + | b | ) . Dividing by h b i and setting a = x − y, b = − y , we obtain the inequality for p = 2 h x i h y i ≤ h x − y i . By taking non-negative powers, this proves the inequality for p ≥
0. On the other hand,when we divided by h b i we could have also set a = x − y, b = x to obtain h y i h x i ≤ h x − y i . Taking strictly non-negative powers here proves the inequality for p < G ǫ with respect to the original function.33 emma 44. For any p ∈ R , we have Z R d h w i p G ǫ ( v − w ) dw ≤ C h v i p , where C > is a constant depending only on | p | and m | p | ( G ) .Proof. We use Peetre’s inequality in Lemma 43 to introduce v − w into the angle brackets Z R d h w i p G ǫ ( v − w ) dw ≤ | p | / h v i p Z R d h v − w i | p | G ǫ ( v − w ) dw = 2 | p | / h v i p Z R d (1 + | w | ) | p | ǫ − d G ( w/ǫ ) dw = 2 | p | / h v i p Z R d (1 + ǫ | w | ) | p | G ( w ) dw ≤ C | p | h v i p (cid:20) ǫ | p | Z R d | w | | p | G ( w ) dw (cid:21) ≤ C | p | (cid:2) ǫ | p | m | p | ( G ) (cid:3) h v i p We stress that Peetre’s inequality 43 is necessary for the estimate of Lemma 44 with non-positive powers p which we apply in the sequel. Finally, the last result we will need isan integration by parts formula for the differential operator associated to the cross Fisherinformation. Lemma 45 (Twisted integration by parts) . Let f, g be smooth scalar functions of R whichare sufficiently integrable. Then, we have the formula Z R ( v × ∇ v g ( v )) f ( v ) dv = − Z R g ( v )( v × ∇ v f ( v )) dv. Here, the meaning of v × ∇ v is v × ∇ v f ( v ) = ( v ∂ f ( v ) − v ∂ f ( v ) , v ∂ f ( v ) − v ∂ f ( v ) , v ∂ f ( v ) − v ∂ f ( v )) . (31) using EDCT 38 We start by decomposing and estimating the integrand of D ǫ . With the help of Lemma 41,we expand the square term of the integrand to see (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H ǫ δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = | v − v ∗ | γ | Π[ v − v ∗ ]( b ǫ ∗ a ǫ − b ǫ ∗ a ǫ ∗ ) | ≤ | v − v ∗ | γ (4 | v × ( b ǫ ∗ a ǫ ) | + 4 | v ∗ × ( b ǫ ∗ a ǫ ∗ ) | + 4 | v × ( b ǫ ∗ a ǫ ∗ ) | + 4 | v ∗ × ( b ǫ ∗ a ǫ ) | ) ≤ | v − v ∗ | γ | v × ( b ǫ ∗ a ǫ ) | | {z } (cid:13) +4 | v − v ∗ | γ | v ∗ × ( b ǫ ∗ a ǫ ∗ ) | | {z } (cid:13) + 4 | v | | v − v ∗ | γ | b ǫ ∗ a ǫ ∗ | | {z } (cid:13) +4 | v ∗ | | v − v ∗ | γ | b ǫ ∗ a ǫ | | {z } (cid:13) , b ǫ = G ǫ and a ǫ = ∇ log( G ǫ ∗ f ) . (33)By using that G ǫ is an approximation of the identity, we know that the integrand of D ǫ converges pointwise a.e. to the integrand of D as ǫ ↓
0. As well, each i (cid:13) for i = 1 , , , (cid:13) → | v × ∇ f | f , (cid:13) → | v ∗ × ∇ ∗ f ∗ | f ∗ , (cid:13) → |∇ ∗ f ∗ | f ∗ , (cid:13) → |∇ f | f . By EDCT 38, to show the integral convergence in (31), it suffices to show, for example,
Z Z R f f ∗ | v − v ∗ | γ (cid:13) dvdv ∗ → Z Z R f f ∗ | v − v ∗ | γ | v × ∇ f | f dvdv ∗ , and similarly for each i (cid:13) for i = 2 , ,
4. By symmetry considerations when swapping thevariables v ↔ v ∗ , the convergence for the terms 1 (cid:13) and 4 (cid:13) controls the convergence for 2 (cid:13) and 3 (cid:13) , respectively. Hence we will focus on the term 4 (cid:13) first and then on term 1 (cid:13) . (cid:13) We seek to show in the limit ǫ ↓ Z Z R f f ∗ | v ∗ | | v − v ∗ | γ | b ǫ ∗ a ǫ | dv ∗ dv = Z R (cid:18)Z R f ∗ | v ∗ | | v − v ∗ | γ dv ∗ (cid:19) f | b ǫ ∗ a ǫ | dv → Z R (cid:18)Z R f ∗ | v ∗ | | v − v ∗ | γ dv ∗ (cid:19) |∇ f | f dv. (34)By the reordering of integrations written above, we now think of the double integral over v, v ∗ of f f ∗ | v ∗ | | v − v ∗ | γ | b ǫ ∗ a ǫ | as a single integral of the function (cid:0)R R d f ∗ | v ∗ | | v − v ∗ | γ dv ∗ (cid:1) f | b ǫ ∗ a ǫ | over v . For this is the single integral convergence that we will use EDCT 38 for. We can useCauchy-Schwarz on the convolution integral to absorb the power term as follows | b ǫ ∗ a ǫ | = (cid:12)(cid:12)(cid:12)(cid:12)Z R b ǫ ( v − w ) a ǫ ( w ) dw (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z R h w i − γ b ǫ ( v − w ) dw (cid:19) (cid:18)Z R b ǫ ( v − w ) h w i γ | a ǫ ( w ) | dw (cid:19) ≤ C h v i − γ b ǫ ∗ [ h·i γ | a ǫ ( · ) | ] , where the last inequality comes from Lemma 44. Continuing with Lemma 42, we have (cid:18)Z R f ∗ | v ∗ | | v − v ∗ | γ dv ∗ (cid:19) f | b ǫ ∗ a ǫ | ≤ Cf b ǫ ∗ [ h·i γ | a ǫ | ] . By EDCT 38, we reduce the problem to showing in the limit ǫ ↓ Z R f b ǫ ∗ [ h·i γ | a ǫ | ] dv → Z R h v i γ |∇ f | f dv. a ǫ and b ǫ (see (33)) yields Z R f b ǫ ∗ [ h·i γ | a ǫ | ] dv = Z R [ b ǫ ∗ f ] h v i γ | a ǫ | dv = Z R h v i γ | b ǫ ∗ ∇ f | b ǫ ∗ f dv. (35)We work with this simplified expression and note that pointwise convergence is still valid | b ǫ ∗ ∇ f | b ǫ ∗ f → |∇ f | f . Next, we notice that the function (
F, f ) | F | f is jointly convex in F ∈ R and f >
0, so wecan use Jensen’s inequality to obtain a further pointwise majorant for the integrand of (35) | b ǫ ∗ ∇ f | b ǫ ∗ f ≤ b ǫ ∗ (cid:20) |∇ f | f (cid:21) . Using EDCT 38 again, we reduce the problem to showing in the limit ǫ ↓ Z R h v i γ b ǫ ∗ (cid:20) |∇ f | f (cid:21) dv → Z R h v i γ |∇ f | f dv. We use SACRE once more and place the convolution onto the weight term Z R h v i γ b ǫ ∗ (cid:20) |∇ f | f (cid:21) dv = Z R [ b ǫ ∗ h·i γ ] |∇ f | f dv. Now, we are in a position to apply the classical Dominated Convergence Theorem. We noticethat we have the pointwise convergence[ b ǫ ∗ h·i γ ] → h v i γ . Furthermore, using Lemma 44, we can estimate b ǫ ∗ h·i γ uniformly in ǫ to find the domination[ b ǫ ∗ h·i γ ] |∇ f | f ≤ C h v i γ |∇ f | f . Using Theorem 40 and the finite entropy-dissipation assumption ( A3 ), we know that theright-hand side belongs to L v a.e. t ∈ (0 , T ). Therefore, for a.e. t ∈ (0 , T ) the conditions ofthe Dominated Convergence Theorem are satisfied so we have the integral convergence Z R [ b ǫ ∗ h·i γ ] |∇ f | f dv → Z R h v i γ |∇ f | f dv. We have closed the argument for the convergence of (34) after retracing the previous esti-mates with EDCT 38. 36 .3.2 Term 1 (cid:13)
We seek to show in the limit ǫ ↓ Z Z R f f ∗ | v − v ∗ | γ | v × ( b ǫ ∗ a ǫ ) | dv ∗ dv = Z R (cid:18)Z R f ∗ | v − v ∗ | γ dv ∗ (cid:19) f | v × ( b ǫ ∗ a ǫ ) | dv → Z R (cid:18)Z R f ∗ | v − v ∗ | γ dv ∗ (cid:19) | v × ∇ f | f dv (36)using the same strategy of nested applications of EDCT 38 like in the previous Section 6.3.1.We will encounter difficulty when trying to use Jensen’s inequality due to the cross Fisherinformation term. As in the previous Section 6.3.1, we have written this double integral over v, v ∗ as a single integral over v . By EDCT 38 and Lemma 42, it suffices to show the integralconvergence of Z R h v i γ f | v × ( b ǫ ∗ a ǫ ) | dv → Z R h v i γ | v × ∇ f | f (37)to obtain the integral convergence of (36). Pointwise, we can make the following manipula-tions v × ( b ǫ ∗ a ǫ ) = v × (cid:18)Z R G ǫ ( v − w ) ∇ log( f ∗ G ǫ ( w )) dw (cid:19) = v × (cid:18)Z R ∇ G ǫ ( v − w ) log( f ∗ G ǫ ( w )) dw (cid:19) = Z R w × ∇ G ǫ ( v − w ) log( f ∗ G ǫ ( w )) dw = Z R G ǫ ( v − w ) w × ∇ log( f ∗ G ǫ ( w )) dw, (38)where we have used the radial symmetry of G ǫ to get the cancellation ( v − w ) ×∇ G ǫ ( v − w ) = 0and the twisted integration by parts Lemma 45 (we note that we not pick any signs in theintegration by parts, as the variable w appears with a minus sign in the arguments of G ǫ ).We apply Cauchy-Schwarz, multiply and divide by h w i γ , and use Lemma 44 to obtain | v × ( b ǫ ∗ a ǫ ) | ≤ (cid:18)Z R G ǫ ( v − w ) h w i − γ dw (cid:19) Z R G ǫ ( v − w ) h w i γ (cid:12)(cid:12)(cid:12)(cid:12) w × ∇ f ∗ G ǫ ( w ) f ∗ G ǫ ( w ) (cid:12)(cid:12)(cid:12)(cid:12) dw ! . γ h v i − γ Z R G ǫ ( v − w ) h w i γ (cid:12)(cid:12)(cid:12)(cid:12) w × ∇ f ∗ G ǫ ( w ) f ∗ G ǫ ( w ) (cid:12)(cid:12)(cid:12)(cid:12) dw ! . Remembering that this majorant holds pointwise on the integrand of (37), we multiply by h v i γ f ( v ) and obtain h v i γ f ( v ) | v × ( b ǫ ∗ a ǫ ) | . f Z R G ǫ ( v − w ) h w i γ (cid:12)(cid:12)(cid:12)(cid:12) w × ∇ f ∗ G ǫ ( w ) f ∗ G ǫ ( w ) (cid:12)(cid:12)(cid:12)(cid:12) dw ! . Z R f Z R G ǫ ( v − w ) h w i γ (cid:12)(cid:12)(cid:12)(cid:12) w × ∇ f ∗ G ǫ ( w ) f ∗ G ǫ ( w ) (cid:12)(cid:12)(cid:12)(cid:12) dw ! dv = Z R h v i γ | v × ∇ f ∗ G ǫ ( v ) | f ∗ G ǫ ( v ) dv. Using EDCT 38, we need to show the convergence of the right-hand side. Here, it is nowpossible to use Jensen’s inequality after some more manipulations.
Claim 46. | v × ∇ f ∗ G ǫ ( v ) | f ∗ G ǫ ( v ) ≤ Z R G ǫ ( v − w ) | w × ∇ f ( w ) | f ( w ) dw. (39) Proof of Claim 46.
We start by repeating a similar argument to (38). Using that G ǫ isradially symmetric and the twisted integration by parts Lemma 45 we obtain v × ∇ f ∗ G ǫ ( v ) = v × (cid:18)Z R ∇ G ǫ ( v − w ) f ( w ) dw (cid:19) = Z R w × ∇ G ǫ ( v − w ) f ( w ) dw = Z R G ǫ ( v − w ) ( w × ∇ w f ( w )) | {z } =: F ( w ) dw. Therefore, since (
F, f ) | F | f is jointly convex in F ∈ R and f >
0, we apply Jensen’sinequality to the left-hand side of (39) to see | v × ∇ f ∗ G ǫ ( v ) | f ∗ G ǫ ( v ) = | F ∗ G ǫ | f ∗ G ǫ ( v ) ≤ | F | f ∗ G ǫ ( v ) = Z R G ǫ ( v − w ) | w × ∇ f ( w ) | f ( w ) dw, which proves the claim.Continuing, by EDCT 38, we seek to establish the integral convergence of Z R h v i γ (cid:20) | F | f ∗ G ǫ (cid:21) ( v ) dv = Z R [ h·i γ ∗ G ǫ ]( v ) | v × ∇ f ( v ) | f ( v ) dv. Finally, the integrand of the right-hand side has a majorant due to Lemma 44[ h·i γ ∗ G ǫ ]( v ) | v × ∇ f ( v ) | f ( v ) . h v i γ | v × ∇ f ( v ) | f ( v ) . Once again using Theorem 40 and Assumption ( A3 ), we obtain that for a.e. t ∈ (0 , T ) theright hand side belongs to L v ( R ). Using Dominated Convergence theorem, we see that theintegral converges. Tracing back the estimates, this takes care of the convergence of the term1 (cid:13) and establishes the convergence in (37).We note that the estimates in the previous subsections not only establish the a.e. point-wise convergence of (31), but also the majorisation Z Z R f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H ǫ δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dv ∗ dv ≤ C Z Z R f f ∗ (cid:12)(cid:12)(cid:12)(cid:12) ˜ ∇ (cid:20) δ H δµ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) dv ∗ dv, a.e. t ∀ ǫ > , C . || h·i − γ f ( t ) || L ∞ (cid:18) ,T ; L ∩ L − η γ − η ( R ) (cid:19) + || h·i − γ f ( t ) || L ∞ (cid:18) ,T ; L ∩ L − η γ − η ( R ) (cid:19) by Lemma 42. Hence, using assumption ( A3 ) and (31) we can apply Lebesgue DCT to passto the limit in the time integral and show the desired chain rule Claim 37. A An auxiliary PDE for Lemma 34
In this section, we study weak solutions to the following PDE (cid:26) ∂ t µ = ∇ · { µφ R R R d φ R ∗ ψ R ( v − v ∗ ) | v − v ∗ | γ +2 Π[ v − v ∗ ]( J ǫ − J ǫ ∗ ) dµ ( v ∗ ) } µ (0) = µ . (40)We assume the initial data µ belongs to P ( R d ). For R , R >
0, the functions 0 ≤ φ R , ψ R ≤ φ R ( v ) = (cid:26) , | v | ≤ R , | v | ≥ R + 1 , ψ R ( z ) = (cid:26) , | z | ≤ /R , | z | ≥ /R . For ǫ > J ǫ is the gradient of first variation of H ǫ applied to µ , meaning J ǫ = ∇ G ǫ ∗ log[ µ ∗ G ǫ ] ∈ C ∞ ( R d ; R d ) . The main result of this section is
Theorem 47.
Fix ǫ, R , R > , γ ∈ R , and µ ∈ P ( R d ) . Then, there exists a T > which depends on ǫ, γ, R , R , and µ such that equation (40) has a unique weak solution µ ∈ C ([0 , T ]; P ( R d )) . By Lemma 30, we know that J ǫ is uniformly (with constant depending on ǫ and µ ). Thepurpose of φ R , φ R ∗ is to cut off the growth of J ǫ , J ǫ ∗ to ensure that the ‘velocity field’ in theright-hand side of (40) is globally Lipschitz (it is, in fact, smooth and compactly supported).The ψ R ( v − v ∗ ) term avoids the possible singularities coming from the weight | v − v ∗ | γ +2 forsoft potentials γ < T > C ([0 , T ]; P ( R d )) which we will equip with the followingmetric d ( µ, ν ) := sup t ∈ [0 ,T ] W ( µ ( t ) , ν ( t )) , µ, ν ∈ C ([0 , T ]; P ( R d )) , where W is the 2-Wasserstein distance on P ( R d ). We have closely followed the procedurein [9] with appropriate modifications for this setting. Remark 48.
Since we are cutting off the ‘velocity’ field at radius R , R , the growth of J ǫ is inconsequential. Hence the results of this section can be applied when replacing theconvolution kernel of J ǫ with general tailed exponential distributions G s,ǫ ( v ) for s > µ ∈ P ( R d ), we will denote by U [ µ ]( v ) the following function U [ µ ]( v ) := − φ R Z R d φ R ∗ ψ R ( v − v ∗ ) | v − v ∗ | γ +2 Π[ v − v ∗ ]( J ǫ − J ǫ ∗ ) dµ ( v ∗ ) , so that the PDE in (40) can be written as a nonlinear transport/continuity equation ∂ t µ ( t ) = −∇ · { µ ( t ) U [ µ ( t )] } . To fix ideas, the weak formulation of (40) is such that the following equality holds for alltest functions τ ∈ C ∞ c ( R d ) and times t ∈ [0 , T ] Z R d τ ( v ) dµ r ( v ) − Z R d τ ( v ) dµ ( v )= Z t Z R d φ R ∇ τ ( v ) · Z R d φ R ∗ ψ R ( v − v ∗ ) | v − v ∗ | γ +2 Π[ v − v ∗ ]( J ǫ − J ǫ ∗ ) dµ s ( v ∗ ) dµ s ( v ) ds. Thanks to all the smooth cutoffs from φ R , φ R ∗ , and ψ R and µ ∈ P ( R d ), we can enlargethe class of test functions to smooth functions with quadratic growth. In particular, bychoosing τ ( v ) = | v | and symmetrising the right-hand side by swapping v ↔ v ∗ , we see thatthe second moment of µ is conserved along the evolution of (40).Our first step is to look at the level of the characteristic equation associated to (40). Lemma 49 (Characteristic equation) . For any
T > , µ ∈ C ([0 , T ]; P ( R d )) and v ∈ R d ,there exists a unique solution v ∈ C ((0 , T ); R d ) ∩ C ([0 , T ]; R d ) to the following ODE dvdt = U [ µ ( t )]( v ) , v (0) = v . Furthermore, the growth rate satisfies | v ( t ) | ≤ max {| v | , R + 1 } , ∀ t ∈ [0 , T ] . Proof. U [ µ ( t )]( · ) is smooth and compactly supported uniformly in t , so classical Cauchy-Lipschitz theory gives existence and uniqueness of solution v with the promised regularity.For the estimate on the growth rate, note that U [ µ ] has support contained in B R +1 .Points outside this ball do not change in time according to this ODE.We will denote by Φ tµ the flow map associated to this ODE, so that ddt Φ tµ ( v ) = U [ µ ( t )](Φ tµ ( v )) , Φ µ ( v ) = v . It is known that, given ν ∈ C ([0 , T ]; P ( R d )), the curve of probability measures µ ( t ) =Φ tν µ is a weak solution to ∂ t µ ( t ) = −∇ · { µ ( t ) U [ ν ( t )] } , µ (0) = µ . Here, Φ tν µ is the push-forward measure of µ defined in duality with τ ∈ C b ( R d ) by Z R d τ ( v ) d (Φ tν µ )( v ) = Z R d τ (Φ tν ( v )) dµ ( v ) . We seek to find a fixed point to the map µ Φ tµ µ as it would weakly solve (40). Tobetter understand the properties of this map, we need to establish estimates on the flow mapthrough U as a function of time and measures.40 emma 50 ( L ∞ estimate for velocity field) . There exists a constant C = C ( ǫ, γ, R , R , µ ) > such that for every T > and ν ∈ C ([0 , T ]; P ( R d )) , we have | U [ ν ( t )]( v ) | ≤ C, ∀ t ∈ [0 , T ] , v ∈ R d . Proof.
Estimate for γ ≥ − | v − v ∗ | γ +2 . γ | v | γ +2 + | v ∗ | γ +2 , || Π[ v − v ∗ ] || ≤ , J ǫ . ǫ,µ γ , boundedness of Π, and Lemma 30, respectively. These three inequalitiesprovide the estimate | U [ ν ( t )]( v ) | . γ,ǫ,µ φ R ( v ) Z R d φ R ( v ∗ )( | v | γ +2 + | v ∗ | γ +2 ) dν t ( v ∗ ) , where we have dropped ψ R altogether. For the integral term, we apply H¨older’s inequalitytaking advantage of the compact support of φ R and the unit mass of ν t to further obtain | U [ ν ( t )]( v ) | . γ,ǫ,µ φ R ( v )( R γ + h v i γ ) Z R d dν t ( v ∗ ) . R φ R ( v ) h v i γ . Again, since φ R has compact support, we can brutally estimate the polynomial to conclude.Estimate for γ < − ψ R ( v − v ∗ ) | v − v ∗ | γ +2 . /R γ +22 , || Π[ v − v ∗ ] || ≤ , J ǫ . ǫ,µ . From these inequalities and the compact support of φ R , we have | U [ ν ( t )]( v ) | . γ,ǫ,µ ,R φ R ( v ) Z R d φ R ( v ∗ ) dν t ( v ∗ ) ≤ , which concludes the proof.The next result follows exactly as in [9]. Lemma 51 (Time continuity of flow map) . Let C = C ( ǫ, γ, R , R , µ ) > be the sameconstant from Lemma 50. Then for any T > , and ν ∈ C ([0 , T ]; P ( R d )) we have || Φ tν − Φ sν || L ∞ ( R d ) ≤ C | t − s | . Our next objective is to establish the regularity of the flow map with respect to themeasures in the subscript. To simplify the subsequent lemmata, let us use the notation inthe following
Lemma 52.
Define F : ( v, w ) ∈ R d × R d φ R ( v ) φ R ( w ) ψ R ( v − w ) | v − w | γ +2 Π[ v − w ]( J ǫ ( v ) − J ǫ ( w )) . The function F is smooth and compactly supported. In particular, for every k, l ∈ N , thereis a constant C = C ( ǫ, γ, R , R , µ , k, l ) > such that || D kv D lw F || L ∞ ( R d × R d ) ≤ C. roof. The compact support property comes from the factor of φ R ( v ) φ R ( w ) in the defini-tion. The regularity comes from the avoidance of v = w due to the factor ψ R ( v − w ). Corollary 53 (Pointwise and measurewise regularity of U ) . Consider the constant C = C ( ǫ, γ, R , R , µ , k, l ) > from Lemma 52 above. We have the following1. Take C = C ( ǫ, γ, R , R , µ , , > . For every T > ν , ν ∈ C ([0 , T ]; P ( R d )); t ∈ [0 , T ]; v ∈ R d we have the estimate | U [ ν ( t )]( v ) − U [ ν ( t )]( v ) | ≤ C W ( ν t , ν t ) .
2. Take C = C ( ǫ, γ, R , R , µ , , > . For every T > ν ∈ C ([0 , T ]; P ( R d )); t ∈ [0 , T ]; v , v ∈ R d we have the estimate | U [ ν ( t )]( v ) − U [ ν ( t )]( v ) | ≤ C | v − v | . Remark 54.
By considering the anti-symmetric property of F when swapping variables v ↔ w , one really obtains C = C . Their distinction in this corollary is artificial. Proof.
Item 1:Firstly, for every t ∈ [0 , T ] take π ( t ) ∈ P ( R d × R d ) the 2-Wasserstein optimal transportationplan connecting ν ( t ) and ν ( t ) which exists, see [46]. We estimate the difference withnotation from Lemma 52 | U [ ν ( t )]( v ) − U [ ν ( t )]( v ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R d F ( v, w ) dν t ( w ) − Z R d F ( v, ¯ w ) dν t ( ¯ w ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Z R d F ( v, w ) − F ( v, ¯ w ) dπ t ( w, ¯ w ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Z Z R d | w − ¯ w | dπ t ( w, ¯ w ) ≤ C W ( ν t , ν t ) . The first inequality uses a mean-value type estimate (in the second variable of F ) and thesecond inequality uses Cauchy-Schwarz or equivalently, that W is stronger than W .Item 2:As with item 1, we estimate the difference using F to find | U [ ν ( t )]( v ) − U [ ν ( t )]( v ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R d F ( v , w ) − F ( v , w ) dν t ( w ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R d | F ( v , w ) − F ( v , w ) | dν t ( w ) ≤ C | v − v | . Once more, a mean-value type estimate is applied (in the first variable of F ) and we recall ν t is a probability measure. 42he next result combines both items of Corollary 53 to estimate the regularity of theflow map with respect to measures and follows exactly as in [9]. Lemma 55 (Continuity of flow map with respect to measures) . For
T > fix any ν , ν ∈ C ([0 , T ]; P ( R d )) and t ∈ [0 , T ] . With C := C = C the same constants in Corollary 53,we have the estimate || Φ tν − Φ tν || L ∞ ( R d ) ≤ ( e Ct − d ( ν , ν ) , recalling that d ( ν , ν ) = sup t ∈ [0 ,T ] W ( ν t , ν t ) . It is by now classical how to obtain Theorem 47 from Corollary 53 and Lemma 55,see [9, 13, 28] for instance. The time of existence can be given by any 0 < T < C log 2 where C >
Acknowledgements
JAC was supported the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Parti-cle Dynamics: Phase Transitions, Patterns and Synchronization) of the European ResearchCouncil Executive Agency (ERC) under the European Union’s Horizon 2020 research andinnovation programme (grant agreement No. 883363). JAC and MGD were partially sup-ported by EPSRC grant number EP/P031587/1. MGD was partially supported by CNPq-Brazil (
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