Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations

In this note we prove that, under some minimal regularity assumptions on the initial datum, solutions to the spatially homogenous Boltzmann and Landau equations for hard potentials uniformly propagate the Fisher information. The proof of such a result is based upon some explicit pointwise lower bound on solutions to Boltzmann equation and strong diffusion properties for the Landau equation. We include an application of this result related to emergence and propagation of exponential tails for the solution's gradient. These results complement estimates provided in the literature.


INTRODUCTION
The Fisher information functional was introduced in [17] (1.1) as a tool in statistics and information theory. It revealed itself a very powerful tool to control regularity and rate of convergence for solutions to several partial differential equations. In particular, in the study of Fokker-Planck equation, the control of the Fisher information along the Orstein-Uhlenbeck semigroup is the key point for the exponential rate of convergence to equilibrium [12] in relative entropy terms. Variants of such an approach can be applied to deal with more general parabolic problems [13]. For these kind of problems, the Fisher information turns out to play the role of a Lyapunov functional. Such techniques have also been applied in the context of general collisional kinetic equation. In particular, for the Boltzmann equation with Maxwell molecules, exploiting commutations between the Boltzmann collision operator and the Orstein-Uhlenbeck semigroup, the Fisher information serves as a Lyapunov functional for the study of the long time relaxation [22,10]. In [8,9,26], the Fisher information was applied for general collision kernels in relation to the entropy production bounds for the Boltzmann equation. Later in [27], ground breaking work related to the Cercignani's conjecture was made using the Fisher information and the ideas preceding such work. We aim however to emphasize that the present contribution, together with [25,26], is to our knowledge the only one dealing with the question of uniform-in-time estimates for the Fisher information in the kinetic context.
The aim of the present contribution is to further investigate the properties of Fisher information along solutions to two important kinetic equations: the Boltzmann equation for hard potentials, under cut-off assumption, and the Landau equation for hard potentials. More specifically, we show here that, along solutions to Boltzmann or Landau equations for hard potentials, the Fisher information will remain uniformly bounded sup t 0 under minimal regularity assumptions on the initial datum. The minimality is understood in terms of smoothness required for the initial datum and not in terms of number of moments, however, we have tried to be as frugal as possible in this latter issue. In remarks below Theorem 1.1 and Theorem 1.2 we expand on the interpretation of the results and the way they are, or are not, optimal. For the Boltzmann equation estimate (1.2) improves, under less restrictive conditions in the model, the local in time estimate obtained in [26] which reads I(f (t)) e c t 2I(f 0 ) + c (1 + t 3 ) , for some explicit c > 0 .
This bound was obtained in the context of Maxwell molecule type of models and does not directly apply to the case of hard potentials, with exception of hard spheres, that we treat here. Interestingly, the bound (1.2) can be used to generalise, to the context of hard potential models, estimates that were designed for a Maxwell gas with respect to propagation of smoothness such as in [10]. Let us mention that for the case of Maxwell model, the Fisher information is in fact non-increasing for both the Boltzmann and Landau equations, see [21,26,25,15]. This explains why the Fisher information is used to prove exponential relaxation towards thermodynamical equilibrium in this case. Furthermore, the Maxwell model can be compared to other models as well to obtain algebraic rate of relaxation towards equilibrium.
As an application of the uniform propagation of the Fisher information, one can deduce that, for any in a relatively simple manner (relatively to [5] for example). The techniques to prove the bound (1.2) seem to differ in nature for the study of Boltzmann (with cutoff) and Landau equations. The reason for this difference is that while Landau's equation is strongly diffusive, the Boltzmann equation is weakly diffusive. For the Boltzmann equation, we exploit the appearance of pointwise exponential lower bounds for solutions obtained in [20] whereas, for the Landau equation, we use the instantaneous regularizing effect to control, for time t t 0 > 0 the Fisher information by Sobolev regularity bounds while, for small time 0 < t < t 0 , the Fisher information is controlled thanks to a new energy estimate for solutions to the Landau equation. These proofs are related in the sense that for positive functions, the models' solutions, regularity imply a particular behaviour near to zero.
1.1. Notations. Let us introduce some useful notations for function spaces. For any p 1 and q 0, we define the space L p q (R d ) through the norm We also define, for k 0, where the operator (1 − ∆) k 2 is defined through its Fourier transform When we write H k + q (R d ), for some k ∈ R, we simply mean that the positive part k + := max{0, k} of k is taken. Also, we define L 1 log (R d ) as

The Boltzmann equation.
Let us now enter into the details by considering the solution f (t, v) to the Boltzmann equation We consider kernels satisfying b L 1 (S d−1 ) < ∞, thus, it is possible to write the collision operator in gain and loss operators where the collision operator is given by We will consider hard potentials γ ∈ (0, 1]. Also, for technical simplicity, we restrict ourselves to d 3.

Remark 1.2.
If the reader is willing to accept more regularity in the initial data, say f 0 ∈ H 2 ν (R d ) for some ν > d 2 , then Theorem 1.1 remains valid for b ∈ L 1 (S d−1 ) using the propagation of regularity given in [5] The collision operator is defined as where the matrix A(z) = (A ij (z)) i,j=1,...,d is given by We concentrate the study in the hard potential case γ ∈ (0, 1]. We refer to [14] for a methodical study of the Landau equation in this setting. The Landau equation can be written in the form of a nonlinear parabolic equation: where the matrix a(v) and the vector b(v) are given by The minimal conditions that will be required on the initial datum f 0 are finite mass, energy and entropy For technical reasons, to assure conservation of energy, a moment higher than 2 is assumed as well. In this situation, [14,Proposition 4] asserts that the equation is uniformly elliptic, that is, for some positive constant a 0 := a 0 (m 0 , E 0 , H 0 ). Under these assumptions, the Cauchy theory, including infinite regularization and moment propagation, has been developed in [14,15]. As in the Boltzmann case, the Fisher information have been used for the analysis of convergence towards equilibrium, see for instance [15,23,24], and also for analysis of regularity, see [16]. Concerning regularisation, the idea is to establish an inequality of the form with constant C depending only on m 0 , E 0 , H 0 , which are the physical conserved quantities, and where D(f ) denotes the entropy production associated to Q L , i.e.
Since, along solutions to the Landau equation such inequality leads to estimate on the time integrated Fisher information. Then, one uses Sobolev inequality to obtain control on the entropy or a higher norm.
For the Fisher information itself, at least for the hard potential case, the following result follows. The theorem is stated for d = 3 because it uses several results given in [14].

Theorem 1.2.
Assume that the initial datum f 0 0 has finite mass m 0 , energy E 0 and entropy H 0 and satisfies in addition for some > 0. Assume moreover that , and the initial Fisher information. [14,Theorem7]. Thus, the assumptions on f 0 in Theorem 1.2 are quite general. Since I(f 0 ) < ∞ is necessary for uniform propagation of the Fisher information, Theorem 1.2 is optimal with respect to the regularity required for f 0 . Furthermore, inspecting the results for existence and regularity of solutions given in [14], the requirement on the moments for f 0 in (1.7) appears very close to optimal.

Remark 1.3. A condition for well-posedness and regularisation of the Cauchy problem for the Landau
The rest of the document is divided in three sections, Section 2 is devoted to the proof of Theorem 1.1 and Section 3 is concerned with the proof of Theorem 1.2. The final section is an Appendix where the reader will find helpful facts about Boltzman (Appendix A.) and Landau (Appendix B.) equations that will be needed along the arguments.

PROOF OF THEOREM 1.1
In order to prove Theorem 1.1, we consider in all this section a solution f (t) = f (t, v) to the Boltzmann equation (1.3) that conserves mass, momentum, and energy. One has first the following lemma.
Multiplying by g i (t, v) and integrating over R d we get Noticing that we get that Using an integration by part in the third integral, and since which yields the desired result after adding in i = 1, 2, . . . , d.
All terms in (2.1) are relatively easy to estimate with exception, perhaps, of the term involving ∆Q + (f, f ). This is the step where the instantaneous appearance of a lower gaussian barrier is important, in particular, for the estimation of the constant c ε (t) in the following lemma.
where c ε (t) := C ε 1 + log + (1/t) for some universal constant C ε > 0, and Proof. Using Theorem A.1, we get that Thus, This results in . Now, using Theorem A.4 we can estimate the last term and get with η 1 , η 2 and s as defined in the statement of the lemma.
Proof of Theorem 1.1. We start with (2.1) and neglect the nonpositive last term in the right side. It follows that d dt Additionally, thanks to (A.1), one has R(f )(v) κ 0 v γ . And due to integration by parts and (A.2) Therefore, where we used, in addition to previous estimates, Lemma 2.2 for the second inequality. Here η 1 , η 2 , and s are those defined in such lemma.
Under our assumptions on f 0 and for a suitable choice of ε > 0 small enough, the L 1 η2 and H 1 η1 norms of f (t) are uniformly bounded, see Theorems A.2 and A.5. Thus, we obtain that, for such choice of ε > 0, it holds d dt Using that the mapping t → 1 + log + (1/t) is integrable at t = 0, a direct integration of this differential inequality implies that sup t 0 I(f (t)) I(f 0 ) + C(f 0 ) < ∞. This proves the result.
A consequence of this result is the exponentially weighted generation/propagation of the solution's gradient. Indeed, one knows thanks to [2] that f (t)e c min{1,t}|v| γ L 1 C(f 0 ) for some sufficiently small c > 0 and constant C(f 0 ) depending only on mass and energy. Then, This proves that exponential moments of the gradient ∇f (t, v) are uniformly bounded by some positive constant depending only on the initial datum f 0 .

PROOF OF THEOREM 1.2
In this section, we prove the uniform in time estimate on the Fisher information for solutions to the Landau equation. The strong diffusion properties of Landau make the Fisher information more suited to this equation than to Boltzmann.
We assume in all this section that f (t) = f (t, v) is a solution to (1.5) with initial datum f 0 (v) with mass m 0 , energy E 0 . We also assume that f 0 has finite entropy H 0 . We shall exploit the parabolic form of the Landau equation that we recall here again for a := a(v) symmetric positive definite matrix and b := b(v) vector. Recall that, according to (B.1), the matrix a = a(t, v) is uniformly elliptic, i.e.
Multiplying the equation by log f and integrating and, using (B.1)

Integrating in time
we just proved the first part of the following proposition.

Proposition 3.1. For a solution f (t) = f (t, v) to the Landau equation one has
Moreover, given k > 0 and > 0, if we assume the initial datum f 0 to be such that

4)
for some positive constant C k depending on the mass m 0 , the energy E 0 , the entropy H 0 and the quantities (3.3).

Note that integrations by parts lead to
The latter inequality follows by using (B.1) and the fact that Similarly, We control the integral with f | log f | using Lemma B.4 with δ > 0 small enough. It follows that for some positive constantC k depending only on sup t 0 f (t) L 1 k+γ+ε for some arbitrary ε > 0. Integrating between 0 and t the previous equation, we get The first integral in the left-hand side has no sign but it can be handled thanks to (B.3). The result follows from here using propagation of the moment k + γ + ε.
One notices that, for solutions of the Landau equation for hard potentials, the Fisher information emerges as soon as t > 0. This result immediately follows from the following lemma.
Proof. The result is a direct consequence of the following link between the Fisher entropy and weighted Sobolev norm, see [23,Lemma 1] and [14,Theorem 5]: there is C > 0 such that We conclude then with Lemma B.3.
With this result at hand, it remains to study the question about the behaviour of the Fisher information at t = 0. To this end, we prove the following lemma.
Proof. With the notations of the lemma and recalling that a = a(t, v) is symmetric, one can compute We also have As a consequence, after some integration by parts, the Dirichlet terms are computed as Here √ a = a(t, v) is the unique positive definite symmetric square root of a(t, v). In addition, Consequently, we can find an energy estimate for g i . Indeed, multiplying the Landau equation (3.1) by 1/ √ f , differentiating in v i , multiplying by g i and integrating in velocity, it follows that We proceed estimating each term, starting for the absorption term For the latter two terms we use Young's inequality 2|ab| We recall that |b i | B(m 0 , E 0 ) v γ , therefore, This gives the result.
Proof of Theorem 1.2. For short time, say t ∈ [0, 1], integrate (3.6) in time and use Proposition 3.1 with k = 2. Then, we can invoke Lemma 3.1 with t 0 = 1 to estimate I(f (t)) for t 1.

Exponential moments for the Landau equation.
In [14, Section 3] emergence and propagation of polynomial moments have been obtained for the Landau equation and, more recently [11, Section 3.2] develops the propagation of exponential moments for soft potentials. The starting point is the weak formulation for the equation Exponential moments can be easily studied in a similar fashion by choosing ϕ(v) = e λ v s with positive parameters λ, s to be determined. We note that, for such a choice, Thus, resuming the computations given in [14, pg. 201 At this point, we choose 0 < s < 2 and thanks to the Young inequality λs v s v 2 Thus, using Lemma B.2, we get where c > 0 depends on m 0 , E 0 . Meanwhile, , and r := r(C, c, γ) .
This proves a propagation result for exponential moments.

Proposition 3.2.
Fix s ∈ (0, γ] and assume that f 0 belongs to L 1 2+γ (R 3 ) ∩ L 1 log (R 3 ). Then, for the solution f (t, v) of the Landau equation with initial datum f 0 given by [14,Theorem 5] there exists some β := β s,γ 1 such that sup Fix s ∈ (0, 2) , λ > 0 , and assume that R 3 f 0 e λ v s dv < ∞. Then, for the solution f (t, v) of the Landau equation with initial datum f 0 given by [14,Theorem 5] it follows that Proof. For the emergence of the exponential tail we assume t ∈ (0, 1) and take ϕ(t, v) = e t β v s with s ∈ (0, 2) and β > 0 to be chosen. We repeat the steps leading to estimate (3.8) to obtain The constant c > 0 depends on m 0 , E 0 whereas C(t) is given by Similarly to the Boltzmann equation, one can prove with the techniques given in [14, Section 3] that f L 1 k t −k/γ . Therefore, choosing we guarantee that C(t) t −β . Thus, This proves the generation of the exponential tail.
As previously expressed for the Boltzmann equation, the propagation/generation of the Fisher information and the exponential moments imply the propagation/generation of the exponential moments for the gradient of solutions. For any s ∈ (0, γ]

APPENDIX A. REGULARITY ESTIMATES FOR THE BOLTZMANN EQUATION
We include here some classical results in the theory of the homogeneous Boltzmann equation. We use them in the core of this note.
be the initial data. Then, the unique solution to (1.3) satisfies: for any ε > 0 there exists C ε > 0 such that , and assume 0 f 0 ∈ L 1 2 (R d ). Then, for every k > 0 there exists a constant C k 0 depending only on k, b, and the initial mass and energy of f 0 , such that If, in addition, m k (0) < ∞ then sup . Let 0 f (t) ∈ L 1 2+ε (R d ), with ε > 0, be such that for some C c > 0 Then, there exists κ 0 depending on C, c, b and sup t 0 f (t) L 1 2+ε such that Moreover, Proof.
For the last inequality we used the Sobolev embedding valid for d 3.
for some positive constant C d depending only on the dimension d.
Proof. Set g(t, v) = ∇f (t, v) so that ∂ t g(t, v) = ∇Q(f, f )(t, v). Applying the inner product of such equation with v 2η g(t, v) and integrating over R d we get that so that, after using (A.1), Thus, Since we estimate this last integral as Using (A.4) and Theorem A.4 in (A.3), we obtain that according to Theorems A.2 and A.3 and our hypothesis on f 0 , it follows that which readily gives that .
This together with the propagation of f L 2 η proves the result.

APPENDIX B. REGULARITY ESTIMATES FOR THE LANDAU EQUATION
We collect here known results, extracted from [14] about the regularity of solutions to the Landau equation (1.5). We begin with classical estimate related to the matrix A(z). For (i, j) ∈ [ [1,3] ] 2 , we recall that A(z) = (A i,j (z)) i,j with A i,j (z) = |z| γ+2 δ i,j − z i z j |z| 2 , and introduce B i (z) = k ∂ k A i,k (z) = −2 z i |z| γ .
Here, f (t, v) will denote a weak solution to (1.5) associated to an initial datum f 0 with mass m 0 , energy E 0 and entropy H 0 . One has then the following result about propagation and appearance of moments, see [14,Theorem 3].

Lemma B.2. For any s 0,
Moreover, for any t 0 > 0 and any s > 0 there exists C > 0 depending only on m 0 , E 0 , H 0 , s and t 0 such that sup t t0 R 3 v s f (t, v)dv C.
We have then the following result about instantaneous appearance and uniform bounds for regularity, see [14,Theorem 5]. Lemma B.3. For any t 0 > 0, any integer k ∈ N and s > 0, there exists a constant C t0 > 0 depending only on m 0 , E 0 , H 0 , k, s and t 0 > 0 such that We end this section with a simple estimate for integral of the type yielding to estimate (3.5). Set, for notational simplicity, Let us emphasize that, contrary to the previous results of this appendix, in the following lemma, the dimension d 2 is arbitrary and the function f is not restricted to a solution to the Landau equation.
Lemma B.4. For any k 0 and any ε > 0, there exists C k (ε) > 0 such that, for any nonnegative f ∈ L 1 k+ε (R d ), one has Furthermore, for any k 0, δ > 0 and any ε > 0, there exist K k (δ) and C k (ε) such that, for any nonnegative f ∈ L 1 k+ε (R d ), one has