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Well-posedness of the spatially homogeneous Landau equation for soft potentials

We consider the spatially homogeneous Landau equation of kinetic theory, and provide a differential inequality for the Wasserstein distance with quadratic cost between two solutions. We deduce some well-posedness results. The main difficulty is that this equation presents a singularity for small relative velocities. Our uniqueness result is the first one in the important case of soft potentials. Furthermore, it is almost optimal for a class of moderately soft potentials, that is for a moderate singularity. Indeed, in such a case, our result applies for initial conditions with finite mass, energy, and entropy. For the other moderatley soft potentials, we assume additionnally some moment conditions on the initial data. For very soft potentials, we obtain only a local (in time) well-posedness result, under some integrability conditions. Our proof is probabilistic, and uses a stochastic version of the Landau equation, in the spirit of Tanaka.


Introduction and results
1.1. The Landau equation. We consider the spatially homogeneous Landau equation in dimension 3 for soft potentials. This equation of kinetic physics, also called Fokker-Planck-Landau equation, has been derived from the Boltzmann equation when the grazing collisions prevail in the gas. It describes the density f t (v) of particles having the velocity v ∈ R 3 at time t ≥ 0 : where ∂ t = ∂ ∂t , ∂ i = ∂ ∂vi and a(z) is a symmetric nonnegative matrix, depending on a parameter γ (we will deal here with soft potentials, that is γ ∈ (−3, 0)), defined by (1.2) a ij (z) = |z| γ (|z| 2 δ ij − z i z j ).
The weak form of (1.1) writes, for any test function ϕ : where the operator L is defined by ∂ j a ij (z) = −2|z| γ z i , for i = 1, 2, 3. (1.4) We observe that the solutions to (1.1) conserve, at least formally, the mass, the momentum and the kinetic energy: for any t ≥ 0, We classically may assume without loss of generality that R 3 f 0 (v)dv = 1. Another fundammental a priori estimate is the decay of entropy, that is solutions satisfy, at least formally, for all t ≥ 0, We refer to Villani [17,18] for many details on this equation, its physical meaning, its derivation from the Boltzmann equation, and on what is known about this equation.

1.2.
Existing results and goals. One usually speaks of hard potentials for γ > 0, Maxwell molecules for γ = 0, soft potentials for γ ∈ (−3, 0), and Coulomb potential for γ = −3. The Landau equation is a continuous approximation of the Boltzmann equation: when there are infinitely many infinitesimally small collisions, the particle velocities become continuous in time, which can be modeled by equation (1.1). The most interesting case is that of Coulomb potential, since then the Boltzmann equation seems to be meaningless. Unfortunately, it is also the most difficult case to study. However, the Landau equation can be derived from the Boltzmann equation with true very soft potentials, that is γ ∈ (−3, −1). The main idea is that the more γ is negative, the more the Landau equation is physically interesting. We refer again to [18] for a detailed survey about such considerations. Another possible issue concerns numerics for the Boltzmann equation without cutoff: one can approximate grazing collisions by the Landau equation.
Let us mention that existence of weak solutions, under physically reasonnable assumptions on initial conditions, has been obtained by Villani [17] for all previously cited potentials.
We study here the question of uniqueness (and stability with respect to the initial condition). This question is of particular importance, since uniqueness is needed to justify the derivation of the equation, the convergence from the Boltzmann equation to the Landau equation, the convergence of some numerical schemes, ... Villani [16] has obtained uniqueness for Maxwell molecules, and this was extended by Desvillettes-Villani [5] to the case of hard potentials. To our knowledge, there is no available result in the important case of soft potentials. We adapt in this paper the ideas of some recent works on the Boltzmann equation, see [7,8] (see also Desvillettes-Mouhot [6] for other ideas). We will essentially prove here that uniqueness and stability hold in the following situations: (a) for γ ∈ (γ 0 , 0), with γ 0 = 1 − √ 5 ≃ −1.236, as soon as f 0 satifies the sole physical assumptions of finite mass, energy and entropy; (b) for γ ∈ (−2, γ 0 ], as soon f 0 has finite mass, energy, entropy, plus some finite moment of order q(γ) large enough; (c) for γ ∈ (−3, −2], as soon f 0 has a finite mass, energy, and belongs to L p , with p > 3/(3 + γ), and the result is local in time.
Observe that (a) is extremely satisfying, and (c) is quite disappointing. Observe also that we obtain some much better results than for the Boltzmann equation [7,8], where well-posedness is proved in the following cases: for γ ∈ (−0.61, 0) and f 0 with finite mass, energy, entropy; for γ ∈ (−1, −0.61) and f 0 with finite mass, energy, entropy, and a moment of order q(γ) sufficiently large, and for γ ∈ (−3, −1] and f 0 ∈ L p with finite mass and energy, with p > 3/(3 + γ), and the result being local in time in the latest case.
In [17], Villani proved several results on the convergence of the Boltzmann equation to the Landau equation. His results work up to extraction of a subsequence. Of course, our uniqueness result allows us to get a true convergence.
1.3. Some notation. Let us denote by C 2 b the set of C 2 -functions ϕ : R 3 → R bounded with their derivatives of order 1 and 2. Let also L p (R 3 ) be the space of measurable functions f with ||f || L p := ( R 3 f p (v)dv) 1/p < ∞, and let P(R 3 ) be the space of probability measures on R 3 . For k ≥ 1, we set For α ∈ (−3, 0), we introduce the space J α of probability measures f on R 3 such that Observe that (see [8, (5.2)]): Remark 1.1. For α ∈ (−3, 0) and p > 3/(3 + α), there exists a constant C α,p > 0 such that for all nonnegative measurable f : For a nonnegative function f ∈ L 1 (R 3 ), we denote its entropy by Finally we denote x ∧ y = min(x, y), x ∨ y = max(x, y) for x, y ∈ R + , and z.z the scalar product of z,z ∈ R 3 and L(X) the distribution of a random variable X. For some set A we write 1 A the usual indicator function of A.

Weak solutions.
We introduce now the notion of weak solution for the Landau equation. Some much more refined definitions were introduced by Villani [17] to allow solutions with only finite mass, energy, and entropy.
We observe here that for This guarantees that all the terms are well-defined in the definition below.
We say that f solves the Landau where the operator L is defined by (1.3).
1.5. The Wasserstein distance and the Monge-Kantorovich problem. Given π ∈ P 2 (R 3 × R 3 ), we respectively denote by π 1 and π 2 its first and second marginals on R 3 . For two probability measures µ, ν ∈ P 2 (R 3 ) and π ∈ P 2 (R 3 × R 3 ), we write π < µ ν if π 1 = µ and π 2 = ν. The Wassertein distance W 2 on P 2 (R 3 ) is defined by The set (P 2 (R 3 ), W 2 ) is a Polish space, see e.g. Rachev and Rüschendorf [13]. The topology is stronger that the usual weak topology (for more details, see Villani, [19] Theorem 7.12). It is well-known (see e.g. Villani [19] Chapter 1 for details) that the infimum is actually a minimum, and that if µ (or ν) has a density with respect to the Lebesgue measure on R 3 , then there is a unique π < µ ν such that is measurable, and if µ λ has a density for all λ, then the function λ → π λ is measurable, see Fontbona-Guérin-Méléard [9].
1.6. A general inequality. Our main result reads as follows.
We also assume that f t (orf t ) has a density with respect the Lebesgue measure on R 3 for each t ∈ [0, T ]. Then there exists an explicit constant C γ > 0, depending only on γ, such that for all t ∈ [0, T ], , so that f t has a density for all times. Thus this result always applies for solutions with finite entropy. This result asserts that uniqueness and stability hold in Using Remark 1.1, we immediately deduce that uniqueness and stability also hold in . Then there exists T * > 0 depending on γ, p, ||f 0 || p such that there is an unique weak solution in L ∞ loc ([0, T * ), L p ).
1.8. Plan of the paper. The paper is organized as follows: we first prove Theorem 1.3 in Section 2, by means of stochastic Landau processes. Next, we prove Corollary 1.4 in Section 3.

A general inequality
We give in this section the proof of Theorem 1.3. In the whole section, T > 0 and γ ∈ (−3, 0) are fixed, and we consider two We also assume that f t has a density with respect to the Lebesgue measure on R 3 for each t ≥ 0, which implies the uniqueness of the minimizer in W 2 (f t ,f t ).
We now introduce two coupled Landau stochastic processes, the first one associated with f , the second one associated withf , in such a way that they remain as close to each other as possible. Our approach is inspired by [11], which was itself inspired by the work of Tanaka [14] on the Boltzmann equation.
For any s ∈ [0, T ], we denote by R s the unique solution of the Monge-Kantorovich transportation problem for the couple . Let us notice that the map s → R s is measurable thanks to Theorem 1.3 of [9]. [20]): W is an orthogonal martingale measure with covariance R s (dv, dṽ)ds. Let us now consider two random variables We finally consider the two following R 3 -valued stochastic differential equations.
We denote (F t ) t≥0 the filtration generated by W and where N is the set of all P-null subsets.
The processes (V t ) t∈[0,T ] and (Ṽ t ) t∈[0,T ] are called Landau processes associated with f andf respectively. We start the proof with a simple remark.
Proof of Proposition 2.1. We just deal with (2.1), the study of (2.2) being of course the same.
Step 1. We start with the proof of (i), that is existence and uniqueness of a solution for (2.1). We consider the map Φ which associates to a continuous adapted process Step 1.1. Let us first prove that which is finite thanks to the conditions imposed on V 0 , f , and X. Using Doob's inequality, we easily get, for some constant C, Using that the first marginal of R s is f s , that |σ(z)| 2 ≤ |z| γ+2 ≤ (1 + |z| 2 + |z| γ ), we observe that by definition of J γ , see (1.5). Next, using that |b(z)| = 2|z| γ+1 ≤ 2 + 2|z| γ + 2|z|, Step 1.2. Let us now consider two adapted processes X, Y ∈ C([0, T ], R 3 ), and show that Arguing as previously and using Remark 2.2, we deduce that We conclude by using the Cauchy-Schwarz inequality.
Step 1.3. The uniqueness of the solution to (2.1) immediately follows from Step 1.2. Indeed, consider two solutions V 1 and V 2 . Then V 1 = Φ(V 1 ) and Step 1.4. Finally, one classically obtains the existence of a solution using a Picard iteraction: consider the process V 0 defined by V 0 t = V 0 , and then define by induction V n+1 = Φ(V n ) (this is well-defined thanks to Step 1.1.). Using Step 1.2 and classical arguments, one easily checks that there exists a continuous adapted process V such that E[sup [0,T ] |V t − V n t | 2 ] tends to 0. It is not difficult to pass to the limit in V n+1 = Φ(V n ), whence V = Φ(V ), and thus V solves (2.1).
Step 2. We now prove (ii). Let V be the unique solution of (2.1) and g s = L(V s ) for all s ∈ [0, T ].
Step 2.1. We first check that the family g solves the linear Landau equation: for any ϕ ∈ C 2 b (R 3 ), Lϕ(x, v)g s (dx)f s (dv)ds.
with L defined by (1.3). Applying the Itô formula, we immediately get Taking expectations (which makes vanish the first integral), using that the first marginal of R s is f s and that L(V 0 ) = f 0 , we obtain from which the conclusion follows, recalling (1.3), since σ.σ t = a. Step Proof. First of all, we observe that since R s (dv, dṽ) has the marginals f s (dv) andf s (dṽ), we may rewrite equations (2.1) and (2.2) as Using the Itô formula and taking expectations, we obtain Using Remark 2.2, we deduce that for some constant C γ , from which the result immediately follows.
We are finally in a position to conclude this section.
By the same way, E(|Ṽ s −ṽ| γ ) ≤ J γ (f s ). On the other hand, since the first marginal of R s is f s , we deduce that Using finally the Gronwall Lemma, we get

Applications
We now want to show that the uniqueness result proved in the previous section is relevant, in the sense that solutions in L ∞ ([0, T ], P 2 ) ∩ L 1 ([0, T ], J γ ) indeed exist. We will use Remark 1.1, which says that L p ∩ L 1 ⊂ J γ for p > 3/(3 + γ).
We first recall that moments of solutions propagate, we give some ellipticity estimate on the diffusion coefficient, and recall the chain rule for the Landau equation. Then Corollary 1.4-(i) is obtained by using the dissipation of entropy. Finally, Corollary 1.4-(ii) is checked, using a direct computation.
3.1. Moments. We first recall the following result, which shows that moments of the solutions to the Landau equation propagate. We will use moments only in the case where γ ∈ [−2, 0), and the proposition below is quite easy. We refer to [18, Section 2.4. p 73]. When γ ∈ (−3, −2], the situation is much more delicate, but Villani [15,Appendix B p 193] has also proved the propagation of moments.
One could easily extend this result to the case where γ ∈ (−3, −2). However, we will not use it.

3.4.
Moderately soft potentials. Using the dissipation of entropy, we will deduce, at least for γ not too much negative, the L p estimate we need. Such an idea was handled in the much more delicate case of the Boltzmann equation by Alexandre-Desvillettes-Villani-Wennberg [2]. We now divide the proof into several steps.
Proof of Corollary 1.4-(ii). We only have to prove the existence, since the uniqueness follows immediately follows from Theorem 1.3 and Remark 1.1. We thus assume that γ ∈ (−3, 0), p > 3/(3 + γ), and consider an initial condition f 0 ∈ P 2 (R 3 ) ∩ L p (R 3 ) (which implies that H(f 0 ) < ∞). Then Villani [17] has shown the existence of a weak solution (f t ) t∈[0,T ] ∈ L ∞ ([0, T ], P 2 ), for arbitrary T . But using the a priori estimate of Proposition 3.4, we deduce that this solution can be built in such a way that it belongs to L ∞ loc ([0, T * ), L p ). This concludes the proof.