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Partial Regularity in Time for the Space Homogeneous Landau Equation with Coulomb Potential

We prove that the set of singular times for weak solutions of the space homogeneous Landau equation with Coulomb potential constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has Hausdorff dimension at most 1/2.


Introduction
We are concerned with the regularity of weak solutions f ≡ f (t, v) ≥ 0 a.e. to the space homogeneous Landau equation with Coulomb interaction potential where the collision kernel a is the matrix field a(z) = ∇ 2 z = Π(z) z , with Π(z) ∶= I − z z ⊗2 .
(In other words, Π(z) is the orthogonal projection on (Rz) ⊥ for all z ∈ R 3 ∖ {0}.) This equation is used in the description of collisions between charged particles in plasma physics (see [24] or §41 in [27]). Equivalently, the Landau equation with Coulomb potential takes the form Villani has proved the global existence of a special kind of weak solutions of the Cauchy problem for (1), known as "H-solutions", for all initial data with finite mass, energy and entropy (Theorem 4 (i) in [35]). Whether H-solutions of (1) with smooth initial data remain smooth for all times or blow up in finite time is one of the outstanding problems in the mathematical analysis of kinetic models: see §1. 3 (2) in chapter 5 of Villani's monograph [36]. The form (2) of the Landau equation suggests that blow-up might occur in finite time, by analogy with the semilinear heat equation ∂ t u(t, x) = ∆ x u(t, x) + u(t, x) 2 : see the last statement in Theorem 1 of [37] (for nonnegative solutions of the initial boundary value problem on a smooth bounded domain of R 3 with homogeneous Dirichlet condition at the boundary), or section 5.4 in [7]. On the other hand, global existence of classical, radially symmetric and nonincresing (in the velocity variable) solutions has been established for the equation ∂ t u(t, x) = ((−∆ x ) −1 u)(t, x)∆ x u(t, x)+αu(t, x) 2 which can be seen as an "isotropic" variant of (2) (i.e. with the diffusion matrix A[f ] replaced with the diffusion coefficient 4π((−∆ v ) −1 f ) and the constant 8π replaced with the (smaller) coefficient 4πα), for all α ∈ [0, 74 74 ) in [12,23], then for α = 1 in [16].
These arguments being somewhat inconclusive, the recent research on the Cauchy problem for (1) has produced mostly conditional results -with one notable exception, which we shall discuss in more detail below. For instance, the uniqueness of bounded solutions of (1) has been proved in [13]; the regularity of radial L p solutions with p > 3 2 has been proved in [16] (see also [17]); the case of nonradial L p solutions with p > 3 2 and moments in v of order larger than 8 is treated in [30], while the large time behavior of H-solutions of (1) is discussed in [5]. Of course, the existence and uniqueness theory for the space inhomogeneous Landau equation is even harder, and most of the existing results on that equation bear on near-Maxwellian equilibrium global solutions [18,6,8], apart from the very general weak stability result in [28]. There are also various local existence and uniqueness results, as well as smoothing estimates for solutions of the space inhomogeneous Landau equation under the assumption of locally (in time and space) bounded moments in v: see [19,21,20] (notice that [21,20] require only that the distribution function has bounded moments in v of order larger than 9 2). Since the present paper is focused on the Coulomb case, which is the most interesting on physical grounds, we have omitted the rather large literature on the generalizations of the Landau equation where the collision kernel a is replaced with z γ+2 Π(z) with γ > −3.
Perhaps the most remarkable recent contribution to the mathematical theory of (1) is the following result by Desvillettes [10]: any (nonnegative) H-solution of (1) on [0, T ] × R 3 with finite mass, energy and entropy satisfies the bound (see Theorem 1 in [10]).This bound implies in particular the propagation of moments in v of arbitrary order for such solutions of (1) (see Proposition 2 in [10]). Both this bound and the propagation of moments are of key importance in the present work. Our main result is the following partial regularity statement, which will be presented and discussed in detail in the next section.
Main Theorem. The set of singular times of any H-solution of (1) constructed by the approximation scheme described in [35] has Hausdorff dimension at most 1 2 .

Main Results
The prototype of all partial regularity results in partial differential equations is Leray's observation [25] that the set of singular times of any Leray solution to the Navier-Stokes equations for incompressible fluid dynamics in three space dimensions has Hausdorff dimension at most 1 2 (see §34 in [25], especially, formula (6.5)). Leray's remark was later considerably refined by Scheffer [29], and by Caffarelli, Kohn and Nirenberg [4] (see also [32,26,34]) One key ingredient in Leray's observation is the energy inequality satisfied by Leray solutions to the Navier-Stokes equations. Our first task is therefore to establish an analogous inequality for solutions to (1). Henceforth we denote by L p k (R 3 ) the set of measurable g ≡ g(v) defined a.e. on R 3 such that We first recall that an H-solution to (1) on the time interval for a.e. t ≥ 0, and . Of course, the notion of H-solution is based on the observation that classical solutions of the Landau equation with appropriate decay as v → +∞ satisfy (6) d dt where The notation H(f ) to designate this quantity comes from the "Boltzmann H Theorem", which is the analogous differential inequality for the Boltzmann equation in the kinetic theory of gases. The positivity of the entropy production − d dt H(f )(t) comes from the symmetries in the Landau collision integral (i.e. the right hand side of (1)), which are hidden in the nonconservative parabolic form (2).
is an H-solution which satisfies, for some negligible set N ⊂ (0, T ), some q ∈ (1, 2) and some C E > 0, the truncated entropy inequality The term "suitable solution" is used here by analogy with the notion of suitable weak solutions of the Navier-Stokes equations defined in [4], which satisfy a local (in space) variant of the Leray energy inequality. Indeed, one can think of (7) as a variant of the entropy inequality for (1) localized in the set of values of the solution.
Our first result is that the approximation scheme used in [35] to construct an H-solution of the Cauchy problem for (1) converges to a suitable solution of (1).
Then, there exists an H-solution of (1) with initial data f in and a negligible set The proof of this proposition is based on the analysis in [35], with the following additional observations (a) the truncated entropy −H + (f κ) is not a increasing function of time as the original entropy −H(f )(t), but combining the symmetries of the collision integral and the fact that div(div a) = ∆ 2 z = −4πδ 0 shows that the negative part of the truncated entropy production involves the depleted nonlinearity κ(f − κ) + = min(f, κ)(f − κ) + instead of the full nonlinearity f 2 (ln(f κ)) + which the nonconservative form (2) seems to suggest; (b) the Desvillettes argument leading to the inequality (3) can be modified to handle the positive part of the truncated entropy production, and (c) the Desvillettes propagation of moments argument can be used to purge the positive part of the truncated entropy production from the weight (1 + v ) −3 at the expense of introducing the exponent q < 2 in the inequality (7).
The notion of relative entropy of f to a Maxwellian M ρ,u,θ for some ρ, θ > 0 and u ∈ R 3 is used traditionally in kinetic theory to measure the distance of f to M ρ,u,θ (see for instance section 3 in [2]). The term −H + (f κ) can be thought of as the truncated variant of the relative entropy −H(f M (2πθ) 3 2 κ,0,θ ) in the large temperature limit θ → +∞; its time derivative benefits therefore from the same cancellations which lead to the nonnegative entropy production term in (6), up to the lower order perturbation term due to the truncation, which lead ultimately to the depleted nonlinearity on the right hand side of (7). The truncated entropy −H(f κ) is therefore the best imaginable tool for applying the Stampacchia truncation method to the Landau equation (1) without loosing the symmetries of the Landau collision integral. This is precisely the reason why the truncated entropy has been introduced in [15] to handle a class of reaction-diffusion systems which is very close to a kinetic equation (in the discrete velocity setting). In particular, the inequality (7) is similar to Lemma 3.1 in [15] -notice however the Remark 3.2 in [15] which states that extensions of these tools outside of the discrete velocity setting is far from straightforward.
Next we study the occurence of blow-up times for suitable solutions of (1). The set of singular times for a suitable solution of (1) is defined by analogy with §33-34 in [25].
Then the set S[f, (0, +∞)] of singular positive times for f satisfies Our approach to Theorem 2.4 departs from the general method for proving partial regularity results outlined in section 1 of [4]. First, we do not use any dimension analysis similar to formula (1.9) of [4] on solutions of the Landau equation. One easily checks that if f is a classical solution of (1), then is a classical solution of (1). But although the Landau equation (1) has a twoparameter family of invariant scaling transformations (richer than the single-parameter family of invariant scaling transformations of the Navier-Stokes equations), the three conserved quantities in (4) and the entropy −H(f )(t) are not simultaneously preserved by any one of these transformations (except for λ = µ = 1). Within this family of scaling transformations, we retain only the case which leaves the inequality (7) invariant (upon transforming κ into λκ). Notice that this scaling transformation depends on the Lebesgue exponent q, which depends itself on the decay in v of f in through (8). The "optimal" scaling corresponds to the limit case k → ∞, i.e. to q → 2 and therefore to γ → 2 in (10). This "optimal" scaling is exactly the same as the invariant scaling for the squared velocity field in the Navier-Stokes equations, which explains why the bound on the Hausdorff dimension of the set of singular times in both equations is the same. However, the "optimal" scaling γ = 2 is only a limit case and cannot be used directly on (1), at variance with the case of the Navier-Stokes equations. A first step towards partial regularity is to prove that any suitable solution of the Landau equation (1) whose truncated entropy has small enough L 1 norm on some time interval is bounded on a smaller time interval.
Proposition 2.5. Let f be a suitable solution to the Landau equation (1) on [0, 1] satisfying (7) for some negligible set N ⊂ (0, 1], some q ∈ ( 6 5 , 2), and some C ′ E > 0. There . This proposition is proved by using the parabolic variant of the De Giorgi method, following the analysis in section 3 of [15]. However, the proof of this result diverges from the classical De Giorgi method in at least one important step. While the key idea in De Giorgi's solution [9] of Hilbert's 19 th problem is to consider the equations satisfied by the partial derivatives of the extremal as a linear elliptic equations with bounded coefficients, our proof of Proposition 2.5 is based on the truncated entropy inequality which makes critical use of the symmetries of the right hand side of (1). It seems very unclear that Proposition 2.5 could follow from applying the De Giorgi strategy to the conservative form of the Landau equation without using cancellations suggested by the integral form (1) of that equation. For the same reason, the De Giorgi method is used in the present paper in a way that differs from earlier applications of the same method to other kinetic equations [14,22]. Notice that the restriction q > 6 5 comes from the Sobolev embedding used in the nonlinearization procedure in the De Giorgi method.
The key result leading to partial regularity is the following proposition involving only the dissipation term in (7), by analogy with the argument in §34 of [25], or Proposition 2 of [4] for the Navier-Stokes equations.
Once Proposition 2.6 is proved, the proof of Theorem 2.4 follows by a Vitali covering argument as in section 6 of [4].
The idea of using the De Giorgi method to prove partial regularity in time of suitable solutions of (1) comes from [34], where the partial regularity result in [4] for the Navier-Stokes equations is recast in terms of the De Giorgi arguments.
The outline of the paper is as follows: the existence proof of suitable solutions (Proposition 2.2) occupies section 3. Then, section 4 contains the proof of Proposition 2.5, while section 5 gives the proof of Proposition 2.6. Finally, the proof of the main theorem (Theorem 2.4) is given in section 6.

Existence of Suitable Solutions
3.1. Proof of Proposition 2.2. The argument is split in several steps, some of which are already used in the construction of a H-solution in [35], but need to be recalled for the sake of clarity.
Step 1: Truncated and regularized initial data. For each integer n ≥ 1, set Obviously Likewise since the function z ↦ z(ln z) + is nondecreasing and convex

Because of the elementary inequality
Let f n ≡ f n (t, v) be the solution of the truncated and regularized Landau equation The truncated and regularized Landau has a unique global smooth solution f n satisfying and the following variant of the H Theorem: with the notation Hence (12) becomes In particular, one has In the limit as n → ∞, one has where f is an H-solution of the Cauchy problem for (1) with initial data f in , according to [35] on p. 297.
Step 4: Truncated entropy inequality For κ > 0, multiplying both sides of the truncated and regularized Landau equation by (ln(f n (t, v) κ)) + and integrating in v shows that In the end, for each κ > 0 and each t 1 , t 2 such that 0 ≤ t 1 ≤ t 2 < ∞, one has Step 5: A local lower bound for the truncated entropy production We begin with the following auxiliary result, whose proof is deferred until the end of the present section. This is an extension of Theorem 3 in [10] to truncated entropies.
Lemma 3.1. The sequence f n constructed in step 1 of the present section satisfies the inequality: Applying Lemma 3.1 shows that for all t 1 , t 2 such that 0 ≤ t 1 < t 2 < ∞. By Proposition 4 [10] recalled in step 3 above, for all T > 0 and all t ∈ [0, T ], one has By Hölder's inequality, choosing p ′ = 2 q (so that q ∈ (1, 2)) Hence Step 6: Passing to the limit in the truncated entropy. By Sobolev embedding, for each T > 0, one has (f n ) 1 q = O(1) L 2 (0,T ;L q * (R 3 )) , and hence f n = O(1) L 2 q (0,T ;L q * q (R 3 )) . Since 1 < q < 2 < q ′ , and is equiintegrable and tight on [0, T ]×R 3 since we have seen that there exists α 1 , α 2 > 0 such that Hence, for all κ > 0, one has ) as n → +∞ , and hence, possibly after extracting a subsequence if necessary The subsequence and the negligible exceptional set of times where the pointwise convergence above is not valid may depend on κ. Henceforth, we restrict our attention to the case of κ being a rational number, and since there are countably many such numbers, the union of the exceptional sets corresponding to each rational κ is another negligible set N . Hence, by diagonal extraction H + (f n (t, ⋅) κ) → H + (f (t, ⋅) κ) for all t ∈ R + ∖ N and all κ ∈ Q ∩ [1, +∞) as n → ∞.
Step 7: Passing to the limit in the truncated entropy production. One has and By continuity of the gradient in the sense of distributions, we already know that On the other hand Since L 2 (0, T ; L q (R 3 )) is reflexive because 1 < q < 2 (see Corollary 2 in Chapter IV, §1 of [11]), the Banach-Alaloglu theorem implies that, possibly after extracting a subsequence, and therefore in D ′ ((0, T ) × R 3 ). Hence and by compactness and uniqueness of the limit For all t 1 < t 2 ∈ R + ∖ N , and all κ ∈ Q ∩ [1, +∞), one has and passing to the limit as n → +∞, one finds that where the first inequality is Proposition III.5 (iii) of [3].
Step 8: Writing (7) for all κ ≥ 1. Let κ > 1, and let κ j be an increasing sequence of rational numbers converging to κ. Since h + is nondecreasing, for all t ∈ R + ∖ N , for all t ∈ R + ∖ N by dominated convergence. By the same token a.e. on (0, +∞) × R 3 , so that as j → +∞ by dominated convergence. Since f satisfies (7) for each k ∈ Q ∩ [1, +∞), and since this set is dense in [1, +∞), we conclude from the argument above that f satisfies (7) for each κ ≥ 1, by passing to the limit as j → +∞ in (7) written for κ j . This concludes the proof of Proposition 2.
which holds for all X, Y ∈ R 3 , with the notation We have denoted By Cramer's formula Using the lower bound for det Γ ji [λ, f n (t, ⋅)] obtained in Lemma 2 of [10], one finds that This concludes the proof of Lemma 3.1.

Proof of Proposition 2.6
The key idea in the proof of Proposition 2.6 is to apply the De Giorgi local boundedness argument (Proposition 2.5) to a scaled suitable solution of (1). As explained in section 2, the 2-parameter group of scaling transformations (9) leaving (1) invariant is not rich enough to contain a subgroup leaving the conserved quantities (4) and the entropy invariant. However, this is unessential for our argument, and we only seek to leave the truncated entropy inequality (7) invariant.
Step 1: Scaling solutions to the Landau equation with Coulomb interaction. Let f be a suitable solution to the Landau equation with Coulomb interaction, satisfying (7) for some Lebesgue negligible set N ⊂ (0, +∞), some q ∈ ( 4 3 , 2), and with a given constant C ′ E > 0. According to Proposition 2.2, the existence of such a solution is known provided that the initial data f t=0 =∶ f in satisfies (8) For each integer n ≥ 0, set ǫ n ∶= 2 −n and (Observe that, up to the translation in time, this is precisely the subgroup of scaling transformations (10)). One easily checks that f n is a H-solution to the Landau equation for each n ≥ 0, that and that dT .
Since f satisfies (7) for all κ ≥ 1 and all t ∈ N , the scaling transformations (21) and (22) imply that for all κ ≥ 1 and all t ∈ N n , where Since N n is the image of the Lebesgue negligible set N by an affine transformation, it is Lebesgue negligible. On the other hand f n satisfies the truncated entropy inequality (23) for all κ ≥ 1, or equivalently whenever κ n ∶= ǫ γ n κ ≥ 2 −nγ . Since q > 4 3 , one has γ > 0, so that This proves that f n is a suitable solution to the Landau equation satisfying (7) with the same Lebesgue exponent q ∈ ( 4 3 , 2), the same constant C ′ E , and with the new Lebesgue-negligible set N n .
From now on, and until the very end of this proof, we forget completely the suitable solution f , and consider only the sequence of scaled solutions f n . The goal is to show that, by choosing η 1 small enough, one can find some n large enough so that the scaled solution f n satisfies the assumption of Proposition 2.5. Applying Proposition 2.5 shows that this f n is locally bounded near t = 1, which implies in turn that f is locally bounded (by a very large number, whose size is unessential) near t = 1 (here again, the size of the domain on which f is bounded is of no interest for the partial regularity result).
Step 2: Replacing the local entropy with Lebesgue norms. In this section we seek to apply (15) to each suitable solution f n with truncation parameter κ n = ǫ γ n κ = 1.
Lemma 5.1. Let X n be a sequence of positive numbers such that where 0 < ρ < 1 2 and X 0 , X 1 ≤ M with M ≥ 1. Then X 2n and X 2n+1 ≤ max 2ρ, (2ρ) Proof. This is proved by an elementary induction argument. The desired conclusion holds for n = 0, i.e.
At this point, we return to the original suitable solution f . The last inequality is equivalent to which completes the proof of Proposition 2.6.