A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equation

We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect implying the ultra-analyticity of both the fluctuation and its Fourier transform for any positive time.


Introduction
In the work [13], we consider the spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules in a close-to-equilibrium framework and study the smoothing properties of the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution. The Boltzmann equation describes the behavior of a dilute gas when the only interactions taken into account are binary collisions [5]. In the spatially homogeneous case with Maxwellian molecules, it reads as the equation for the density distribution of the particles f = f (t, v) ≥ 0, t ≥ 0, v ∈ R d , with d ≥ 2, where the non-linear term stands for the Boltzmann collision operator whose cross section is a non-negative function satisfying to the assumption for some 0 < s < 1. The notation a ≈ b means that a/b is bounded from above and below by fixed positive constants. The term (1.3) is not integrable in zero This non-integrability plays a major role regarding the qualitative behaviour of the solutions to the Boltzmann equation and this feature is essential for the smoothing effect to be present, see the discussion in [13] and all the references herein.
In [13], we consider the spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules (1.1) in the radially symmetric case with initial density distributions (1.4) f 0 = µ d + √ µ d g 0 , g 0 ∈ L 2 (R d ) radial, g 0 L 2 ≪ 1, close to the Maxwellian equilibrium distribution where | · | is the Euclidean norm on R d , in the physical 3-dimensional case d = 3. The main result in [13] shows that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution , enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to the fractional harmonic oscillator where 0 < s < 1 is the positive parameter appearing in the assumption (1.3). More specifically, we prove that under the assumption (1.4), the Cauchy problem (1.6) admits a unique global radial solution g ∈ L ∞ (R + t , L 2 (R 3 v )), which belongs to the Gelfand-Shilov class S 1/2s 1/2s (R 3 ) for any positive time . The definition of the Gelfand-Shilov regularity is recalled in appendix (Section 4.2).
In the present work, we study the spatially homogeneous Landau equation with Maxwellian molecules The Landau collision operator Q L (f, f ) is understood as the limiting Boltzmann operator in the grazing collision limit asymptotic [1,2,6,7,18], when s tends to 1 in the singularity assumption (1.3). In the physical 3-dimensional case, the linearized non-cutoff Boltzmann operator with Maxwellian molecules was actually showed to be equal to the fractional linearized Landau operator with Maxwellian molecules [12] (Theorem 2.3), commuting with the harmonic oscillator H = −∆ v + |v| 2 4 and the Laplace-Beltrami operator on the unit sphere S 2 . In view of this link between the linearized Boltzmann and Landau operators, and in analogy with the Gelfand-Shilov smoothing result proven in [13] for the spatially homogeneous non-cutoff Boltzmann equation, we may therefore expect that the spatially homogeneous Landau equation also enjoys specific Gelfand-Shilov smoothing properties. The purpose of this note is to confirm this insight and to check that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution associated to the spatially homogeneous Landau equation with Maxwellian molecules actually enjoys a Gelfand-Shilov regularizing effect in the class S 1/2 1/2 (R d ), implying the ultra-analyticity of both the fluctuation and its Fourier transform, for any positive time.

The Landau equation
The Landau equation written by Landau in 1936 [11] is the equation for the density distribution of the particles f = f (t, x, v) ≥ 0 at time t, having position x ∈ R d and velocity v ∈ R d , with d ≥ 2. The term Q L (f, f ) is the Landau collision operator associated to the Landau bilinear operator where a = (a i,j ) 1≤i,j≤d stands for the non-negative symmetric matrix In this work, we study the spatially homogeneous case when the density distribution of the particles does not depend on the position variable for Maxwellian molecules, that is, when the parameter γ = 0 in the assumption (2.2). At least formally, it is easily checked that the mass, the momentum and the kinetic energy are conserved quantities by this evolution equation 3) associated to the spatially homogeneous Landau equation with Maxwellian molecules and some quantitative features of the solutions were thoroughly studied by Villani [17]. The propositions 4 and 6 of the work [17] show that, for each non-negative measurable initial density distribution f 0 having finite mass and finite energy the Cauchy problem (2.3) admits a unique global classical solution f (t, v) defined for all t ≥ 0. Furthermore, this solution is showed to be a non-negative bounded smooth function , for any positive time t > 0.
In this work, we study a close-to-equilibrium framework. To that end, we consider the linearization of the spatially homogeneous Landau equation around the Maxwellian equilibrium distribution the original spatially homogeneous Landau equation (2.3) is reduced to the Cauchy problem for the fluctuation An explicit computation [12] (Proposition 2.1) shows that the linearized Landau operator with Maxwellian molecules acting on the Schwartz space is equal to where H = −∆ v + |v| 2 4 is the harmonic oscillator, stands for the Laplace-Beltrami operator on the unit sphere S d−1 and P k are the orthogonal projections onto the Hermite basis defined in Section 4.1. The linearized Landau operator is a non-negative operator By elaborating on the solutions constructed by Villani [17], the purpose of this note is to study the Gelfand-Shilov regularizing properties of the Cauchy problem (2.8) for the fluctuation around the Maxwellian equilibrium distribution. For the sake of simplicity, we may assume without loss of generality that the density distribution satisfies (2.4) with V = 0. Furthermore, by changing the unknown function f tof as we may reduce our study to the case when , be a non-negative initial density distribution having finite mass and finite energy such that Such an initial density distribution is rapidly decreasing with a finite temperature tail . The analysis of the evolution of the temperature tail led in [17] (Section 6, p. 972-974) shows that This implies that the fluctuation f = µ d + √ µ d g ≥ 0, around the Maxwellian equilibrium distribution defined by and therefore remains a tempered distribution for all t > 0. The following statement is the main result contained in this note: , be a non-negative measurable function having finite mass and finite energy such that when t > 0, be the unique global classical solution of the Cauchy problem associated to the spatially homogeneous Landau equation with Maxwellian molecules constructed by Villani [17]. Then, there exists a positive constant δ > 0 such that where · L 2 stands for the L 2 (R d v )-norm and P k are the orthogonal projections onto the Hermite basis defined in Section 4.1. In particular, this implies that the fluctuation belongs to the Gelfand-Shilov space S 1/2 1/2 (R d ) for any positive time Remark. The orthogonal projection P k : is well-defined on tempered distributions since the Hermite functions are Schwartz functions.
This result shows that the Cauchy problem (2.8) enjoys an ultra-analytic regularizing effect in the Gevrey class G 1/2 (R d ) both for the fluctuation and its Fourier transform in the velocity variable for any positive time Let us recall that the existence, uniqueness, the Sobolev regularity and the polynomial decay of the weak solutions to the Cauchy problem (2.3) have been studied by Desvillettes and Villani for hard potentials [8] (Theorem 6), that is, when the parameter satisfies 0 < γ ≤ 1 in the assumption (2.2). Under rather weak assumptions on the initial datum, e.g. f 0 ∈ L 1 2+δ , with δ > 0, they prove that there exists a weak solution to the Cauchy problem such that f ∈ C ∞ ([t 0 , +∞[, S (R d v )), for all t 0 > 0, and for all t 0 > 0, s > 0, m ∈ N, sup t≥t 0 The Gevrey regularity f (t, ·) ∈ G σ , for any σ > 1, for all positive time t > 0 of the solution to the Cauchy problem (2.3) with an initial datum f 0 with finite mass, energy and entropy satisfying was later established by Chen, Li and Xu for the hard potential case and the Maxwellian molecules case [3]. Under the same assumptions on the solution, this result was later extended to analytic regularity [4]: in the hard potential case and the Maxwellian molecules case. Regarding specifically the Maxwellian molecules case γ = 0, Morimoto and Xu established in the ultra-analyticity [14] (Theorem 1.1), The result of Theorem 2.1 allows to specify further the property of ultraanalytic smoothing proven by Morimoto and Xu in the close-to-equilibrium framework [14]. This result points out the specific decay of the fluctuation both in the velocity and its dual Fourier variable. As for the Boltzmann equation, the Gelfand-Shilov regularity seems relevant to describe the regularizing properties of the Landau equation in the close-toequilibrium framework.

Proof of Theorem 2.1
The proof of Theorem 2.1 is elementary and relies only on spectral arguments following the results established by Villani [17].
, be a non-negative measurable function having finite mass and finite energy such that Following [17] (p. 966), we may choose an orthonormal basis of R d diagonalizing the nonnegative symmetric quadratic form when t ≥ 0. These conditions imply that the fluctuation satisfies when t ≥ 0. The equation (3.2) may be rewritten for the fluctuation as It follows that By using that d j=1 α j = 0, we notice that where A +,j is the creation operator defined in Section 4.1. We consider the orthogonal projection onto the n + 1 lowest energy levels of the harmonic oscillator, where P k stands for the orthogonal projection onto the Hermite basis defined in Section 4.1.
As mentioned above, the orthogonal projection S n is well-defined on tempered distributions since the Hermite functions are Schwartz functions. This gives a sense for the orthogonal projection of the fluctuation S n g(t) ∈ S (R d v ) as a Schwartz function. Then, a direct computation shows that for all t ≥ 0, δ > 0, n ≥ 2, 1 2 ∂ t ( e tδH S n g 2 L 2 ) − δ(H(e tδH S n g), e tδH S n g) L 2 = Re(∂ t S n g, e 2δtH S n g) L 2 = − (d − 1)(H(e tδH S n g), e tδH S n g) L 2 − ((−∆ S d−1 )(e tδH S n g), e tδH S n g) L 2 since the harmonic oscillator and the Laplace-Beltrami operator on S d−1 are commuting selfadjoint operators. We deduce from (4.5), (4.6) and (4.7) that (e tδH S n (A +,j ) 2 g, e tδH S n g) L 2 = e 2δt ((A +,j ) 2 e tδH S n−2 g, e tδH S n g) L 2 = e 2δt (A +,j e tδH S n−2 g, A −,j e tδH S n g) L 2 and It follows that |α j | A +,j e tδH S n−2 g L 2 A −,j e tδH S n g L 2 + e −(4− δ 2 )dt 2e 4δt + 1 d j=1 |α j | e tδH S n g L 2 .
This implies that −1 < α j < d − 1, because d ≥ 2. We may choose the positive constant 0 < δ ≤ 1 such that sup It follows that We obtain that for all t ≥ 0, n ≥ 2, which implies that for all t ≥ 0, It follows that there exists a positive constant C > 0 such that where a + is the creation operator The family (φ n ) n≥0 is an orthonormal basis of L 2 (R). We set for n ≥ 0, with |α| = α 1 + · · ·+ α d . The family (Ψ α ) α∈N d is an orthonormal basis of L 2 (R d ) composed by the eigenfunctions of the d-dimensional harmonic oscillator where P k is the orthogonal projection onto E k whose dimension is k+d−1 d−1 . The eigenvalue d/2 is simple in all dimensions and E 0 is generated by the function where µ d is the Maxwellian distribution defined in (1.5). Setting where (e 1 , ..., e d ) stands for the canonical basis of R d . In particular, we readily notice that for all t ≥ 0, δ > 0, (4.7) e tδH A +,j = e δt A +,j e tδH , e tδH A −,j = e −δt A −,j e tδH .

4.2.
Gelfand-Shilov regularity. We refer the reader to the works [9,10,15,16] and the references herein for extensive expositions of the Gelfand-Shilov regularity. The Gelfand-Shilov spaces S µ ν (R d ), with µ, ν > 0, µ + ν ≥ 1, are defined as the spaces of smooth functions f ∈ C ∞ (R d ) satisfying to the estimates or, equivalently These Gelfand-Shilov spaces S µ ν (R d ) may also be characterized as the spaces of Schwartz functions f ∈ S (R d ) satisfying to the estimates In particular, we notice that Hermite functions belong to the symmetric Gelfand-Shilov space S 1/2 1/2 (R d ). More generally, the symmetric Gelfand-Shilov spaces S µ µ (R d ), with µ ≥ 1/2, can be nicely characterized through the decomposition into the Hermite basis (Ψ α ) α∈N d , see e.g. [16] (Proposition 1.2), where (Ψ α ) α∈N d stands for the Hermite basis defined in Section 4.1, and where is the d-dimensional harmonic oscillator. The Cauchy problem defined by the evolution equation associated to the harmonic oscillator (4.9) ∂ t f + Hf = 0, f | t=0 = f 0 ∈ L 2 (R d ), enjoys nice regularizing properties. The smoothing effect for the solutions to this Cauchy problem is naturally described in term of the Gelfand-Shilov regularity. The characterization (4.8) proves that there is a regularizing effect for the solutions to the Cauchy problem (4.9) in the symmetric Gelfand-Shilov space S 1/2 1/2 (R d ) for any positive time, whereas the smoothing effect for the solutions to the Cauchy problem defined by the evolution equation associated to the fractional harmonic oscillator (4.10) with 0 < s < 1, occurs for any positive time in the symmetric Gelfand-Shilov space S 1/2s 1/2s (R d ).