Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules
aa r X i v : . [ m a t h . A P ] A p r PROPAGATION OF STRETCHED EXPONENTIAL MOMENTSFOR THE KAC EQUATION AND BOLTZMANN EQUATIONWITH MAXWELL MOLECULES
Milana Pavi´c- ˇColi´c
Department of Mathematics and InformaticsFaculty of Sciences, University of Novi SadTrg Dositeja Obradovi´ca 4, 21000 Novi Sad, Serbia
Maja Taskovi´c
Department of MathematicsUniversity of PennsylvaniaDavid Rittenhouse Lab.209 South 33rd Street, Philadelphia, PA 19104
Abstract.
We study the spatially homogeneous Boltzmann equation for Ma-xwell molecules, and its 1-dimensional model, the Kac equation. We provepropagation in time of stretched exponential moments of their weak solutions,both for the angular cutoff and the angular non-cutoff case. The order of thestretched exponential moments in question depends on the singularity rate ofthe angular kernel of the Boltzmann and the Kac equation. One of the maintools we use are Mittag-Leffler moments, which generalize the exponential ones. Introduction
In this paper we study exponential tails (exponentially weighted L norms) ofweak solutions to the Kac equation and the spatially homogeneous Boltzmann equa-tion for Maxwell molecules. We show propagation in time of such tails, both in theso-called cutoff and the non-cutoff case.Both the Kac equation and the Boltzmann equation model the evolution of aprobability distribution of particles inside a gas interacting via binary collisions. Themodels we consider are spatially homogeneous, which means that the probabilitydistribution f ( t, v ) depends only on time t , velocity v , but not of the spatial variable x . The Kac equation is a model for a 1-dimensional spatially homogeneous gas inwhich collisions conserve the mass and the energy, but not the momentum. Onthe other hand, the spatially homogeneous Boltzmann equation describes a gas ina d -dimensional space, with d ≥
2, in which particles collisions are elastic, meaningthey conserve the mass, momentum and energy.The probability distribution function f ( t, v ), for time t ∈ R + and velocity v ∈ R d (with d = 1 for the Kac equation and d ≥ ∂ t f ( t, v ) = Z R Z π − π ( f ′ f ′∗ − f f ∗ ) b K ( | θ | ) d θ d v ∗ . (1) The spatially homogeneous Boltzmann equation on the other hand reads ∂ t f ( t, v ) = Z R d Z S d − ( f ′ f ′∗ − f f ∗ ) | v − v ∗ | γ b B (cid:16) v − v ∗ | v − v ∗ | · σ (cid:17) d σ d v ∗ , (2)which in the case of Maxwell molecules ( γ = 0) reduces to ∂ t f ( t, v ) = Z R d Z S d − ( f ′ f ′∗ − f f ∗ ) b B (cid:16) v − v ∗ | v − v ∗ | · σ (cid:17) d σ d v ∗ . (3)Details about the notation employed are contained in Section 2 for the Kacequation and in Section 3 for the Boltzmann equation. For now we only remarkthat for both equations we consider angular kernels b K and b B that may of may notbe integrable. When the angular singularity is non-integrable, our results dependon the singularity rate of the kernels.The Kac equation (1) and the corresponding Boltzmann equation for Maxwellmolecules (3) share many properties (one notable difference is that the Kac equa-tion does not conserve the momentum). In particular, both equations propagatepolynomial and exponential moments, whose definitions we now recall. Definition 1.1.
The polynomial moment of order q of the distribution function f is defined by m q ( t ) := Z R d f ( t, v ) h v i q d v. (4) Definition 1.2.
The stretched exponential moment of order s and rate α of thedistribution function f is defined by M α,s ( t ) := Z R d f ( t, v ) e α h v i s d v, α > . (5)In this paper, the special case when s = 2 is referred to as the Maxwellian moment.When f solves the Kac equation, the dimension d in these formulas is one. Wealso remark that we use the following notation h x i := p x + . . . x d , for any x = ( x , . . . , x d ) ∈ R d , d ≥
1. The results presented in this paper are also validwhen the moments are defined with absolute values | v | in place of h v i .In the case of the Kac equation, the study of stretched exponential moments goesback to [8]. There, the constant angular kernel is considered, and the propagationof stretched exponential moments of orders s = 1 and s = 2 is proved.For the Boltzmann equation, propagation of Maxwellian moments was proved inthe case of Maxwell molecules γ = 0 in [3, 4] and recently in [6] via Fourier trans-form techniques. The theory was later extended to hard potentials γ ∈ (0 ,
1] inthe context of Maxwellian moments in [5, 7, 10, 6], and in the context of stretchedexponential moments in [12, 1]. Finally, the propagation of stretched exponentialmoments for the non-integrable angular kernels are studied in [11, 13].In this paper, we generalize results of [8] to include more general orders ofstretched exponentials, namely s ∈ (0 , γ = 0, thus extending the result of [13]. ROPAGATION OF STRETCHED EXPONENTIAL MOMENTS 3
We point out that the method we employ in this paper differs from the approachin [8]. Elegant calculations for exponential moments (5) of order s = 1 and s = 2 in[8] are done directly at the level of exponential moments. In this manuscript, we takea different route and express exponential moments as infinite sums of polynomialmoments and then strive to show that such infinite sums are finite. Such approachhas been first developed in the context of the Boltzmann equation in [5], where thefollowing fundamental relation was noted M α,s ( t ) = Z R d f ( t, v ) ∞ X q =0 α q h v i sq q ! = ∞ X q =0 α q m sq ( t )Γ( q + 1) . (6)Finiteness of such sums can be studied by proving term-by-term geometric decay,or by showing that partial sums are uniformly bounded.Our proof is inspired by the works [1, 13], where the partial sum approach isdeveloped. Moreover, motivated by [13], we exploit the notion of Mittag-Lefflermoments, which serve as a generalization of stretched exponential moments andwhich are very flexible for the calculations at hand. We recall the definition andthe motivation for Mittag-Leffler moments in Section 5.The paper is organized as follows. A brief review of the Kac equation in providedin Section 2, while the review of the Boltzmann equation is contained in Section 3.In Section 4 we state our main result. Section 5 recalls the notion of Mittag-Lefflerfunctions and moments, one of the main tools in the proof of the main theorem.Section 6 contains another key tool - an angular averaging lemma with cancellation.In Section 7, the angular averaging lemma is used to derive differential inequalitiessatisfied by polynomial moments of the solution to the Cauchy problem under theconsideration. Finally, in Section 8 we provide the proof of the main theorem. TheAppendix lists auxiliary Lemmas.2. The spatially homogeneous Kac equation
The Kac model statistically describes the state of the gas in one dimension. Themain object is the distribution function f ( t, x, v ) ≥ t ≥ x ∈ R and velocity v ∈ R , and which changes in time due to the freetransport and collisions between gas particles. Assuming that collisions are binaryand that they conserve mass and energy, but not momentum, the evolution of thedistribution function is determined by the Kac equation.In this paper we assume that the distribution function does not depend on thespace position x , i.e. f := f ( t, v ). In that case, f satisfies the spatially homogeneousKac equation ∂ t f ( t, v ) = K ( f, f )( t, v ) , (7)where the collision operator K ( f, f ) is defined by K ( f, f )( t, v ) = Z R Z π − π ( f ′ f ′∗ − f f ∗ ) b K ( | θ | ) d θ d v ∗ , (8)with the standard abbreviations f ∗ := f ( t, v ∗ ), f ′ := f ( t, v ′ ), f ′∗ := f ( t, v ′∗ ).The velocities v ′ , v ′∗ and v, v ∗ denote the pre and post-collisional velocities forthe pair of colliding particles, respectively. A collision conserves the energy of thetwo particles v ′ + v ′ ∗ = v + v ∗ , MILANA PAVI´C-ˇCOLI´C AND MAJA TASKOVI´C so by introducing a parameter θ ∈ [ − π, π ], the collision rules read v ′ = v cos θ − v ∗ sin θ, (9) v ′∗ = v sin θ + v ∗ cos θ. Note that a 2-dimensional vector ( v ′ , v ′∗ ) can be viewed as a rotation of the 2-dimensional vector ( v, v ∗ ) by the angle θ .In this paper we assume that the angular kernel b K ( | θ | ) ≥ Z π − π b K ( | θ | ) sin κ θ d θ < ∞ , for some κ ∈ [0 , . (10)The case κ = 0 corresponds to the so called Grad’s cutoff case, when the angularkernel is integrable on [ − π, π ]. Otherwise, when κ is strictly positive, i.e. thenon-cutoff case, b K ( | θ | ) is allowed to have κ more degrees of singularity at θ = 0.2.1. Weak formulation of the collision operator.
Since the Jacobian of thetransformation (9) is unit, for a test function φ ( v ), the weak formulation of thecollision operator K ( f, f ) reads Z R K ( f, f ) φ ( v ) d v = 12 Z R Z R Z π − π f f ∗ ( φ ( v ′ ) + φ ( v ′∗ ) − φ ( v ) − φ ( v ∗ )) b K ( | θ | ) d θ d v ∗ d v. (11)2.2. Weak solutions of the Kac equation.
We recall the definition of a weaksolution to the Cauchy problem for the Kac equation ( ∂ t f ( t, v ) = K ( f, f )( t, v ) t ∈ R + , v ∈ R ,f (0 , v ) = f ( v ) , (12)whose existence was proved in [8] for the cutoff case, i.e. κ = 0, and in [9] for thenon-cutoff case, i.e. κ ∈ (0 , Definition 2.1.
Let f ≥ R d with finite mass, energyand entropy, i.e. Z R f ( v ) (cid:0) h v i + | log f ( v ) | (cid:1) d v < ∞ . (13)Then we say f ≥ weak solution to the Cauchy problem (12) with K ( f, f )given by (8) if f ( t, v ) ∈ L ∞ (cid:0) [0 , + ∞ ) ; L (cid:1) , and for all test functions φ ∈ W , ∞ ( R v )we have ∂ t Z R f φ ( v )d v = Z R Z R K φ ( v, v ∗ ) f f ∗ d v d v ∗ , where K φ ( v, v ∗ ) = Z π − π ( φ ( v ′ ) − φ ( v )) b K ( | θ | )d θ. For these solutions conservation of mass holds Z R f ( t, v ) d v = Z R f ( v ) d v. (14)while the energy decreases in time. However, the energy is conserved, that is Z R f ( t, v ) v d v = Z R f ( v ) v d v, ROPAGATION OF STRETCHED EXPONENTIAL MOMENTS 5 if there exists
C > Z R f ( v ) (cid:0) | v | p (cid:1) d v < C, for some p ≥
2. For details, see [9].3.
The spatially homogeneous Boltzmann equation
The state of gas particles which at a time t ∈ R + have a position x ∈ R d and velocity v ∈ R d , d ≥
2, is statistically described by the distribution function f = f ( t, x, v ) ≥
0. The evolution of such distribution function is modeled by theBoltzmann equation, which takes into account the effects of the free transport andcollisions on f . The collisions are assumed to be binary and elastic, that is, theyconserve mass, momentum and energy for any pair of colliding particles.When the distribution function is independent of the spatial variable x , that is f := f ( t, v ) ≥
0, which is the so called spatially homogeneous case, the Boltzmannequation reads ∂ t f ( t, v ) = Q ( f, f )( t, v ) . (15)The collision operator Q ( f, f ) is defined by Q ( f, f )( t, v ) = Z R d Z S d − ( f ′ f ′∗ − f f ∗ ) | v − v ∗ | γ b B (ˆ u · σ ) d σ d v ∗ , (16)with the standard abbreviations f ∗ := f ( t, v ∗ ), f ′ := f ( t, v ′ ), f ′∗ := f ( t, v ′∗ ). For apair of particles, vectors v ′ , v ′∗ denote pre-collisional velocities, while vectors v, v ∗ denote their post-collisional velocities. Local momentum and energy are conserved,i.e. v ′ + v ′∗ = v + v ∗ | v ′ | + | v ′∗ | = | v | + | v | ∗ . Thus by introducing a parameter σ ∈ S d − , the collision laws can be expressed as v ′ = v + v ∗ | v − v ∗ | σ, (17) v ′∗ = v + v ∗ − | v − v ∗ | σ. The unit vector σ ∈ S d − has the direction of the relative velocity u ′ = v ′ − v ′∗ ,while the normalization of the relative velocity u = v − v ∗ is denoted by ˆ u := u | u | .The angle between these two directions, denoted by θ , is called the scattering angleand it satisfies ˆ u · σ = cos θ .Due to physical considerations, the parameter γ is a number in the range ( − d, γ = 0 . (18)The angular kernel b B (ˆ u · σ ) = b B (cos θ ) is a non-negative function that encodesthe likelihood of collisions between particles. It has a singularity for σ that satisfiesˆ u · σ = 1, i.e. θ = 0, which may or may not be integrable in σ ∈ S d − . Itsintegrability is often referred to as the angular cutoff, while its non-integrability isreferred to as the non-cutoff case. In this paper we assume that Z π b B (cos θ ) sin β θ sin d − θ dθ < ∞ , for some β ∈ [0 , . (19) MILANA PAVI´C-ˇCOLI´C AND MAJA TASKOVI´C
The case β = 0 corresponds to b B (ˆ u · σ ) being integrable in σ ∈ S d − , i.e. itcorresponds to the cutoff case. When β >
0, then the angular kernel b B is allowedto have β more degrees of singularity compared to the cutoff case.In particular, in the case of inverse power-law potentials for the Maxwell molecules,the interaction potential in 3 dimensions is of the form V ( r ) = r − . Then a nonin-tegrable singularity of the function b B is known b B (cos θ ) sin θ ∼ θ − , θ → . Therefore, β should satisfy β > .3.1. Weak formulation of the collision operator.
Since the Jacobian of thepre to post collision transformation is unit and due to the symmetries of the kernel,for any sufficiently smooth test function φ ( v ), the weak formulation of the collisionoperator Q ( f, f ) reads Z R d Q ( f, f ) φ ( v ) d v = 12 Z R d f f ∗ | v − v ∗ | γ Z S d − ( φ ( v ′ ) + φ ( v ′∗ ) − φ ( v ) − φ ( v ∗ )) b B (ˆ u · σ ) d σ d v ∗ d v. (20)3.2. Weak solutions to the Boltzmann equation.
We recall the definition ofa weak solution to the Cauchy problem for the Boltzmann equation ( ∂ t f ( t, v ) = Q ( f, f )( t, v ) t ∈ R + , v ∈ R d ,f (0 , v ) = f ( v ) , (21)whose existence in three dimensions and for the angular kernel (19) with β ∈ [0 , Definition 3.1.
Let f ≥ R d with finite mass, energyand entropy Z R d f ( v ) (cid:0) | v | + log(1 + f ( v )) (cid:1) dv < + ∞ . (22)Then we say f is a weak solution to the Cauchy problem (21) if it satisfies thefollowing conditions • f ≥ , f ∈ C ( R + ; D ′ ( R d )) ∩ f ∈ L ([0 , T ]; L γ ) • f (0 , v ) = f ( v ) • ∀ t ≥ R f ( t, v ) ψ ( v ) dv = R f ( v ) ψ ( v ) dv , for ψ ( v ) = 1 , v , ..., v d , | v | • f ( t, · ) ∈ L log L and ∀ t ≥ R f ( t, v ) log f ( t, v ) dv ≤ R f ( v ) log f dv • ∀ φ ( t, v ) ∈ C ( R + , C ∞ ( R )), ∀ t ≥ Z R d f ( t, v ) φ ( t, v ) dv − Z R d f ( v ) φ (0 , v ) dv − Z t dτ Z R d f ( τ, v ) ∂ τ φ ( τ, v ) dv = Z t dτ Z R d Q ( f, f )( τ, v ) φ ( τ, v ) dv. ROPAGATION OF STRETCHED EXPONENTIAL MOMENTS 7 The main results
Our main result establishes propagation of stretched exponential moments for theKac equation and for the Boltzmann equation corresponding to Maxwell molecules.
Theorem 4.1.
Suppose initial datum f ≥ has finite mass, energy and entropy,i.e. (13) in case of the Kac equation and (22) in the case of the Boltzmann equation.(a) Kac equation: Let f ( t, v ) be an associated weak solution to the Cauchyproblem (12) , with (8) and with the angular kernel satisfying (10) with κ ∈ [0 , . If s ≤
42 + κ , (23) then for every α > there exists < α ≤ α and a constant C > (depending only on the initial data and κ ) so thatif Z R f ( v ) e α h v i s d v ≤ M < ∞ , then Z R f ( t, v ) e α h v i s d v ≤ C, ∀ t ≥ . (24) (b) Boltzmann equation for Maxwell molecules: Let f ( t, v ) be an associated weaksolution to the Cauchy problem (21) with the angular kernel satisfying (19) with β ∈ [0 , . If s ≤
42 + β , (25) then for every α > there exists < α ≤ α and a constant C > (depending only on the initial data and β ) so thatif Z R d f ( v ) e α h v i s d v ≤ M < ∞ , then Z R d f ( t, v ) e α h v i s d v ≤ C, ∀ t ≥ . (26) Remark 1.
We make several remarks about this result.(i) The order s of the stretched exponential moment that propagates in timedepends on the singularity rate of the angular kernel. According to (23)and (25), the more singular the kernel is, the smaller the s is.(ii) The Maxwellian moment s = 2 can be reached only in the cutoff case i.e. κ = 0 for the Kac equation or β = 0 for the Boltzmann equation.(iii) The cutoff Kac equation was studied in [8], where propagation of momentsof order s = 1 and s = 2 was proved. We extend this result by allowing s ∈ [0 , s = 2) is proved. We extendthis result by allowing s ∈ [0 , γ > s = γ was proved, and [13] where propagation of stretched exponential momentswas proved depending on the singularity rate of the angular kernel. Weextend the work of [13] to include the case γ = 0. MILANA PAVI´C-ˇCOLI´C AND MAJA TASKOVI´C Mittag-Leffler moments
In this section, we recall the definition of Mittag-Leffler moments, first intro-duced in [13]. They are a generalization of stretched exponential moments, andthey are convenient for the study of exponential decay properties of a function f .Namely, these moments are the L norms weighted with Mittag-Leffler functionswhich asymptotically behave like exponentials. More precisely, a Mittag-Lefflerfunction with parameter a > E a ( x ) := ∞ X q =0 x q Γ( aq + 1) , a > , x ∈ R . Note that E ( x ) is simply the Maclaurin series of e x , while it is well-known that for a > E a ( x ) ∼ e x /a , as x → + ∞ . Therefore, e α h v i s ∼ E /s ( α /s h v i ) = ∞ X q =0 α qs Γ( s q + 1) h v i q , when v → + ∞ . This motivated the definition of Mittag-Leffler moment [13]
Definition 5.1.
The Mittag-Leffler moment of a rate α > s > M α,s ( t ) := Z R d f ( t, v ) E /s ( α /s h v i s ) d v = ∞ X q =0 α qs Γ( s q + 1) m q ( t )for any t ≥ Remark 2.
Due to the asymptotic behavior of Mittag-Leffler functions, the finite-ness of the stretched exponential moment M α,s ( t ) at any time t > M α,s ( t ).6. Angular averaging lemmas with cancellation
Before proving Theorem 4.1, we provide an estimate of the angular part of theweak formulation (11) and (20) when the test function is a monomial φ ( v ) = h v i q .These bounds will be later used to derive a differential inequality for polynomialmoment in Lemma 7.1. Lemma 6.1.
Let q ≥ .(a) Kac equation: Suppose that the angular kernel of the Kac equation b K sat-isfies the assumption (10) . Then Z π − π (cid:0) h v ′ i q + h v ′∗ i q − h v i q − h v ∗ i q (cid:1) b K ( | θ | ) d θ ≤ − C (cid:0) h v i q + h v ∗ i q (cid:1) + C (cid:0) h v i h v ∗ i q − + h v i q − h v ∗ i (cid:1) + C q ( q − ε κ,q h v i h v ∗ i (cid:0) h v i + h v ∗ i (cid:1) q − , where C = Z π − π sin (2 θ ) b K ( | θ | ) d θ < ∞ (27) ROPAGATION OF STRETCHED EXPONENTIAL MOMENTS 9 and ε κ,q = 2 C Z π − π sin (2 θ ) b K ( | θ | ) Z t (cid:18) − t θ (cid:19) q − d t d θ ≤ . (28) (b) Boltzmann equation for Maxwell molecules: Suppose that the angular kernel b B satisfies the assumption (19) . Z S d − (cid:0) h v ′ i q + h v ′∗ i q − h v i q − h v ∗ i q (cid:1) b B (ˆ u · σ ) d σ ≤ − C (cid:0) h v i q + h v ∗ i q (cid:1) + C (cid:0) h v i h v ∗ i q − + h v i q − h v ∗ i (cid:1) + C q ( q − ε β,q h v i h v ∗ i (cid:0) h v i + h v ∗ i (cid:1) q − , where C = | S d − | Z π b B (cos θ ) sin d θ d θ < ∞ (29) and ε β,q = 2 C | S d − | Z π sin d ( θ ) b B (cos θ ) Z t (cid:18) − t θ (cid:19) q − d t d θ ≤ . (30) Remark 3.
The sequences { ε κ,q } q and { ε β,q } q are decreasing to zero with a certaindecay rate depending on the angular singularity rate κ ∈ [0 ,
2] in the case of theKac equation and β ∈ [0 ,
2] in the case of the Boltzmann equation, [11], ε κ,q q − κ → , as q → ∞ , (31) ε β,q q − β → , as q → ∞ . (32) Proof of Lemma 6.1.
The proof of part (b) can be found in [13, Lemma 2.3]. Thus,here we provide only the proof of part (a).If E ( θ ) denotes the following convex combination of particle energies E ( θ ) := h v i cos θ + h v ∗ i sin θ, (33)then, using the collision rules (9), we obtain h v ′ i = E ( θ ) − vv ∗ sin θ cos θ, h v ′∗ i = E ( π − θ ) + 2 vv ∗ sin θ cos θ. (34)Taylor expansion of h v ′ i q around E ( θ ) up to the second order yields h v ′ i q = ( E ( θ ) − vv ∗ sin θ cos θ ) q = E ( θ ) q − qE ( θ ) q − vv ∗ sin θ cos θ + 4 q ( q − v v ∗ sin θ cos θ Z (1 − t ) ( E ( θ ) − t vv ∗ sin θ cos θ ) q − d t. Analogous expression can be written for h v ′∗ i as well.The first order term in the above expression is an odd function in θ , which nullifiesby integration over the even domain [ − π, π ]. Therefore, we can write Z π − π (cid:0) h v ′ i q + h v ′∗ i q − h v i q − h v ∗ i q (cid:1) b K ( | θ | ) d θ = I + I , where I = Z π − π (cid:0) E ( θ ) q + E ( π − θ ) q − h v i q − h v ∗ i q (cid:1) b K ( | θ | ) d θ,I = 4 q ( q − v v ∗ Z π − π sin θ cos θ b K ( | θ | ) × Z (1 − t ) h ( E ( θ ) − t vv ∗ sin θ cos θ ) q − + ( E ( π − θ ) + 2 t vv ∗ sin θ cos θ ) q − i d t d θ. We now proceed to estimate the terms I and I separately. Term I . The term I is estimated by an application of Lemma A.1. Indeed, for t = cos θ , so that 1 − t = sin θ , and a = h v i , b = h v ∗ i , recalling (33) we obtain: I ≤ − Z π − π cos θ sin θ (cid:0) h v i q + h v ∗ i q (cid:1) b K ( | θ | ) d θ + 2 Z π − π cos θ sin θ (cid:0) h v i h v ∗ i q − + h v i q − h v ∗ i (cid:1) b K ( | θ | ) d θ = − (cid:0) h v i q + h v ∗ i q (cid:1) Z π − π sin θ b K ( | θ | ) d θ + 12 (cid:0) h v i h v ∗ i q − + h v i q − h v ∗ i (cid:1) Z π − π sin θ b K ( | θ | ) d θ. Therefore, recalling (27), we have I ≤ − C (cid:0) h v i q + h v ∗ i q (cid:1) + C (cid:0) h v i h v ∗ i q − + h v i q − h v ∗ i (cid:1) . Term I . By Cauchy-Schwartz inequality, − t vv ∗ sin θ cos θ ≤ t E ( π − θ ). Thus, E ( θ ) − t vv ∗ sin θ cos θ ≤ E ( θ ) + t E ( π − θ )= h v i + h v ∗ i − (1 − t ) E ( π − θ ) ≤ (cid:0) h v i + h v ∗ i (cid:1) (cid:18) − (1 − t ) sin θ (cid:19) , where the last inequality follows from E ( π − θ ) = h v i sin θ + h v ∗ i cos θ ≥ (cid:0) h v i + h v ∗ i (cid:1) min (cid:8) sin θ, cos θ (cid:9) ≥ (cid:0) h v i + h v ∗ i (cid:1) sin θ . The same estimate holds for E ( π − θ ) + 2 tvv ∗ sin θ cos θ . Therefore, recalling (28)the definition of ε κ,q , we have I ≤ q ( q − v v ∗ (cid:0) h v i + h v ∗ i (cid:1) q − ε κ,q . Adding estimates for terms I and I completes the proof of lemma. (cid:3) ROPAGATION OF STRETCHED EXPONENTIAL MOMENTS 11 Bounds on polynomial moments
Lemma 7.1.
With assumptions and notations of Lemma 6.1, we have the followingdifferential inequalities for a polynomial moment m q (a) In the case of the Kac equation, m ′ q ≤ − C m m q + C m m q − + C q ( q − ε κ,q ⌊ q +12 ⌋ X k =1 (cid:18) q − k − (cid:19) m k m q − k . (35) (b) In the case of the Boltzmann equation for Maxwell molecules, m ′ q ≤ − C m m q + C m m q − + C q ( q − ε β,q ⌊ q +12 ⌋ X k =1 (cid:18) q − k − (cid:19) m k m q − k . (36) Proof.
Multiplying the Kac equation (7) with h v i q and integrating with respect to v , we obtain an equation for the polynomial moment m q m ′ q = Z R h v i q K ( f, f )( t, v ) d v. Using the weak formulation (11) one has m ′ q = Z R Z R f f ∗ Z π − π (cid:0) h v ′ i q + h v ′∗ i q − h v i q − h v ∗ i q (cid:1) b K ( | θ | ) d θ d v ∗ d v. (37)Applying Lemma 6.1 and Lemma A.2 yields m ′ q ≤ Z R Z R f f ∗ (cid:20) − C (cid:0) h v i q + h v ∗ i q (cid:1) + C (cid:0) h v i h v ∗ i q − + h v i q − h v ∗ i (cid:1) + C q ( q − ε κ,q v v ∗ (cid:0) h v i + h v ∗ i (cid:1) q − (cid:21) d v ∗ d v ≤ Z R Z R f f ∗ (cid:20) − C (cid:0) h v i q + h v ∗ i q (cid:1) + C (cid:0) h v i h v ∗ i q − + h v i q − h v ∗ i (cid:1) + C q ( q − ε κ,q h v i h v ∗ i × k q − X k =0 (cid:18) q − k (cid:19) (cid:16) h v i k h v ∗ i q − − k ) + h v i q − − k ) h v ∗ i k (cid:17) d v ∗ d v = − C m m q + C m m q − + C q ( q − ε κ,q k q − X k =0 (cid:18) q − k (cid:19) m k +2 m q − k − . It remains to change index k in the sum and the part (a) is proven. The proof ofthe part (b) can be done in an analogous way. (cid:3) Lemma 7.2 (Propagation of polynomial moments ) . Suppose f is a weak solutionto the Cauchy problem of either Kac equation (12) or the Boltzmann equation forMaxwell molecules (21) . Then for every q ≥ , we have m q (0) < ∞ ⇒ m q ( t ) ≤ C ∗ q , (38) where the constant C ∗ q > , is uniform in time, and depends on q and the first q moments of the initial data. Proof.
Applying the inequality (57) to the differential inequality (35) yields m ′ q ≤ − C m m q + C q m q − , (39)where C q = C m (0) + C q ( q −
1) 2 q − . Therefore, m q ( t ) ≤ max (cid:26) m q (0) , C q C m (0) m q − ( t ) (cid:27) . (40)Applying this inequality inductively, for an integer q ∈ N , we have m q ( t ) ≤ max (cid:26) m q (0) , C q C m (0) m q − (0) , C q C m (0) C q − C m (0) m q − ( t ) (cid:27) .... ≤ max (cid:26) m q (0) , C q m q − (0) C m (0) , C q C q − m q − (0)( C m (0)) , . . . , C q C q − . . . C m ( t )( C m (0)) q − (cid:27) ≤ max (cid:26) m q (0) , C q m q − (0) C m (0) , C q C q − m q − (0)( C m (0)) , . . . , C q C q − . . . C m (0)( C m (0)) q − (cid:27) . Therefore, every even moment is bounded uniformly in time.Moments whose order is not an even integer can be interpolated by even moments.For example, if 0 < q − ≤ p ≤ q , then m p ≤ m q − p q − m p − q +22 q . Hence, polynomial moments of non-even orders are bounded uniformly in time aswell. (cid:3)
Remark 4.
We also note that derivatives of polynomial moments are uniformlybounded in time. Namely, applying the inequality (57) to the differential inequality(35) yields m ′ q ( t ) ≤ C q m q − ( t ) , (41)where C q = C m (0) + 4 q ( q − ε κ,q q − . Lemma 7.2 implies that m q − ( t ) isbounded uniformly in time by a constant C ∗ q − . Thus, m ′ q ( t ) ≤ C q C ∗ q − . (42)8. Proof of Theorem 4.1
Proof of Theorem 4.1 (a).
Recall from Remark 2 that finiteness of the stretchedexponential moment M α,s ( t ) is equivalent to finiteness of the Mittag-Leffler momentof the same rate and order. Therefore, we set out to prove finiteness of Mittag-Lefflermoment of order s and rate α that will be determined later: M α, a ( t ) = ∞ X q =0 α aq Γ( aq + 1) m q ( t ) , (43)where a = 2 s > . The case a = 1 corresponds to s = 2 i.e. Maxwellian moments. Propagation ofsuch moments can be esablished according to (23) only in the cutoff case κ = 0.This result (propagation of Maxwellian moments in the cutoff case) was alradyestablished in [8]. Thus, we here focus on the case when a > ROPAGATION OF STRETCHED EXPONENTIAL MOMENTS 13
The goal is to prove that partial sums of (43) E n ( t ) := n X q =0 α aq Γ( aq + 1) m q ( t ) . are bounded uniformly in time and n .From the differential inequality for polynomial moments (35), and by denoting S = q − X q =0 m ′ q α aq Γ( aq + 1) , S = n X q = q m q α aq Γ( aq + 1) , S = n X q = q m q − α aq Γ( aq + 1) ,S = n X q = q q ( q − ε κ,q α aq Γ( aq + 1) k q X k =1 (cid:18) q − k − (cid:19) m k m q − k , for any q ≥
3, we obtain the following differential inequality for the partial sum E n : dd t E n ≤ S − A m S + A m S + 4 S . (44)We proceed to estimate each S i , i = 0 , , , q − c q = max q =1 ,...,q − (cid:8) C ∗ q , C q C ∗ q − (cid:9) , (45)where C ∗ q is the constant from Lemma 7.2, and C q C ∗ q − is from Remark 4. Then m q ( t ) ≤ c q and m ′ q ( t ) ≤ c q for q = 1 , , ..., q − . (46) Term S . Since the mass is conserved (14), i.e. m ′ = 0, the first term in the sum S is equal to zero. Hence, by (46) we have S = α a q − X q =1 m ′ q α a ( q − Γ( aq + 1) ≤ c q α a q − X q =1 α a ( q − Γ( aq + 1) . Reindexing the sum and using the monotonicity of the Gamma function Γ( aq + a +1) ≥ Γ( q + 1), for a > q = 0 , , , . . . , we obtain S ≤ c q α a q − X q =0 α aq Γ( aq + a + 1) ≤ c q α a q − X q =0 ( α a ) q Γ( q + 1) ≤ c q α a e α a ≤ c q α a , (47)for α small enough so that e α a ≤ , or α ≤ (ln2) /a . (48) Term S . Using the bound (46) and parameter α chosen so that (48) holds, we have S = E n − m − q − X q =1 m q α aq Γ( aq + 1) ≥ E n − m − c q α a . (49) Term S . Using again the monotonicity of the Gamma function, we obtain S = α a n X q = q m q − α a ( q − Γ( aq + 1) ≤ α a n X q = q m q − α a ( q − Γ( a ( q −
1) + 1) ≤ α a E n − ≤ α a E n . (50) Term S . Using the property of the Beta function B ( x, y ) = Γ( x )Γ( y )Γ( x + y ) , the term S can be rearranged S = n X q = q q ( q − ε κ,q α aq Γ( aq + 1) k q X k =1 (cid:18) q − k − (cid:19) m k α ak Γ( ak + 1) m q − k α aq − ak Γ( aq − ak + 1) × B ( ak + 1 , aq − ak + 1)Γ( aq + 2) ≤ n X q = q q ( q − aq + 1) ε κ,q k q X k =1 (cid:18) q − k − (cid:19) B ( ak + 1 , aq − ak + 1) × k q X k =1 m k α ak Γ( ak + 1) m q − k α aq − ak Γ( aq − ak + 1) ≤ aC a n X q = q q − a ε κ,q k q X k =1 m k α ak Γ( ak + 1) m q − k α aq − ak Γ( aq − ak + 1) , where the last estimate follows by an application of Lemma A.4.Since by (23) we have 2 − a = 2 − s ≤ − κ , Remark 3 implies that (cid:8) q − a ε κ,q (cid:9) q is a decreasing sequence and q − a ε κ,q → , as q → ∞ . (51)If we denote c a = aC a , then the monotonicity of (cid:8) q − a ε κ,q (cid:9) q yields S ≤ c a q − a ε κ,q n X q = q ⌊ q +12 ⌋ X k =1 m k α ak Γ( ak + 1) m q − k α aq − ak Γ( aq − ak + 1) ≤ c a q − a ε κ,q ⌊ n +12 ⌋ X k =1 m k α ak Γ( ak + 1) n − X ℓ =1 m ℓ α aℓ Γ( aℓ + 1) ! ≤ c a q − a ε κ,q E n E n (52) ROPAGATION OF STRETCHED EXPONENTIAL MOMENTS 15
Going back to (44) and applying the bounds (47), (49), (50), (52) we obtain adifferential inequality for the partial sum E n dd t E n ( t ) ≤ − A m ( t ) E n ( t ) + c q α a + A m ( t ) + A m ( t ) c q α a + A m ( t ) α a E n ( t ) + 4 c a q − a ε κ,q ( E n ( t )) . Due to the conservation of mass, i.e. m ( t ) = m (0), and the dissipation of energy,i.e. m ( t ) ≤ m (0) for the weak solution f , we havedd t E n ( t ) ≤ − A m (0) E n ( t ) + c q α a + A m (0) + A m (0) c q α a + A m (0) α a E n ( t ) + 4 c a q − a ε κ,q ( E n ( t )) . (53)To show that such E n ( t ) is uniformly bounded in time and n , we define T n := sup { t ≥ E n ( τ ) < M , ∀ τ ∈ [0 , t ) } , where M is the bound on the initial data in (4.1), with the goal of proving that T n = ∞ for all n ∈ N .The number T n is well-defined and positive. Indeed, since α < α , at time t = 0we have E n (0) = n X q =0 α aq Γ( aq + 1) m q (0) < ∞ X q =0 α aq Γ( aq + 1) m q (0)= M α , a (0) < M , uniformly in n , by (4.1). Since E n ( t ) are continuous functions of t , E n ( t ) < M for t on some positive time interval [0 , t n ), t n >
0. Therefore, T n > E n ( t ) ≤ M holds on the time interval [0 , T n ], from (53) we obtainthe following differential inequalitydd t E n ( t ) ≤ − A m (0) E n ( t ) + c q α a + A m (0) + A m (0) c q α a + A m (0) α a M + 4 c a q − a ε κ,q ( M ) . We conclude that E n ( t ) ≤ M + c q α a + A m (0) + A m (0) c q α a + A m (0) α a M + 4 c a q − a ε κ,q ( M ) A m (0)= M + m (0) + α a (cid:18) c q A m (0) + c q + m (0) M m (0) (cid:19) + 4 c a q − a ε κ,q ( M ) A m (0) . (54)First, since q − a ε κ,q converges to zero as q tends to infinity, we can choose q largeenough so that 4 c a q − a ε κ,q ( M ) A m (0) ≤ M . (55)Then, we choose α sufficiently small so that α a (cid:18) c q A m (0) + c q + m (0) M m (0) (cid:19) ≤ M . (56) Therefore, applying estimates (55), (56) and m (0) ≤ M to the differential inequal-ity (54) yields E n ( t ) ≤ M < M , for any t ∈ [0 , T n ]. Therefore, the strict inequality E n < M holds on the closedinterval [0 , T n ] for each n . But, since E n ( t ) is continuous function in t , the inequality E n ( t ) < M holds on a slightly larger interval [0 , T n + µ ), µ >
0. This contradictsdefinition of T n unless T n = + ∞ for all n . Therefore, E n ( t ) < M for any t ∈ [0 , + ∞ ) and for all n ∈ N . Hence, letting n → ∞ , we conclude that M α,s ( t ) < M for all t ≥ ε k , q and ε β , q depends in the exact same way on the the singularity rate ofthe angular kernel ( κ for the Kac equation and β for the Boltzmann equation). (cid:3) Appendix A. Auxiliary results
Lemma A.1.
Let a, b ≥ , t ∈ [0 , and p ∈ (0 , ∪ [2 , ∞ ) . Then ( ta + (1 − t ) b ) p + ((1 − t ) a + tb ) p − a p − b p ≤ − t (1 − t )( a p + b p ) + 2 t (1 − t )( ab p − + a p − b ) . Lemma A.2.
Assume p > and let k p = ⌊ p +12 ⌋ . Then, for all x, y > thefollowing inequality holds: ( x + y ) p ≤ k p X k =0 (cid:18) pk (cid:19) (cid:0) x k y p − k + x p − k y k (cid:1) . Lemma A.3.
Let b ≤ a ≤ p . Then for any x, y ≥ x a y p − a + x p − a y a ≤ x b y p − b + x p − b y b . Remark 5.
The above lemma is useful for comparing products of moments whosetotal homogeneity is the same. Namely, m m p ≤ m m p − ≤ m m p − ≤ ... ≤ m ⌊ p ⌋ m p −⌊ p ⌋ . (57) Lemma A.4.
Let p ≥ and k p = ⌊ p +12 ⌋ . Then for any a > we have k p X k =1 (cid:18) p − k − (cid:19) B ( ak + 1 , a ( p − k ) + 1) ≤ C a q − (1+ a ) , where the constant C a depends only on a . Acknowledgments
Authors are grateful to Laurent Desvillettes for suggesting us to study the prob-lem considered in this paper. We also thank Ricardo J. Alonso for discussionson the subject. We would like to thank Irene M. Gamba and Nataˇsa Pavlovi´cfor their valuable remarks. The work of M. P.- ˇC. was partially supported by theNSF grant NSF-DMS-RNMS-1107465 and by Project No. ON174016 of the Ser-bian Ministry of Education, Science and Technological Development. The work of
ROPAGATION OF STRETCHED EXPONENTIAL MOMENTS 17
M.T. has been supported by NSF grants DMS-1413064, NSF-DMS-RNMS-1107465and DMS-1516228. Support from the Institute of Computational Engineering andSciences (ICES) at the University of Texas Austin is gratefully acknowledged.
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