Global well-posedness of a binary-ternary Boltzmann equation

In this paper we show global well-posedness near vacuum for the binary-ternary Boltzmann equation. The binary-ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary interactions of particles, and may serve as a more accurate description model for denser gases in non-equilibrium. Well-posedness of the classical Boltzmann equation and, independently, the purely ternary Boltzmann equation follow as special cases. To prove global well-posedness, we use a Kaniel-Shinbrot iteration and related work to approximate the solution of the nonlinear equation by monotone sequences of supersolutions and subsolutions. This analysis required establishing new convolution type estimates to control the contribution of the ternary collisional operator to the model. We show that the ternary operator allows consideration of softer potentials than the one binary operator, consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution. These results are novel for collisional operators of monoatomic gases with either hard or soft potentials that model both binary and ternary interactions.


Introduction
We study global in time well-posedness near vacuum of the Cauchy problem for an extension of the classical Boltzmann transport equation (BTE) for monoatomic binary interactions gases that includes ternary interactions.This equation, which can be viewed as a model of a denser gas dynamics, has been recently introduced by two of the authors in [5], who rigorously derived, from finitely many particle dynamics, the purely ternary model for the case of hard potential interactions zone for short times.Moreover, it is seen in [3] that the ternary collisional operator derived in [5] has the same conservation laws and entropy production properties as the classical binary operator, which justifies that the introduced ternary term can serve as a higher order correction to the Boltzmann equation.Such rigorous derivation of the full binary-ternary model is a work in progress [4].Let us also mention that Maxwell models with multiple particle interactions have been studied in [7,8], for the space homogeneous case via Fourier Transform methods.
In this paper, we provide the first rigorous analytical result that shows global in time existence and uniqueness of mild solutions near vacuum to the binary-ternary model and the purely ternary model on its own.By mild solutions we mean that the x-space dependence of the solution is evaluated along the characteristic curves given by the Hamiltonian evolution of the particle system in between collisions (denoted by f # , being introduced in Subsection 2.2).The analytical techniques we use are inspired by the works [14,13,6,19,20,16] and the more recent work of [1,2].These techniques rely on finding convergent supersolutions and subsolutions to the strong form of (1.1) in the associated strong topology of space-velocity Maxwellian weighted in L ∞ -functions.
The binary-ternary Boltzmann transport equation we focus on is given by and describes the evolution of the probability density f of a dilute gas in non-equilibrium in R d , d ≥ 2, given an initial condition f0 : R d × R d → R, when both binary and ternary interactions among particles can occur.The operator Q2(f, f ) is the classical binary collisional operator, which expresses binary elastic interactions between particles, and is of quadratic order, while the operator Q3(f, f, f ) is the ternary collisional operator which expresses ternary interactions among particles, and is of cubic order.For the exact forms of the operators Q2(f, f ), Q3(f, f, f ) used in this paper, see (2.1), (2.14) respectively.We should mention that the purely ternary model, rigorously derived for short times in [5], is given by We refer to (1.2) as the ternary Boltzmann transport equation.
For the classical Boltzmann transport equation (1.3) one way to obtain global well-posedness near vacuum is by utilizing an iterative scheme which constructs monotone sequences of supersolutions and subsolutions that converge to the global solution of (1.3).This has been carried out for the first time by Illner and Shinbrot [13], who were motivated by the work of Kaniel and Shinbrot [14], who in turn showed local in time well-posedness for (1.3) following this program.Later, this work was extended to include wider range of potentials and to relax assumptions on initial data by Bellomo and Toscani [6] , Toscani [19,20] and Palczewski and Toscani [16].Alonso and Gamba [2] used Kaniel-Shinbrot iteration to derive distributional and classical solutions to (1.3) for soft potentials for large initial data near two sufficiently close Maxwellians in position and velocity space, while Alonso [1] used this technique to study the inelastic Boltzmann equation for hard spheres.Strain [18] remarks that the estimates he derives can be combined with the Kaniel-Shinbrot iteration to obtain existence of unique mild solution for the relativistic Boltzmann equation.
Kaniel-Shinbrot iteration is also an important tool for proving non-negativity of solutions, see for example [17,10,9].Also, when initial data has decay in the direction of x − v as opposed to x and v separately, Kaniel-Shinbrot iteration can be used to construct solutions with infinite energy, see for example [15,24,23].
Certain problems have been solved by considering modifications of the Kaniel-Shinbrot iteration.For example, Bellomo and Toscani [21] adapted the iteration to the Boltzmann-Enskog equation.Ha, Noh and Yun [12] and Ha and Noh [11] also modified the iteration to prove global existence of mild solutions to the Boltzmann system for gas mixtures in the elastic and the inelastic cases respectively.Also, Wei and Zhang [22] used another modified iteration to obtain eternal solutions for the Boltzmann equation.
The goal of this paper is to establish global existence and uniqueness of a mild solution near vacuum to the binary-ternary Boltzmann equation (1.1) in spaces of non-negative functions bounded by a Maxwellian.Moreover, solution of (1.2) follows as a special case.Inspired by [13,14,2], we devise an iterative scheme which constructs monotone sequences of supersolutions and subsolutions to (1.1).For small enough initial data, the beginning condition of the iteration holds globally in time and the two sequences can be shown to converge to the same limit, namely a function f which solves equation (1.1) in a mild sense.This strategy requires new ideas given the fact that ternary interactions are also taken into account in (1.1).
In particular, due to the presence of the ternary correction term, one needs to properly adapt the iteration, so that the corresponding supersolutions and subsolutions remain monotone and convergent.One of the main tools is stated in Lemma 3.2 which provides important exponentially weighted convolution estimates.This Lemma not only recovers the estimates developed in [2] for the binary interaction case, but also develops a new approach in order to treat the ternary interaction case.Lemma 3.2 is crucially used to obtain uniform in time, space-velocity L 1 -bounds that control the ternary gain and loss terms (L ∞ L 1 estimates).In fact, using Lemma 3.2 one first obtains asymmetric estimates (see Lemma 3.3) because of the asymmetry introduced by the ternary collisional operator which is not present in the binary case.However, to obtain convergence, it is essential to have symmetry with respect to the inputs of the gain and loss operators.We were able to achieve this symmetrization in Proposition 3.4.Finally, we also use Lemma 3.2 to prove a global estimate for the time average of the gain and loss operators along the characteristics of the Hamiltonian, see Proposition 3.7.With this, we were able to extend the argument for controlling the binary time integrals of both, gain and loss terms, (see [2]), to the ternary case by invoking properties of ternary interactions and a 2d-analog of the time integration bound for a traveling Maxwellian.
With these tools in hand, for small initial data, the constructed iteration scheme is proved to converge to the unique, global in time mild solution of (1.1).For more details see Section 4 and Section 5.
Organization of the paper.In Section 2, we review the binary and ternary collisional operators and decompose them into gain and loss forms.We then introduce some necessary notation and state our main result (Theorem 2.10).In Section 3, we prove the convolution estimate and derive essential bounds for the gain and loss operators.In Section 4, we inductively construct monotone sequences of supersolutions and subsolutions which are shown to converge to a common limit which solves the binary-ternary Boltzmann equation (1.1), as long as a beginning condition is satisfied.Finally, in Section 5 we provide the proof of our main result (Theorem 2.10).

Towards the statement of the main result
The goal of this section is to present the precise statement of the main result of this paper.In order to do so, we first review the collisional operators and decompose them to gain and loss form in Subsection 2.1, introduce necessary notation and the notion of a solution in Subsection 2.2, and then state the main result in Subsection 2.3 (Theorem 2.10).

Collisional operators.
2.1.1.Binary collisional operator.The binary collisional operator is given by where is the relative velocity of a pair of interacting particles centered at x, x1 ∈ R d , with velocities v, v1 ∈ R d before the binary interaction with respect to the impact direction and are the outgoing velocities after the binary interaction.
One can easily verify the binary energy-momentum conservation system is satisfied ) (2.6) (2.7) In addition, equation (2.4) yields the specular reflection with respect to the impact direction In fact it is not hard to show that, given v, v1 ∈ R d , expression (2.4) gives the general solution of the system (2.5)-(2.6),parametrized by ω ∈ S d−1 .The factor B2 in the integrand of (2.1) is referred as the binary interaction differential cross-section which depends on relative velocity u and the impact direction ω.It expresses the transition probability of binary interactions, and we assume it is of the form where û = u |u| ∈ S d−1 is the relative velocity direction and b2 : [−1, 1] → [0, ∞) is the binary angular transition probability density.It is worth mentioning that the case γ2 ∈ (0, 1] corresponds to hard potentials, the case γ2 ∈ (−d + 1, 0) corresponds to soft potentials and the case γ2 = 0 corresponds to Maxwell molecules.
We assume that the binary angular transition probability density b2 satisfies the following properties (2.10) which, due to property from (2.8), yields the binary micro-reversibility condition where û′ = u ′ |u| ∈ S d−1 is the scattering direction.In addition, relations (2.7), (2.9) and (2.11) yield (2.12) • The probability density is integrable on the sphere, i.e. for any fixed û we have b2(û where |S d−2 | is the volume of the (d − 2)-dimensional sphere.
Remark 2.1.The integrability condition on b2 is weaker than the classical Grad cut-off assumption which assumes b2 is a bounded function of z = û • ω.So our result is valid for a broader class of angular transition probability measures.
Remark 2.2.One can see that the usual hard sphere model is a special case of the form (2.9) 2.1.2.Ternary collisional operator.The ternary collisional operator is given by (see [5] for details) where is the relative velocity of some colliding particles centered at x, x1, x2 ∈ R d , with velocities v, v1, v2 ∈ R d before the ternary interaction with respect to the impact directions vector and are the outgoing velocities of the particles after the ternary interaction.It can be easily seen that 2 are given by (2.17), the ternary energy-momentum conservation system ) (2.20) For the postcollisional relative velocity, we will write (2.23) Then (2.20) can be written as which is the ternary analog of the binary expression (2.7).Defining , where is a (2d − 1)-dimensional ellipsoid.The vectors ū, ū * are the ternary analogs of the relative velocity direction and the scattering direction of the binary interaction.Because of the assymetry of the ternary interaction they are not unit vectors, they lie on the ellipsoid E 2d−1 instead.However, for convenience we will refer to ū, ū * as relative velocity direction and scattering direction respectively.
The collisional formulas (2.17) also imply which is the ternary analog to specular reflection with respect to the impact directions vector ω = (ω1, ω2) ∈ S 2d−1 .Indeed, one has which is equivalent to (2.27) due to (2.25).
In addition, we assume that b3 satisfies the following properties • b3 is even with respect to the first argument i.e. (2.30) In addition, due to (2.27), the ternary micro-reversibility condition holds and relations (2.28), (2.25) and (2.31) imply the total ternary collision kernel satisfies (2.32) • The probability density b3 is integrable on S 2d−1 i.e.
b3 L 1 (S 2d−1 ) := sup ( (2.34) (2.35) Due to the assumptions (2.13), (2.33), the binary-ternary operator Q2(f, g) + Q3(f, g, h) can be decomposed into a gain and a loss term as follows (2.36) where (2.38) The binary gain and loss operators G2, L2 are given respectively by (2.41) and are clearly bilinear.The ternary gain and loss operators L3, G3 are given respectively by and are clearly trilinear.Notice the loss term can be factorized as where R is given by R2 is the linear operator and R3 is the bilinear operator (2.47) 2.2.Some notation and the notion of a solution.Throughout the paper, the dimension d ≥ 2, the binary and ternary integrability assumptions (2.13), (2.33) respectively, and the cross-section exponents (2.48) appearing respectively in (2.9), (2.28) will be fixed.Moreover, C d denotes a general constant depending on the dimension d and can change values.

Functional spaces.
Let us introduce the functional spaces used in this paper.First, in order to point out the dependence in positions and velocities, we will use the notation (2.49) (2.50) We also define the sets of space-velocity functions (2.53) Same notation will hold for equality as well.
Given α, β > 0, we define the corresponding (non-normalized (2.54) We also define the corresponding Banach space of functions essentially bounded by M α,β as We will write fn We also define the set of a.e.non-negative functions essentially bounded by M α,β as (2.57) Given 0 < T ≤ ∞, we define the sets of time dependent functions and given f, g ∈ FT , we will write f ≥ g iff f (t) ≥ g(t) for all t ∈ [0, T ).Same notation will hold for equalities as well.
Finally, we define the following subsets of functional spaces (2.60) and given α, β > 0, we define the Banach space of time dependent functions uniformly essentially bounded by with norm (2.64) Notice that in definition (2.63), the supremum is taken with respect to all t ∈ [0, T ).We also write (2.65) 2.2.2.Transport operator.We now introduce the transport operator which will be crucial to define mild solutions to (1.1).Let us recall from (2.51)-(2.52) the sets of functions Consider a positive time 0 < T ≤ ∞ (we can have T = ∞) and recall from (2.58)-(2.59) the sets of time dependent functions (2.66) Clearly, the operators # : FT → FT and −# : FT → FT are linear and invertible and in particular Remark 2.5.Let f, g ∈ FT .Since the maps (x, v) → (x + tv, v) and (x, v) → (x − tv, v) are measurepreserving, for all t ∈ [0, T ), we have In particular (2.67) (2.68) 68) and linearity of the transport operator imply (2.69) Throughout the manuscript, we will often define f # ∈ FT directly, implying that f is defined by 2.2.3.Transported gain and loss operators.In order to define mild solutions to (1.1) , it is important to understand the action of the transport operator on the gain and loss operators.More specifically, given f, g, h ∈ FT , for the gain operators we write and for the loss operators we write (2.71) Under this notation, it is straightforward to verify that ( where the gain term G(f, f, f ) and the loss term L(f, f, f ) are given by (2.38)-(2.37)respectively.
Here is where the importance of the transport operator will become clear.Indeed, using the chain rule, the initial value problem (2.74) can be formally written as (2.75) Motivated by (2.75), we aim to define solutions of (1.1) up to time 0 < T ≤ ∞, with respect to a given Maxwellian M α,β , where α, β > 0.

2.3.
Statement of the main result.Now we are ready to state the main result of the paper.
Theorem 2.10. where (2.78) and C d is an appropriate constant depending on the dimension d, the binary-ternary Boltzmann equation (1.1) has a unique mild solution f satisfying the bound (2.79) Remark 2.11.As we will see, the uniqueness claimed above holds in the class of solutions of (1.1) satisfying (2.79).
Remark 2.12.According to the assumptions on b2, b3 made in Remark 2.4, Theorem 2.10 applies as well to the end point cases where either b2 = 0 or b3 = 0 (but not both).In the case b3 = 0, one recovers the solution of the classical Boltzmann equation (1.3) constructed in [13], while in the case b2 = 0, one obtains well-posedness of the ternary Boltzmann equation (1.2), introduced in [5].

Properties of the transported gain and loss operators
In this section, we investigate properties of the transported gain and loss operators which will be important for proving global well-posedness of (1.1).
3.1.Monotonicity and L 1 -norms.As we will see, the transported gain and loss operators are monotone increasing when acting on non-negative functions.These monotonicity properties will allow us to construct monotone sequences of supersolutions and subsolutions to (1.1).Moreover, we show that the L 1 -norm of the gain is equal to the L 1 -norm of the loss.This equality will allow us to reduce estimates on the norm of the gain term to estimating the norm of the loss term.In the following, saying that an operator is bilinear/trilinear, we mean it is linear in each argument, and saying it is monotone increasing, we mean it is increasing in each argument.Proposition 3.1.Let 0 < T ≤ ∞.Then the following hold T is linear and monotone increasing.
T are bilinear and monotone increasing.
T are trilinear and monotone increasing.
T are monotone increasing.(v) For any f, g, h ∈ F + T , the following identities hold Let us now prove (v).We first prove (3.1) for the binary case.By (2.68), we have Therefore, for any t ∈ [0, T ), using (2.12) and involutionary substitution (v ′ , v ′ 1 ) → (v, v1), we obtain We now prove (3.1) for the ternary case.By (2.68), we have Therefore, for any t ∈ [0, T ), using (2.32) and the involutionary substitution we obtain We finally prove (3.1) for the mixed case.By positivity, for any t ∈ [0, T ), we have Equality (3.1) for the mixed case immediately follows from the corresponding binary and ternary equalities.

Convolution estimates.
We now present a general convolution-type result, which will be essential for the control of the binary and the ternary collisional operators.These estimates will be of fundamental importance in the proof of the L ∞ L 1 estimates (see Subsection 3.3) and the global estimate on the time average of the transported gain and loss operators appearing in Proposition 3.7, which in turn will be crucial for the proof of global well-posedness of (1.1).For the binary case one can find similar convolution estimates in [14,13,2].Here, our contribution is the derivation of these estimates for the ternary case, since this is the first time global well-posedness is studied for such a ternary correction of the Boltzmann equation.The estimates of the ternary term illustrate that consideration of softer potentials is allowed for the ternary collisional operator.where u = v1 − v, q + 2 := max{0, q2}, K 2 β,q 2 is given by and C d is an appropriate constant depending on the dimension d.
(ii) For any v ∈ R d , we have where | u| is given by (2.22), q + 3 := max{0, q3}, K 3 β,q 3 is given by and C d is an appropriate constant depending on the dimension d.
since we have assumed q2 > −d.
Therefore, Fubini's Theorem and estimates (3.6)-(3.7)applied for q = q3 imply (3.10) • q3 ∈ (−2d, 0]: Recalling (2.22) and using the fact that q3 ≤ 0, Fubini's Theorem and estimates 3.3.L ∞ L 1 estimates.Here we prove uniform in time, space-velocity L 1 estimates on the transported gain and loss operators.These estimates will be of fundamental importance for the convergence of the iteration to the global solution.As we will see, the ternary collisional operator introduces some asymmetry which is not present in the binary case.For this reason, when we use Lemma 3.2, we first obtain estimates in asymmetric form (see Lemma 3.3).However, we will need a symmetric version of this estimate which we derive in Proposition 3.4.To achieve that, we crucially rely on properties of the ternary interactions.

Estimate (3.16) follows by the fact that
Proof of (ii): We first prove the claim for the loss operators.Positivity follows immediately from the monotonicity of x,v by (3.13).For the gain operators, positivity follows immediately from the monotonicity of  Notice that bounds (3.17)- (3.19) are only with respect to the first argument f .Although this is not an issue in the binary case where the gain and loss collisional operators are symmetric with respect to the inputs in the L 1 -norm, this is not the case for the ternary operators.In order to treat this assymetry, we need to derive estimates with respect to all three inputs of the ternary gain and loss collisional operators.This is achieved in the following result Proposition 3.4.Let 0 < T ≤ ∞ and α, β > 0. Consider f1, f2, f3 Then, there is a constant C β > 0 such that, for any permutation π : {1, 2, 3} → {1, 2, 3}, the following estimates hold for any t ∈ [0, T ) Proof.By (3.1), triangle inequality and part (ii) of Lemma 3.3, the proof of (3.24)-(3.26)for the loss term reduces to showing the following estimates (3.29) • Proof of (3.27): Performing the involutionary change of variables (v, v1) → (v1, v) and using (2.10), for any t ∈ [0, T ), we have The claim comes from part (ii) of Lemma 3.3.
• Proof of (3.28):Here the proof is subtler because the inner product ū • ω is not symmetric upon renaming the velocities.However, we will strongly rely on the fact that the expression given in (2.22) is symmetric with respect to the inputs v, v1, v2.
By (3.32) and the Dominated Convergence Theorem, each of the terms in (3.36) goes to zero as n → ∞ and (3.34) is proved.
• Proof of (3.35):Using trilinearity of L # 3 , bound (3.33) and monotonicity of L # 3 , we have The gain operator convergence follows with a similar argument.

A global estimate on the time average of the transported gain and loss operators.
Here, we prove Proposition 3.7, which provides upper global bounds for the time average of the transported operators.These estimates will be essential to prove that the necessary beginning condition (4.49) for the convergence of the iteration holds globally in time for small enough initial data (see Section 5).For the binary case and soft potentials, these bounds were established in [2].However, the presence of the ternary collisional operator requires new treatment which strongly relies on the properties of ternary interactions.
Before stating Proposition 3.7, we provide the following auxiliary estimate for the time integral of a traveling Maxwellian which will be used in the proof of the result for n = d in the binary case and n = 2d in the ternary case.Lemma 3.6.Let n ∈ N, x0, u0 ∈ R n , with u0 = 0 and α > 0.Then, the following estimate holds Proof.By triangle inequality, we have Therefore integrating in τ , we obtain and the estimate is proved.
Proof of (3.38):As mentioned above, these bounds were established for the soft potential case in [2].Here we also treat the hard potential case.Since L # 2 , G # 2 are bilinear, we may assume without loss of generality that Let us first prove it for the loss term.For any t ∈ [0, T ) and a.e.(x, v) ∈ R 2d , relation (3.42), followed by an application of Lemma 3.6 for n = d, x0 = x, u0 = u, the fact that −d + 1 < γ2 ≤ 1, and an application of part (i) of Lemma 3.2 for q2 = γ2 − 1 imply where C d is an appropriate constant depending on the dimension d.To obtain (3.43), we used (3.3) and the fact that q2 = γ2 − 1 ≤ 0. Estimate (3.38) for the loss term follows.
To prove (3.38) for the gain term, we will use the identity which follows from the binary conservation of momentum and energy Combining (3.46) with an identical argument to the one used for the loss term, we obtain and estimate (3.38) for the gain term follows.

The Kaniel-Shinbrot iteration scheme and the associated linear problem
In this section, we present the Kaniel-Shinbrot iteration scheme which will then be used as the heart of the construction of a global solution in Section 5.This scheme is motivated by the works of [13,14].However the presence of the ternary collisional operator, in addition to the binary collisional operator, required a modification of the original construction.
In particular, we outline the construction of the Kaniel-Shinbrot iteration that we will use in this paper.Formally speaking, given an initial data f0, we construct an increasing sequence (ln) n∈N and a decreasing sequence (un) n∈N , with ln ≤ un, through the iteration  We will see that that the sequences ln, un converge to the same limit, namely a function f , which will be the solution of the binary-ternary Boltzmann equation (1.1).
To make things rigorous, we first study an associated linear problem, and then inductively apply these results, together with the estimates derived in Section 3, to establish that the Kaniel-Shinbrot iteration converges to a solution of (1.1), provided that an appropriate beginning condition is satisfied.This solution will be unique in the class of functions uniformly bounded by a Maxwellian.4.1.The associated linear problem.Here, we prove well-posedness for a linear problem associated to the iteration scheme (4.1)-(4.2).More precisely, given some functions of time g, h, we show well-posedness up to time 0 < T ≤ ∞ of the linear problem We say that a function (iii) f # is weakly differentiable and satisfies is a mild solution of (4.3) in [0, T ) with initial data f0 ∈ L 1,+ x,v .Remark 4.2.The differential equation of (4.4) is interpreted as an equality of distributions since all terms involved belong to L 1 loc ([0, T ), L 1,+ x,v ).
Finally, by (4.5), (4.14), representation (4.8) and the Dominated Convergence Theorem, we conclude that f # is weakly differentiable and satisfies thus it is a mild solution of (4.3).
Integrating (4.15), the Fundamental Theorem of Calculus and the fact that f # ∈ C 0 ([0, T ), L 1,+ x,v ), Using non-negativity of all terms involved in (4.16) and Fubini's Theorem, we obtain (4.6) and absolute continuity of f (t) L 1 x,v follows.The proof is complete.
Since the gain operator does not satisfy (4.5), it will be convenient to relax assumption (4.5) to As in [14], the idea is to approximate f0, h # in the L 1 x,v -norm with a monotone sequence of solutions occurring from a repeated application of Lemma 4.4.We obtain the following well-posedness result x,v ).Then, there exists a unique mild solution f of (4.3).In particular f # is given by and It is clear that f0,n, hn satisfy condition (4.5) for all n ∈ N and that Then the Monotone Convergence Theorem yields that ∀t ∈ [0, T ) : for a.e.
∀t ∈ [0, T ) : and is given by the formula Also by (4.6), given t ∈ [0, T ), we have the bound where to obtain the last bound we use (4.22)-(4.23), the fact that R # (g, g) ≥ 0 (by monotonicity of Since the sequences (f0,n)n, (h # n (t))n are increasing and non-negative for all t ∈ [0, T ), formula (4.29) implies that the sequence (f # n (t))n is increasing for all t ∈ [0, T ).Let us define Clearly f ≥ 0. By the Monotone Convergence Theorem and bound (4.30) we obtain that f # (t) ∈ L 1,+ x,v , ∀t ∈ [0, T ).Then, the Dominated Convergence Theorem implies .32) since R # (g, g)(t) ≥ 0 by monotonicity of R # and the fact that g ≥ 0. By the Monotone Convergence Theorem, we obtain Therefore, for any t ∈ [0, T ), (4.6), implies x,v , for a.e.t ∈ [0, T ), (4.36) and by (4.37) and another application of the Dominated Convergence Theorem, we obtain Since f # n satisfies (4.28), the Fundamental Theorem of Calculus and the fact that f ), f # is weakly differentiable and satisfies (4.4).We conclude that f is a mild solution of (4.3).Moreover, since g ≥ 0, we may take the limit as n → ∞ in both sides of (4.29) to obtain (4.17).
Uniqueness: Since the problem is linear it suffices to show that if f is a solution of (4.3) with f0 = 0 and h = 0, then f = 0.
Assume f is a mild solution of (4.3) with f0 = 0 and h x,v ) and f # is weakly differentiable and satisfies (4.43) We claim the following Proof of the claim: Fix any compact set K ⊆ R d × R d .By (4.43), Fubini's Theorem, part (i) of Lemma 3.3 and the fact that pγ 2 ,γ 3 is continuous, we obtain is continuous, thus (4.44) and Gronwall's inequality imply that f # (t) L 1 x,v (K) = 0, ∀t ∈ [0, T ).
The claim is proved.
Consider now a sequence of compact sets (Km)m ր R d × R d .By the claim above, and the Monotone Convergence Theorem, we have Since f # ≥ 0, we obtain f # = 0 and hence f = 0. Uniqueness is proved.
Proof.We have g # Moreover, since for any t ∈ [0, T ) we have as n → ∞, Corollary 3.5 implies that for any t ∈ [0, T ), we have           Then, either (4.51) or (4.52) implies Recalling Definition 2.7, we conclude that f is a mild solution to the binary-ternary Boltzmann equation (1.1) with initial data f0.Bound (4.53) directly follows from (4.63).
Uniqueness of solutions satisfying (4.53) follows similarly to the proof of (iii) using a bilinearitytrilinearity argument and Proposition 3.7.Clearly, condition (4.53) is needed to have a contraction.

Global well-posedness near vacuum
In this final section, we prove the main result of this paper, stated in Theorem 2.10, which gives global well-posedness of (1.1) near vacuum in the interval [0, T ), where 0 < T ≤ ∞.To prove this result we will rely on the time average bound of the gain term from Proposition 3.7.

Proof of Theorem 2.10
Ioakeim Ampatzoglou, Courant Institute of Mathematical Sciences, New York University.
42), the Fundamental Theorem of Calculus and the facts f