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Existence of global weak solutions for the Navier-Stokes-Vlasov-Boltzmann equations

A moderately thick spray can be described by a coupled system of equations consisting of the incompressible Navier-Stokes equations and the Vlasov-Boltzmann equation. We investigate this kind of mathematical model in this paper. In particular, we study the initial value problem for the Navier-Stokes-Vlasov-Boltzmann equations. The existence of global weak solutions is established by a weak convergence method. The interesting point of our main result is to handle the model with some breakup effects while the velocity of particles is in the whole space.


Introduction
A spray is a process in which drops dispersed in a gas. The sprays can help people to distribute material over a cross-section and to generate liquid surface area. There are various applications in different fields, the examples include but not limit fuel injectors for gasoline and Diesel engines, atomizers for jet engines, atomizers for injecting heavy fuel oil into combustion air in steam boiler injectors, and rocket engine injectors. It also has a big impact on crop yield, plant health, efficiency of pest control and of course, profitability. Here we will focus on the model of the moderately thick spray. In general, we refer the reader to [4] for physical backgrounds.
For the moderately thick spray, we can assume that the volume fraction occupied by the droplets is small enough to be neglected. Thus, we are able to apply the Vlasov-Boltzmann equation to model the liquid phase, which can be performed by the use of a particle density function. In particular, a function f (t, x, ξ) denotes a number density of droplets of which at time t and physical position x with velocity ξ, which is a solution of the following Vlasov-Boltzmann equation where F is an acceleration resulting from the drag force exerted by the gas, and Q(f, f ) is an operator taking into account the complex phenomena happening at the level of the droplets (collisions, coalescences, breakup). The incompressible Navier-Stokes equations could be used to describe the motion of gas when instigating the transport of sprays in the upper airways of the human lungs: where u is velocity of gas, and F e denotes the external force with F denotes the acceleration, it is given by µ is the viscosity constant of the incompressible Navier-Stokes equations, ρ is the density of gas, r is the radios of the droplet. We can assume that 9µ 2ρr 2 = 1 in the whole paper, thus One of the typical forms of the collision kernel is given by The constant λ > 0 is the breakup frequency. The kernel T (ξ, ξ ′ ) is the probability of a change with respect to velocity from ξ ′ to ξ, and ξ is the velocity of individuals before the collision while ξ ′ is the velocity immediately after the collision. Given that a reorientation occurs, the probability function T (ξ, ξ ′ ) is a non-negative function and after normalization we have By the acceleration of the Vlasov-Blotzmann equations and the external force of the incompressible Navier-Stokes equations, the above equations (1.1)-(1.5) can be coupled with each other. As a result, it is a model for the moderately thick spray, namely the Navier-Stokes-Vlasov-Blotzmann equations: The main goal of this paper is to investigate the existence of weak solutions globally in time t for equations (1.7) with the following initial data 8) and the initial data f 0 (x, ξ) is suitable decay condition as |ξ| → ∞.
The mathematical analysis for spray models is very challengingly because the coupled term for unknowns that does not depend on the same set of variables. Concerning to the existence theory of global weak solution for such models, we should date back to late of 1990's. The first work in this field was done in [8] where the author proved the global weak existence and its large-time behavior for the Vlasov-Stokes equations. The existence theorem of weak solutions has been extended in [1,3,10,12,14], where the authors did not neglect the convection term and considered the Navier-Stokes equations, including incompressible and compressible ones; Among which, Anoschchenko, Khruslov and Stephan obtained the global existence of weak solutions to the Navier-Stokes-Vlasov-Poisson system. In [5], the existence and uniqueness of global smooth solutions near equilibrium are proved under smallness conditions for the Navier-Stokes system coupled with the Vlasov-Fokker-Planck equation in 3D. In the mean time, there are a lot of works in hydrodynamic limits, we refer the reader to [6,7,11]. In these works, the authors rely on convergence methods, like compactness and relative entropy method, to investigate hydrodynamic limits. It is natural to treat the mathematical analysis for the models with the effects of collisions or the effects of breakup, but more challenging. In [9], Legar-Vasseur established the existence theory of the weak solutions for a system coupled by the Navier-Stokes equations and Vlasov-Blotzmann equation where they restricted the velocity ξ in a bounded domain. Benjelloun-Desvillettes-Moussa in [2] introduced the Navier-Stokes-Vlasov-Blotzman model for spray theory, and a typical operator is given when the droplets after breakup have the same velocities as before breakup. Finally, they established an existence result of weak solutions for a simple model which derived from the Navier-Stokes-Vlasov-Blotzman model. In particular, they assume that the aerosol is bidispersed, in other words, there are only two possible radius r 1 > r 2 exist for the droplets, and the result of the breakup of particles of radius r 1 are particles of radius r 2 . Under this assumption, the density function will split into as follows therefor, the Navier-Stokes-Vlasov-Blotzman model will reduced to the Navier-Stokes-Vlasov equations with damping terms. Motivated by the work of Yu [15], of particular interests in this paper is to establish the existence of weak solutions of the Navier-Stokes-Vlasov-Blotzman equations with a typical breakup operator given in [2], for the velocity ξ ∈ R 3 . However, the restriction of the same velocities is not necessary in our paper.
It is not necessary to assume that when the droplets after breakup have the same velocities as before breakup in this paper. In fact, we deduce the following ones from the conservation of kinetic energy, We will assume (1.9) in the whole paper. In other word, different from the work of [2], we only need to assume that the droplets after breakup have the same speeds as before breakup, not the same velocities.
As the same in the work of [9], we assume T (ξ, ξ ′ ) satisfies a self-similarity property, namely,  x, ξ * ) for x ∈ ∂Ω, ξ · n(x) < 0, where ξ * = ξ − 2(ξ · n(x))n(x) is the specular velocity, n(x) is the outward normal to Ω, and Ω ⊂ R 3 is a bounded domain. Meanwhile, the incompressible Navier-Stokes equations of system (1.7) can be replaced by the inhomogeneous ones, the extension of our result, in this context, is considered in the forthcoming paper [13].
The interesting point of our main result is to handle the model with some breakup effects while the velocity of particles is in the whole space. As we mentioned before, we need to assume that the droplets after breakup have the same speeds as before breakup, not the same velocities. This is different from the work of [2]. A key observation of our proof is Lemma 2.2, which gives us some uniform control on density function f (t, x, ξ). By the Fubini Theorem, we are able to show the following a priori estimate which allows us to obtain further bounds and compactness of smooth solutions. Thus, with a suitable approximation, a weak solution could be recovered. Our idea of the approximation is to construct an iteration for the kinetic part, and to adopt the Galerkin's method for the fluid part. However, the breakup operator in iteration is given by for any integers n ≥ 1. Note that, {f n m } is an increasing sequence with respect to n. This allows us to obtain the bounds on f n m and R 3 |ξ| k f n m dξ, which yields the weak stability. By the weak convergence method, the existence of weak solutions could be done.
Notations: In the following, C from line to line denote the generic positive constants depending on the initial data, T and the physical coefficients; C(E, B) denotes the positive constants depending on the initial data, T , the physical coefficients and E, B.
We organize the rest of the paper as follows. In Section 2, we deduce a prior estimates, state some useful lemmas. In Section 3, we construct a smooth solution of an approximation scheme of (1.7)-(1.8). In Section 4, we recover the weak solutions from approximation by the weak convergence method.

A priori estimates and some useful lemmas
In this section, we derive a priori estimates of the problem (1.7)-(1.8), which will help us to derive the weak stability of the solutions. Firstly, we derive the energy inequality for any smooth solutions of (1.7)-(1.8).
Lemma 2.1. For any smooth solution (u, f ) to the the problem (1.7)-(1.8), the following equality holds Proof. Multiplying by u on both sides of (1.7) 2 , and integrating over T 3 , we have Multiplying by 1 + 1 2 |ξ| 2 on both sides of (1.7) 1 , and integrating over By using the Fubini's theorem, (1.9) and (1.6), the last term on the right hand side of (2.3) can be estimated as follows Adding (2.2) to (2.3) and using (2.4) together, (2.1) follows.
To develop further estimates, we will rely on the following lemma.

An approximation scheme
In this section, we construct smooth solutions of an approximation system. To that purpose, we define a finite dimensional space We propose the following approximation scheme associated with the Navier-Stokes-Boltzmann equations (1.7)

The weak solutions of kinetic part
The first step of solving (3.1)-(3.2) is to investigate the global existence of weak solutions of the following problem: To this end, we construct a sequence of solutions in n verifying The characteristic method gives us a solution of (3.4) for any given f n−1 m . Indeed, when f n−1 m is given, we are able to define the trajectories x(τ ) = x(τ, t, x, ξ) and ξ(τ ) = ξ(τ, t, x, ξ) with the following ODE systems Along the trajectories above, the solutions of (3.4) satisfies By the standard theory of ODE, there exists a smooth solution of (3.5) as follows It is also a smooth solution of (3.4) for any given f n−1 m . Thanks to (3.6), we find f n m ≥ 0, for all n ≥ 0, and {f n m } ∞ n=1 is an increasing sequence of measure functions with respect to n. For this proof, we refer the reader to [9].
Next we shall apply the compactness argument to recover the weak solutions of problem (3.3) by passing to the limits of f n m as n goes to infinity. To this end, we need to derive uniform estimates of f n m with respect to n. Note that, {f n m } ∞ n=1 is an increasing nonnegative function. The Gronwall inequality yields the unform estimates of f n m with respect to n.
The following Lemma 3.1 provides that f n m is bounded in L ∞ (0, T ; L p (T 3 ×R 3 )), for any p ≥ 1, with respect to n > 0. , Proof. Thanks to Lemma 2.2, we derive the following bound from (3.6) where we used {f n m } is an increasing sequence with respect to n. Applying the Gronwall inequality to (3.8), one obtains where we used facts T (ξ, ξ ′ ) ≥ 0, f n m ≥ 0 for all n ≥ 0, and λ > 0.

Lemma 3.2. For any
Proof. We derive the following bound from (2.5) and (3.7): In the meanwhile, by (1.6), (3.7) and Fubini's theorem, we find here C > 0 depends only on the initial data, λ and T . This completes the proof of Lemma 3.2.
In order to pass to the limit as n goes to large, we need the bounds of R 3 f n m dξ, R 3 ξf n m dξ and R 3 |ξ| 2 f n m dξ in the following lemma.
for some k ≥ 1, then the following estimates hold
Similar, we can show the following lemma: , for all 0 ≤ t ≤ T.
Proof. Multiplying by |ξ| k on both sides of (3.4), one finds where we have used f n m is an increasing sequence with respect to n. Thus, we have Using Hölder's inequality, the right hand side of (3.17) can be estimated as follows: , substituting this into (3.17), we complete the proof of Lemma 3.4.
Thus, we are ready to show the following existence of weak solutions of approximation (3.3).
Lemma 3.5. For any T > 0 and fixed m > 0, there exists a weak solution of approximation (3.3) in the following sense: In particular, the solution satisfies the following bounds: Proof. In order to pass to the limits as n → ∞, we investigate the convergence of operator x, ξ ′ )dξ ′ in the suitable space. To this end, we need the following convergence as n goes to infinity. Since f n m is an increasing sequence with respect to n, then by Lemma 3.1 and monotone convergence theorem, as n → ∞, we have f n m ⇀ f m , weakly( * ) in L ∞ (0, T ; L q (T 3 × R 3 )), f or any q > 1; f n m → f m , strongly in L p (0, T ; L p (T 3 × R 3 )), f or any 1 ≤ p < ∞; (3.24) Meanwhile, for any smooth test function ψ(x), we find as n → ∞, thanks to (3.24). By Lemma 3.2, this yields (3.26) for any q > 1. On the other hand, from (3.12), (3.13) and (3.14), we deduce the following convergence Since f n m given by (3.6) is a smooth solution of problem (3.4), it satisfies the following weak formulation Letting n tend to infinity in (3.30), and using (3.24), (3.26), (3.27) and (3.28), one obtains Here we should give a remark that the solution f m satisfies the following bounds f m (t, x, ξ) L ∞ (0,T ;L p (T 3 ×R 3 )) ≤ C, f or any p ≥ 1, (3.31) ≤ C, f or any p ≥ 1, (3.32)

The Navier-Stokes part
In this subsection, we shall study the solution of the Navier-Stokes part, and the energy inequality for the whole approximation (3.1). First, we are able to view the right hand side of (3.1) 2 as a external force of the Navier-Stokes equations. From (3.21) and (3.22), the right hand side of (3.1) 2 can be estimated as follows: Thus, we can apply the classical theory of Navier-Stokes equations to solve (3.1) 2 when f m is a solution of (3.3).
Next we consider the following weak formulation of the Navier-Stokes equation (3.1) 2 : where ϕ ∈ X m . Since X m is a finite dimensional space, we can write u m as follows By the standard Galerkin method, approximation (3.1) 2 yields the following ODE: From (3.36), then by using the classical ODE theory, there exists a unique solution α im (t)(i = 1, · · · , m) of (3.38) for any 0 ≤ t ≤ T m , where 0 < T m ≤ T. This gives us a unique solution u m ∈ Y m of weak formulation (3.37) for any 0 ≤ t ≤ T m . Next, we can derive the energy inequality for u m . Indeed, taking ϕ i for (3.37) and multiplying the equation by α im , then summing the resulting equality from i = 1 to m, we have 1 2 where we used a fact that all norms in X m are equivalent. This yields ds.  Proof. Thanks to (3.40), u m is bounded in the set A. Meanwhile, taking ϕ in (3.37) to be ϕ i , multiplying the equation by α ′ im (t), then summing the resulting equality from i = 1 to m, we have Note that all norms in X m are equivalent to each other, (3.36) and (3.40), one obtains that , ũ L ∞ (0,T ;L 2 (T 3 )) ), (3.41) for any t ∈ [0, T m ]. Thanks to (3.40) 2 and (3.41), the Aubin-Lions Lemma, u m = S(ũ) is compact in A. On the other hand, it is easy to verify that S is sequentially continuous, see [9,12] for the details. Thus the Schauder's fixed point theorem gives that S has a fixed point u m in A.
We have shown that there exists (f m , u m ) on a small time interval [0, T m ]. In an effort to extend T m to T , we rely on the following uniform bounds on u m . Lemma 3.7. For any t ∈ [0, T m ], we have where we used the Fubini's theorem, (1.9) and (1.6) in the last equality of (3.45). Next, by the convergence (3.24)-(3.29), we are able to pass to the limits in (3.45) as n → ∞. In fact, we can recover the following inequality Applying Lemma 3.6 to (3.46), one obtains

(3.49)
Moreover, for any 0 ≤ t ≤ T , the weak solution (u m , f m ) satisfies the following energy inequality

Recover the weak solutions
The main goal of this section is to recover the weak solutions of problem (1.7)-(1.8) by passing to the limits of (u m , f m ) which constructed in Proposition 3.1. In particular, we shall pass to the limits as m goes to infinity and ǫ tends to zero, and show that the limit function is a weak solution of problem (1.7)-(1.8). In the following, we will investigate the weak limits with respect to m in Step 1 and pass to the limits with respect to ǫ in Step 2.
Step 1. Passing to the limit as m → ∞.
In this step, we keep ǫ > 0 fixed, deducing from Proposition 3.1, we have the following estimates independent of m: f m L ∞ (0,T ;L p (T 3 ×R 3 )) ≤ C, for any 1 ≤ p ≤ +∞, (4.1) Meanwhile, as the same in Section 3, the solution satisfies the following estimate: ≤ C, for any 1 ≤ p ≤ +∞. With the above estimates (4.1)-(4.9), we are ready to investigate the limits as m goes to large. To this end, we shall rely on the Aubin-Lions lemma for the Navier-Stokes part and the L p average velocity lemma for the kinetic part. Using the same arguments as that in [3], we prove that ∂ t u m is bounded in L   Lemma 4.2. The following energy inequality holds as m → ∞: