Strong solutions to the inhomogeneous Navier–Stokes–BGK system

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Abstract

In this paper, we are concerned with the local-in-time well-posedness of a fluid-kinetic model in which the BGK model with density dependent collision frequency is coupled with the inhomogeneous Navier–Stokes equation through drag forces. To the best knowledge of authors, this is the first result on the existence of local-in-time smooth solution for particle–fluid model with nonlinear inter-particle operator for which the existence of time can be prolonged as the size of initial data gets smaller.

Introduction

Sprays are complex flows consisting of dispersed particles in underlying gas, for instances, spray in the air, fuel-droplets suspended in the cylinder in the combustion process of engines, pollutants floating in the air or water. The evolution of such particle–fluid system can be described in various ways according to the corresponding physical situation and the modeling assumptions. In this paper, we consider the case where the relaxation through inter-particle collisions and the drag of the surrounding fluid compete, which is described by the BGK model coupled with the inhomogeneous Navier–Stokes equations through drag forces: tf+vxf+v((uv)f)=ρf(M(f)f),tρ+x(ρu)=0,t(ρu)+x(ρuu)+xpμΔxu=R3(uv)fdv,xu=0,subject to initial data: (f(x,v,0),ρ(x,0),u(x,0))(f0(x,v),ρ0(x),u0(x)),(x,v)T3×R3.Here, f=f(x,v,t) denotes the number density function of the immersed particles on the phase space of position xT3 and velocity vR3 at time t>0, and ρ=ρ(x,t) and u=u(x,t) are the local density and bulk velocity of the fluid, respectively. For simplicity, we assume that the viscosity coefficient μ=1 throughout the paper. The local Maxwellian M(f) is defined by M(f)(x,v,t)=ρf(x,t)(2πTf(x,t))3exp|vUf(x,t)|22Tf(x,t),where the macroscopic fields of local particle density ρf, local particle velocity Uf, and local particle temperature Tf are given by ρf(x,t)R3f(x,v,t)dv,ρf(x,t)Uf(x,t)R3vf(x,v,t)dv,and3ρf(x,t)Tf(x,t)R3|vUf(x,t)|2f(x,v,t)dv. An explicit computation gives the following cancellation property: R3(M(f)f)1v|v|2dv=0.

Particle–fluid models have received immense attention recently since the situation of particles drafting in fluid arises very often in nature or engineering, and the coupling of kinetic equations and fluid equations addresses various interesting mathematical problems and modeling issues. We can roughly divide the literature on the mathematical theory of such kinetic-fluid model into two categories according to whether the collisional interactions between the immersed particles are taken into account or not. In the absence of collisional interactions, Vlasov or Vlasov–Fokker–Planck type equations coupled with various fluid equations are investigated. For the existence of the weak solutions of such collisionless particle–fluid models, we refer to [1], [2], [3], [4], [5], [6]. Results on the strong solutions can be found in [7], [8]. Particle-kinetic models involving local-alignment phenomena between the immersed particles can be found in [9], [10], [11]. We now turn to literature including particle–particle collisions. In [12], [13] the existence of weak solutions for Vlasov–Navier–Stokes equations with a linear particle operator that explains the break-up of droplets is considered. In [14], Mathiaud obtained the existence of local-in-time classical solution for the Navier–Stokes–Boltzmann equation when the initial data is a small perturbation of a global Maxwellian. In [15], the authors obtained the existence of global-in-time existence of weak solutions under the condition of finite mass, energy and entropy. In [16], [17], large-time behavior of solutions and finite-time blow-up phenomena of particle–fluid systems are considered.

A brief review on the BGK model is also in order. The BGK models [18] have been very popularly employed in physics and engineering as a satisfactory relaxational approximation of the Boltzmann equation which suffers severely from high computational cost. The existence theory for the BGK model is first established by Perthame [19] in which the weak solution is obtained under the condition of finite mass momentum and energy. For the initial data with appropriate decay in the velocity space, a unique existence is established in [20]. These results are adapted and extended, for example, to Lp problem [21], gases under the influence of external forces or mean-fields [22], gas mixture problem in which the gas consists of more than one type of gas molecules [23], ellipsoidally generalized BGK model introduced to better calibrate fluid coefficients [24], and polyatomic molecules formed by bonds of more than one atom [25], [26]. The existence of classical solution near equilibrium and their asymptotic equilibrization can be found in [27], [28]. For the studies on the stationary problems for the BGK model, see [29], [30]. BGK model is also fruitfully employed in the derivation of various macroscopic or hydrodynamic models [31], [32], [33], [34], [35], [36], [37], [38]. The literature on the numerical applications of the BGK model are immense, we refer to [39], [40], [41], [42], [43], [44], [45], [46] and references therein for interested readers.

The inhomogeneous incompressible flow arises when two immiscible incompressible flows with different densities are blended. It also describes incompressible flows in which other substances are melted. We refer to [47] for the general background on the incompressible flows. Besides its own physical relevance, this model has been attracting interest of researchers as a mathematical bridge toward compressible fluid models. Kazhikhov first considered the existence of (1.1) in [48] where he established the existence of global weak solutions and the local strong solutions to this model. The uniqueness was then obtained by Ladyženskaja and Solonnikov in [49]. Danchin and Mucha [50] consider the well-posedness in the framework of strong solution in critical spaces. Lions [47] obtained the global existence of weak solutions for the inhomogeneous incompressible Navier–Stokes equations that allows the vacuum condition together with the density-dependent viscosity coefficient. Strong solutions for the initial data with vacuum was addressed in [51] with the introduction of a compatibility condition. Results on blow-up criteria can be found in [52], [53]. For the study of the particle–fluid model using the inhomogeneous incompressible fluid equations, we refer to [3], [6].

To the best knowledge of the authors, the only result on the existence of classical solutions for particle-kinetic models involving collisional interactions between immersed particle is established in [14] (for weak solutions, see [15]), in which Mathiaud considers a local-in-time existence for a fluid-kinetic model constructed from the coupling of the Navier–Stokes equation with the Boltzmann equation near a global Maxwellian under the assumption that the high order energy functional is sufficiently small. In [14], however, the exchange between the length of the life span and the size of the initial data does not occur. That is, no matter how small an initial perturbation we take in the energy norm, the life span of the solution cannot be extended over a certain fixed time. In this paper, we show that such restriction can be removed, at least for the case of the BGK type relaxation operator. We also mention that the global-in-time existence of strong solution for the relaxation operator with nontrivial collision frequency remains open even for the non-coupled classical BGK model.

To precisely state our main result, we first define the notion of a strong solution.

Definition 1.1

For a given time T(0,), we say that (f,ρ,u) is a strong solution to system (1.1)–(1.2) if it satisfies the system in the sense of distributions with the following regularity: (i)fC([0,T];Wq1,(T3×R3)) with q>5,(ii)ρC([0,T];H3(T3)),and(iii)uC([0,T];H2(T3))L2(0,T;H3(T3)).

Our main results read as follows (see Notation below the statement of the theorem for the definitions of function spaces):

Theorem 1.1

Fix T(0,). Then, there exists ε>0, which depends only on T, such that for any initial data (f0,ρ0,u0) satisfying the following conditions: (i)infxT3ρ0(x)>0,ρ0H3(T3),(ii)|ν|1esssupx,v(1+|v|)q|νf0(x,v)|+u0H2(T3)<ε,and(iii)f0>ε1(1+|v|)(q+3+a),for someε1>0 and a>0, the system (1.1)(1.2) admits the unique strong solution (f,ρ,u).

Remark 1.1

The initial positivity condition (iii) is necessary to guarantee the positivity of macroscopic field ρf, see Lemma 3.3.

Remark 1.2

The solution space for the kinetic equation requires the boundedness of fluid velocity xu due to the coupling through the drag force, see Lemma 3.4 for more details. To handle this regularity for the fluid equation, we considered uC([0,T];H2(T3)), and this subsequently gives uL2(0,T;H3(T3)) because of the viscosity in the fluid equations. Notice that Sobolev embedding theorem asserts H2(T3)L(T3). The function spaces we are considering for the fluid part may not be optimal. In view of the review given in the introduction, it might be extended to weaker regularity spaces, especially the critical ones as in [50]. The removal of the strict positivity of the initial fluid density would also be an interesting problems [51]. We leave them for our future work.

Notation

Throughout the paper, k denotes any partial derivative α with multi-index α, |α|=k. We often omit x-dependence of differential operators for simplicity of notation. We denote by C a generic, not necessarily identical, positive constant. The relation AB denotes the inequality ACB for such a generic constant. Below we introduce the norms and function spaces to be used in the paper.

  • For functions f(x,v),g(x), fLp and gLp denote the usual Lp(T3×R3)-norm and Lp(T3)-norm, respectively.

  • We use the following weighted norms for f(x,v): fqfLqesssupx,v(1+|v|)qf(x,v),fWq1,|ν|1νfq. Lq(T3×R3) and Wq1,(T3×R3) naturally denote the spaces of functions with finite corresponding norms.

  • Hs(T3) denotes the sth order L2(T3) Sobolev space.

The rest of the paper is organized as follows. In Section 2, we introduce several lemmas regarding boundedness properties of the macroscopic fields (ρf,Uf,Tf) and the local Maxwellian M(f), which will be heavily used throughout the paper. In Section 3, a sequence of approximation systems to (1.1)–(1.2) is constructed. In Section 4, we prove that the sequence of solutions constructed in Section 3 is indeed a Cauchy sequence and the limit is the solution of the system (1.1) in the sense of Definition 1.1.

Section snippets

Preliminaries

We present a series of lemmas that will be crucially used throughout the paper.

Lemma 2.1

[20]

There exists a positive constant Cq, which depends only on q, satisfying

  • (i)

    ρfCqfqTf32(q>3orq=0),

  • (ii)

    ρf(Tf+|Uf|2)(q3)2Cqfq(q>5orq=0), and

  • (iii)

    ρf|Uf|q+3((Tf+|Uf|2)Tf)32Cqfq.(q>1orq=0),

for almost everywhere xT3.

We now show that the q-norm of a generalized local Maxwellian Mγ(f) with γ>0 can be controlled by that of f. Although the proof is essentially given in [20], we provide it here for the completeness of our

Global existence and uniqueness of approximation system

We construct the sequence of approximation solutions to linearized systems of (1.1). We consider following linearized NS–BGK system: tfn+1+vxfn+1+v((unv)fn+1)=ρfn(M(fn)fn+1),tρn+1+unxρn+1=0,ρn+1tun+1+ρn+1unxun+1Δxun+1+xpn+1=R3(unv)fn+1dv,xun+1=0,with the initial data and the first iteration step: (fn+1(x,v,0),ρn+1(x,0),un+1(x,0))=(f0(x,v),ρ0(x),u0(x))and(f0(x,v,t),ρ0(x,t),u0(x,t))=(f0(x,v),ρ0(x),u0(x))for n0 and (x,v,t)T3×R3×(0,T).

We now consider the backward

Proof of Theorem 1.1

In this section, we first prove that the approximation sequence (fn,ρn,un) is a Cauchy sequence. Subsequently, we show that the corresponding limit (f,ρ,u) is the solution to the system (1.1), and moreover it has the desired regularity (1.3).

Acknowledgments

The authors would like to thank the anonymous reviewers for the invaluable comments and advice. Young-Pil Choi was supported by National Research Foundation of Korea (NRF), South Korea grant funded by the Korea government (MSIP) (Nos. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCO TJ Park Foundation. Seok-Bae Yun is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-02.

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