We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.

1. Introduction

The Boltzmann equation (BE) is a fundamental equation in kinetic theory. Half-space problems for the BE are of great importance in the study of the asymptotic behavior of the solutions of boundary value problems of the BE for small Knudsen numbers [1, 2] and have been extensively studied both for the full BE [3, 4] and for the discrete Boltzmann equation (DBE) [5–8]. The half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-type equations in a neighborhood of the boundary. In [8] nonlinear boundary layers for the DBE, the general discrete velocity model (DVM) was considered. Existence of weakly nonlinear boundary layers was proved. Here we exemplify the theory in [8] for a simplified model, the Broadwell model [9], where the whole machinery is actually not really needed, even if it helps out. For the nonlinear Broadwell model, we obtain explicit expressions for boundary layers near a wall moving with a constant speed. The number of conditions, on the assigned data for the outgoing particles at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian at infinity) solution of the problem is in complete agreement with the results in [8] for the DBE and [3] for the full BE. Here we also want to mention a series of papers studying initial boundary value problems for the Broadwell model using Green’s functions [10–16].

We also consider the question of existence of shock profiles [17, 18] for the same model [9, 19]. The shock profiles can then be seen as heteroclinic orbits connecting two singular points (Maxwellians) [20]. In [20] existence of shock profiles for the DBE in the case of weak shocks was proved. We exemplify the theory in [20] for the Broadwell model, where again the whole machinery is not really needed, even if it helps out. In this way we, in a new way, obtain similar explicit solutions, not only for weak shocks, as the ones obtained in [19] for the same problem.

The paper is organized as follows. In Section 2 we introduce the Broadwell model and find explicit expressions for the nonlinear boundary layers near a wall moving with a constant speed, and in Section 3 we find explicit expressions for the shock profiles for the Broadwell model.

2. Nonlinear Boundary Layers for the Broadwell Model Near a Moving Wall

In this section we study boundary layers for the nonlinear Broadwell model near a wall moving with a constant speed b. In [8] the nonlinear boundary layers for the DBE, the general discrete velocity model (DVM) was considered. Existence of nonlinear boundary layers was proved. Here we exemplify the theory in [8] in the case of a simplified model, where the whole machinery is actually not really needed, even if it helps out. The same problem was considered in [21] for a mixture model, where one of the two species was modelled by the Broadwell model.

We consider the classical Broadwell model [9] in space (with velocities ξ1=(1,0,0), ξ2=(-1,0,0), ξ3=(0,1,0), ξ4=(0,-1,0), ξ5=(0,0,1), and ξ6=(0,0,-1))(1)∂f~1∂t+∂f~1∂x=2σ3f~3f~4+f~5f~6-2f~1f~2,∂f~2∂t-∂f~2∂x=2σ3f~3f~4+f~5f~6-2f~1f~2,∂f~3∂t=-2σ3f~3f~4-f~1f~2,∂f~4∂t=-2σ3f~3f~4-f~1f~2,∂f~5∂t=-2σ3f~5f~6-f~1f~2,∂f~6∂t=-2σ3f~5f~6-f~1f~2,where σ is the mutual collision cross section. For a flow axially symmetric around the x-axis we can reduce system (1) to (with f1=f~1, f2=f~3=f~4=f~5=f~6, and f3=f~2) [9](2)∂f1∂t+∂f1∂x=4σ3f22-f1f3,4∂f2∂t=-8σ3f22-f1f3,∂f3∂t-∂f3∂x=4σ3f22-f1f3.

The collision invariants are(3)ϕ=α1,1,1+β1,0,-1,α,β∈R,and the Maxwellians (equilibrium distributions) are(4)M=s4a4,a2,1,with s=eα-β/4>0,a=eβ/4>0,α,β∈R.The density, momentum, and internal energy can be obtained by(5)ρ=f1+4f2+f3,ρu=f1-f3,2ρe=f1+f3.

Let b, the speed of the wall, be a real number such that(6)b∉-1,0,1.We define the projections R+:R3→Rn+ and R-:R3→Rn-, n-=3-n+, by(7)R+h=h+=h1,…,hn+,R-h=h-=hn++1,…,h3,where(8)n+=0if b>11if 0<b<12if -1<b<03if b<-1,h=h1,h2,h3.

Consider the problem(9)D∂f∂t+B0∂f∂x=Qf,f,x>bt,t>0,f+bt,t=C~f-bt,t+φ~0,fx,0=f0x,f0x⟶Mas x⟶∞,where f=f1,f2,f3, D=(1,4,1), B0=(1,0,-1), M=s4(a4,a2,1), C~ is a given n+×n- matrix, φ~0∈Rn+, and Qf,f is defined by the bilinear expression:(10)Qf,g=2σ32f2g2-f1g3-g1f31,-2,1.

After the change of variables y=x-bt and the transformation(11)f=M+M1/2h,with M=s4a4,a2,1,we obtain the new system(12)∂h∂t+B∂h∂y+Lh=Sh,h,y>0,t>0,h+0,t=Ch-0,t+φ0,hy,0=h0y,h0y⟶0as y⟶∞, where h=h1,h2,h3, B=B0-bD, C is an n+×n- matrix, φ0∈Rn+, and(13)Lh=-2M-1/2QM,M1/2h=4s4σ3h1-2ah2+a2h31,-2a,a2,Sh,g=M-1/2QM1/2h,M1/2g=2s2σ32h2g2-h1g3-g1h31,-2a,a2.

Similar initial boundary value problems have been studied in a series of papers using Green’s functions (with s4=1/6, a=1) for 1/3<b<1 in [10] (with C=(c1c2)) and [14], for -1<b<-1/3 in [11], for 0<b<1/3, with C=0 in [12, 13, 15], and for diffuse boundary conditions in [16].

Here we consider the stationary nonlinear system(14)Bdhdy+Lh=Sh,h,h=hy,y>0,h+0=Ch-0+φ0,hy⟶0as y⟶∞.The linearized collision operator(15)L=4s4σ31-2aa2-2a4a2-2a3a2-2a3a4is symmetric and semipositive and have the null-space(16)NL=spana2,a,1,a2,0,-1=spane1,e2,for some e1 and e2, such that(17)e1,e2B=γiδij,i,j=1,2.Note also that(18)Sh,h,e=0,∀e∈NL.Here and below, ·,· denotes the Euclidean scalar product and we denote ·,·B=·,B·. If b≠a2/(1+a2) we can choose(19)e1=a,12,0,e2=a2b,a321-b,a21-b-b,and then(20)γ1=a2-b1+a2,γ2=a2-b1+a2-a2+1-a4b+1+a2+a4b2.

We let m± denote the number of the positive and negative eigenvalues of the matrix B-1L. The numbers n±, with n++n-=3, defined above, denote the numbers of the positive and negative eigenvalues of the matrix B. Moreover, we let k+, k-, and l denote the number of positive, negative, and zero eigenvalues of the 2×2-matrix(21)K=y1,y1By1,y2By2,y1By2,y2B,where y1=a2,a,1 and y2=a2,0,-1. Then m±=n±-k±-l [22, 23]. The eigenvalues of K are(22)λ±=η±η2-4a2κ,where(23)η=1+a2a2-1-b1+a2,κ=-a2+1-a4b+1+a2+a4b2.We find that sgnλ±=sgnη if κ>0, sgnλ±=±1 if κ<0, λ+=2η, and λ-=0 if κ=0, but κ<0 if η=0, sgn(η)=-sgn(b) if κ>0, and (24)κ=0⟺b=b±=a4-1±1+4a2+2a4+4a6+a821+a2+a4,κ>0if b>b+, or b<b-,κ<0if b-<b<b+.Hence, we obtain the following number of positive and negative eigenvalues for different values of b(25)b-1b-0b+1n+32221110n-01112223k+22111000k-00011122l00100100m+10010010m-01001001for(26)b±=a4-1±1+4a2+2a4+4a6+a821+a2+a4. Particularly, if a=1 then b±=±1/3.

Explicitly, the eigenvalues of the matrix B-1L are 0 (of multiplicity 2) and(27)λ=4s4σκ3b-b3,with κ=-a2+1-a4b+1+a2+a4b2.For b≠b± an eigenvector corresponding to the nonzero eigenvalue λ is(28)v=b+b2,a21-b2,a2b2-b;that is(29)B-1Lv=λv,v,eB=0,∀e∈NL.Furthermore,(30)B-1Sv,v=kv,with k=s2a2σ3b1+3b2.

Example 1.

If a=1, corresponding to a nondrifting Maxwellian M, then we get that(31)b±=±13,λ=4s4σ3b2-13b-b3,v=b+b2,1-b22,b2-b,k=s2σ3b1+3b2.

Note that if φ0=0, then we always have the trivial solution h=0, and if -1<b≤b- (φ0=(φ01,φ02), where both φ01 and φ02 must be zero, i.e., φ01=φ02=0), 0<b≤b+ (φ0∈R, where φ0 must be zero, i.e., φ0=0), or 1<b (no boundary conditions at all at the wall), then we have no other solutions. Otherwise, we have solutions if and only if h0∈span(v+-Cv-).

Below we consider the remaining different cases.

If b+<b<1 then C=(c1c2) and φ0∈R. Hence, if φ0≠0 we obtain the unique solution(32)hy=λk+Deλyb+b2,a21-b2,a2b2-b,

with(33)D=2b+b2-c1a1-b2-2c2a2b2-b2h0λ-k.

If b-<b<0 then C=c1c2 and φ0=(φ01,φ02)∈R2. Hence, if φ01≠0, c1≠2b/a(1-b), and(34)φ02=a1-b2-2c2a2b2-b2b+b2-2c1a2b2-bφ01,

then we obtain the unique solution (32) with(35)D=b+b2-c12ab2-bφ01λ-k,

and if c1=2b/a(1-b), c2≠-(1+b)/2ab, φ01=0, and φ02≠0, then we obtain the unique solution (32) with(36)D=a1-b2-2c2a2b2-b2φ02λ-k.

If b<-1 then C=0 and φ0=(φ01,φ02,φ03)∈R3. Hence, if φ01≠0 and(37)φ02=a1-b2bφ01,φ03=a2b-11+bφ01,

then we obtain the unique solution (32), where(38)D=b+b2φ01λ-k.

We note that in each of the above cases k+ conditions on the assigned data φ0 are implied to have a unique solution. This is in good agreement with the results for the DBE in [8] and for the continuous BE in [3].

Remark 2.

Similar results can be obtained for the (reduced) plane Broadwell model(39)1-bdf1dy=σf22-f1f3,-2bdf2dy=-2σf22-f1f3,-1+bdf3dy=σf22-f1f3. Particularly, with a=1 we have(40)b±=±12,λ=2s4σ2b2-13b-b3,v=b+b2,1-b2,b2-b,k=s2σb.

3. Shock Profiles

In this section we are concerned with the existence of shock profiles [17, 18](41)F=Fx1,ξ,t=fx1-bt,ξ for the Boltzmann equation(42)∂F∂t+ξ·∇xF=QF,F. Here x=(x1,…,xd)∈Rd, ξ=(ξ1,…,ξd)∈Rd, and t∈R+ denote position, velocity, and time, respectively. Furthermore, b denotes the speed of the wave. The profiles are assumed to approach two given Maxwellians(43)M±=ρ±2πT±d/2e-ξ-u±2/2T±(ρ, u, and T denote density, bulk velocity, and temperature, resp.) as x→±∞, which are related through the Rankine-Hugoniot conditions.

The (shock wave) problem is to find a solution f=f(y,ξ) (y=x1-bt) of the equation(44)ξ1-b∂f∂y=Qf,f,such that(45)f⟶M±as y⟶±∞.

In [20] the shock wave problem (44), (45) for the DBE was considered. Existence of shock profiles in the case of weak shocks was proved. Here we exemplify the theory in [20] in the case of a simplified model, where the whole machinery is actually not really needed, even if it helps out. In this way we, in a different way, obtain similar results as is obtained in [19] for the same problem.

We study the reduced system (2) of the classical Broadwell model in (1) [9] in space. The collision invariants are given by (3) and the Maxwellians (equilibrium distributions) by (4).

The shock wave problem for the Broadwell model reads(46)Bdfdy=Qf,f,where f⟶M± as y⟶±∞,where B=B(b)=diag(1-b,-4b,-1+b), f=f1,f2,f3, and Qf,f is defined by the bilinear expression (10).

The density ρ, momentum ρu, and internal energy 2ρE can be obtained by (5). The Maxwellians M-=s-4(a-4,a-2,1) and M+=s+4(a+4,a+2,1) must fulfill the Rankine-Hugoniot conditions(47)ρ+u+-b=ρ-u--b,ρ+2E+-bu+=ρ-2E--bu-,with(48)ρ±=s±41+4a±2+a±4,ρ±u±=s±4a±4-1,2ρ±E±=s±41+a±4.After some manipulations we obtain that(49)2E±=1321+3u±2-1.

We consider(50)Bdfdy=Qf,f,where f⟶M+ as y⟶∞and denote(51)F=M+M1/2h,with M=M+=s+4a+4,a+2,1=s4a4,a2,1.Then we obtain(52)Bdhdy+Lh=Sh,h,where h⟶0 as y⟶∞,with the linearized operator L and the quadratic part S(h,h) given by (13). The linearized collision operator is given by (15) and then fulfills properties (16)–(20).

We assume that B is nonsingular; that is b∉-1,0,1. Then by (52) we obtain the system(53)dhdy+B-1Lh=B-1Sh,h.

In (25) we obtain that(54)b±=a4-1±1+4a2+2a4+4a6+a821+a2+a4=2u+±22E+1+6E+, and the eigenvalues of the matrix B-1L are 0 (of multiplicity 2) and (55)λ=4s4σκ3b-b3=ρ+σ3b-b32E+-1-4u+b+1+6E+b2,with κ=-a2+1-a4b+1+a2+a4b2.

Let(56)h=ϑe1+χe2+μv,where e1 and e2 are eigenvectors (19) corresponding to the zero eigenvalue and v is eigenvector (28) corresponding to the nonzero eigenvalue λ. Then(57)dϑdy=dχdy=0, which implies that(58)ϑ=χ=0, since limy→∞ϑ=limy→∞χ=0. Therefore (59)dμdy+λμ=kμ2, where k is given in (30). We obtain that(60)μ=λk+Deλy. Assume that D≠0 and let(61)b+<b<1,or b-<b≤-11+2a2. Then(62)limy→∞μ=0,limy→-∞μ=λk,and therefore(63)hy=λk+Deλyv,D≠0.We conclude that the solution of system (50) is of the form (64)fy=M++λk+DeλyM+1/2v,where λ=4s4σ3b-b3-a2+1-a4b+1+a2+a4b2,k=s2a2σ3b1+3b2,v=b+b2,a21-b2,a2b2-b,D≠0.It follows that(65)M-=M++λkM+1/2v=s41+3b2p21+b1-b,pq,q21-b1+b,with p=2+a2b-a2,q=1+1+2a2b,which is a Maxwellian. Formally we can allow b<-1 and -1/(1+2a2)<b<0. However, then, the equilibrium distribution (65) will not be nonnegative and, hence, not a Maxwellian.

We note that(66)fy=ΘyM++1-ΘyM-,with Θy=11+Ce-λy,where C=k/D≠0 is an arbitrary nonzero constant. The structure coincides with the one for the Mott-Smith approximation [24] in [25]. However, λ is obtained in different ways.

Remark 3.

We can instead of system (50) consider(67)Bdfdy=Qf,f,where f⟶M- as y⟶-∞,with(68)a22+a2<b<b+,or -1<b<b-,and in a similar way as above, we obtain(69)fy=M-+λk+DeλyM-1/2v,M+=M-+λkM-1/2v.

Example 4.

If a=1 then we have (70)ρ+=6s+4,u+=0,E+=16,b±=±13,λ=4s4σ3b2-13b-b3,v=b+b2,1-b22,b2-b,k=s2σ3b1+3b2.Furthermore,(71)fy=M++λk+DeλyM+1/2v=s41,1,1+4s43b2-11+3b2+D~eλyb1-b,12,-b1+b=r3b-121+b1-b,9b2-1,3b+121-b1+b+D~eλy1,1,1,where r=s41+3b2+D~eλy, and the other Maxwellian is(72)M-=s41+3b23b-121+b1-b,3b+13b-1,3b+121-b1+b.

Example 5.

Similar results can be obtained for the (reduced) plane Broadwell model(73)1-bdf1dy=σf22-f1f3,-2bdf2dy=-2σf22-f1f3,-1+bdf3dy=σf22-f1f3. Particularly, with a=1 we have (74)ρ+=4s+4,u+=0,E+=14,c±=±12,λ=2s4σ2b2-13b-b3,v=b+b2,1-b2,b2-b,k=s2σb. The other Maxwellian is then(75)M-=s42b-121+b1-b,2b+12b-1,2b+121-b1+b.

The shock strength (cf. [19]) is given by the density ratio(76)σρ=ρ--ρ+ρ+=2a2ρ+λk=8s2ρ+b+b2+a2b2-1+a4b2-b1-b21+3b2, if b>0 and if b<0 by(77)σρ=ρ--ρ+ρ-=2a2ρ-λk=8s2ρ-b+b2+a2b2-1+a4b2-b1-b21+3b2.Then the shock strength σρ tends to infinity as b approaches 1 and to zero as b approaches b±; that is(78)σρ⟶∞as b⟶1,σρ⟶0as b⟶b±.

The shock width (cf. [19]) is given by the density ratio(79)dρ=ρ--ρ+maxydρ/dy=4λ=3b1-b2s4σb+b2+a2b-1+a4b2-bor by the velocity ratio(80)du=u--u+maxydu/dy=4λ1-u-/bρ+.We conclude that the shock widths dρ and du tend to zero as b approaches 1 and to infinity as b approaches b±; that is(81)dρ⟶0,du⟶0as b⟶1,dρ⟶∞,du⟶∞as b⟶b±.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

SoneY.SoneY.UkaiS.YangT.YuS.-H.Nonlinear boundary layers of the Boltzmann equation: I. ExistenceBardosC.GolseF.SoneY.Half-space problems for the Boltzmann equation: a surveyUkaiS.KawashimaS.YanagisawaT.On the half-space problem for the discrete velocity model of the Boltzmann equationKawashimaS.NishibataS.Existence of a stationary wave for the discrete Boltzmann equation in the half spaceKawashimaS.NishibataS.Stationary waves for the discrete Boltzmann equation in the half space with reflective boundariesBernhoffN.On half-space problems for the weakly non-linear discrete Boltzmann equationBroadwellJ. E.Shock structure in a simple discrete velocity gasLanC.-Y.LinH.-E.YuS.-H.The Green's functions for the Broadwell model in a half space problemDengS.WangW.YuS.-H.Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundaryDengS.WangW.YuS.-H.Pointwise convergence to Knudsen layers of the Boltzmann equationLanC.-Y.LinH.-E.YuS.-H.The Green's function for the Broadwell model with a transonic boundaryDengS.WangW.YuS.-H.Broadwell model and conservative supersonic boundaryLinH.-E.Nonlinear stability of the initial-boundary value problem for the Broadwell model around a MaxwellianDengS.WangW.YuS.-H.Bifurcation on boundary data for linear Broadwell model with conservative boundary conditionCaflischR. E.NicolaenkoB.Shock profile solutions of the Boltzmann equationLiuT.-P.YuS.-H.Boltzmann equation: micro-macro decompositions and positivity of shock profilesCaflischR. E.Navier-Stokes and Boltzmann shock profiles for a model of gas dynamicsBernhoffN.BobylevA.Weak shock waves for the general discrete velocity model of the Boltzmann equationBernhoffN.Boundary layers and shock profiles for the discrete Boltzmann equation for mixturesBobylevA. V.BernhoffN.BellomoN.GatignolR.Discrete velocity models and dynamical systemsBernhoffN.On half-space problems for the linearized discrete Boltzmann equationMott-SmithH. M.The solution of the Boltzmann equation for a shock waveBobylevA. V.BisiM.CassinariM. P.SpigaG.Shock wave structure for generalized Burnett equations