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Revista de la Unión Matemática Argentina

versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.48 n.1 Bahía Blanca ene./jun. 2007

 

The Boltzmann equation with Force Term near the Vacuum

Rafael Galeano Andrades

Abstract. We prove a theorem of existence, uniqueness and positivity of the solution for the Boltzmann equation with force term and initial data near the Vacuum.

Key words and phrases. Force Term, Boltzmann equation, Near the vacuum

This article is a result of the project of investigation ”Evolution of Equations of Kinetic Type”, financed by the University of Cartagena.

1. INTRODUCTION

The aim in this article it is prove global existence of the problem for small data (near to the vacuum) in the case of a solid sphere. For the density f = f(t,x,v), t ≥ 0; x, v ∈ ℝ3 ,we write the equation of Boltzmann:

ft + v.∇xf + F.∇vf = Q (f,f )} f(0,x,v) = f0(x,v )
(1)

Let us also consider the following problem:

 ∂F ) ft + v.∇xf + F.∇vf + t---.∇vf = Q (f,f)} ∂t ) f (0,x,v) = f0(x,v)
(2)

where

pict

here S2+ = {w ∈ S2 : wv ≥ wu } , σ is a constant proportional to the area of the sphere and

 ′ } v = v - aw u′ = u + aw

a = w (v - u) .

The Conservation momentum is given by

u′ + v ′ = u + v

and the Energy of conservation is

|u′|2 + |v′|2 = |u|2 + |v|2.

We write Ql(f,f) = f R(f ) , where

 ∫ ∫ ∫ R (f)(t,x,v) = σ w (v - u)f(t,x,u )dudw = πσ |v - u|f(t,x,u )du S2 ℝ3 ℝ3 +

Let F be a vectorial field not depending on f such that:

pict

and for a given β > 0 , let

 { 0 3 3 -β(|x|2+|v|2)} M = f ∈ C ([0,∞ ) × ℝ × ℝ ) : exists c > 0 such that |f(t,x,v )| ≤ ce

with norm

 β(|x|2+|v|2) ||f|| = supe |f(t,x,v)|. t,x,v

This makes M into a Banach Space [8].

To solve problem (2) we introduce the following notation:

f# (t,x,v) = f(t,x + vt,v + tF(t,x,v))
(3)

with F differentiable with regard to time. Therefore

pict

we write the equation (1) as

-df #(t,x,v) = Q (f#, f#)(t,x,v) dt
(4)

Integrating (3) with respect to time, we obtain

 ∫ t f# (t,x, v) = f0(x,v) + Q (f#,f #)dτ 0
(5)

We observe that if F is constant in time, (2) reduces to (1). We will prove the following theorems:

Theorem 1. There exists a constant R0 , such that if ||f0|| and σ β-2R0 are sufficiently small and F is differentiable with regard to the time t , then the equation (4) has a unique solution  # f ∈ MR0 = {f ∈ M : ||f || ≤ R0 } .

Theorem 2. Let us consider

 ∫ t f#(t,x,v ) = f (x, v) + Q (f #,f #)dτ. 0 0
(6)

There are constants c0 and R0 such that if ||f0|| ≤ c0R0 and  -2 α β R0 are sufficiently small, then the equation (5) has a unique non-negative solution f# ∈ MR0 .

To solve problem (1) we consider:

 ∫ t # f (t,x,v) = f (t,x + vt,v + 0 F(τ,x, v)dτ)
(7)

Assuming F integrable with regard to the time and therefore

 # df--(t,x,v) = Q (f#, f#) dt
(8)

and integrating we obtain again (5) and the theorems are completed, but with F integrable with regard to the time.
This topic is developed in [4], as well as in the articles [11], [9], [5] and [15].  

Article [11] refers to a gas with a strong sphere and to a initial condition which tends exponentially to zero at infinity in phase space. Article [9] generalizes results of [11] to the Boltzmann equation with interaction potential between two particles. In both articles, the mass of the gas is infinite. In another direction the result of [5] refers to a gas with infinite mass where the initial condition is assumed to decay in a physical space with behavior of power inverse. In this case for sufficiently small decay the mass of the gas can be infinite. Another generalization is given in the article [15] where global existence is proven for initial conditions decaying with behavior of power inverse in the whole phase space. The articles [5] and [13] refer both to Boltzmann equation with interaction of general potential for the couples of internal forces of particles. Besides these previously mentioned articles, we also have to mention two results given by Polewczak. In the first one [13], the author generalizes the theory of previously limited existence to the case of mild solution, also to the case of classic solution. In the second [14] the mathematical results are generalized to the case of initial conditions with more general decay at infinity in phase space.

On the Problem of Cauchy with field of External force

First, let us consider the spatially inhomogeneous equation for neuter particles in a force field F = F (t,x, v) . In relationship with this topic we should mention the article of Asano [1] in which local existence is proven for general initial conditions. The formulation of Asano is the starting point of all the studies developed in this article.  

Some studies of the problem can be seen in the articles [2], [7], [10] and [6]. In particular, the articles of Asano [2] and Grunfeld [7] provide a global existence proof for the solution with initial condition close to equilibrium and a conservative force field. Hamdache [10] gives us a result with initial conditions decaying exponentially to zero at infinity in phase space and trajectories prescribed by a oscillating field. In [6] it is given a result of global existence for decaying conditions and for a force field acting in an interval of time [0,T ] with T big, but finite. Recently in [12] Lions developed the problem of Cauchy in L1 in the presence of an external field. The important aspect of the result in [12] is that refers to force fields depending on the distribution function, this case is entirely different from the cases previously prescribed.  

This article divides into two parts, the first one has to do with a theorem which, as opposed to the articles previously mentioned, solves the problem for fields differentiable with regard to time, and subsequently we extend it to a theorem of positivity of the solutions.

Demonstration of the theorem 1. Indeed, let us consider :

Q(f #,f# )(t,x, v) = Q (f #,f# )(t,x, v) - Q (f#,f #)(t,x,v), g l

where

(a)

pict

(b)

pict

now

pict

therefore

pict

(c)

pict

(d)

pict

here ρ(x,v)-1 = e-β(|x|2+|u|2) , therefore

pict

now ∫∞ 2 ∘ -- 0 e-β|x+τ(v- u)| dτ ≤ πβ|v-1u| ; [8, pag. 28]

This is,

pict

(e) We define the operator H on M by

 ∫ # t # # Hf = f0(x,v) + Q (f ,f )|dτ 0

and let MR = {f ∈ M : ||f|| ≤ R }

The previous estimates show that if  R- ||f0|| ≤ 2 and  # f ∈ MR , then

pict

We choose 2 π3β-2σR ≤ 1 2 . Therefore H applies M R into itself for R sufficiently small. Now H is a contraction on MR . Since the elements of MR are continuous, by the fixed point theorem the statement is proven. □

Demonstration of the theorem 2. If σβ -2R and ||f0|| are sufficiently small, then

0 ≤ l0(t) ≤ l1(t) ≤ u1(t) ≤ u0 (t), 0 ≤ t ≤ T
(9)

and therefore the system

 ∫ t ∫ t ) l#(t) + l#R# (u)(τ)dτ = f0 + Qg(l#,l#)(τ)dτ ||} 0 0 # ∫ t # # ∫ t # # || u (t) + u R (l)(τ)dτ = f0 + Qg (u ,u )(τ)dτ ) 0 0
(10)

has a unique global solution (l,u) with (l#, u#) ∈ MR × MR . Indeed, if l0 = 0

 ∫ t u#1 (τ ) = f0 + Qg(u#0 ,u#0 )dτ. 0

This is , with q = w ⋅ (v - u)

pict

That is, condition (6) is true if

pict

If we define  # -β|x-tv|2 u0 (τ,x, v) = e w (u) , we have that

 ′ ) u# (τ,x + τ(v - v′),v′) = e- β|x+t(v-v )|2w (v ′)} 0 ′ u#0 (τ,x + τ (v - u′),u ′) = e-β|x+t(v-u )|2w(u ′))

and therefore

pict

We write  2 ψ (σ) = supx eβ|x|f0(x,v ) . This is

 ∫ t∫ ∫ ′ ′ β|x|2 -β|x+τ(v- v′)|2 -β|x+ τ(v-u′)|2 ψ(v)+ σ w⋅(v- u)w (v )w (u )e e +e dwdud τ ≤ w(v) 0 S2+ ℝ3
(11)

We want a non negative solution w for (8), and a sufficient condition is:

 ∫ t∫ ∫ ′ ′ β|x+t(v-u)|2 ψ(v) + σ 0 S2 ℝ3 w ⋅ (v - u )w(v )w(u )e dwdud τ = w(v). +

This is

 ∘ --∫ ∫ π- ′ ′ ψ (v ) + σ β 2 3 w w (v)w (u )dwdu = w(v) S+ ℝ
(12)

To prove the existence of a solution w ≥ 0 of (9 ) let us consider the space

 { 0 3 -β|v|2} G = g ∈ C (ℝ ) : It exists c > 0 con |g(v)| ≤ ce

with norm  β|v|2 ||g||G = supv e |g(v)|

Let us consider the operator F defined on G by:

 ∫ ∫ ′ ′ F(w )(v) = ψ(v) + ε w(v )w (u)dudw S2+ ℝ3

with  ∘ π- ε = σ β .
F applies a sufficiently small ball in G into itself. Indeed, let w ≥ 0 , since 0 ≤ f0 ∈ M , F (w )(v) ≥ 0 and

pict

This is, for such w , ||F w ||G ≤ c0 + 2σπ3 β-2||w ||2G . Therefore for w1, w2 ∈ G

 3 -2( ) ||F w1 - F w2||G ≤ 2σπ β ||w1||G + ||w2||G ||w1 - w2||G.

Namely, F applies non-negative functions in the ball of radius R0 into itself, and it is a contraction there if ||f0|| and αβ -2R0 are sufficiently small.

As F is a contraction,  n w = limn → ∞ F (w ) , we can take  0 F (w ) = f0 , f0 ≥ 0 and the solution for (9) is non-negative.

It remains to show that u = l . By definition

 ∫ ∫ # t# # t # # )| l (t) + l R (u)(τ)dτ = f0 + Qg(l ,l )(τ)dτ |} ∫ 0 ∫ 0 # t # # t # # ||) u (t) + u R (l)(τ)dτ = f0 + Qg (u ,u )(τ)dτ 0 0

then

 ∫ # # t # # # # # # # # # # # # (u - l )(t) = (Qg (u ,u - l )+Qg (u - l ,l )+l R (u - l )- (u - l )R (l ))dτ. 0

This is ||u# - l#||M ≤ cσ β-2(||u# ||||u# - l# || + ||l#||||u# - l# ||) . Now u# , l# both are in M R and in this way ||u#|| and ||l# || are bounded by cR 0 ; therefore the conclusion follows if  - 2 σβ R is sufficiently small. □

Bibliography

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[2]    ASANO K. Global Solutions to the initial boundary value Problem for the Boltzmann equation with an external force. Transp. Theory Statist. Phys., 16 (1987), 735-761.        [ Links ]

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[9]    HAMDACHE R.Existence in the Large and Asymptotic behaviour for the Boltzmann equation. Japan J. Appl. Math., 2 (1985),1-15.        [ Links ]

[10]    HAMDACHE R. Thése, Univ.Pierre et Marie Curie. Paris 1988.        [ Links ]

[11]    ILLNER R., SHIMBROT M. The Boltzman equation: Global existence for a rare gas in an Infinite Vacuum. Commun. Math. Phys. 95 (1984), 117-126.        [ Links ]

[12]    LIONS P. L. Compactness in Boltzmann equation via Fourier integrals operator and applications III. J. Math. Kyoto Univ.,34(1994), 539-584.        [ Links ]

[13]    POLEWCZAK J. Classical solution of the nonolinear Boltzmann equation in all ℝ3 : asymptotic behaviour of solution. J. Stat. Phys,50 (1988), 611-632.        [ Links ]

[14]    POLEWCZAK J. New estimates of the nonlinear Boltzmann operator and their application to existence theorem. Transp. Theory Statist. Phys., 18(1988), 235-247.        [ Links ]

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Rafael Galeano Andrades Programa de Matemáticas, Universidad de Cartagena,
rgaleanoa@unicartagena.edu.co

Recibido: 9 de noviembre de 2006
Aceptado: 15 de agosto de 2007