Diffusion Limit of 1-D Small Mean Free Path of Radiative Transfer Equations in Bounded Domain

CHEN Xiaoand WANG Shu

College of Applied Science,Beijing University of Technology,Beijing 100124,China.

Abstract.In this paper,we consider the diffusion limit of the small mean free path for the radiative transfer equations,which describe the spatial transport of radiation in material.By using asymptotic expansions,we prove that the nonlinear transfer equation has a diffusion limit as the mean free path tends to zero,and moreover we study the boundary layer problem and mixed layer problem in bounded domain[0,1].Then we show the validity of their asymptotic expansions relies only on the smoothness of boundary condition,and remove the Fredholm alternative and centering condition.

Key Words:Transferequations;asymptotic analysis;diffusion limit;boundary layer;mixed layer.

1 Introduction

The radiative transport in space and the interaction with material medium plays an important role in physical systems(see[1–3]).We ignore the thermal diffusion and material motion,and assume that the material is in local thermodynamic equilibrium.The spatial transport of radiation in a material medium is described by the following equation

and the associated energy balance equation

Here u(t,x,ω)de fines the speci fic intensity at time t≥0,location x∈[0,1].Among them ω =cosθ,θ is the angle between the direction of travel of the photon and the positive x-axis.The scalar density is de fined as

where c is the vacuum speed of light.σ=σa+σsis the transport coef ficient,σs＞0 is the scattering coef ficient,σa＞0 is the absorption coef ficient,∈/σ＞0 is the mean free path,β＞0 is called the Stefan-Boltzmann constant,v is material temperature,Ch＞0 is the pseudo-heat capacity.

The computational methods,asymptotic analysis and approximate models of radiative transport equations(1.1)-(1.2)are the focus of scholars such as Pomraning[3],Larsen[2,4,5],Bowers and Wilson[1],Anistratov and Larsen[6],Adams and Larsen[7],Morel[8],Roberts and Anistratov[9].However,since the mathematically results are dif ficult to solve,relevant results are few.To our knowledge,there are only two works as follows:

Guo and Han[10]study the well-posedness and initial layer of equations(1.1)-(1.2)in R3with initial conditions

and then they also[11]study the well-posedness and initial-boundary layer of equations(1.1)-(1.2)in[0,1]with absorbing boundary conditions

Motivated by their results,we consider two different initial boundary conditions,which cause different kinds of layers.Firstly,we introduce the following initial boundary condition:

Secondly,we focus on the initial boundary condition as follows

In both case,we assume u0(x,ω)≥0,v0(x)≥0 and AL(t,ω)≥0,AR(t,ω)≥0 which represents photon entering the area at the left boundaries and right boundaries,respectively.

Obviously,u0,uL,uRsatisfy the following compatible condition:

This article will concentrate on the formally diffusion limit of small mean free path∈of equations(1.1)-(1.2),(1.4)-(1.6)and(1.1)-(1.2),(1.7)-(1.9).In Sections 2-3,in order to study diffusion limit,we first construct asymptotic expansions in ordinary way,and then verify the error analysis.For this end,we find u should be the sum of three parts:u represents inner part,ubrepresents left-boundary layer,uBrepresents right-boundary layer.For simplicity,here we only consider the boundary layers near the left boundary x=0.The case near the right boundary x=1 can be dealt with in a similar way,so we will directly give the equations of right boundary layer later.And in Sections 4-5,for equations(1.1)-(1.2)with initial boundary conditions(1.7)-(1.9),we find u should be the sum of six parts:u represents inner part,uIrepresents initial layer,ubrepresents leftboundary layer,uBrepresents right-boundary layer,uIbrepresents left-mixed layer,and uIBrepresents right-mixed layer.Here we only consider the boundary layers and mixed layers near the left boundary x=0 also.The case near the right boundary x=1 can be dealt with in a similar way,so we will directly give the equations of right boundary layer later.

2 Asymptotic analysis and construction of the boundary layers

In this section,we mainly discuss the asymptotic expansion of the equation(1.1)-(1.2)with initial boundary conditions(1.4)-(1.6).Firstly,we state global existence of solution(u,v)of(1.1)-(1.2)as Lemma 2.1.

Lemma 2.1.Assume that u0,v0satisfy(1.4),uL,uRsatisfy(1.5)and(1.6).Let 0≤uL∈Then for any T≥0,there exists global solution(u,v)of equations(1.1)-(1.2)with initial condition(1.4)and boundary conditions(1.5)and(1.6).Moreover,

The proof of Lemma 2.1 can be referred from[11],and we omit the details here for the sake of simplicity.

We want to describe the asymptotic analysis of the solution(u,v)of equations(1.1)-(1.2)as∈→0.First,let Ch=Ch∈.Now we construct an asymptotic expansion for(u,v)under liberal smoothness assumptions,and then the expansions will be veri fied.

Let us introduce the variable

where∈is the length of the boundary layer.Next we seek an expansion of the form

and

where f and g are smoothC1cut-offfunction satisfying f(0)=g(1)=1 and f(1)=g(0)=0.

In order to get the asymptotic limit with an error of∈,we require that the initial and boundary conditions are also satisfying with O(∈)as follows

Inserting(2.2)-(2.3)into equations(1.1)-(1.2)and balancing the terms order of∈,we can get inner functions uiand boundary layer functionsand

Applying(2.5),(2.7)in(2.9)and integrating equation(2.9)respect to ω over[?1,1],we get

By(2.6),(2.12),we have

Then we obtain

Now balancing o(∈1),we have

Then we apply(2.7),(2.9)in(2.16)and integrating equation(2.16)respect to ω over[?1,1],we obtain

Next,we apply following initial condition with

which implies a unique solution

The compatible conditions are

Employing equation(2.7),we obtain

Then we balance the o(∈2)terms to get

Inserting(2.7),(2.9),(2.16)in(2.24)and integrating equation(2.24)respect to ω over[?1,1],we have

And by equation(2.9),we have

Similarly,we can get

The following two lemmas will be used to state the global existence and uniqueness of solutions to equations(2.13),(2.10)and(2.28),(2.25),respectively.

Lemma 2.2.Set

where u0s,v0sare the stationary state of u0,v0.Moreover,lettingand u0L,u0R∈W3,∞(R+),we have

Lemma 2.3.Set

which satisfy the compatible conditions

The proof of Lemmas 2.2 and 2.3 can be referred from[11],and we omit the details here for the sake of simplicity.

3 Energy estimates

Now we want to verify the expansion(2.2)-(2.3),when the mean free path∈tends to zero.To this end,let u0∈W3,∞(0,1;L∞(?1,1)),u0∈w4,∞(0,1),v0∈w3(0,1),u0,v0,uL,uRsatisfy(1.4)-(1.6)and assumptions of Lemma 2.3.

Set

The same way,u2is established by(2.29)and

After simple calculation,

Among them,

Now,we prove the solution(u,v)of equations(1.1)-(1.2)with initial boundary conditions(1.4)-(1.6)converges toas∈→0.

Theorem 3.1.Let Ch=Ch∈.Assume u0,v0≥0,u0,v0satisfy(1.4),uL,uRsatisfy(1.5)-(1.6),Then the solution of equations(1.1)-(1.2)with initial boundary conditions(1.4)-(1.6)converges to(u0,v0)as mean free path∈→0,where(u0,v0)is the solution of equations(2.13)and(2.10),with initial boundary conditions(2.35)-(2.36).

Moreover,for any 0≤t≤T,we have the following estimates

where constant C is independent of∈and

Proof.We rewrite equations(3.9)as integral equations by characteristics methods:

Next,for any T＞0 and 0≤t≤T,equations yields

By Gronwall inequality,we have

4 Asymptotic analysis and construction of the mixed layers

In this section,we focus on the asymptotic expansion of the equation(1.1)-(1.2)with initial boundary conditions(1.7)-(1.9).

Now,we want to describe the asymptotic analysis of the solution(u,v)of equations(1.1)-(1.2)as ∈→0.First,let Ch=Ch∈.We construct an asymptotic expansion for(u,v)under liberal smoothness assumptions,and then the expansions will be veri fied.

Let us introduce the variable

where ∈2is the length of the initial layer,∈is the length of the boundary layer.Next we seek an expansion of the form

Here f and g are smooth C1cut-offfunction satisfying f(0)=g(1)=1,f(1)=g(0)=0.

In order to get the asymptotic limit with an error of∈,we require that the initial and boundary condition are also satisfy with O(∈).

Inserting(4.2)-(4.3)into equations(1.1)-(1.2)and balancing the terms order of∈,we can get inner functions ui,initial layer functionsboundary layer functionsandand mixed layer functionsand

First,we balance the terms o(∈?2),we have

By Balancing o(∈?1),we get

Next we balance the o(∈0)terms to get

By balancing o(∈1),we have

We balance the o(∈2)terms to get

Firstly,by(4.5c),(4.7f),we have

Then we can obtain

And insert(4.8f),(4.11)into(4.6c),we can get

Next,we apply(4.7f),(4.9f),(4.12)into(4.7e),similarly we have

Then by(4.5d),(4.7h)and characteristic method

And we apply(4.8h)into(4.6d),

Next,we apply(4.8h),(4.9d)into(4.7g),we have

Similarly,

5 Energy estimates

Now we want to verify the expansion(4.2)-(4.3)as the mean free path∈tends to zero.Also we letsatisfy(1.7)-(1.9)and assumptions of Lemma 2.3.

Set

The same way,u2is established by(2.29)and

After simple calculation,

where

Now,we prove the solution(u,v)of equations(1.1)-(1.2)with initial boundary conditions(1.4)-(1.6)converges toas∈→0.

It must be mentioned that the F′in this section equals to F+h5(F is error of u in Section 3)and obviously h5→0 as∈→0,so the convergence is same with the situation in Section 3.

Theorem 5.1.Let Ch=Ch∈.Assume u0,v0≥0,u0,v0satisfy(1.7),uL,uRsatisfy(1.8)-(1.9),Then the solution of equations(1.1)-(1.2)with initial boundary conditions(1.7)-(1.9)converges to(u0,v0)as mean free path∈→0,where(u0,v0)is the solution of equations(2.13)and(2.10),with initial boundary conditions(2.35)-(2.36).

Moreover,for any 0≤t≤T,we have

where constant C is independent of∈and

Acknowledgments

The research of X.Chen was supported in part by Beijing Natural Science Founds under Grant No.1164010.The research of S.Wang was supported in part by National Natural Science Foundation of China under Grants No.11371042.