Global weak solutions to the non-relativistic radiation hydrodynamical equations with isothermal fluids

J. Differential Equations 253 (2012) 2191–2223

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Global weak solutions to the non-relativistic radiation hydrodynamical equations with isothermal fluids Peng Jiang a,b,∗ a b

Department of Mathematics, College of Science, Hohai University, Nanjing 210098, PR China Institute of Applied Physics and Computational Mathematics, Beijing 100088, PR China

a r t i c l e

i n f o

Article history: Received 18 January 2012 Revised 23 May 2012 Available online 4 July 2012 MSC: 41A63 78A40 76N15 76X05 74G25 54D30

a b s t r a c t We prove the existence of global entropy solutions in L ∞ to the one-dimensional Euler–Boltzmann equations for compressible isothermal fluids and multidimensional Euler–Boltzmann equations for compressible isothermal fluids with spherical symmetry in radiation hydrodynamics. The global weak entropy solutions are constructed using the shock capturing scheme of Godunov type with cut-off technique. The global existence of weak entropy solutions in L ∞ with arbitrarily large initial data even containing the vacuum is established with the aid of the compensated compactness method. © 2012 Elsevier Inc. All rights reserved.

Keywords: Radiation hydrodynamics Euler–Boltzmann equation Isothermal gas dynamics Global weak entropy solution Compensated compactness Spherical symmetry

1. Introduction The radiation hydrodynamics is mainly concerned with the propagation of thermal radiation through a fluid or gas, and the effect of this radiation on the dynamics, see, for example, [1,28,30, 32], and the references cited therein. The theory of radiation hydrodynamics finds a very broad range of applications, including such diverse astrophysical phenomena as waves and oscillations in stellar atmospheres and envelopes, nonlinear stellar pulsation, supernova explosions, stellar winds, and so on

*

Correspondence to: Department of Mathematics, College of Science, Hohai University, Nanjing 210098, PR China. E-mail address: [email protected].

0022-0396/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2012.06.014

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(cf. [21]). It has also direct applications in other areas, for example, the physics of laser fusion, the reentry of vehicles [29], the high temperature combustion phenomena [35], and many others. As in ordinary fluid mechanics, the equations of motion in radiation hydrodynamics are derived from the conservation laws for macroscopic quantities. However, due to the presence of radiation, the classical “material” flow has to be coupled with radiation which is an assembly of photons (the photons are massless particles traveling at the speed of light). Hence, the whole problem to consider is then a coupling between the standard hydrodynamics for the matter and radiative equation for the photon distribution. So the mathematical equations governing the radiation hydrodynamics are the Euler equations of compressible fluids coupled with the Boltzmann equation of particle transport. In this paper, we are interested in the global weak solution to the one-dimensional Euler–Boltzmann equations and multidimensional Euler–Boltzmann equations with spherical symmetry structure in radiation hydrodynamics. It is very important to understand the solutions of the Euler–Boltzmann equations in order to obtain insights into the radiation hydrodynamics and related physical phenomena as well as applications. However, solving the Euler–Boltzmann equations is challenging because of the complexity and mathematical difficulty. We use I (x, t , ν , Ω) to denote the specific intensity of radiation (at time t) at spatial point x ∈ R, with frequency ν > 0 in a direction Ω ∈ S 2 , where S 2 is the unit ball in R3 , then the system of partial differential equations of one-dimensional non-relativistic radiation hydrodynamics (cf. [3,28, 30]) consists of the following Boltzmann equation:

1 ∂ I (ν , Ω) c

∂ I (ν , Ω) + Ω1 ∂t ∂x = S (ν ) − σa (ν , ρ ) I (ν , Ω)  ∞         ν      − + dν  σ ν → ν , Ω · Ω, ρ I ν , Ω σ ν → ν , Ω · Ω , ρ I ( ν , Ω) dΩ  , s s 

ν

0

(1.1)

S2

and the Euler equations:

⎧ + (ρ u )x = 0, ⎨ρ  t ⎩ ρu +

1

c2

Fr t

  + ρ u 2 + P r + P m x = 0,

(1.2)

where c denotes the light speed, Ω1 denotes the projection of Ω along the x-axis, S (ν ) = S (x, t , ν , Ω) denotes the rate of energy emission due to spontaneous processes, σa (ν , ρ ) = σa (x, t , ν , Ω, ρ ) denotes the absorption coefficient. Similar to absorption, a photon can undergo scattering interactions with matter, and the scattering interactions change the photon’s characteristics ν  and Ω  to a new set of characteristics ν and Ω , which leads to the definition of the “differential scattering coefficient” σs (ν  → ν , Ω  · Ω, ρ ) = σs (x, t , ν  → ν , Ω  · Ω, ρ ). In the Euler equations (1.2), ρ = ρ (x, t ) is the density, u = u (x, t ) is the velocity, P m (ρ ) is the pressure which is governed by the γ -law: P m (ρ ) = ρ γ /γ with γ  1 being the adiabatic index, F r and P r represent the radiative flux and the radiative pressure respectively defined by

⎧ ∞  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ F r = dν Ω1 I (ν , Ω) dΩ, ⎪ ⎨ 2 0

S

∞  ⎪ ⎪ 1 ⎪ ⎪ ⎪ P = dν Ω12 I (ν , Ω) dΩ. ⎪ r ⎪ c ⎩ 0

In this paper we consider the isothermal case

S2

γ = 1, i.e., P m (ρ ) = ρ .

(1.3)

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Using (1.1) and (1.3), we can rewrite (1.1)–(1.2) as

⎧ 1 ⎪ ⎪ I t + Ω1 I x = F (ρ , I ), ⎪ ⎪ ⎨c ρt + mx = 0,   2 ⎪ ⎪ m ⎪ ⎪ + ρ = G (ρ , I ), ⎩ mt +

ρ

(1.4)

x

where m = ρ u and

∞

 

0

S2

dν 

F (ρ , I ) = S (ν ) − σ (ν , ρ ) I (ν , Ω) +

G (ρ , I ) = −

1

 ∞

  Ω1 S (ν ) − σ (ν , ρ ) I (ν , Ω) dΩ

dν

c 0

S2

∞ +





dν 0



   ν   σs ν → ν , Ω  · Ω, ρ I ν  , Ω  dΩ  , ν

∞ dν

dΩ 0

S2







   ν  Ω1  σs ν  → ν , Ω  · Ω, ρ I ν  , Ω  dΩ  , ν

S2

with

∞

σ (ν , ρ ) = σa (ν , ρ ) +





σs ν  → ν , Ω  · Ω, ρ dΩ  dν  .

0 S2

We consider the Cauchy problem of system (1.4) with the following initial data:

I |t =0 = I 0 (x, ν , Ω), x ∈ R, ν > 0, Ω ∈ S 2 , ρ |t =0 = ρ0 (x), m|t =0 = m0 (x), x ∈ R,

(1.5)

and then construct the global weak entropy solutions in L ∞ . Next, for multidimensional Euler equation and some other system consisting of Euler equation (e.g. Euler–Poisson equations), the global existence of weak entropy solutions is still open question. So for the three-dimensional radiation hydrodynamical equations

1 ∂ I (ν , Ω) c

+ Ω · ∇ I (ν , Ω) ∂t = S (ν ) − σa (ν , ρ ) I (ν , Ω) ∞ +

dν 0



 



     ν   σs ν → ν , Ω  · Ω, ρ I ν  , Ω  − σs ν → ν  , Ω · Ω  , ρ I (ν , Ω) dΩ  , (1.6) ν

S2

⎧  = 0, ρt + ∇ · m ⎨  + ⎩ m

1

c

F 2 r

+ ∇ρ + ∇ · t



 ⊗m  m

ρ

 + P r = 0,

(1.7)

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with

⎧ ∞  ⎪ ⎪ ⎪ ⎪ F = dν Ω I (ν , Ω) dΩ, ⎪ r ⎪ ⎪ ⎨ 0

S2

(1.8)

∞  ⎪ ⎪ 1 ⎪ ⎪ ⎪ Pr = dν Ω ⊗ Ω I (ν , Ω) dΩ, ⎪ ⎪ c ⎩ 0

S2

where all the functions in system (1.6)–(1.8) are similar to those defined in (1.1)–(1.3) only using x instead of x, we consider the symmetric solutions to the above system with the form:



   x  (x, t ) = I (r , t , ν , Ω), ρ (r , t ), m(r , t ) , I (x, t , ν , Ω), ρ (x, t ), m r

Let

r = |x|.

α be the angle between x and Ω ; we define μ ≡ cos α , μ ∈ [−1, 1]. Then (1.6)–(1.7) becomes ⎧ 1 ⎪ ⎪ I t (ν , Ω) + μ I r (ν , Ω) = F 1 (ρ , I ), ⎪ ⎪ c ⎪ ⎪ ⎨ 2 ρt + mr = − m, ⎪ ⎪  2 r  ⎪ ⎪ m 2 m2 ⎪ ⎪ ⎩ mt + +ρ =− + G 1 (ρ , I ),

ρ

r

r

(1.9)

ρ

where

∞

 

0

S2

dν 

F 1 (ρ , I ) = S (ν ) − σ (ν , ρ ) I (ν , Ω) +

G 1 (ρ , I ) = −

1

 ∞ 0

∞



S2



dν 0



μ S (ν ) − σ (ν , ρ ) I (ν , Ω) dΩ

dν

c

+





   ν   σs ν → ν , Ω  · Ω, ρ I ν  , Ω  dΩ  ,  ν

∞ dν

dΩ S2

0







   ν  μ  σs ν  → ν , Ω  · Ω, ρ I ν  , Ω  dΩ  . ν

S2

For spherical flow, we impose reflecting boundary conditions at r = 1 to exclude the singularity at the origin, so we consider the initial–boundary value problem of system (1.9) in the domain {(r , t ) ∈ [1, +∞) × R+ } with the following data:

⎧ I| = I 0 (r , ν , Ω), ⎪ ⎨ t =0 ρ |t =0 = ρ0 (r ), m|t =0 = m0 (r ), ⎪ ⎩ I |r =1 = 0 if μ > 0, m|r =1 = 0.

(1.10)

Then we transform system (1.6)–(1.8) to a one-dimensional system (1.9) with initial–boundary data (1.10). Then we can construct the global weak entropy solutions in L ∞ with the aid of compensated compactness method. This result can help us to well understand the global existence of weak entropy solutions for the general three-dimensional radiation hydrodynamical equations (1.6)–(1.8),

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and gives us some useful ideals to prove the global existence of weak entropy solutions for the multidimensional case. The existence of global weak entropy solutions in L ∞ for the one-dimensional compressible Euler equations for isentropic fluids and some related applications has been established, see [4,9,11,12,23, 24] and [8,10,33,36] as well as the references therein. For isothermal fluids, see [13,22]. For multidimensional compressible Euler equations and other related models with spherical symmetry, the local existence of such a weak solution for 1 < γ < 53 was constructed in [25]. The global existence for γ > 1 was established in [5,7] with the aid of shock capturing scheme and compactness theorem. For isothermal case, see [6,26,27,34]. For radiation hydrodynamical equations, due to its complexity, there are few mathematical results. For the local existence of C 1 solutions and finite-time formation of singularities in solutions to the system of radiation hydrodynamics (1.6)–(1.8), see [14,16,17]. For the existence of global weak entropy solutions of the one-dimensional radiation hydrodynamics equations (1.1)–(1.3) with isentropic fluids, see [15]. For the studies of some simplified systems of radiation hydrodynamic models, see [2,18–20,31]. In this paper, we study the initial value problem of (1.4)–(1.5) and establish the existence of global weak entropy solutions in L ∞ with arbitrarily large data. Different from the isentropic case in [15], when γ = 1, the velocity u may tend to infinity near vacuum. This may lead to the Courant– Friedrichs–Lewy stability condition fail for the standard fractional-step Godunov scheme. In order to overcome this difficulty, we use shock capturing scheme with a cut-off technique to modify the approximate density functions and adjust the ratio of the space and time mesh size to construct the approximate solutions of system (1.4)–(1.5). For the initial–boundary problem (1.6)–(1.8), the geo2

metrical source term (− 2r m, − 2r mρ ) induces the resonance between the characteristic fields and the stationary geometrical source, such a nonlinear resonance causes extra difficulties, one may obtain local existence only, see [25]. To avoid the difficulty of resonance arising from the geometric source term, we introduce new variable = r 2 ρ , = r 2 m, the weighted density and momentum so that the resulting hyperbolic system has less geometrical source effect. Another main obstacle of this problem is that the system (1.4) and (1.9) are Euler–Boltzmann equations, we should use different methods to construct the approximate solutions. For the Euler equation, we use the shock capture scheme and for Boltzmann type equation, we use the classic characteristic method to construct the approximate solutions. This modified for the shock capture scheme can be useful to deal with the system (1.4) and (1.9) which consists of different type equations. Then we apply the compensated compactness framework to prove the convergence of the approximate solutions. In particular, we show the L ∞ uniform estimates, the H −1 compactness, and the entropy conditions. To obtain the global existence, we need to overcome the complex structure of the system, especially from the radiation terms in the Euler equations (1.2) and (1.7), which requires some new ideas and techniques are thus required in the proof. The rest of this paper is organized as follows. In Section 2, we reformulate the problem and state the main result. In Section 3, we develop a shock capturing scheme of Godunov type to construct the approximate solutions of (1.4)–(1.5). In Section 4, we achieve the uniform boundedness in L ∞ of approximate solutions by estimating the corresponding Riemann invariants of the approximate solutions and by identifying invariant regions for Riemann solutions at each time step. Then, one uses the compactness embedding technique to prove that the family of entropy dissipation measure is compact −1 in H loc . In Section 5, we employ the compensated compactness framework to obtain a convergent subsequence, and then we show that the limit of this subsequence is a weak entropy solution. Finally, in Section 6, we give the global existence of weak entropy solution to multidimensional case with spherical symmetry structure by the approaches and ideals developed in Sections 3–5. 2. Reformulation and main result In this section, we give some useful assumptions of source term with radiation F (ρ , I ), G (ρ , I ) and then give the global existence of weak entropy solutions for system (1.4)–(1.5). We set V := (ρ , m) and R T := R × [0, T ].

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Definition 2.1. The bounded measurable functions ( I , V ) = ( I (x, t , ν , Ω), ρ (x, t ), m(x, t )), x ∈ R, t > 0, is a global weak solution of (1.4)–(1.5) if the following integral identities hold for any fixed T > 0, ν > 0, Ω ∈ S 2 :

T 

1 c

0 R





I ϕt + Ω1 I ϕx dx dt + R

1 c

T I 0 ϕ (x, 0) dx +

T 0 R



mϕt +

m2

ρ

0 R

for all

(2.1)

 (ρϕt + mϕx ) dx dt +

T 

F (ρ , I )ϕ dx dt = 0, 0 R

ρ0 ϕ (x, 0) dx = 0,

(2.2)

R

  T  + ρ ϕx dx dt + G (ρ , I )ϕ dx + m0 ϕ (x, 0) dx = 0, 0 R

(2.3)

R

ϕ ∈ C 0∞ ( R T ) satisfying ϕ (x, T ) = 0 for x ∈ R.

The main assumptions of this paper are the following: (A1) The source terms F (ρ , I ), G (ρ , I ) satisfy the following conditions:

   F (ρ , I )  C 1 | I | + C 2 |ln ρ |,

    G (ρ , I )  ρ C 3 | I | + C 4 |ln ρ |

(2.4)

where ρ = max(ρ , lβ ), with 2  β  3, l ∈ (0, l0 ) for some sufficiently small l0 ∈ (0, 1). (A2) For bounded ρi , I i , i = 1, 2,

   F (ρ1 , I 1 ) − F (ρ2 , I 2 )  C ( I 1 , I 2 )|ρ1 − ρ2 |, if ρ1 = ρ2 ,    F (ρ1 , I 1 ) − F (ρ2 , I 2 )  C | I 1 − I 2 |, if ρ1 = ρ2 ,   G (ρ1 , I 1 ) − G (ρ2 , I 2 )  C ( I 1 , I 2 )|ρ1 − ρ2 |, if ρ1 = ρ2 ,   G (ρ1 , I 1 ) − G (ρ2 , I 2 )  C | I 1 − I 2 |, if ρ1 = ρ2 ,

(2.5)

(2.6)

where C denotes a positive constant, C ( I 1 , I 2 ) denotes a positive constant C depending on I 1 and I 2 . Remark 2.1. These assumptions hold for some class of functions σ (in terms of σa , σs ) and are required for the approach of compensated compactness in this paper. In particular, for σ satisfying the following Lipschitz conditions (cf. [17,30]):

  σ (ρ1 ) − σ (ρ2 )  C |ρ1 − ρ2 |, ∞

   ν     σ ( ρ ) − σ ( ρ ) s 2  dΩ  C |ρ1 − ρ2 |,  ν s 1

0

S2

dν 

one has

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

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   F (ρ1 , I 1 ) − F (ρ2 , I 2 )    σ (ρ1 ) I 1 − σ (ρ2 ) I 2  +

∞

   ν     σ ( ρ ) I − σ ( ρ ) I s 2 2  dΩ  ν s 1 1

0

S2

dν 

   C ( I 1 , I 2 )σ (ρ1 ) − σ (ρ2 ) + C ( I 1 , I 2 )

∞

   ν    dΩ  σ ( ρ ) − σ ( ρ ) s 1 s 2  ν 

0

S2

dν 

 C ( I 1 , I 2 )|ρ1 − ρ2 |. Similarly, (2.6) holds. Consider the following family of entropy pairs (η, q) for the isothermal case: 1



ξ

η = ρ 1−ξ 2 e 1−ξ 2 ,

q=

m

ρ

 + ξ η for ξ ∈ R.

(2.7)

It is easy to check that η is a convex weak entropy for any ξ ∈ (−1, 1). Now we state the main result of this paper as follows. Theorem 2.1. Suppose that the initial data ( I 0 , ρ0 , m0 ) satisfy the conditions:

   I 0 (x, ν , Ω)  C 0 ,

0  ρ0 (x)  C 0 ,

     m0 (x)  ρ0 (x) C 0 + ln ρ0 (x)

for some positive constant C 0 , and F (ρ , I ), G (ρ , I ) satisfy (A1), (A2). Then, for γ = 1, the Cauchy problem (1.4)–(1.5) has a global weak entropy solution ( I (x, t , ν , Ω), ρ (x, t ), m(x, t )), x ∈ R, t > 0, for any fixed ν > 0, Ω ∈ S 2 , satisfying the following estimates and entropy condition: on R T for any fixed T > 0,

| I |  C ( T ),

0  ρ  C ( T ),

  |m|  ρ C ( T ) + |ln ρ | ,

for a constant C ( T ) > 0 which depends on T , and

T



T



η(ρ , m)ψt + q(ρ , m)ψx dx dt +

0 R

ηm (ρ , m)G (ρ , I )ψ dx dt  0

(2.8)

0 R

for any entropy pair (η, q) in (2.7) with ξ ∈ (−1, 1) and for all nonnegative test functions ψ ∈ C 0∞ ( R T ), with ψ(x, 0) = ψ(x, T ) = 0. Next, we will divide the proof of Theorem 2.1 into three steps. First, we develop a shock capturing scheme of Godunov type to construct approximate solutions of (1.4)–(1.5). The second step is to show the convergence by the compensated compactness framework, and the final step is to prove that the limit function is a weak entropy solution. 3. Construction of approximate solutions We first recall that the homogeneous system:

⎧ x = 0, ⎨ ρt + m   2 m +ρ =0 ⎩ mt +

ρ

x

(3.1)

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has two eigenvalues

λ1 =

m

ρ

− 1,

λ2 =

m

ρ

+1

and Riemann invariants

w=

m

ρ

+ ln ρ ,

m

z=

ρ

− ln ρ .

Now, we construct the approximate solutions ( I l , V l ) = ( I l , ρ l , ml ) with V l = (ρ l , ml ) of (1.4)–(1.5) in the strip 0  t  T for any fixed T > 0, where l is the space mesh length defined as in (2.4). To ensure the Courant–Friedrichs–Lewy stability condition, we choose the time mesh length

h=

l 10(1 + |ln l|)

(3.2)

.

We now use the shock capturing scheme of Godunov type to construct a sequence of approximate solutions of (1.4)–(1.5). To ensure the CFL condition, we employ a cut-off technique (cf. [6]) so that the approximate density functions stay away from vacuum by lβ , starting from the initial data:

I 0 = I0,





ρ 0 = max ρ0 , lβ ,

m0 = m0 .

Namely, we solve the Riemann problems of (3.1) in the region R 1j ≡ {(x, t ): x j −1/2  x  x j +1/2 , 0  t < h}, x j ±1/2 = ( j ± 1/2)l:

⎧    ⎪ ρ0l t + ml0 x = 0, ⎪ ⎪ ⎪   l 2 ⎪ ⎪  l  l (m0 ) ⎪ ⎪ + ρ0 = 0, ⎨ m0 t + ρ0l x ⎧ ⎪  ⎪ ⎨ 1 jl (ρ 0 (x), m0 (x)) dx, x < jl, ⎪ ⎪ l ( j −1)l ⎪  l l  ⎪ ρ ,m = ⎪  ⎪ ⎩ 0 0 t =0 ⎩ 1 ( j +1)l (ρ 0 (x), m0 (x)) dx, x > jl, l jl for all integers j; and we also solve the following problem in the strip: x ∈ R, 0  t  h,

⎧   ⎨ 1 I l + Ω  I l  = 0, 1 0 x c 0 t ⎩ l  I 0 t =0 = I 0 (x, ν , Ω). Then, we set



I l (x, t , ν , Ω) = I l0 (x, t , ν , Ω) + c F



V l (x, t ) = V 0l (x, t ) +  G







ρ0l , I l0 t ,

ρ0l , I l0 t ,

for x ∈ R, 0 < t < h, where V 0l = (ρ0l , ml0 ) and  G = (0, G ) . We continue the construction to the next time step recursively as follows. Suppose that we have defined approximate solutions ( I l , V l ) for 0  t < ih, with i  1 an integer. Then we set

I l (x, t , ν , Ω) = I l (x, t , ν , Ω),





ρ l (x, t ) = max ρ l (x, t ), lβ ,

ml (x, t ) = ml (x, t )

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for 0  t < ih. We then define



I l (x, t , ν , Ω) = I l0 (x, t , ν , Ω) + c F



V l (x, t ) = V 0l (x, t ) +  G







ρ0l , I l0 (t − ih),

ρ0l , I l0 (t − ih),

for ih  t < (i + 1)h, where V 0l (x, t ) are piecewise smooth functions defined as a solution of the

Riemann problem in the region R ij+1 ≡ {(x, t ): x j −1/2  x  x j +1/2 , ih  t < (i + 1)h}:

⎧    ⎪ ρ0l t + ml0 x = 0, ⎪ ⎪ ⎪   l 2 ⎪ ⎪   (m0 ) ⎪ l ⎪ ⎨ ml0 t + + ρ = 0, 0 ρ0l x ⎧  ⎪ ⎪ ⎨ 1 jl (ρ l (x, ih − 0), ml (x, ih − 0)) dx, x < jl, ⎪ ⎪  l l  l ( j −1)l ⎪  ⎪ ρ ,m = ⎪  ⎪ ⎩ 0 0 t =ih ⎩ 1 ( j +1)l (ρ l (x, ih − 0), ml (x, ih − 0)) dx, x > jl; l jl and I l0 (x, t , ν , Ω) is the solution of the following problem in the region x ∈ R, ih  t < (i + 1)h:

⎧   ⎨ 1 I l + Ω  I l  = 0, 1 0 x c 0 t ⎩ l  l I 0 t =ih = I (x, ih − 0, ν , Ω). Finally, we set

I l (x, t , ν , Ω) = I l (x, t , ν , Ω),





ρ l (x, t ) = max ρ l (x, t ), lβ ,

ml (x, t ) = ml (x, t )

for 0  t < ih. Then the approximate solutions ( I l (x, t , ν , Ω), V l (x, t )) are well-defined for x ∈ R, 0  t  T and any fixed ν > 0, Ω ∈ S 2 . 4. Convergence via compensated compactness method We now show the convergence of the approximate solutions constructed in Section 3 with the aid of the compensated compactness method. The following compactness theorem for γ = 1 was established in Huang and Wang [13]: Theorem 4.1. If a sequence of measurable functions ( I l (x, t , ν , Ω), V l (x, t )) = ( I l (x, t , ν , Ω), ρ l (x, t ), ml (x, t )), x ∈ R, t > 0, ν > 0, Ω ∈ S 2 , satisfies the following conditions, for each fixed pair of parameters (ν , Ω): (1) there is a constant C > 0 such that, for (x, t ) ∈ R2+ := R × (0, ∞),

l   I (x, t , ν , Ω)  C ,

 l     m (x, t )  ρ l (x, t ) C + ln ρ l (x, t ) ;

0  ρ l (x, t )  C ,

(2) the measure sequence







η Vl t +q Vl

 x

−1 is compact in H loc (Ω  ), for any entropy pair (η, q) in (2.7) with ξ ∈ (−1, 1), where Ω  ⊂ R2+ is any bounded and open set,

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then, for γ = 1, there exist functions ( I , ρ , m) ∈ L ∞ (R2+ ) and a convergent subsequence (still labeled) ( I l , V l ), such that as l → 0, I l (x, t , ν , Ω)  I (x, t , ν , Ω) weakly-∗ and V l (x, t ) → V (x, t ) = (ρ (x, t ), m(x, t )) almost everywhere in R2+ for each fixed pair of parameters (ν , Ω). We now prove that the approximate solutions ( I l , V l ) satisfy the above framework, i.e., the L ∞ estimate and the H −1 estimate. 4.1. L ∞ estimate First, we prove that the approximate solutions ( I l , V l ) have a uniform bound. Theorem 4.2. Suppose that the initial data ( I 0 , ρ0 , m0 ) satisfy the following conditions:

   I 0 (x, ν , Ω)  C 0 ,

    m0 (x)  ρ0 (x) C 0 + |ln ρ0 | ,

0  ρ0 (x)  C 0 ,

x ∈ R, ν > 0, Ω ∈ S 2 , for some constant C 0 > 0, and F (ρ , I ), G (ρ , I ) satisfy (2.4). Then, there exists a positive constant C ( T ), independent of l and h, such that

 l  I   C ( T ),

 l    m   ρ l C ( T ) + ln ρ l  ,

lβ  ρ l (x, t )  C ( T ),

(x, t ) ∈ R T .

(4.1)

Proof. Let







w l0 (x, t ) = w V 0l (x, t ) ,







w l (x, t ) = w V l (x, t ) ,





zl0 (x, t ) = z V 0l (x, t ) ,



w l (x, t ) = w V l (x, t ) ,





zl (x, t ) = z V l (x, t ) ,



zl (x, t ) = z V l (x, t ) .

By the assumption that

   I 0 (x, ν , Ω)  C 0 ,

    m0 (x)  ρ0 (x) C 0 + |ln ρ0 | ,

0  ρ0 (x)  C 0 ,

there exists β0 > 0 such that

   I 0 (x, ν , Ω)  β0 ,

sup w



x



ρ0 (x), m0 (x)  β0 ,

inf z x





ρ0 (x), m0 (x)  −β0 .

It is easy to check that, for the Riemann invariants corresponding to the modified functions by the cut-off technique, we have

   I 0 (x, ν , Ω)  β0 ,

w





ρ 0 (x), m0 (x)  β0 ,

z



Now, we divide two cases to discuss the uniform boundedness of ( I l , V l ). Case 1. If

ρ0l  1 > lβ , ρ l = ρ0l , for x ∈ R, ih  t < (i + 1)h with i  0 integers,

   l  I (x, t , ν , Ω) =  I l (x, t , ν , Ω)        I l0 L ∞ + C  F ρ0l , I l0  L ∞ h          I l0 L ∞ + C  I l0 L ∞ + ln ρ0l  L ∞ h          I l0 L ∞ + C  I l0 L ∞ +  w l0 − zl0  L ∞ h



ρ 0 (x), m0 (x)  −β0 .

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

2201

    I l0 (x, ih + 0, ν , Ω) L ∞    + C  I l0 (x, ih + 0, ν , Ω) L ∞ + sup w l0 (x, ih + 0) − inf zl0 (x, ih + 0) h. x

x

For the Riemann invariants, we have the following estimates:

w l (x, t ) = w l0 (x, t ) +

G

ρ0l



sup w l0 (x, ih x



sup w l0 (x, ih x

(t − ih)

 l  ρ      + 0) + C  l   I l0 L ∞ + ln ρ0l  L ∞ h ρ 0

  + 0) + C  I l 

0 L∞

   +  w l0 − zl0  L ∞ h

 sup w l0 (x, ih + 0) x

   + C  I l0 (x, ih + 0, ν , Ω) L ∞ + sup w l0 (x, ih + 0) − inf zl0 (x, ih + 0) h x

x

and

zl (x, t ) = zl0 (x, t ) +

G

ρ0l

(t − ih)

    inf zl0 (x, ih + 0) − C  I l0 (x, ih + 0, ν , Ω) L ∞ + sup w l0 (x, ih + 0) − inf zl0 (x, ih + 0) h. x

x

x

Let







αi = max  I l0 (x, ih + 0, ν , Ω) L ∞ , sup w l0 (x, ih + 0), − inf zl0 (x, ih + 0) . x

x

Then

 









max  I l0 x, (i + 1)h − 0, ν , Ω  L ∞ , sup w l x, (i + 1)h − 0 , − inf zl x, (i + 1)h − 0 x

x

 C αi (1 + h). It follows that

αi+1  C (1 + h)αi . We can deduce that

αi  C (1 + h)n α0  C 1 ( T ), 0  i  n = where

T h

,

α0 = max{ I 0 L ∞ , supx w (ρ 0 (x), m0 (x)), − infx z(ρ 0 (x), m0 (x))}. Then  l  l  I  =  I   C 1 ( T ), w l (x, t )  C 1 ( T ), − zl (x, t )  C 1 ( T ).

Performing the same procedure, we conclude that, for 0  t  T ,

w l (x, t )  C 1 ( T ),

− zl (x, t )  C 1 ( T ).



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P. Jiang / J. Differential Equations 253 (2012) 2191–2223

Case 2. If lβ < ρ0l < 1, then |ln ρ l | = |ln ρ0l | < |ln lβ |. If these, we have if

ρ0l < lβ < 1, then |ln ρ l | = |ln lβ |. Combining ρ0l < 1, |ln ρ l |  |ln lβ |. According to (3.2), we get

l       I (x, t , ν , Ω) =  I l (x, t , ν , Ω)   I l 

0 L∞

    I l0  L ∞     I l0  L ∞

   + C  F ρ0l , I l0  L ∞ h     + C  I l0  L ∞ + ln lβ  h   + C  I l0  L ∞ h + Cl.

Similarly, we have

 

w l (x, t )  sup w l0 (x, ih + 0) + C  I l0  L ∞ h + Cl, x

 

zl (x, t )  inf zl0 (x, ih + 0) − C  I l0  L ∞ h − Cl. x

Then, we can deduce that

αi  C (1 + h)n α0 + Cnl  C 2 ( T ). Thus

 l  l  I  =  I   C 2 ( T ),

w l (x, t )  C 2 ( T ),

− zl (x, t )  C 2 ( T ).

Performing the same procedure, we conclude that, for 0  t  T ,

w l (x, t )  C 2 ( T ),

− zl (x, t )  C 2 ( T ).

Then, there is a constant C ( T ) = max(C 1 ( T ), C 2 ( T )) > 0 independent of l and h such that

l   I (x, t , ν , Ω)  C ( T ),

0  ρ l (x, t )  C ( T ),

On the other hand, by the definition of

 l     m (x, t )  ρ l (x, t ) C ( T ) + ln ρ l (x, t ) .

ρ l , we have ρ l  lβ . This completes the proof. 2

4.2. H −1 estimate In order to prove the H −1 estimate, we need some useful lemmas. Lemma 4.1. Let V l (x, t ) = (ρ l (x, t ), ml (x, t )) be the approximate solution. Then, for any given R > 0 and T > 0, there is a positive constant C = C ( R , T ) independent of l such that jl      V l (x, ih − 0) − V i 2 dx  C , j

i , j ( j −1)l

where V ij =

1 l



jl

l ( j −1)l V (x, ih − 0) dx.

 for 1 −

R l



  R  j , l

(4.2)

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

η with ξ ∈ (− 12 , 12 ). For ih  t  (i + 1)h, Green’s formula implies

Proof. Fix a strictly convex entropy that jl    

η

2203

 i +1 V 0l

−η



V ij



(i+1)h

dx +

j ( j −1)l



 δ[η] − [ g ] dt = 0,

(4.3)

ih



where ( V 0l )i +1 = V 0l (x, (i + 1)h − 0), the summation is taken over all the shock waves in V 0l at a fixed t between ih and (i + 1)h, δ is the propagating speed of the shock wave, and [η], [q] denote the jumps of η, q across the shock wave from the left to the right, respectively. That is, if S = (x(t ), t ) denotes a shock wave in V 0l , then

      [η] = η V 0l x(t ) + 0, t − η V 0l x(t ) − 0, t ,

      [q] = η V 0l x(t ) + 0, t − q V 0l x(t ) − 0, t .

Summing over all i in (4.3) gives jl    

η

i V 0l

−η



V ij



dx +

i , j ( j −1)l

=



T  

 δ[η] − [ g ] dt

0

 jl

 







η V 0l (x, 0) − η V 0l (x, T − 0) dx.

(4.4)

i , j ( j −1)l

From the uniform bound of V 0l , the right side of (4.4) is dominated by a constant C > 0 independent of l; i.e., jl    

η

i V 0l

−η



V ij



dx +

i , j ( j −1)l

T  

 δ[η] − [q] dt  C .

(4.5)

0

We decompose the first term of (4.5) into three parts: jl    

i





η V 0l − η V ij dx =

i , j ( j −1)l

jl    

i



η V 0l − η V l

i 

dx +

i , j ( j −1)l

+

jl    

i





η V l − η V ij dx

= I1 + I2 + I3.

ηm =

ξ 1−ξ 2

1

ρ 1−ξ 2

−1

ξ

e 1−ξ 2

m

ρ

=

ξ 1−ξ 2

1 ρ η , we have

  jl   jl    l i   l i  l i      η V − η V l i  dx η V 0 − η V dx   0   ( j −1)l

( j −1)l

i



η Vl −η Vl

i , j ( j −1)l

i , j ( j −1)l

For I 1 , noting that

jl    

i 

dx

2204

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

  jl  1  l i  l i   l i  l i  l i   = ∇η V + θ V 0 − V dθ V0 − V   dx   ( j −1)l

0

 1    l i  l i  l i   l i  l i   ∇η V + θ V 0 − V dθ  G ρ0 , I 0  dx   

 jl =h ( j −1)l

0

 jl  1 h ( j −1)l

0

   G ((ρ0l )i , ( I l0 )i )   l i  l i  |ξ |   dx,  θ dθ  η ρ , m  1 − ξ2 (ρ l )i ( ml )i

where ( ml )θi = (1 − θ)(ml )i + θ(ml0 )i . Notice that, using Theorem 4.2, | l θi |  C + |ln(ρ l )i | and, hence (ρ )

 i  i 

l η ρl , m Since

1−|ξ | 1−ξ 2

>

1 2

θ

=

 l i 

ρ

1 1−ξ 2

ξ

e 1−ξ

( ml )i θ 2 (ρ l ) i

 i ( 1 2 − |ξ | 2 )  i  1−|ξ | 1−ξ 1−ξ  C ρl = C ρ l 1−ξ 2 .

when ξ ∈ (−1, 1), both functions

(4.6)

η((ρ l )i , ( ml )θi ) and |ln(ρ )i |η((ρ l )i , ( ml )θi ) are

bounded. With the help of (2.4), then we conclude

  jl     l i  l i    η V 0 − η V dx  Clh.    ( j −1)l

Summing over all cells, we have

|I1|  C . When

(4.7)

ρ < lβ , ξ

1

If ξ ∈ (−1, 1),

1−|ξ | 1−ξ 2

ξ

m

η = ρ 1−ξ 2 e 1−ξ 2

C

1−|ξ |

ξ

C β

 e 1−ξ 2 ρ 1−ξ 2  e 1−ξ 2 l

ρ

1−|ξ | 1−ξ 2

.

| ∈ (− 12 , 1]. Since β  2 > 1 + |ξ |, then β 11−|ξ > 1. By (3.2), we have −ξ 2

    |I2| =  i, j

0+



 +

[( j −1)l, jl]∩{(ρ l )i >lβ } jl  

ξ

C β

2e 1−ξ 2 l



 

η V

 l i



−η V

 l i

  dx

[( j −1)l, jl]∩{(ρ l )i lβ }

1−|ξ | 1−ξ 2

dx

i , j ( j −1)l

  β 1−|ξ | −1  C 1 + |ln l| l 1−ξ 2 .

(4.8)

Thus, we have

I 2 → 0 as l → 0.

(4.9)

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

2205

Since η is convex, the entropy inequality δ[η] − [q]  0 holds across the shock waves. So we can T  deduce that 0 {δ[η] − [ g ]} dt  0. For I 3 , we take the Taylor expansion for η( V l )i to get



i







η V l = η V ij + ∇ η V ij



Vl

i

 1  l i    i  − V ij + V − V ij ∇ 2 η ζ ji V l − V ij ,

(4.10)

2

where ζ ji is a mean value. Integrating (4.10) on the cell (( j − 1)l, jl) and summing over all cells we have

I3 =

 jl

1 2



Vl

i

− V ij



  i  ∇ 2 η ζ ji V l − V ij dx.

(4.11)

i , j ( j −1)l

By (4.5), (4.7), (4.9), and (4.11), we have jl   

Vl

i

− V ij



  i  ∇ 2 η ζ ji V l − V ij dx + 2

i , j ( j −1)l

T  

 δ[η] − [ g ] dt  C .

0

Thus jl   

Vl

i

− V ij



  i  ∇ 2 η ζ ji V l − V ij dx  C ,

i , j ( j −1)l

In particular,

T  

 δ[η] − [ g ] dt  C .

(4.12)

0

η is strictly convex for ξ ∈ (− 12 , 12 ); i.e., there exists a constant α > 0 such that V ∇ η2 V  α | V |2 .

It follows from (4.12) that (4.2) is true.

2

The proof of the following lemma can be found in [4,9]. Lemma 4.2. Let Ω ⊂ R N be a bounded open set. Then,













−1 compact set of W −1, p (Ω) ∩ bounded set of W −1,r (Ω) ⊂ compact set of H loc (Ω) ,

for some constants p and r satisfying 1 < p  2 < r < ∞. Now, we can prove that the sequence of entropy dissipation measures −1 . in H loc

η( V l )t + q( V l )x is compact

Theorem 4.3. For the approximate solution { V l }, the measure sequence







η Vl t +q Vl





x

−1 Ω is a compact subset of H loc



for any weak entropy pair (η, q) in (2.7) with ξ ∈ (−1, 1), where Ω  is any open bounded subset in R T .

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P. Jiang / J. Differential Equations 253 (2012) 2191–2223

Proof. For any

ϕ ∈ C 0∞ (Ω  ), we consider

T

 









η V l ϕt + q V l ϕx dx dt = A (ϕ ) + R (ϕ ) + B (ϕ ) + Σ(ϕ ) + S (ϕ ),

(4.13)

0 Ω

where

A (ϕ ) =

jl    

i





η V l − η V ij ϕ (x, ih) dx,

i , j ( j −1)l

R (ϕ ) =

jl    

η

i V 0l

i , j ( j −1)l



−η V

 l i

ϕ (x, ih) dx +

jl    

i



η Vl −η Vl

i 

ϕ (x, ih) dx,

i , j ( j −1)l



 







 





η V 0l (x, T ) ϕ (x, T ) − η V 0l (x, 0) ϕ (x, 0) dx,

B (ϕ ) = Ω

Σ(ϕ ) =

T  

σ [η] − [q] ϕ x(t ), t dt ,

0

S (ϕ ) = S 1 (ϕ ) + S 2 (ϕ ), with

T

 







 









 







 









η V l − η V 0l ϕt + q V l − q V 0l ϕx dx dt ,

S 1 (ϕ ) = 0 Ω

T

η V l − η V l ϕt + q V l − q V l ϕx dx dt .

S 2 (ϕ ) = 0 Ω

We now make estimates for each of these terms. (1) We decompose A (ϕ ) into two parts:

A (ϕ ) =

 i, j

 jl

ϕ

 

i j

η V

 l i

−η



V ij



dx +

jl    

i



η V l − η V ij



i , j ( j −1)l

( j −1)l

≡ A 1 (ϕ ) + A 2 (ϕ ), where

ϕ ij = ϕ ( jl, ih) and ϕ i = ϕ (x, ih). For A 1 (ϕ ), using (4.11) and (4.12), we have    A 1 (ϕ ) =



1  



2

i, j

 jl

ϕ

i j

( j −1)l



V

 l i

−

 V ij ∇ 2



ϕ i − ϕ ij dx

  i  l i   i η ζ j V − V j dx 

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

 C ϕ L ∞

jl   

Vl

i

− V ij



2207

  i  ∇ 2 η ζ ji V l − V ij dx

i , j ( j −1)l

 C ϕ L ∞ .

(4.14)

For A 2 (ϕ ), we have

 jl

    A 2 (ϕ ) 

  l i    η V − η( V )i ϕ i − ϕ i  dx j

j

i , j ( j −1)l

 C ϕ

C α l

jl     1    l i   1   ρ − ρi   1 − ξ m θ ηθ  d θ j  ρθ  1 − ξ 2 ρθ

α

0

i , j ( j −1)l

 i  +  ml − mi  j

0

1    ηθ    dθ dx, ρ  θ

0

1 1−ξ 2

where 0 < α < 1, V θ = V ij + θ(( V l )i − V ij ), ηθ = ρθ By Theorem 4.2 and Lemma 4.1, we have

ξ

e 1−ξ 2

mθ

ρθ

.

jl     1    l i   ξ 1  mθ i  ρ −ρ   1− ηθ  dθ dx j  2 ρθ 1 − ξ ρθ i , j ( j −1)l

=

0

 i, j





 l i   ρ − ρi 

+ [( j −1)l, jl]∩{ρθ >1}

 C



jl    i l

ρ

j

[( j −1)l, jl]∩{ρθ 1}



−ρ

 i 2 dx j

1 2

h

− 12

 Ch

0

    mθ  1−ξ ηθ  dθ dx  ρθ ρθ 1 

jl  1    l i  − 34 i  +C ρ −ρj ρθ dθ dx

i , j ( j −1)l

− 12

1

i , j ( j −1)l

0

jl     i 1  ρ l − ρ i  4 dx  Ch− 78 . +C j

i , j ( j −1)l

From the uniform boundedness of approximate solutions, we get

 l  i              3   3   m − mi   ρ l i C + ln ρ l i  + ρ i C + ln ρ i   C ρ l i 4 + ρ i 4 . j j j j 3

We now consider four cases. If 0  (ρ l )i − ρ ij  ρ ij , then |(ml )i − mij |  C (ρ ij ) 4 , that is,

  l i   3  m − mi −1  C ρ i − 4 . j

Thus, by (4.15), we have

j

(4.15)

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P. Jiang / J. Differential Equations 253 (2012) 2191–2223

jl  1      l  i    i  m − m  +  ηθ  dθ dx j ρ  i , j ( j −1)l

=

 i, j

θ

0





 +

[( j −1)l, jl]∩{ρθ >1}

  l i  m − mi  j

[( j −1)l, jl]∩{ρθ 1}

1    ηθ    dθ dx ρ  0

θ

jl jl    i    i    1 l i   ml − mi  ρ i − 2 dx C m − m j dx + C j j i , j ( j −1)l

i , j ( j −1)l

jl jl    i    i  1 l i   ml − mi  3 dx C m − m j dx + C j i , j ( j −1)l 1

i , j ( j −1)l 5

5

 Ch− 2 + Ch− 6  Ch− 6 . The case ρ ij < (ρ l )i − ρ ij , 0  Therefore, by (4.1), we obtain

ρ ij − (ρ l )i  (ρ l )i and (ρ l )i  ρ ij − (ρ l )i can be estimated similarly.

   7  A 2 (ϕ )  C ϕ C α lα h− 78  C ϕ C α lα − 78 1 + |ln l| 8  Clα − 89 ϕ C α 0 0 0

for

8 9

< α < 1.

(2) For R (ϕ ), with the help of (4.7), (4.9), we deduce that

 jl   jl      i     i    l i    l i   l l  R (ϕ )  ϕ L ∞ η V 0 − η V dx + ϕ L ∞ η V − η V dx       i, j

i, j

( j −1)l

( j −1)l

 C ϕ L ∞ .

(4.16)

(3) For B (ϕ ), we have

   B (ϕ )  ϕ L ∞



  l     η V (x, T )  + η V l (x, 0)  dx  C ϕ L ∞ . 0

0

(4.17)

Ω

(4) Using (4.12) yields

  Σ(ϕ )  ϕ L ∞

T    σ [η] − [q] dt  C ϕ L ∞ . 0

lθ = ml0 + θ(ml − ml0 ). Then (5) Set m

 T  1       l  l    l l l  S 1 (ϕ )   ηm V 0 + θ V − V 0 dθ m − m0 ϕt dx dt    0 Ω

0

(4.18)

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

2209

 T  1   l      l   l  l θ m 1  l  l l + η V0 + θ V − V0 + + ξ ηm V 0 + θ V − V 0 dθ  ρl ρl 0 Ω



l

× m

0

− ml0



  ϕx dx dt  

 T  1      l    l   l ξ   l l η V 0 + θ V − V 0 dθ  I 0  + ln ρ0  ϕt dx dt   Ch 2   1−ξ 0 Ω

0

 T  1   l      θ m ξ  1+ +ξ η V 0l + θ V l − V 0l dθ + Ch 2 l  1−ξ ρ 0 Ω

0

     l    ×  I 0  + ln ρ0l  ϕx dx dt   T  Ch



 |ϕt | + |ϕx | dx dt  Clϕ  H 1 (Ω  ) , 0

0 Ω

where we used the fact that

η( V 0l + θ( V l − V 0l ))|ln ρ l | is bounded for any s  0 by (4.6). Also,

 T        l   l   l     l  S 2 (ϕ )   η V − η V ϕt + q V − q V ϕx dx dt    0 Ω

 Cl

β 1−|ξ2|

T |ϕt | dx dt + Cl

1−ξ

β

T |ϕx | dx dt

2

0 Ω

0 Ω

 Clϕ  H 1 . 0

Therefore,

   S (ϕ )  Clϕ 

(4.19)

H 01 .

Since C 0∞ (Ω  ) is dense in H 01 (Ω  ), it follows that

 S  H −1 (Ω  )  Cl → 0 as l → 0. loc

−1 Thus S is compact in H loc (Ω  ). From (4.14) and (4.16)–(4.18), we have

 A 1 + R + B + Σ(C 0 )∗  C . By the embedding theorem, (C 0 (Ω  ))∗ → W −1, p 0 is compact for 1 < p 0 < 2. Thus, A 1 + R + B + Σ is 1, p compact in W −1, p 0 (Ω  ). By the Sobolev theorem, W 0 1 (Ω  ) → C 0α (Ω  ), p 1 > 1−2 α and the estimate

   A 2 (ϕ )  Clα − 89 ϕ C α , 0

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P. Jiang / J. Differential Equations 253 (2012) 2191–2223

we have

   A 2 (ϕ )  Clα − 89 ϕ 

1, p 1

W0

(Ω  )

.

It follows from duality that

2

8

 A 2 W −1, p2 (Ω  )  Clα − 9 → 0 as l → 0 for 1 < p 2 <

1+α

< 2.

Then, A 2 is compact in W −1, p 2 (Ω  ). So A + R + B + Σ is compact in W −1, p (Ω  ), where 1 < p < min( p 0 , p 2 ) < 2. Next, from the uniform boundedness of V l , we have the following fact:













η V l t + q V l x − S is bounded in W −1,∞ Ω  . Since Ω  is bounded, the above statement implies that









η V l t + q V l x − S is bounded in W −1,r (Ω  ), for r > 1. That is, A + R + B + Σ is bounded in W −1,r (Ω  ), r > 1. It follows from Lemma 4.2, that

A+ R + B +Σ





−1 Ω . is compact in H loc

So













−1 Ω . η V l t + q V l x − S is compact in H loc −1 Since S is also compact in H loc (Ω  ), we conclude that ξ ∈ (−1, 1). 2

−1 η( V )t + q( V l )x is compact in H loc (Ω  ) for

Therefore, Theorem 4.1 follows from Theorems 4.2 and 4.3. 5. Existence of weak solution By Theorem 2.1, we obtain that as l → 0 the approximate solutions I l (x, t , ν , Ω)  I (x, t , ν , Ω) weakly-∗, and V l (x, t ) → V (x, t ) = (ρ (x, t ), m(x, t )) almost everywhere in R2+ for each fixed pair of parameters (ν , Ω). Now, we will show that ( I , V ) is a weak entropy solution of (1.4)–(1.5). We first recall the following useful lemma (cf. [25,33]). Lemma 5.1. For Riemann solution V 0l defined in Section 3, there is a positive constant C ( T ), such that (i +1)h  jl    i, j

ih



V 0l − V 0l

 i 2

dx dt  C ( T )h.

(5.1)

( j −1)l

Theorem 5.1. Suppose that the conditions of Theorem 2.1 are satisfied, then the bounded measurable functions ( I , V ) is a weak entropy solution of (1.4)–(1.5), i.e., ( I , V ) satisfies (2.1)–(2.3) and (2.8).

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

ϕ ∈ C 0∞ ( R T ) satisfying ϕ (x, T ) = 0, we consider the following inte-

Proof. (1) For every test function gral identity:

T  0 R

1 c

 I

l

2211



1

l

ϕt + Ω1 I ϕx dx dt +

c

R

T l

I (x, 0)ϕ (x, 0) dx +

F





ρ l , I l ϕ dx dt = A (ϕ ) + R (ϕ ),

0 R

where

A (ϕ ) =

jl  1   i l i, j

c

I0

 l i 

− I

i

ϕ dx +

jl  

jl    1

c

i , j ( j −1)l

+





ρ0l , I l0 ϕ dx dt ,

i , j ( j −1)l

( j −1)l

R (ϕ ) =

F

I l − I l0

jl    

F









ϕt + Ω1 I l − I l0 ϕx dx dt







ρ l , I l − F ρ0l , I l0 ϕ dx dt .

i , j ( j −1)l

First for R (ϕ ), by (2.5) we have

R (ϕ )  Ch

jl   

   |ϕt | + |ϕx |  F ρ0l , I l0  dx dt

i , j ( j −1)l

  + C I l , I l0





i, j

 Ch



 +

[( j −1)l, jl]∩{ρ l
   l ρ − ρ l |ϕ | dx dt 0

[( j −1)l, jl]∩{ρ l lβ }

jl   



    |ϕt | + |ϕx |  F ρ0l , I l0  dx dt + Clβ

i , j ( j −1)l

i, j

|ϕ | dx dt [( j −1)l, jl]∩{ρ l
 Clϕ C 1 + Clβ ϕ L ∞ → 0 as l → 0.

(5.2)

To estimate A (ϕ ), we decompose A (ϕ ) into three parts:

A (ϕ ) =

jl  1   i l

c

i, j

+

I0

 l i 

− I

i j



ϕ − ϕ dx +

F

i, j

(i +1)h    i, j

( j −1)l

(i +1)h     





i  i 

ρ0l , I l0 − F ρ0l , I l0

ih

≡ A 1 (ϕ ) + A 2 (ϕ ) + A 3 (ϕ ). For A 1 (ϕ ), we have

i

ih

ϕ ij dx dt

F







ρ0l , I l0 ϕ − ϕ ij dx dt

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P. Jiang / J. Differential Equations 253 (2012) 2191–2223

       l i  l i   i   i  A 1 (ϕ ) =   F ρ , I h ϕ − ϕ dx 0 0 j   i, j

 Ch

 

F

2



 12    jl

  l i  l i dx 0 , I0

ρ

i, j



12

i

ϕ −ϕ

 i 2 dx j

i , j ( j −1)l

    1 1  C T  F ρ0l , I l0  L ∞ 1 + |ln l| 2 l 2 ϕ C 1  1 1  C 1 + |ln l| 2 l 2 ϕ C 1 → 0 as l → 0,

(5.3)

and for A 2 (ϕ ), one has

 jl

    A 2 (ϕ ) 

i , j ( j −1)l

    l l  |ϕ i − ϕ ij | |ϕ − ϕ i | F ρ , I  l + h dx dt 0 0 |x − jl| |t − ih|

 Cl F L ∞ ϕ C 1  Clϕ C 1 → 0 as l → 0.

(5.4)

By (2.5) and Lemma 5.1, we have

       A 3 (ϕ )  C I l , I l i ϕ L ∞ 0

 jl

 l  l i  ρ − ρ  dx dt 0 0

0

i , j ( j −1)l

  C ϕ L ∞

jl   

l 0



 l i 2 dx dt 0

12

ρ − ρ

i , j ( j −1)l 1

 Cl 2 ϕ L ∞ → 0 as l → 0.

(5.5)

It follows from (5.2)–(5.5) and the assumption (2.5), that, as l → 0,

T  0 R

1 c

 I ϕt + Ω1 I ϕx dx dt +

R

(2) For every test function identity:

T 0 R

where







1 c

T I 0 ϕ (x, 0) dx +

F (ρ , I )ϕ dx dt = 0. 0 R

ϕ ∈ C 0∞ ( R T ) satisfying ϕ (x, T ) = 0, we consider the following integral 

ρ l ϕt + ml ϕx dx dt +

ρ l (x, 0)ϕ (x, 0) dx = A 1 (ϕ ) + A 2 (ϕ ) + R 1 (ϕ ) + R 2 (ϕ ), R

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

A 1 (ϕ ) =

jl   

i





ρ0l − ρ ij ϕ i dx,

A 2 (ϕ ) =

i , j ( j −1)l

R 1 (ϕ ) =

l

m

ih



ρ l (x, 0) − ρ0l (x, 0) ϕ (x, 0) dx,

R

(i +1)h     i, j



2213

− ml0





T

R 2 (ϕ ) =

ϕx dx dt ,

 jl





ρ l − ρ0l ϕt dx dt .

i , j 0 ( j −1)l

R

We claim that A 1 (ϕ ), A 2 (ϕ ), R 1 (ϕ ) and R 2 (ϕ ) → 0, as l → 0. For A 1 (ϕ ), one has

     jl  jl     l i  l i  i     l i  i  i  A 1 (ϕ )   ρ0 − ρ ϕ dx +  ρ − ρ j ϕ dx     i , j ( j −1)l

 C ϕ L ∞

i , j ( j −1)l

     jl    jl    i     l i  i    l i i i i + ρ − ρ j ϕ j dx +  ρ − ρ j ϕ − ϕ j dx     h

lβ

i , j ( j −1)l

i , j ( j −1)l

   C ϕ L ∞ 1 + |ln l| lβ−1 + 0  +

12  12 jl jl        i   2 2 ϕ i − ϕ i  dx  ρ l − ρ i  dx j j i , j ( j −1)l

i , j ( j −1)l

   1 1  C ϕ L ∞ 1 + |ln l| lβ−1 + C ϕ C 1 1 + |ln l| 2 l 2 → 0 as l → 0.

(5.6)

For A 2 (ϕ ), we have

   jl       l   l  A 2 (ϕ )   ρ (x, 0) − ρ0 (x, 0) ϕ (x j , 0) dx   j ( j −1)l

     jl      l l ρ (x, 0) − ρ0 (x, 0) ϕ (x, 0) − ϕ (x j , 0) dx +   j ( j −1)l

jl     ρ l (x, 0) − ρ l (x, 0) dx  Clϕ  1 → 0 as l → 0.  0 + lϕ C 1 C 0

(5.7)

j ( j −1)l

It follows from the uniform boundedness for ( I l , V l ) that

      R 1 (ϕ )  ChG ρ l , I l  0

0

T |ϕx | dx dt

L∞ 0 Ω

 Clϕ C 1 → 0 as l → 0.

(5.8)

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P. Jiang / J. Differential Equations 253 (2012) 2191–2223

Also,

  T  jl      l  l  R 2 (ϕ ) =  ρ − ρ0 ϕt dx dt   Clβ ϕ C 1 → 0 as l → 0.  

(5.9)

i , j 0 ( j −1)l

Then (5.6)–(5.9) imply that

T



lim

l→0





ρ l ϕt + ml ϕx dx dt +

0 R

ρ l (x, 0)ϕ (x, 0) dx = 0.

(5.10)

R

Notice that

         ρ l (x, 0)ϕ (x, 0) dx − ρ0 (x)ϕ (x, 0) dx  Clβ ϕ (x, 0) dx → 0,   R

R

R

that is,





ρ l (x, 0)ϕ (x, 0) dx =

lim

l→0

R

ρ0 (x)ϕ (x, 0) dx. R

Applying the dominated convergence theorem to (5.10), we have

T

 (ρϕt + mϕx ) dx dt +

0 R

(3) For every test function identity:

T



ρ0 ϕ (x, 0) dx = 0. R

ϕ ∈ C 0∞ ( R T ) satisfying ϕ (x, T ) = 0, we consider the following integral



ml ϕt + f V l







 

ϕx + G ρ l , I l ϕ dx dt +

0 R

ml (x, 0)ϕ (x, 0) dx R

= A 1 (ϕ ) + A 2 (ϕ ) + R 1 (ϕ ) + R 2 (ϕ ), 2

where f ( V ) = mρ + ρ , and

T

 





f Vl − f Vl

A 1 (ϕ ) =



ϕx dx dt ,

0 R

A 2 (ϕ ) =

jl    

ml − ml0



 







 





 

ϕt + f V l − f V 0l ϕx + G ρ l , I l − G ρ0l , I l0 ϕ dx dt ,

i , j ( j −1)l







ml (x, 0) − ml0 (x, 0)

R 1 (ϕ ) = R

ϕ (x, 0) dx,

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

R 2 (ϕ ) =

jl   

i ml0

− mij



i

ϕ dx +

i , j ( j −1)l

jl  

G



2215



ρ0l , I l0 ϕ dx dt .

i , j ( j −1)l

We estimate A 1 (ϕ ) first,

   A 1 (ϕ ) 

T



0 {ρ l lβ }

T

   l 2   (m ) (ml )2   l l   ρ l − ρ l  + ρ − ρ |ϕx | dx dt



2



 l



ρ 1 + ln ρ l  + lβ |ϕx | dx dt

C 0 {ρ l lβ }

 2  C ϕ C 1 lβ 1 + |ln l| → 0 as l → 0.

(5.11)

For A 2 (ϕ ), by (2.6) we have

    A 2 (ϕ ) 

jl   jl  l      m + ml0  l   l ρ l − ρ l  dx dt   m − m ϕ C + x 0 0  ρl

i , j ( j −1)l

i , j ( j −1)l

jl       G ρ l , I l ϕt  dx dt +h 0

0

i , j ( j −1)l jl      G (ρ l , I l )    0 0 l l  h m + m0 ϕx  dx dt + Clβ + Chϕ C 1  ρl 0

i , j ( j −1)l

 Clϕ C 1 + Clβ → 0 as l → 0.

(5.12)

For R 1 (ϕ ),

  jl    jl           l l 0 l l 0 R 1 (ϕ ) =  m − m0 (x, 0)ϕ j dx + m − m0 (x, 0) ϕ (x, 0) − ϕ j dx   j ( j −1)l

j ( j −1)l

 0 + Clϕ C 1 → 0 as l → 0.

(5.13)

We decompose R 2 (ϕ ) into three parts:

R 2 (ϕ ) =

jl   

i ml0

− mij



i

i j



ϕ − ϕ dx +

i , j ( j −1)l

+ ≡

i, j

(i +1)h     

G

i, j

R 12 (

(i +1)h   





ih

ϕ ) + R 22 (ϕ ) + R 32 (ϕ ).

ih

i  i 

ρ0l , I l0 − G ρ0l , I l0

G

ϕ ij dx dt







ρ0l , I l0 ϕ − ϕ ij dx dt

2216

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

For R 12 , using Lemma 4.1, we have

   jl  1      l i  l i   l i  i   i i  R (ϕ ) =  hG ρ0 , I 0 + m − m j ϕ − ϕ j dx 2   i , j ( j −1)l

 h

jl  

G

2





12

  l i  l i dx 0 , I0

ρ

+

m

i , j ( j −1)l

 ×

jl    i l

12

2 − mij dx

i , j ( j −1)l

jl   

12

i

ϕ −ϕ

 i 2 dx j

i , j ( j −1)l

 1    1  Cl 2 ϕ C 1 l 2 G ρ0l , I l0  L ∞ + C → 0 as l → 0,

(5.14)

and for R 22 (ϕ ), one has

 2    R (ϕ ) 

    l l  |ϕ i − ϕ ij | |ϕ − ϕ i | G ρ , I  l + h dx dt 0 0 |x − jl| |t − ih|

 jl

2

i , j ( j −1)l

 ClG L ∞ ϕ C 1  Clϕ C 1 → 0 as l → 0.

(5.15)

By (2.6) and Lemma 5.1, we have

  3      R (ϕ )  C I l , I l i ϕ L ∞ 0

2

 jl

 l  l i  ρ − ρ  dx dt 0 0

0

i , j ( j −1)l

  C ϕ L ∞

jl   

 l i 2



ρ0l − ρ0

12

1

 Cl 2 ϕ L ∞ → 0 as l → 0. (5.16)

dx dt

i , j ( j −1)l

It follows from (5.11)–(5.16) that

T

l



m ϕt + f V

lim

l→0



l





l

l

 

 ml (x, 0)ϕ (x, 0) dx = 0.

ϕx + G ρ , I ϕ dx dt +

0 R

R

Using the dominated convergence theorem, we have

T  mϕt + 0 R



m2

ρ

  T  + P m (ρ ) ϕx dx dt + G (ρ , I )ϕ dx + m0 ϕ (x, 0) dx = 0. 0 R

R

(4) Now we show the entropy inequality. For every weak and convex entropy pair (η, q) and every nonnegative smooth function ψ that has compact support in region R T satisfying ψ(x, 0) = ψ(x, T ) = 0, we consider the following integral identity:

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

T

 







2217



η V l ψt + q V l ψx dx dt = A (ψ) + R (ψ) + Σ(ψ) + S (ψ),

0 R

where A (ψ), R (ψ), Σ(ψ), S (ψ) are similar to those of (4.13). Since (η, q) is a convex entropy pair for ξ ∈ (−1, 1) and ψ  0, we have

Σ(ψ)  0, 1

A (ψ) =

2

 jl



ψ ij

i, j

Vl

i

− V ij



  i  ∇ 2 η ζ ji V l − V ij dx

( j −1)l

jl    

i



η V l − η V ij

+

(5.17)

 i  ψ − ψ ij dx

i , j ( j −1)l



jl    

i



η V l − η V ij

 i  8 ψ − ψ ij dx  −Clα − 9 ψC 0α

for

i , j ( j −1)l

8 9

< α < 1.

(5.18)

As in (4.19), we have

S (ψ)  −Clψ H 1 .

(5.19)

0

By (4.8), we have

R (ψ) = −h

 1





ηm V

i, j

 l i

+θ



i V 0l



  l i



 1

− V

dθ G



i  i 

ρ0l , I l0

ψ i dx

0



 β 1−|ξ | −1 − C 1 + |ln l| l 1−ξ 2 ψL ∞ 

 β 1−|ξ2| −1

= −C 1 + |ln l| l

1−ξ

ψL ∞ − h

ηm V

i, j

 

  l i  l i − 0 , I0

× G ρ −h

 i, j

 1





 l i

+θ



i V 0l



− V

  l i

dθ

0

 i  i  i ψ dx G ρl , I l

ηm V

 l i

+θ



i V 0l



− V

  l i

dθ G

 l i  l i  i ρ , I ψ dx

0

  β 1−|ξ | −1  −C 1 + |ln l| l 1−ξ 2 ψL ∞ − ClψL ∞ −h

 i, j

 1



 l i

ηm V θ 0

It follows from (5.17)–(5.20) that



i V 0l



− V

  l i

dθ G

 l i  l i  i ρ , I ψ dx.

(5.20)

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P. Jiang / J. Differential Equations 253 (2012) 2191–2223

T

 









η V l ψt + q V l ψx dx dt

0 R

+h

jl  1   i , j ( j −1)l



ηm V

 l i

+θ



i V 0l



− V

  l i

dθ G

 l i  l i  i ρ , I ψ dx

0

  β 1−|ξ | −1 8  −Clα − 9 ψC 0α − C 1 + |ln l| l 1−ξ 2 ψL ∞ − ClψL ∞ − Clψ H 1 . 0

(5.21)

Letting l → 0 in (5.21) and using the fact that I l  I weakly-∗ in R2+ and V l → V a.e. for each fixed pair of parameters (ν , Ω), we obtain the following entropy condition:

T



T



η(ρ , m)ψt + q(ρ , m)ψx dx dt +

0 R

ηm (ρ , m)G (ρ , I )ψ dx dt  0. 0 R

This completes the proof of the main result in Theorem 2.1.

2

6. Global entropy solutions to the spherical symmetry case In this section, we prove the existence of global entropy solutions to initial–boundary problem (1.9)–(1.10). First, we introduce new variables

= r2ρ ,

= r 2 m,

to transform system (1.9)–(1.10) into

⎧ 1 ⎪ ⎪ I t (ν , Ω) + μ I r (ν , Ω) = F 1 ( , I ), ⎪ ⎪ ⎨c t + r = 0,   2 ⎪ ⎪

2 ⎪ ⎪ ⎩ t + + = + r 2 G 1 ( , I ),



r

(6.1)

r

with initial–boundary data

⎧ I| = I 0 (r , ν , Ω), ⎪ ⎪ t =0 ⎨ |t =0 = 0 (r ), |t =0 = 0 (r ), ⎪ I |r =1 = 0 if μ > 0, ⎪ ⎩

|r =1 = 0.

(6.2)

According to assumptions (A1), (A2) in Section 2, the source terms F 1 ( , I ), G 1 ( , I ) satisfy the following conditions:

   F 1 ( , I )  C 1 | I | + C 2 |ln |, where

    G 1 ( , I )  C 3 | I | + C 4 |ln |

= max( r 2 , lβ ), with 2  β  3, l ∈ (0, l0 ) for some sufficiently small l0 ∈ (0, 1).

(6.3)

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

For bounded

2219

i , I i , i = 1, 2, ⎧ ⎨  F ( , I ) − F ( , I )  C ( I , I ) 1 | − |, 1 1 1 1 2 2 1 2 2 1 2 r  ⎩   F 1 ( 1 , I 1 ) − F 1 ( 2 , I 2 )  C | I 1 − I 2 |, ⎧ ⎨ G ( , I ) − G ( , I )  C ( I , I ) 1 | − |, 1 1 1 1 2 2 1 2 2 1 2 r  ⎩   G 1 ( 1 , I 1 ) − G 1 ( 2 , I 2 )  C | I 1 − I 2 |,

if 1 = 2 ,

(6.4)

if 1 = 2 , if 1 = 2 ,

(6.5)

if 1 = 2 .

Then we give the main result in this section. Theorem 6.1. Suppose that the initial data ( I 0 , 0 , 0 ) satisfy the conditions:

   I 0 (r , ν , Ω)  C 0 ,

      0 (r )  0 (r ) C 0 + ln 0 (r )

0  0 (r )  C 0 ,

for some positive constant C 0 , and F 1 ( , I ), G 1 ( , I ) satisfy (6.3)–(6.5). Then, for γ = 1, the initial–boundary problem (6.1)–(6.2) has a global weak entropy solution ( I (r , t , ν , Ω), (r , t ), (r , t )), r > 1, t > 0, for any fixed ν > 0, Ω ∈ S 2 , satisfying the following estimates and entropy condition: on Π T = [1, +∞) × [0, T ] for any fixed T > 0,

| I |  C ( T ),

  | |  C ( T ) + |ln | ,

0   C ( T ),

for a constant C ( T ) > 0 which depends on T , and

T∞ 



T∞



η( , )φt + q( , )φr dr dt +

0 1

η ( , )

2 r



+ r 2 G 1 ( , I ) φ dr dt  0

0 1

for any entropy pair (η, q) in (2.7) with ξ ∈ (−1, 1) and for all nonnegative test functions φ ∈ C 0∞ (Π T ), with φ(r , 0) = φ(r , T ) = φ(1, t ) = 0. To prove Theorem 6.1, first we use the shock capturing scheme of Godunov type in Section 3 to construct the approximate solutions ( I l , V l ) = ( I l , l , l ) with V l = ( l , l ) of (6.1)–(6.2) in the strip 0  t  T for any fixed T > 0, where l > 0 is the space mesh length. To ensure the Courant–Friedrichs– Lewy stability condition, we choose the time mesh length

h=

l 10(1 + |ln l|)

(6.6)

.

As for the one-dimensional case (1.4)–(1.5), to ensure the CFL condition, we employ a cut-off technique so that the approximate density functions stay away from vacuum by lβ , starting from the initial data:

I 0 = I0,





0 = max 0 , lβ ,

0 = 0 .

Namely, we solve the Riemann problems of (3.1) in the region R 1j ≡ {(r , t ): 1 + r j −1/2  r  1 + r j +1/2 , 0  t < h}, r j ±1/2 = ( j ± 1/2)l:

2220

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

⎧    ⎪ 0l t + 0l r = 0, ⎪ ⎪ ⎪   ⎪ ⎪  l ( 0l )2  l  ⎪ ⎪ ⎨ 0 t + + 0 = 0, 0l r ⎧ ⎪  ⎪ ⎨ 1 1+ jl ( 0 (r ), 0 (r )) dr , r < 1 + jl, ⎪ ⎪  l 1+( j −1)l ⎪ l l  ⎪ ⎪ ⎪ 0 , 0 t =0 = ⎩ 1  1+( j +1)l ⎩ ( 0 (r ), 0 (r )) dr , r > 1 + jl, l 1+ jl for all integers j  1; and the lateral Riemann problem in R 11 ≡ {1  r  1 + r1/2 , 0  t < h},

⎧ l   l + 0 r = 0 , ⎪ ⎪ ⎪ 0 t ⎪   ⎪ ⎪ ( 0l )2  l  ⎪ l ⎪

+ + = 0, ⎪ 0 t 0 ⎪ ⎪ 0l ⎨ r 1+l ⎪  l    1 ⎪ l  ⎪ ⎪ 0 , 0 t =0 = 0 (r ), 0 (r ) dr , ⎪ ⎪ l ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ l 

0 r = 1 = 0 ,

r > 1,

and we also solve the following problem in the strip: r > 1, 0  t  h,

⎧  l 1 l  ⎪ ⎪ ⎪ I 0 t + μ I 0 r = 0, ⎨ c I l0 t =0 = I 0 (r , ν , Ω), ⎪ ⎪ ⎪ ⎩ l  I 0 r =1 = 0 if μ > 0. Then, we set



I l (r , t , ν , Ω) = I l0 (r , t , ν , Ω) + c F



V l (r , t ) = V 0l (r , t ) +  G







0l , I l0 t ,

0l , I l0 t ,

for r > 1, 0 < t < h, where V 0l = ( 0l , 0l ) and  G = (0, 2r + r 2 G 1 ) . We continue the construction to the next time step recursively as follows. Suppose that we have defined approximate solutions ( I l , V l ) for 0  t < ih, with i  1 an integer. Then we set

I l (r , t , ν , Ω) = I l (r , t , ν , Ω),





l (r , t ) = max l (r , t ), lβ ,

l (r , t ) = l (r , t )

for 0  t < ih. We then define



 l l 0 , I 0 (t − ih),   l l l l V (r , t ) = V 0 (r , t ) +  G 0 , I 0 (t − ih), I l (r , t , ν , Ω) = I l0 (r , t , ν , Ω) + c F

for ih  t < (i + 1)h, where V 0l (r , t ) are piecewise smooth functions defined as solution of the Riemann

problem in the region R ij+1 ≡ {(r , t ): 1 + r j −1/2  r  1 + r j +1/2 , ih  t < (i + 1)h}:

P. Jiang / J. Differential Equations 253 (2012) 2191–2223

2221

⎧    ⎪ 0l t + 0l r = 0, ⎪ ⎪ ⎪   ⎪ ⎪  l ( 0l )2 ⎪ l ⎪ + 0 = 0, ⎨ 0 t + 0l r ⎧ ⎪  ⎪ 1 1+ jl ⎨ ⎪ ( l (r , ih − 0), l (r , ih − 0)) dr , r < 1 + jl, ⎪  l 1+( j −1)l ⎪ l l  ⎪ ,

= ⎪  ⎪ ⎩ 0 0 t =ih ⎩ 1 1+( j +1)l ( l (r , ih − 0), l (r , ih − 0)) dr , r > 1 + jl; l 1+ jl and the solution of the lateral Riemann problem in region R 1i +1 ≡ {1  r  1 + r1/2 , ih  t < (i + 1)h},

⎧ l   l ⎪ ⎪ 0 t + 0 r = 0, ⎪ ⎪   ⎪ ⎪  l ( 0l )2  l  ⎪ ⎪

+ + = 0, ⎪ 0 t 0 ⎪ ⎪ l ⎨ r 0

1+l ⎪   l  l  1 ⎪ l  ⎪ ⎪ (r , ih − 0), l (r , ih − 0) dr , ⎪ 0 , 0 t =0 = l ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎩ l 

= 0,

r > 1,

0 r =1

and I l0 (r , t , ν , Ω) is the solution of the following problem in the region r > 1, ih  t < (i + 1)h:

⎧  l 1 l  ⎪ ⎪ ⎪ ⎨ c I 0 t + μ I 0 r = 0,  I l0 t =ih = I l (r , ih − 0, ν , Ω), ⎪ ⎪ ⎪ ⎩ l  I 0 r =1 = 0 if μ > 0. Finally, we set

I l (r , t , ν , Ω) = I l (r , t , ν , Ω),





l (r , t ) = max l (r , t ), lβ ,

l (r , t ) = l (r , t )

for 0  t < ih. Then the approximate solutions ( I l (r , t , ν , Ω), V l (r , t )) are well-defined for r > 1, 0  t  T and any fixed ν > 0, Ω ∈ S 2 . Theorem 6.2. Suppose that the initial data ( I 0 , 0 , 0 ) satisfy the following conditions:

   I 0 (r , ν , Ω)  C 0 ,

0  0 (r )  C 0 ,

     0 (x)  0 (r ) C 0 + |ln 0 | ,

r > 1,

ν > 0, Ω ∈ S 2 ,

for some constant C 0 > 0, and F 1 ( , I ), G 1 ( , I ) satisfy (6.3). Then, there exists a positive constant C ( T ), independent of l and h, such that

 l  I   C ( T ),

lβ  l (r , t )  C ( T ),

 l       l C ( T ) + ln l  ,

(r , t ) ∈ ΠT .

The proof of uniform boundedness of ( I l , l , l ) is similar to that of Theorem 4.2, only using

l0 r2

ρ0l which was discussed in Theorem 4.2. We also notice that for the geometric source term we have 2r  2 , the rest part of the proof is similar to the proof of Theorem 2.1 so that we omit it instead of here.

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