Lp -estimates of the Boltzmann equation around a traveling local Maxwellian

J. Differential Equations 251 (2011) 45–57

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Journal of Differential Equations www.elsevier.com/locate/jde

L p -estimates of the Boltzmann equation around a traveling local Maxwellian Seok-Bae Yun Department of Mathematical Sciences, KAIST (Korea Advanced Institute of Science and Technology), 373-1 Guseong-dong, Yuseong-gu Daejeon, 305-701, South Korea

a r t i c l e

i n f o

a b s t r a c t In this paper, we are interested in the L p -estimates of the Boltzmann equation in the case that the distribution function stays around a traveling local Maxwellian. For this, we divide both sides of the Boltzmann equation by the velocity distribution function with a fractional exponent and reformulate the Boltzmann equation into a regularized one. This amounts to endowing additional integrability on the collision kernel, which in turn enables us to apply simple Hölder type inequalities. Our results cover the whole range of Lebesgue exponents: 0 < p  ∞. © 2011 Elsevier Inc. All rights reserved.

Article history: Received 21 March 2010 Revised 9 September 2010 Available online 16 March 2011 MSC: 82C40 35Q20 35B35 35B30 Keywords: Boltzmann equation L p -stability L p -estimate Traveling local Maxwellian

1. Introduction In the kinetic theory of gases, it is postulated that all the relevant information is encoded in a velocity distribution function f (x, v , t ) representing the number density of particles located at position x with velocity v at time t. For non-ionized monatomic rarefied gas, the time evolution of f is governed by the celebrated Boltzmann equation:

∂t f + v · ∇x f = Q ( f , f ),

(x, v , t ) ∈ R3 × R3 × R+ .

E-mail address: [email protected]. 0022-0396/$ – see front matter doi:10.1016/j.jde.2011.03.001

© 2011

Elsevier Inc. All rights reserved.

(1.1)

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S.-B. Yun / J. Differential Equations 251 (2011) 45–57

The left-hand side of (1.1) describes the free transport of non-interacting particles, whereas the collision operator Q ( f , f ) captures collisions or interaction between particles. It can be written down explicitly as follows:

Q ( f , f )( v ) ≡



1





B ( v − v ∗ , ω) f  f ∗ − f f ∗ dω dv ∗ .

κ

(1.2)

R3 ×S2+

Here κ is the Knudsen number which is the ratio between the mean free path of molecules and the characteristic length of the flow and S2+ = {ω ∈ S2 | ( v − v ∗ )· ω  0}. For the simplicity of presentation, we adopt the following handy notations:







f  ≡ f x, v  , t ,



f ∗ ≡ f x, v ∗ , t ,

f ≡ f (x, v , t )

and

f ∗ ≡ f (x, v ∗ , t ),

where the pair ( v  , v ∗ ) denotes the post-collisional velocities which can be calculated explicitly from the pre-collisional pair of velocities ( v , v ∗ ) by



v  = v − (v − v ∗) · ω







ω and v ∗ = v ∗ + ( v − v ∗ ) · ω ω.

(1.3)

The collision kernel B ( v − v ∗ , ω) is determined by types of interaction between gas particles. For the precise form and relevant structural assumptions imposed on the collision kernel, see (A1) below. For more detailed survey of mathematical and physical results of the Boltzmann equation, we refer to [4,6,7,21,22,26]. In this paper, we study the stability problem of the Boltzmann equation in L p spaces when the velocity distribution function is bounded from above and below by a traveling local Maxwellian:

am Mα ,β (x, v )  f  (x, v , t )  a M Mα ,β (x, v ),

(1.4)

where am , a M denote positive constants and Mα ,β (x, v ) is a traveling local Maxwellian solution:

Mα ,β (x, v ) ≡ e −α |x|

2

−β| v |2

for positive constants α , β > 0.

(1.5)

For the stability problem of kinetic equations, L 1 space is the most natural setting in that it corresponds to the total mass of the system. The study of stability in L 1 space for the Boltzmann equation near vacuum was initiated by Ha [11,12] who introduced a nonlinear functional approach motivated by the stability theory of hyperbolic conservation laws, and was studied extensively by Ha and his coworkers [8,10,14,16]. See also [3,19]. It is then quite natural to ask whether the stability results in L 1 can be extended to general L p space. Considering that the asymptotic behavior of the Boltzmann equation in this regime is largely governed by the free transport equation:

∂f + v · ∇ f = 0, ∂t for which the uniform L p -stability estimate trivially holds, it is reasonable to expect similar estimates to hold true for general L p spaces. In this vein, there have been several results on the L p -stability estimates of the Boltzmann equation near vacuum. In [15], Ha’s nonlinear functional approach was extended to summational L p setting. Then the Gronwall type argument also became available in [13] to obtain weighted L p -stability estimates. Recently, Alonso and Gamba [1] resolved the uniform L p stability problem for the Boltzmann equation with soft potential in the affirmative. The usual difficulty encountered in the study of L p type estimates of the collision operator is that even the simple Hölder inequality cannot be directly applied due to the singularity of the collision kernel. In [13], this difficulty was overcome by introducing polynomial weights in the velocity fields.

S.-B. Yun / J. Differential Equations 251 (2011) 45–57

47

1 1−μ In this paper, we attack this problem by dividing both sides of (1.1) by μ f and reformulating the Boltzmann equation into the following form (see (3.2)):

  ∂ fμ + v · ∇ f μ = Q μ f μ, f μ . ∂t In this way, the reformulated collision operator Q μ gains additional integrability, and we are now able to apply Hölder type inequalities to obtain the following L p -estimate:

 μ    f   C μ, p  f μ  , p p which, upon adjusting the value of μ and p properly, leads to the main results. (See Theorem 1.1 below.) We mention that the parameter μ provides greater degree of freedom in determining the Lebesgue exponent, which is a key element in obtaining L p -estimates for 0 < p < 1. Before we state our assumptions and main results, we introduce the notion of mild solutions. Definition 1.1. We say that a nonnegative function f ∈ L ∞ (0, T ; L 1 (R3 × R3 )) is a mild solution if it satisfies the mild form:

t f (x, v , t ) = f 0 (x, v ) + 

Q  ( f , f )(x, v , s) ds,

(x, v , t ) ∈ R3 × R3 × R+ ,

(1.6)

0

where the operator  is defined by

f  (x, v , t ) ≡ f (x + t v , v , t ). The global existence of mild solutions for the Boltzmann equation in infinite vacuum was first established by Illner and Shinbrot [17] in the case that the solution decays exponentially in phase space by combining fixed point arguments with the celebrated Kaniel–Shinbrot scheme [18]. Their result was then extended to more general settings including algebraically decaying data by several authors [5,24,25]. In [20,23] the smallness assumption imposed on the upper traveling Maxwellian bound was replaced by a closedness condition to resolve the Cauchy problem of the Boltzmann equation close to a local Maxwellian regime, which is relevant to our case. We remark, however, that our stability analysis in this paper does not require any closedness nor smallness restrictions on the solutions. The main structural assumptions of this paper are as follows.

• (A1) The collision kernel satisfies an inverse power potential and an angular cut-off assumption: B ( v − v ∗ , ω) = | v − v ∗ |γ bγ (θ),

−3 < γ  1,

and

 bγ (θ) dω = B γ < ∞, S2+

where θ is the angle between v − v ∗ and ω . • (A2) Mild solution f satisfies

am Mα ,β (x, v )  f  (x, v , t )  a M Mα ,β (x, v ), for some strictly positive constants a M , am .

a.e. (x, v ),

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S.-B. Yun / J. Differential Equations 251 (2011) 45–57

Remark 1.1. The existence of mild solution satisfying (A2) with additional condition that a M − am is sufficiently small was established in [20,23]. Recently, this result was extended to the classical solutions for soft potentials in [1]. We are now in a position to state our main results. Below G p denotes constants which depend on the Lebesgue exponent p, but not on x, v and t. Theorem 1.1. Suppose that main assumption (A1) holds with −3 < γ  1 and let f be a mild solution of (1.1) satisfying (A2) corresponding to an initial datum f 0 . Then we have

   f (t )

Lp

 G p  f 0 L p ,

0 < p  ∞.

(1.7)

Remark 1.2. 1. Alonso et al. [1,2] have resolved L p -stability problem of the Boltzmann equation with soft potentials for spatially decaying solutions. Our result is weaker in the sense that we cannot consider the difference of the two distribution functions: f − ¯f , but stronger in that it covers the hard potential case and the whole range of exponent: 0 < p  ∞. 2. We do not impose any smallness condition neither on a M nor on a M − am . Although the existence result was established only when the distribution functions lie close to a local Maxwellian regime in the sense that a M − am is sufficiently small. The rest of this paper is organized as follows. In Section 2, we present several estimates which will be crucial for the later sections. Through Section 3 to Section 4, we prove our main results. In the last section, we consider the stability problem of the difference of two distribution functions. 2. Preliminaries 2.1. Basic estimates In this part, we present several estimates to be used in later sections. For the proof, we refer the readers to [11,13,16,19]. Lemma 2.1. Let x ∈ R3 , V = 0 and a > 0. Then we have

∞

 e −a|x+τ V | dτ  2

0

π 1 a |V |

.

Lemma 2.2. For −3 < γ  0, we have







B ( v − v ∗ , ω)Mα ,β x + t ( v − v ∗ ), v ∗ dω dv ∗  C (γ , α , β) · R3 ×S2

1

(t + 1)γ +3

,

+

π + where C (γ , α , β) = B γ [ γ2+ 3

( πα )3 + ( πβ )3 ].

3. The proof of Theorem 1.1 (−3 < γ  0) Let f be a mild solution of the Boltzmann equation satisfying the structural assumption (A2). We then have from (1.6)

∂ f 1 1 = Q  ( f , f )  Q + ( f , f ). ∂t κ κ

(3.1)

S.-B. Yun / J. Differential Equations 251 (2011) 45–57

49

1 We divide both sides of (3.1) by μ ( f ε )1−μ (0 < μ < 1) to get



μ ∂( f  )μ 1  Q +( f , f ) ∂t κ ( f  )1−μ

     1 −μ       μ μ f f∗ = B ( v − v ∗ , ω) f f∗ dω dv ∗ .  κ

f

(3.2)

R3 ×S2

+

We observe from the lower and upper bound estimate of (A2) 

f   f∗ f

 = =

 2  2  2  2 a2M e −α |x−t ( v − v )| −β| v ∗ | e −α |x−t ( v − v ∗ )| −β| v ∗ | 2 2 am e α |x| −β| v | 2 2 2 2 a2M e −α |x| +β| v | e −α |x−t ( v − v ∗ )| −β| v ∗ | 2 2 am e −α |x| −β| v |

a2M −α |x+t ( v − v ∗ )|2 −β| v |2 e . am

We substitute the above estimate into (3.2) to obtain

2 1−μ  aM ∂( f  )μ α  μe ∂t am



  μ

A μ,α ,β ( v − v ∗ )b(θ) f   f ∗

dω dv ∗ ,

(3.3)

R3 ×S2

+

where A μ,α ,β ( v − v ∗ ) denotes the regularized collision kernel defined by 2 2 A μ,α ,β ≡ A μ,α ,β ( v − v ∗ ) ≡ | v − v ∗ |γ e −(1−μ)(α |x−t ( v − v ∗ )| +β| v ∗ | ) .

Note that A μ,α ,β now is an integrable function, which is a crucial ingredient in estimating the reformulated collision operator in L p . We then multiply (3.3) by p ( f  )μ( p −1) and integrate over R3 × R3 × R+ with respect to (x, v , t ) to obtain

2  1 −μ    μ  p    μ  p  f  + p μe α a M (t )   f 0

p

p

∞  ×

am

  μ   μ ( p − 1 )



A μ,α ,β b(θ) f   f ∗

f

dω dv ∗ dv dx dt .

(3.4)

0 R3 ×S2 +

For brevity, we put

 N1 (t ) ≡



  μ   μ ( p − 1 )

A μ,α ,β ( v − v ∗ )b(θ) f   f ∗

f

dω dx dv dv ∗ .

R3 ×S2+

Lemma 3.1. Let γ ∈ (−2, 0]. Then for p  1, N1 satisfies the following pointwise estimate:

N1 (t )  for some constant C N1 = C N1 (μ, α , β).

C N1 (a M )μ   μ  p  f (t ) p , (t + 1)3+γ

(3.5)

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S.-B. Yun / J. Differential Equations 251 (2011) 45–57

Proof. We apply Hölder inequality to N1 to obtain

  p−p 1 − (1p−−μ1) p (α |x+t ( v − v ∗ )|2 +β| v ∗ |2 )   μ p γ b(θ) |v − v ∗| e f dv dv ∗ dx

 N1 

R9

S2+



 ×





N1 A

 1p   pμ     pμ | v − v ∗ |γ f   f∗ dv dv ∗ dx dω .

R9





(3.6)



N1B

(i) The estimate of N1 A : We observe from Lemma 2.2



| v − v ∗ |γ e −

N1 A ≡

(1−μ) p 2 2 p −1 (α |x+t ( v − v ∗ )| +β| v ∗ | )

R9





=

μ p



f  (x, v , t )

R6

 ≡

R3



1

(t

+ 1)3+γ





+

γ +3

μ p

f  (x, v , t )

dv dv ∗ dx

(1−μ) p 2 2 p −1 (α |x+t ( v − v ∗ )| +β| v ∗ | ) dv

π ( p − 1) α (1 − μ) p



3 +

π ( p − 1) β(1 − μ) p

3 

 ∗

dx dv

  μ  p  f (t )  p

   μ  p  f (t )  .

C N1 A

(t

2π

| v − v ∗ |γ e −



+ 1)3+γ

(3.7)

p

(ii) The estimate of N1B : Applying a series of standard changes of variables, we have

 N1B =

   p μ    p μ | v − v ∗ |γ f x + t v , v  f x + t v , v ∗ dx dv dv ∗

R9



=

   p μ    p μ | v − v ∗ |γ f x, v  f x, v ∗ dx dv dv ∗

R9



=

  pμ   pμ | v − v ∗ |γ f (x, v ) f (x, v ∗ ) dv dv ∗ dx

R9



=

  p μ  pμ    | v − v ∗ |γ f  (x, v ) f x + t ( v − v ∗ ), v ∗ dx dv dv ∗ .

R9

We then use Lemma 2.2 to see

 N1B =



 pμ



f  (x, v )

R6



 (a M ) p μ R6

    p μ | v − v ∗ |γ f  x + t ( v − v ∗ ), v ∗ dv ∗ dv dx

R3



 pμ 



f  (x, v )

R3

| v − v ∗ |γ e − p μ(α |x−t ( v − v ∗ )|

2

+β| v ∗ |2 )

 dv ∗ dv dx

S.-B. Yun / J. Differential Equations 251 (2011) 45–57

51

  



  2π π 3 π 3   μ  p (a M ) p μ  + + f (t ) p α pμ β pμ (t + 1)3+γ γ + 3    μ  p C N1B  f ≡ (a M ) p μ (t ) p . (t + 1)3+γ

(3.8)

Substituting (3.7) and (3.8) into (3.6), we obtain 1 1   p −(3+γ )( p − + 1p ) p N1  (a M )μ C N1 A p−1 (C N1B ) p (t + 1)





b(θ) f 

μ  p  dω p

S2+ p

1

 (C N1 A ) p−1 (C N1B ) p

   (a M )μ B γ   f  μp . p (t + 1)3+γ

We set p

1

C N1 (α , β, μ) = (a M )μ (C N1 A ) p−1 (C N1B ) p B γ to complete the proof.

2

We now substitute the estimate (3.5) of Lemma 3.1 into (3.4) to obtain

  μ  p   μ  p  f  + μ p D μ, p (t )   f p

0

t

p

0

1

(t + 1)3+γ

   μ  p  f (t ) dt , p

(3.9)

where μ



D μ, p = a M C N1 B γ

a2M

1−μ

am

.

By Gronwall’s lemma, this yields

  p  f (t )μ p  e 2μ p D μ, p  f 0 μ μp μp

or, equivalently,

   f (t )

D μ, p  f 0 μ p . μp  e

We now adjust μ and p to complete the proof. For this, assume we are given a Lebesgue exponent P ∈ (0, ∞). We divide the argument into the following two cases: P to obtain (i) P ∈ [1, ∞): we fix μ between 0 and 1 and set p = μ

   f (t )  e D μ, p  f 0  P . P Letting P → ∞, we get

   f (t )

∞

 e D μ,∞  f 0 ∞ .

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S.-B. Yun / J. Differential Equations 251 (2011) 45–57

Here D μ,∞ denotes

D μ,∞ = lim D μ, P < ∞. μ P →∞ (ii) P ∈ (0, 1): we fix p in [1, ∞) and set

μ=

P p

to obtain

   f (t )  e D μ, p  f 0  P . P Note that in both cases 0 < μ < 1 and 1  p < ∞ hold, which guarantee the relevance of the preceding argument. 4. The proof of Theorem 1.1 (−2 < γ  1) If the intermolecular force is governed by hard potentials (0 < γ  1), most of the crucial estimates in the previous sections are not relevant anymore due to the unboundedness of the collision kernel at infinity. We overcome this difficulty by incorporating the idea of Cho and Yu [9] into the reformulated setting. More precisely, we introduce a maximal distribution function supt f  and interchange the order of integration between time and velocity, to resolve the singularity of the collision kernel at infinity. We mention that the proof of this section is not restricted to the hard potential case and can be applied to the soft potential case either for −2 < γ  1. We again start from the following inequality:

2  1 −μ  a ∂( f  )μ μ M ∂t am



  μ

A μ,α ,β ( v − v ∗ )b(θ) f   f ∗

dω dv ∗ .

(4.1)

R3 ×S2

+

We integrate from 0 to t to obtain



μ

f  (t )

2  1 −μ  t   μ a  f0 + μ M am



 μ

 f0

+μ

a2M





  μ



  μ

A μ,α ,β b(θ) f   f ∗

dω dv ∗ dt

0 R9 ×S2 +

1−μ ∞ 

am

A μ,α ,β b(θ) f   f ∗

dω dv ∗ dt .

(4.2)

0 R9 ×S2 +

We then take the supremum in time to obtain



sup f t

  μ



 μ

 f0

+μ

a2M am

1−μ ∞ 



 μ

A μ,α ,β b(θ) f  f ∗

dω dv ∗ dt .

(4.3)

0 R3 ×S2 +

The reason why we do this will be clear in Lemma 4.1. We now take L p -norm directly on both sides, instead of multiplying both sides of (4.3) by p f  p −1 and integrating with respect to (x, v ) as in the previous sections, to see

S.-B. Yun / J. Differential Equations 251 (2011) 45–57

2  1 −μ   ∞          aM μ μ        +μ  sup f   f0 p  p am t

53

       μ  A μ,α ,β b(θ) f f ∗ dω dv ∗ dt  

0 R3 ×S2 +

p

2  1 −μ    μ  a ≡  f 0 p + μ M N2 .

(4.4)

am

In the following lemma, we estimate N2 . Note that N2 is bounded by the L p -norm of supt ( f  ). Lemma 4.1. Let γ ∈ (−2, 1]. Then for p  1 and μ ∈ (0, 1), we have

 μ    N2  C μ, p  sup f   t

(4.5)

p

for some positive constant C μ, p . Proof. By Hölder inequality, we have

  ∞  p−p 1   − p (p1−−1μ) (α |x+t ( v − v ∗ )|2 +β| v ∗ |2 ) γ N2   b(θ) |v − v ∗| e dv ∗ ds  0 R3

S2+



 ∞ ×





N2 A

    pμ | v − v ∗ |γ f   ) p μ ( f ∗ dv ∗ ds

 1p

0 R3

   dω  

(4.6)

. L p ( dx,dv )

We use Lemmas 2.1 and 2.2 to see

∞ | v − v ∗ |γ e

N2 A ≡

−μ) − p (p1− (α |x+t ( v − v ∗ )|2 +β| v ∗ |2 ) 1

0 R3

 =

− | v − v ∗ |γ e

p (1−μ) β| v ∗ |2 p −1

 ∞ e

R3



    

dt dv ∗

− p (p1−−1μ) α |x+t ( v − v ∗ )|2

 dt dv ∗

0

π ( p − 1) α p (1 − μ) π ( p − 1) α p (1 − μ)



| v − v ∗ |γ − 1 e −

p (1−μ) β| v ∗ |2 p −1

dv ∗

R3



| v − v ∗ |γ −1 dv ∗ +

| v ∗ |1



π ( p − 1) 2π + α p (1 − μ) γ + 2



 e

− p (p1−−1μ) β| v ∗ |2

 dv ∗

| v ∗ |>1



p−1 β p (1 − μ)

3 

≡ C N2 A . Note that we performed integration in time first before the velocity integration. We plug the above estimate of N2 A into (4.6) to obtain

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S.-B. Yun / J. Differential Equations 251 (2011) 45–57

p −1 p

N2  (C N2 A )

 ∞       p μ  | v − v ∗ |γ f  x − t v − v  , v  , t   0 R3



× f







x−t v −

v ∗



, v ∗ , t

 p μ

 1p    dv ∗ dt  

. L p (dx,dv )

Applying a series of changes of variables: x + t v → x, ( v  , v ∗ ) → ( v , v ∗ ) and x → x + t v gives

N2  (C N2 A )

p −1 p

 ∞

  pμ | v − v ∗ |γ f  (x, v , t )

0 R9



× f





x − t ( v − v ∗ ), v ∗ , t

 p μ

 1p dx dv dv ∗ dt

.

We now introduce the maximal distribution supt f  (x, v ) as follows p N2

pμ  a M (C N2 A ) p −1

  ∞ × R3

 

R6

 pμ 



sup f  (x, v ) t



2 2 | v − v |γ e − p μ(α |x−t ( v − v ∗ )| +β| v ∗ | ) dt dv ∗

0

 a M (C N2 A ) p −1 pμ



π αμ p

  R6

pμ  a M (C N2 A ) p −1





sup f 

  pμ 

t





π 2π + αμ p γ + 2

dv dx

 2 | v − v |γ −1 e − p μβ| v ∗ | dv ∗ dv dx

R3



1

β pμ

3      μ  p sup f  , p

t

where we used





f  (x, v , t )  sup f  (x, v )

and

t

2 2 f  (x, v ∗ , t )  a M e −α |x−t ( v − v ∗ )| −β| v ∗ | .

Finally we put

μ

C μ, p ≡ a M (C N2 A ) to obtain the desired result.

p −1 p





π 2π + αμ p γ + 2





1

3  1p

β pμ

2

We now go back to the proof of the main theorem of this section. Substituting (4.5) into (4.4) and recalling

   μ     f  =  f  μ , μp p

S.-B. Yun / J. Differential Equations 251 (2011) 45–57

55

we have

   μ    μ   μ     sup f     f 0 μ p + C¯ μ, p sup f   , μp

t

(4.7)

μp

t

where

C¯ μ, p ≡ μa M (C N2B ) μ

p −1 p



a2M

1−μ 

am

As in the previous section, we first fix observe that

P

μ



aM Therefore, we can take

p −1 p

a2M



1

3  1p

β pμ

.

μ p = P for a given Lebesgue exponent 0 < P < ∞. We then



1 C¯ μ, p  μO (1) 1+

where we used the fact that (C N2 A )





π 2π + αμ p γ + 2



3  μP 1

,

P

is uniformly bounded for p  1, 0 < μ < 1, and

 1 −μ

am

= aM

aM

 1 −μ <

am

a2M am

.

μ sufficiently small (with P fixed) such that C¯ μ, p < 1, which gives from (4.7)    μ    μ  f (t )   sup f    P

t

P

1 1 − C¯ μ, p

  μ  f  . 0

P

This implies the desired result. 5. On the stability of f − f¯ Let f , ¯f be two mild solutions of (1.1) which satisfy the upper bound estimate (but not necessarily lower bound estimate) of the main assumption (A2):

(A2) 0  f  (x, v , t ),

¯f  (x, v , t )  a M Mα ,β (x, v ),

a.e. (x, v ),

for some strictly positive constant a M . Since the difference f − ¯f does not satisfy the lower bound estimate of (A2) in general, the arguments given in Sections 3 and 4 are not directly applicable to the difference of two distribution functions. One way to circumvent this problem is to consider 1   g  (x, v , t ) ≡ M− α ,β f (x, v , t ) instead of f (x, v , t ). Substituting this into (1.1), we obtain

∂ g = ∂t















A α ,β ( v − v ∗ , ω) g   g ∗ − g  g ∗ dω dv ∗ ,

(5.1)

R3 ×S2+

∂ g¯  = ∂t



R3 ×S2+

A α ,β ( v − v ∗ , ω) g¯   g¯ ∗ − g¯  g¯ ∗ dω dv ∗ ,

(5.2)

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S.-B. Yun / J. Differential Equations 251 (2011) 45–57

where A α ,β denotes the regularized collision kernel as before: 2 2 A α ,β ( v − v ∗ , ω) = | v − v ∗ |γ e −α |x−( v − v ∗ )t | −β| v ∗ | .

We subtract (5.2) from (5.1) and multiply both sides by sgn( f  − ¯f  ) to see

∂ G  ∂t











A α ,β ( v − v ∗ , ω) G   D ∗ + D   G ∗ + G  D ∗ + D  G ∗ dω dv ∗ , 

R3 ×S2+

where G = | g − g¯ | and D = | g + g¯ |. Then the exactly same arguments as in the previous sections yield

G  p  C p G 0  p (−3 < γ  1),

(5.3)

where θ = 1 for sufficiently small a M . We now introduce the following notation for simplicity:

   f (t )

 p

LM

≡



1 f  (x, v , t )M− α ,β

p

 1p dx dv

,

R6

then (5.3) leads to the following theorem. Theorem 5.1. Suppose that main assumption (A1) holds with −3 < γ  1. Let f and ¯f be mild solutions satisfying (A2) corresponding to initial data f 0 , ¯f 0 respectively. Then we have

   f (t ) − ¯f (t ) p  G p  f 0 − ¯f 0  p , L L M

M

t  0.

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