Global weak solutions and uniform Lp -stability of the Boltzmann–Enskog equation

J. Differential Equations 251 (2011) 1–25

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Global weak solutions and uniform L p -stability of the Boltzmann–Enskog equation Seung-Yeal Ha a,∗,1 , Se Eun Noh b a b

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan, ROC

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 11 September 2009 Revised 22 March 2011

We present the regularity theory of renormalized solutions and uniform L p -stability estimates of the Boltzmann–Enskog model. We use a multi-dimensional Bony-type functional to control the time–phase space integral of a truncated collision operator. We employ Cercignani’s arguments and a Bony-type functional to show that the renormalized solutions to the Boltzmann–Enskog model are weak solutions in usual distributional sense. For the uniform L p -stability estimate, we use Lu’s trick together with the Hölder inequality. © 2011 Elsevier Inc. All rights reserved.

Keywords: Enskog–Boltzmann equation Lyapunov functional L p -stability

1. Introduction The transport phenomena of moderately dense gases with hard sphere molecules can be modeled by the Boltzmann–Enskog equation, which is an evolution equation for the one-particle distribution function f = f (x, ξ, t ) with some geometric effects due to the overall dimensions of hard sphere molecules. In the absence of external forces, the distribution function f satisfies an integro-differential equation [17]:



∂t f + ξ · ∇x f = Q E ( f , f ), f (x, ξ, 0) = f 0 (x, ξ ).

(x, ξ ) ∈ R6 , t ∈ R+ ,

*

(1)

Corresponding author. E-mail addresses: [email protected] (S.-Y. Ha), [email protected] (S.E. Noh). URL: http://www.math.snu.ac.kr/~syha (S.-Y. Ha). 1 The work of S.-Y. Ha is partially supported by Korea Research Foundation grant (2008-C00060) and a research grant SNUCNS. 0022-0396/$ – see front matter doi:10.1016/j.jde.2011.03.021

© 2011

Elsevier Inc. All rights reserved.

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S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

Here Q E ( f , f ) denotes a collision operator measuring binary collisions between particles, and its explicit form will be addressed later. Following Kaniel–Shinbrot’s notations in [25], we introduce auxiliary functions as the functions f and Q E ( f , f ) evaluated along the particle trajectories:

f  (x, ξ, t ) ≡ f (x + t ξ, ξ, t )



Q E ( f , f )(x, ξ, t ) ≡ Q E ( f , f )(x + t ξ, ξ, t ).

and

We integrate Eq. (1) along the particle path (x + sξ, ξ, s) to find a mild form:

t f  (x, ξ, t ) = f 0 (x, ξ ) +



Q E ( f , f )(x, ξ, s) ds,

(x, ξ ) ∈ R6 , t  0.

(2)

0

We first recall several concepts of definitions for solutions to (1) as follows. Definition 1.1. Let T be a constant in (0, ∞]. Then several concepts of solutions are as follows. 1. A nonnegative function f ∈ C (R6 × [0, T )) is a classical solution of (1) with a nonnegative initial datum f 0 if and only if f is continuously differentiable with respect to (x, t ) and f satisfies Eq. (1) pointwise. 2. A nonnegative function f ∈ C ([0, T ); L 1 (R6 )) is a mild solution of (1) with a nonnegative initial datum f 0 if and only if for all t ∈ [0, T ) and a.e. (x, ξ ) ∈ R6 , f satisfies the integral equation (2). 3. A nonnegative function f ∈ L 1loc (R6 × (0, ∞)) is a renormalized solution of (1) if and only if f satisfies the following two conditions:

(i)

Q E( f , f ) 1+δf

  ∈ L 1loc R6 × (0, ∞) ,

  Q E( f , f ) , (ii) (∂t + ξ · ∇x ) δ −1 log(1 + δ f ) = (1 + δ f )

δ > 0,

in the sense of distribution and f is independent of δ . We next review the existence theory of solutions to (1). Local existence theory to the Enskog equation was first studied in [26], while global existence theory of classical and mild solutions when initial data are small perturbations of a vacuum was established in [13,32]. In contrast, global large mild solutions were obtained in [1,14,18,28–30]. [18,28,30] obtained the renormalized solutions in L 1 using the compactness method originally applied to the Boltzmann equation by Diperna and Lions [16]. On the other hand, asymptotic equivalence between the Enskog equation and the Boltzmann equation was investigated in [3–5]. For further references, we refer to [6]. In this paper, we address two separate issues (regularity of renormalized solution and L p -stability estimates) on Eq. (1): First, we discuss the existence of weak solutions to (1) using the Bony-type nonlinear functional. This nonlinear functional approach in kinetic context was first initiated by Bony [9] for one-dimensional discrete velocity Boltzmann models, and was further generalized to other one-dimensional continuous kinetic models in [10–12,15]. We use a multi-D Bony-type functional in [22] to control the phase space–time integral of the collision operator, which is crucial to prove that renormalized solutions are weak solutions in distribution sense. Because the Bony-type functional is a priori bounded by mass and energy, we can apply this approach to classical solutions with finite initial mass and energy and hence we do not need the explicit decay of solutions in the phase space to get the global weak solution. Secondly, we investigate L p -stability theory of continuous mild solutions. This issue was first addressed in [2] for the spatially homogeneous Boltzmann equation. Recently in [23] the uniform L p -stability estimate was studied for the spatially inhomogeneous Boltzmann equation with external forces adopting Lu’s trick in [27]. We apply Lu’s approach to the Boltzmann– Enskog equation to get the L p -stability of continuous mild solutions.

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

3

The rest of the paper after this introduction is organized as follows. In Section 2, we present a functional setting for L p -stability, and discuss the main results. We also briefly review the basic properties and a priori estimates for the Boltzmann–Enskog equation. In Section 3, we introduce a multi-dimensional Bony-type functional and study its time-decay estimate to get a priori estimates leading to the regularity of renormalized solutions. Finally, Section 4 is devoted to the weighted L p stability estimates of mild solution. Notations. Throughout the paper, we use simplified notations for local and global norms: For any measurable function h = h(x, ξ, t ) and g = g (x, ξ, t ) defined on R6 × (0, ∞),

  h(t )

Lp

  := h(·, ·, t ) L p (R6 ) , x,ξ   h(t ) ∞ Lx

  h(x, t )

  := h(x, ·, t ) L p (R3 ) , ξ   p := sup h (x, ·, t ) p . (L ) L (R3 ) ξ

p

Lξ

1  p  ∞,

ξ

x∈R3

2. Preliminaries In this section, we review the Enskog equation, and present basic a priori estimates and summary of main results of this paper. In [17], Enskog heuristically introduced a new Boltzmann type kinetic equation based on the following two principles:

• Pairs of particles with their centers at a distance a will collide elastically so that the momentum and energy are conserved.

• The collision frequency is increased by a factor Y depending only on the local mass density at the contact point of two colliding hard sphere molecules. Under the above two principles, pairs of pre-collisional and post-collisional velocities (ξ, ξ∗ ) and (ξ  , ξ∗ ) satisfy a collision transformation:

    ξ  = ξ − (ξ − ξ∗ ) · ω ω and ξ∗ = ξ∗ + (ξ − ξ∗ ) · ω ω,

ω ∈ S2+ ,

(3)

where v · w is the standard inner product between v and w in R3 and S2+ = {ω ∈ S2 : (ξ − ξ∗ ) · ω  0} and the Enskog equation takes the form

∂t f + ξ · ∇x f = Q E ( f , f ),

(4)

with the collision operator:

 Q E( f , f ) ≡ a

    (ξ − ξ∗ ) · ω Y − f  f ∗− − Y + f f ∗+ dω dξ∗ ,

2

(5)

R3 ×S2

+

where

ρ=



R3 f dξ is a local mass density and we have used abbreviated notations:

Y

±





 =Y ρ x± ω , 2



f  = f x, ξ  , t , f = f (x, ξ, t )

a





 f ∗− = f ∗ x − aω, ξ∗ , t ,

and

f ∗+ = f (x + aω, ξ∗ , t ).

Although the Enskog equation (4)–(5) gives a consistent calculation for the transport coefficients such as viscosity, heat conductivity for real gases, it is well known [24,33] that the Enskog equation is

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S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

not consistent with irreversible thermodynamics for mixtures of gases. Hence several revised Enskog equations were introduced in [28,30,31,33] for the consistency with irreversible thermodynamics. In the revised Enskog equations the equilibrium pair correlation functions Y ± are modified to be consistent with irreversible thermodynamics, for example, in [30] Y ± was modified to depend on the local density at the centers of the colliding particles, i.e.,

Y+ = Y+





ρ (x, t ), ρ (x + aω, t )

and

Y− = Y−





ρ (x, t ), ρ (x − aω, t ) .

The Boltzmann–Enskog equation corresponds to the ideal case:

Y + = Y − = Y ∞,

 Q E ( f , f ) := a2 Y ∞

where Y ∞ is a constant,      (ξ − ξ∗ ) · ω f f ∗− − f f ∗+ dω dξ∗ = Q E+ ( f , f ) − Q E− ( f , f ).

R3 ×S2+

2.1. Basic a priori estimates In this part, we present some basic a priori estimates on the Boltzmann–Enskog equation. Lemma 2.1. (See [22].) Let f ∈ L 1 (R6 ) be rapidly decaying at infinity in the phase space. Then for fixed (x, t ) ∈ R3 × R+ , we have

 Q E ( f , f )(ξ ) dξ = 0,

(i) R3



 ξ Q E ( f , f )(ξ ) dξ = −a2 Y ∞

(ii) R3



2 (ξ − ξ∗ ) · ω ω f f ∗+ dω dξ∗ dξ.

R6 ×S2+

Proof. We first claim: For ψ ∈ L ∞ (R3ξ ),







Q E ( f , f )(ξ )ψ(ξ ) dξ = a2 Y ∞ R3

  (ξ − ξ∗ ) · ω ψ  − ψ f f ∗+ dω dξ∗ dξ,

R6 ×S+ 2

where ψ = ψ(ξ ), ψ  = ψ(ξ  ). Proof of claim. Note that

 Q E ( f , f )(ξ )ψ(ξ ) dξ R3

 = a2 Y ∞



  (ξ − ξ∗ ) · ω ψ f  f ∗− d ω d ξ∗ d ξ − a 2 Y ∞

R6 ×S2+



  (ξ − ξ∗ ) · ω ψ f f ∗+ dω dξ∗ dξ

R6 ×S2+

:= I1 + I2 . We now consider I1 . We use a transformation (ξ, ξ∗ ) ↔ (ξ  , ξ∗ ) and note that

  (ξ − ξ∗ ) · ω = − ξ  − ξ∗ · ω,

  ξ = ξ and dξ∗ dξ = dξ  dξ∗ ,

(6)

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

5

to estimate





I1 = −a2 Y ∞

   ξ  − ξ∗ · ω ψ f  f ∗− dω dξ∗ dξ 

R6 ×S2−



  (ξ − ξ∗ ) · ω ψ  f f ∗− dω dξ∗ dξ

2

= −a Y ∞ R6 ×S2−



  (ξ − ξ∗ ) · ω ψ  f f ∗+ dω dξ∗ dξ.

= a2 Y ∞ R6 ×S2+

Finally we combine I1 and I2 to get the claim. We set ψ = 1 in (6) to find

 Q E ( f , f ) d ξ = 0. R3

On the other hand, we set ψ = ξ and use the relation in (3) ξ  − ξ = −[(ξ − ξ∗ ) · ω]ω to see





 2 (ξ − ξ∗ ) · ω ω f f ∗+ dω dξ∗ dξ.

2

ξ Q E ( f , f ) dξ = −a Y ∞ R3

2

R6 ×S2

+

We next study the time decay of gain collision operator Q E+ ( f , f )(x, ξ, t ) when f is a continuous function decaying exponentially, for that we need the following simple lemma. Lemma 2.2. Let (ξ, ξ∗ ) and (ξ  , ξ∗ ) be the pre- and post-collisional velocities given by the relation (3) satisfying (ξ − ξ∗ ) · ω  0. Then we have

    x + t ξ − ξ  2 + x − aω + t ξ − ξ  2  |x|2 + x − aω + t (ξ − ξ∗ ) 2 . ∗ Proof. Note that momentum and energy conservation yield

    x + t ξ − ξ  2 + x − aω + t ξ − ξ  2 ∗   2 2   = 2|x + t ξ |2 − 2t (x + t ξ ) · ξ  − aω + ξ∗ + t 2 ξ  + ξ∗ + 2at ω · ξ∗ + a2 |ω|2   = 2|x + t ξ |2 − 2t (x + t ξ ) · (ξ − aω + ξ∗ ) + t 2 |ξ |2 + |ξ∗ |2   + 2at ω · ξ∗ + 2at (ξ − ξ∗ ) · ω + a2 |ω|2 2 2   = (x + t ξ ) − t ξ + (x + t ξ ) − aω − t ξ∗ + 2at (ξ − ξ∗ ) · ω 2   = |x|2 + x − aω + t (ξ − ξ∗ ) + 2at (ξ − ξ∗ ) · ω 2  |x|2 + x − aω + t (ξ − ξ∗ ) , where we used the fact that [(ξ − ξ∗ ) · ω]  0 to get the desired inequality.

2

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S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

Proposition 2.1. Let f be a continuous exponentially decaying function with a pointwise ansatz: 2 2 f  (x, ξ, t ) < δ e −α |x| −β|ξ | ,

δ > 0.

(7)

Then we have the following estimate: +

2

Q E ( f , f )(x, ξ, t )  O (1)a Y ∞ δ

2e

− α2 |x|2 − β2 |ξ |2

.

(1 + t )4 +

Proof. We plug the pointwise ansatz (7) in the definition of Q E

to obtain

+

Q E ( f , f )(x, ξ, t )





= a2 Y ∞

     (ξ − ξ∗ ) · ω f x, ξ  , t f x − aω, ξ∗ , t dω dξ

R3 ×S2+



   2  2  2  2 (ξ − ξ∗ ) · ω δ 2 e −α |x+t (ξ −ξ )| e −β|ξ | e −α |x−aω+t (ξ −ξ∗ )| e −β|ξ∗ | dω dξ

2

 a Y∞ R3 ×S2+

 a2 Y ∞ δ 2 e −α |x|

2



−β|ξ |2

  2 2 (ξ − ξ∗ ) · ω e −α |x−aω+t (ξ −ξ∗ )| e −β|ξ∗ | dω dξ ,

R3 ×S2+





:= A

where we used Lemma 2.2. We claim:

A  O (1)

(1 + |x| + |ξ |) . (1 + t )4

To derive the time-decay rate of A, we divide its estimate “small” time and “large” time, i.e.: 2 Case 1 (0  t  1). In this case, we use e −α |x−aω+t (ξ −ξ∗ )|  1 to find

 A  2π

|ξ − ξ∗ |e −β|ξ∗ | dξ 2

R3





 2π |ξ |

e

−β|ξ∗ |2

 dξ +

R3

|ξ∗ |e

−β|ξ∗ |2

 dξ

R3

   O(1) 1 + |ξ | .

Case 2 (t  1). We use the change of variable x − aω + t (ξ − ξ∗ ) =: ξ¯ to obtain

A

1



t4 R3 ×S2+

¯

|ξ¯ − x + aω|e −α |ξ | dω dξ¯ 2

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25





1 t4

 2π

¯

|ξ¯ |e −α |ξ | dξ¯ + 2π 2

R3

 O(1)





7

 ¯2 |x| + a e −α |ξ | dξ¯



R3

(1 + |x|) t4

.

We combine Cases 1 and 2 to obtain the desired result.

2

2.2. Summary of the main results In this part, we briefly summarize the main results in this paper. The first theorem of this paper is the existence of weak solutions to (1) with the modified kernel (8). For this, we employ the nonlinear functional approach to get the bound of time–phase space integral of collision operator together with Cercignani’s arguments in [11,12]. For the global existence of weak solution, we consider the truncated collision kernel for the Boltzmann–Enskog collision operator Q E ( f , f ), i.e.,







 qε (ξ − ξ∗ , ω) f  f ∗− − f f ∗+ dω dξ∗ ,

Q Eε ( f , f )(x, ξ, t ) := a2 Y ∞

(8)

R3 ×S2+

where the truncated collision kernel qε is defined as follows: For 0 < ε 1,

 qε (ξ − ξ∗ , ω) =

(ξ − ξ∗ ) · ω, if |(ξ − ξ∗ ) · ω| > ε ; if |(ξ − ξ∗ ) · ω|  ε .

0,

Theorem 2.1. Let f 0 ∈ L 1 (R6 ) be such that





√



 |x|2 + |ξ |2 f 0 dξ dx < ∞,

f 0 dξ dx <

R6

R6

2

4Y ∞

 f 0 |ln f 0 | dξ dx < ∞.

and R6

Then there is a weak solution to (1) with the modified collision operator (8). Remark 2.1. The truncation assumptions have been used in previous literature: For some constant ε > 0, q(ξ − ξ∗ , ω) = 0, |ξ − ξ∗ |  ε in [8,11,12,15], q(ξ − ξ∗ , ω) = 0, |(ξ − ξ∗ ) · ω|  ε in [10,18]. We now discuss a framework for the weighted L p -stability of mild solutions in Section 4. We 2 2 choose an exponentially decaying function as standard bounding functions e −α |x| and e −β|ξ | and define a set of functions S (α , β, δ):

    2 2 S (α , β, δ) := f ∈ C R6 × R+ : sup e α |x| e β|ξ | f  (x, ξ, t ) < δ , x,ξ,t





r 2

ϕr (ξ ) := 1 + |ξ |2 ,

   f (t )

p

Lr

  :=  f (·, ·, t ) L p (ϕ

r

dx dξ )

.

The second main result is on the following weighted L p -stability estimate of continuous mild solution to (1).

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S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

Theorem 2.2. Let f and ¯f be continuous mild solutions to (1) in S (α , β, δ) corresponding the initial data f 0 and ¯f 0 respectively. Then for a pair ( p , r ) satisfying

1 < p < 3,

r>

3( p − 1) p

,

we have





sup  f (t ) − ¯f (t ) L p  C  f 0 − ¯f 0 θ p , Lr

r

0t <∞

for some θ ∈ (0, 1).

(9)

Remark 2.2. 1. The existence of mild solutions in S (α , β, δ) was studied in [32]. 2. The L 1 -stability of the above type (9) was first obtained by Lu for the Boltzmann equation in [27]. 3. Global existence of distributional weak solutions In this section, we basically adopt the Cercignani’s idea [10] to show that DiPerna–Lions’ renormalized solutions to (1) with the truncated kernel are weak solutions in distribution sense. To show the regularity of the renormalized solutions, it is crucial to obtain the uniform bound of time–phase space integral of a collision operator, for which we employ a nonlinear functional approach to get the a priori estimates. First, we obtain the a priori estimate for classical solutions using the multi-D Bony-type functional in [22] and then extend this estimate to the renormalized solution by standard mollification arguments. Finally, we prove the regularity of renormalized solutions in the second part of this section. 3.1. A multi-dimensional Bony-type functional In this part, we review the nonlinear functional which is a multi-dimensional analogue of the Bony functional in [9], and study its time-evolution estimates along the classical solutions of (1). Let f be a classical solution to (1) and (8), and consider hard sphere test particles at B a (x) with a velocity ξ and their neighboring field particles located at y ∈ B a (x)c := R3 − B a (x) with a velocity ξ∗ . In this case, local net interactions between mass and momentum flux due to the field particles are measured by

B(x, ξ, t ) := f (x, ξ )

ω(x − y ) · (ξ∗ − ξ ) f ( y , ξ∗ ) dξ∗ dy , Ba

where



 (x)c ×R3

ω(x, y ) is the unit vector in the direction of x − y, i.e.,

ω(x, y ) =

x− y , |x − y |

y ∈ B a (x)c .

We now define the multi-dimensional Bony-type functional B and its production rates:

  B f (t ) :=







R6

  a2 Λ1 f (t ) := √

2

 R9 ×S2



ω(x, y ) · (ξ∗ − ξ ) f ( y , ξ∗ ) dξ∗ dy dξ dx,

f (x, ξ ) B a (x)c ×R3

 2 (ξ∗ − ξ ) · ω f (x, ξ ) f (x + aω, ξ∗ ) dω dξ∗ dξ dx,

(10)

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

9

Fig. 1. Schematic diagram of collisions.

  Λ2 f (t ) :=



R9 × B a (x)c

|(ξ − ξ∗ ) × (x − y )|2 f (x, ξ ) f ( y , ξ∗ ) dξ∗ dξ dy dx, | y − x|3 

  a2 Y ∞ Λε3 f (t ) := √





qε (ξ − ξ∗ , ω) (ξ∗ − ξ ) · ω f (x, ξ ) f (x + aω, ξ∗ ) dω dξ∗ dξ dx.

2

R9 ×S2+

Remark 3.1. 1. We note that (see Fig. 1)

 B(x, ξ, t )

> 0, two particles are approaching; < 0, otherwise.

2. It is easy to see that

    1  B f (t )  2 f (t )31/2 |ξ |2 f (t ) 21 < ∞, L L

(11)

which is clearly bounded by the total mass and energy of initial datum f 0 . We calculate the time-evolution estimate of this multi-D Bony-type functional following [22] in which the case of the whole collision kernel was showed. For the simplicity of notations, we suppress t-dependence of f and Q ( f , f ), i.e.,

f (x, ξ ) := f (x, ξ, t ),

Q ( f , f )(x, ξ ) := Q ( f , f )(x, ξ, t ).

Lemma 3.1. (See [22].) Let f be a classical solution to (1) and (8) with a finite energy and mass. Then we have

d dt

√         B f (t )  −Λ1 f (t ) − Λ2 f (t ) + 2 2 f 0 Λε3 f (t ) ,

t  0.

Proof. Consider equations for f (x, ξ ) and f ( y , ξ∗ ):

∂t f (x, ξ ) + ξ · ∇x f (x, ξ ) = Q Eε ( f , f )(x, ξ ),

(12)

∂t f ( y , ξ∗ ) + ξ∗ · ∇ y f ( y , ξ∗ ) = Q Eε ( f , f )( y , ξ∗ ).

(13)

Then (ξ∗ − ξ ) · ω(x, y )[(12) · f ( y , ξ∗ ) + (13) · f (x, ξ )] yields

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S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

  ∂t (ξ∗ − ξ ) · ω(x, y ) f (x, ξ ) f ( y , ξ∗ )   = − div(x, y ) (ξ, ξ∗ )(ξ∗ − ξ ) · ω(x, y ) f (x, ξ ) f ( y , ξ∗ )     + ∇(x, y ) (ξ∗ − ξ ) · ω(x, y ) · (ξ, ξ∗ ) f (x, ξ ) f ( y , ξ∗ )   + (ξ∗ − ξ ) · ω(x, y ) Q Eε ( f , f )(x, ξ ) f ( y , ξ∗ ) + Q Eε ( f , f )( y , ξ∗ ) f (x, ξ ) . We now integrate the above equation over R9 × B a (x)c to get

d dt



  B f (t ) = −





div(x, y ) (ξ, ξ∗ )(ξ∗ − ξ ) · ω(x, y ) f (x, ξ ) f ( y , ξ∗ ) dξ∗ dξ dy dx

R9 × B a (x)c





+

   ∇(x, y ) (ξ∗ − ξ ) · ω(x, y ) · (ξ, ξ∗ ) f (x, ξ ) f ( y , ξ∗ ) dξ∗ dξ dy dx

R9 × B a (x)c



+

(ξ∗ − ξ ) · ω(x, y )

R9 × B

a

(x)c

  × Q Eε ( f , f )(x, ξ ) f ( y , ξ∗ ) + Q Eε ( f , f )( y , ξ∗ ) f (x, ξ ) dξ∗ dξ dy dx =: J1 (t ) + J2 (t ) + J3 (t ). Because J1 (t ) and J2 (t ) can be calculated the same as done in [22], we omit the proof,



a2

J1 (t )  − √

2



(ξ∗ − ξ ) · ω

2

f (x, ξ ) f (x + aω, ξ∗ ) dω dξ∗ dξ dx,

R9 ×S2

 J2 (t )  −

R9 × B a (x)c

|(ξ − ξ∗ ) × ( y − x)|2 f (x, ξ ) f ( y , ξ∗ ) dξ∗ dξ dy dx. | y − x|3

We now estimate J3 (t ). We first rewrite

 

 

 J3 (t ) =

ω(x, y ) ·

R3 × B a (x)c

R3



−



R3

 

ξ∗ Q Eε ( f , f )( y , ξ∗ ) dξ∗

+ R3

Q Eε ( f , f )( y , ξ∗ ) dξ∗ R3

We now use Lemma 2.1: For fixed (x, t ) ∈ R3 × R+ ,

 f (x, ξ ) dξ

 

 −



ξ Q Eε ( f , f )(x, ξ ) dξ

f ( y , ξ∗ ) d ξ∗ R3

R3

 

 Q Eε ( f , f )(x, ξ ) dξ

ξ∗ f ( y , ξ∗ ) d ξ∗

R3

ξ f (x, ξ ) dξ R3

 dy dx.

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

11

 Q Eε ( f , f )(x, ξ ) dξ = 0 and







ξ Q Eε ( f , f )(x, ξ ) dξ = −a2 Y ∞

qε (ξ − ξ∗ , ω) (ξ − ξ∗ ) · ω



ω f f ∗+ dω dξ∗ dξ,

R6 ×S2

+

to see

 

 J3 (t ) = R3 × B a (x)c

R3



R3

ω(x, y ) ·

R3 × B

a

 f (x, ξ ) dξ

R3

 



= −2

dy dx

 

ξ Q E ( f , f )(x, ξ ) dξ R3

(x)c



 2a2 Y ∞

f ( y , ξ∗ ) d ξ∗ R3

 

ξ∗ Q Eε ( f , f )( y , ξ∗ ) dξ∗

+



  ξ Q Eε ( f , f )(x, ξ ) dξ

ω(x, y ) · −

 f ( y , ξ∗ ) d ξ∗

dy dx

R3

ω(x, y )

R3 × B a (x)c





·

qε (ξ − ξ∗ , ω) (ξ − ξ∗ ) · ω R6 ×S2+

 ×





ω f (x, ξ ) f (x + aω, ξ∗ ) dω dξ∗ dξ



f ( y , ξ∗ ) dξ∗ dy dx R3





 2a2 Y ∞

f ( y , ξ∗ ) dξ∗ dy



R6

×

   qε (ξ − ξ∗ , ω) (ξ − ξ∗ ) · ω f (x, ξ ) f (x + aω, ξ∗ ) dω dξ∗ dξ dx

R9 ×S2+

√   = 2 f 0  2Λε3 f (t ) .

2

Remark 3.2. The existence of classical solutions was obtained in [32] in the class of decaying solution in the phase space, i.e.,

 Mβ =





f ∈ C [0, T ]; C b R6

    2 f (x, ξ, t )  h |x| e −β|ξ | , h ∈ C b (R+ ) ∩ L 1 (R+ ),

√

sup x0

h(x/ 2) h(x)

< ∞ and hσ = sup

h(|x − 2σ |)

x0

h(x)

 <∞ .

For example, we can choose h(x) = (1+σ 21x2 ) p/2 , p > 1. Equipped with Lemma 3.1, we can have the following a priori estimate.

12

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

Proposition 3.1. Let√f be a classical solution to (1) and (8) corresponding to initial data f 0 with finite energy

and mass  f 0  L 1 <

2 . 4

Then we have the uniform time–phase space estimate:

∞  2

qε (ξ − ξ∗ , ω) f (x, ξ ) f (x + aω, ξ∗ ) dω dξ∗ dξ dx dt

a Y∞ 0 R9 ×S2

∞



+ 0 R9 × B a (x)c

|(ξ − ξ∗ ) × ( y − x)|2 f (x, ξ ) f ( y , ξ∗ ) dξ∗ dξ dy dx dt  C ( f 0 , ε ). | y − x|3

Here C ( f 0 ) is a positive constant only depending on initial datum. Proof. Noting that Λ3 ( f (s))  Y ∞ Λ1 ( f (s)), it follows from Lemma 3.1 that

d ds

      B f ( s )  − C 2 Λ3 f ( s ) − Λ2 f ( s ) ,

(14)

√

where C 2 = Y1 − 2 2 f 0  L 1 is a positive constant independent of time t. We now integrate (14) ∞ from s = 0 to s = t to get

t B(t ) + C 2

  Λ3 f (s) ds +

0

t

  Λ2 f (s) ds  B(0).

0

This yields

a2 Y ∞ C 2

∞  qε (ξ∗ − ξ, ω) f (x, ξ ) f (x + aω, ξ∗ ) dω dξ∗ dξ dx dt

√

ε 2

0 R9 ×S2

∞



+ 0 R9 × B a (x)c

|(ξ − ξ∗ ) × ( y − x)|2 f (x, ξ ) f ( y , ξ∗ ) dξ∗ dξ dy dx dt | y − x|3

   B(0) + B f (t ) 1 3/2   4 f 0 L 1 |ξ |2 f 0  L21 < ∞, This completes the proof.

by (11).

2

3.2. Existence of global weak solutions In this part, we show the global existence of weak solutions to (1) and (8) in distribution sense. To obtain crucial a priori estimates for renormalized solutions, we need to construct the sequence of approximate classical solutions (see [16,30]). Before we prove Theorem 2.1, we present two preparing lemmas.

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

13

Lemma 3.2. Let f ∈ L 1loc (R6 × (0, ∞)) be a renormalized solution to (1) and (8) corresponding to initial data f 0 such that









1 + |x|2 + |ξ |2 f 0 dξ dx < ∞,

R6

f 0 |log f 0 | dξ dx < ∞. R6

Then there exists a subsequence f n satisfying

Q Eε± ( f n , f n )



1 + δ R3 f n d ξ



Q Eε± ( f , f )



1 + δ R3 f d ξ





weakly in L 1 (0, T ); R6 .

Proof. This proof follows from the arguments in the proof to DiPerna–Lions’ classical paper [16, pp. 341–342]. 2 We now prove that the time–phase space integral of a truncated collision operator for the renormalized solution is uniformly bounded. For the simplicity of presentation we introduce notation for the inner product on the phase space:



f , g (t ) :=

f (x, ξ, t ) g (x, ξ, t ) dξ dx. R6

Lemma 3.3. Let f ∈ L 1loc (R6 × (0, ∞)) be a renormalized solution to (1) and (8) corresponding to initial data f 0 with finite energy and mass  f 0  L 1 <

√

2 . 4Y ∞

Then we have the following estimate

∞  ε±  Q E ( f , f ), ϕ (t ) dt  C ,

  ∀ϕ ∈ C c∞ R6 × R+ ,

0

where C is a positive constant independent of δ . Proof. Note that for the renormalized solution f , there exists a sequence of approximate solutions f n satisfying Lemma 3.2 and recall that a priori estimate of Proposition 3.1 holds for the approximate solution f n :

∞  2

qε (ξ − ξ∗ , ω) f n (x, ξ ) f n (x + aω, ξ∗ ) dω dξ∗ dξ dx dt  C ,

a Y∞ 0

R9 ×S2

and this estimate gives a uniform bound of approximate collision operators:

∞ 0

Q Eε± ( f n , f n )



1 + δ R3 f n d ξ

 , ϕ (t ) dt

∞  

qε (ξ − ξ∗ , ω) f n (x, ξ ) f n (x + aω, ξ∗ )ϕ (x, ξ ) dω dξ∗ dξ dx dt 0

R9 ×S2

(15)

14

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

∞ 



 

qε (ξ − ξ∗ , ω) f n x, ξ  f n x − aω, ξ∗

+



ϕ (x, ξ ) dω dξ∗ dξ dx dt

0 R9 ×S2

 C ( f 0 , ε)

ϕ L ∞ a2 Y ∞

.

On the other hand, by monotone convergence theorem, we can obtain

t 

Q Eε± ( f , f )



1 + δ R3 f d ξ

0

 t  ε±  , ϕ (s) ds → Q E ( f , f ), ϕ (s) ds. 0

Therefore we can have

∞ 0



Q Eε± ( f n , f n )



1 + δ R3 f n d ξ

∞  ε±  Q E ( f , f ), ϕ (t ) dt as δ → 0 and n → ∞, , ϕ (t ) dt → 0

and it is easy to see that their limit is also uniformly bounded, i.e.,

∞  ε±  Q E ( f , f ), ϕ (t ) dt  C , 0

which completes the proof.

2

We now return to the proof of Theorem 2.1. Proof of Theorem 2.1. Note that the renormalized solution is a distributional weak solution in L 1loc (R6 × (0, ∞)) of the renormalized form

  Q ε( f , f ) (∂t + ξ · ∇x ) δ −1 log(1 + δ f ) = E , 1+δf We now take a test function



1

δ



T 

1

δ



1

δ

T  0

 log(1 + δ f 0 ), ϕ



T 

log(1 + δ f ), ϕ (t ) dt +

0

=

(16)

ϕ ∈ C c∞ (R6 × R+ ), and rewrite (16) as an integral form

log(1 + δ f ), ϕ ( T ) −

+

δ > 0.

1

δ

 log(1 + δ f ), ξ · ∇x ϕ (t ) dt

0

Q Eε ( f , f ) 1+δf

 , ϕ (t ) dt .

(17)

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

15

log(1+δ f )

We can easily see that is bounded by f ∈ L 1loc (R6 × (0, ∞)), which is independent of δ , δ therefore we use the dominated convergence theorem and let δ → 0 of the left-hand side of (17):

T L.H.S. of (17) → f , ϕ ( T ) − f 0 , ϕ  +

T

f , ϕ (t ) dt +

0

f , ξ · ∇x ϕ (t ) dt ,

(18)

0

while the right-hand side is bounded by Lemma 3.3

T  0

Q Eε ( f , f ) 1+δf



T

, ϕ (t ) dt 

 ε+  Q E ( f , f ) + Q Eε− ( f , f ), ϕ (t ) dt < C .

(19)

0

We now combine (18) and (19) and let δ → 0 to get the distributional form of the Boltzmann–Enskog equation:

T

f , ϕ ( T ) − f 0 , ϕ  +

T

f , ϕ (t ) dt +

0

T

f , ξ · ∇x ϕ (t ) dt =

0

 ε  Q E ( f , f ), ϕ (t ) dt .

0

In this way, we get the global existence of distributional weak solution with initial data whose mass and momentum are finite. 2 Remark 3.3. 1. As mentioned in Section 1, there are two results [1,14] for large-data existence theory, however both results require the higher velocity moment of initial data. For example, in [1] we need ( 1 + |ξ |)n f 0 dξ dx, for all n, whereas R6 (1 + |ξ |)n f 0 dξ dx, for some n  4 in [14]. 6 R 2. Our methodology employed in Theorem 2.1 can be applied to the revised Enskog equation with a bounded equilibrium pair correlation functions Y ± < ∞. The Boltzmann–Enskog equation is a special example satisfying a boundedness assumption. 3. For the modified Enskog equation with Y ± :

sup

τ ,σ 0

τ Y ± (τ , σ ) < ∞,

the existence of renormalized solutions has been studied using the arguments of the Lyapunov functional approach and DiPerna–Lions method for the Boltzmann equation in [30]. Note that the constant Y ± = 1 does not satisfy the above condition, hence Polewczak’s result does not seem to be applicable for our framework in the present form. 4. In [10–12], Cercignani studied the regularity of DiPerna–Lions’ renormalized solution to the one-dimensional Botlzmann equation with a suitably cut-off collision kernel:

∂t f + ξ1 ∂x1 f = Q ( f , f ), by establishing the uniform time–phase space integral of Q ( f , f ) using the one-dimensional continuous Bony functional. Hence our result is a multi-dimensional analogue for the Boltzmann–Enskog equation. Of course, it will be very interesting if the similar approach works for the Boltzmann equation. At this moment, our approach does not seem to be applicable for the Boltzmann equation as it is.

16

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

4. A uniform L p -stability estimate In this section, we study the weighted L p -stability of mild solutions to the Boltzmann–Enskog equation. Recently, the first author has developed a nonlinear functional approach in the study of large-time behavior and the L 1 -stability of (1) in [19] (see corresponding results [20,21] for the Boltzmann equation). For L p -stability estimate of continuous mild solutions we employ the approach given in [23], which basically uses the Hölder inequality successively and Lu’s trick in [27]. Lemma 4.1. (See [27].) Let ρ be a nonnegative function in L 1 ([0, T )), T ∈ (0, ∞]. Suppose that a nonnegative function u ∈ L 1+ ([0, T )) satisfies

u (t )  u 0 + κ e

−R

t +R

ρ (s)u (s) ds, for every R > 0 and t ∈ [0, T ), 0

where u 0 and κ are positive constants. Then we have Θ( T ) 



u (t )  u 0 + Ξ ( T )u 0

,

t ∈ [0, T ).

Here we set

1

Θ( T ) :=

2

 T  exp − ρ (s) ds ,



2

Ξ (t ) := e κ

1−Θ(T )



T

1+

0



ρ (s) ds . 0

2

Proof. For the proof, we refer to [27].

In order to study a weighted L p -stability, we set

g (x, ξ, t ) := ϕr (ξ ) f (x, ξ, t ).

(20)

p Then L r -distance between f and ¯f corresponds to the L p -distance between g and g¯ , and we see that g satisfies

∂t g + ξ · ∇x g = Q E ( g , g ),

(21)

where Q E ( g , g ) is given by the following formula







 





q1 g x, ξ  , t g x − aω, ξ∗ , t − q2 g (x, ξ, t ) g (x + aω, ξ∗ , t ) dω dξ∗ ,

Q E ( g , g ) = a2 Y ∞ R3 ×S2+

and q1 and q2 have the forms: r

q1 (ξ, ξ∗ , ω, r ) :=

[(ξ − ξ∗ ) · ω](1 + |ξ |2 ) 2 r

r

(1 + |ξ  |2 ) 2 (1 + |ξ∗ |2 ) 2

,

q2 (ξ, ξ∗ , ω, r ) :=

[(ξ − ξ∗ ) · ω] r

(1 + |ξ |2 ) 2

.

Remark 4.1. For the stability analysis, we use the original collision kernel [(ξ − ξ∗ ) · ω], which does not need to be truncated. We now study the time-decay estimate of g as follows.

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

17

Lemma 4.2. Let f be in S (α , β, δ). Then g defined by (20) satisfies

   g (t )

3

L∞ x (Lξ ) p

 O(1)δ p (1 + t )− p ,

t  0.

r

Proof. Note that g (x, ξ, t ) = (1 + |ξ |2 ) 2 f (x, ξ, t ) and we calculate

   g (x, t ) pp  L





ξ

1 + |ξ |2

R3



 O(1)δ p



 rp2

f p (x, ξ, t ) dξ  δ p



1 + |ξ |2

 rp2

2 2 e −α p |x−t ξ | e −β p |ξ | dξ

R3 βp 2 2 e −α p |x−t ξ | e − 2 |ξ | dξ =: B .

R3

We now separate this estimate into two parts.



Case 1: 0 < t < 1. B  O (1)δ p R3 e −

βp 2

|ξ |2

p

dξ  O (1) (β pδ )3/2 .

βp 2 Case 2: t > 1. We use the fact e − 2 |ξ |  1 and change the variables x − t ξ → ξ¯ to see

 B  O (1)δ

p

e

−α p |x−t ξ |2

dξ  O (1)



δp t3

R3

¯

e −α p |ξ | dξ¯  O (1) 2

R3

δp 1 , (α p )3/2 t 3

2

and we combine two cases to complete the proof.

Now we return to the stability estimates. Let g and g¯ be mild solutions to (21) with the initial data g 0 and g¯ 0 respectively. Then we have

t g  (x, ξ, t ) = g 0 (x, ξ ) +

Q E ( g , g ) (x, ξ, s) ds, 0

t ¯

g (x, ξ, t ) = g¯ 0 (x, ξ ) +

Q E ( g¯ , g¯ ) (x, ξ, s) ds. 0

We subtract the equations and multiply the resulting equation by sgn( g (t ) − g¯ (t ))| g (t ) − g¯ (t )| p −1 , integrate the resulting relation over R6x,ξ and use the Fubini theorem to find

   g (t ) − g¯ (t ) pp  L



p −1 | g 0 − g¯ 0 | g (t ) − g¯ (t ) dξ dx

R6

+

a2

t 

2 0 R9 ×S2 +

  p −1   q1 g (s) − g¯ (s) g (s) + g¯ (s) ∗− g (t ) − g¯ (t )

  p −1   + g (s) − g¯ (s) ∗− g (s) + g¯ (s) g (t ) − g¯ (t ) dω dξ∗ dξ dx ds

18

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

+

a2

t 

2











p −1 q2 g (s) − g¯ (s) g (s) + g¯ (s) ∗+ g (t ) − g¯ (t )

0 R9 ×S2 +

 p −1   + g (s) − g¯ (s) ∗+ g (s) + g¯ (s) g (t ) − g¯ (t ) dω dξ∗ dξ dx ds,

(22)

where we suppress the space and velocity coordinates for simplicity. To take care of the hard sphere collision kernel, we now adopt Lu’s trick in [27] as follows,

|ξ − ξ∗ |  R + e |ξ −ξ∗ |− R ,

for all R  0,

and plug this into (22) to obtain

   g (t ) − g¯ (t ) pp  L



p −1 | g 0 − g¯ 0 | g (t ) − g¯ (t ) dξ dx

R6

+

a2 R

t 

  p −1   q˜ 1 g (s) − g¯ (s) g (s) + g¯ (s) ∗− g (t ) − g¯ (t )

2 0 R9 ×S2 +

  p −1   + g (s) − g¯ (s) ∗− g (s) + g¯ (s) g (t ) − g¯ (t ) dω dξ∗ dξ dx ds +

a2 R

t 











p −1 q˜ 2 g (s) − g¯ (s) g (s) + g¯ (s) ∗+ g (t ) − g¯ (t )

2 0 R9 ×S2 +

 p −1   + g (s) − g¯ (s) ∗+ g (s) + g¯ (s) g (t ) − g¯ (t ) dω dξ∗ dξ dx ds +

a2 e − R

t 

2

  p −1   q˜ 1 e |ξ −ξ∗ | g (s) − g¯ (s) g (s) + g¯ (s) ∗− g (t ) − g¯ (t )

0 R9 ×S2 +

  p −1   + g (s) − g¯ (s) ∗− g (s) + g¯ (s) g (t ) − g¯ (t ) dω dξ∗ dξ dx ds +

a2 e − R

t 

2











p −1 q˜ 2 e |ξ −ξ∗ | g (s) − g¯ (s) g (s) + g¯ (s) ∗+ g (t ) − g¯ (t )

0 R9 ×S2 +

 p −1   + g (s) − g¯ (s) ∗+ g (s) + g¯ (s) g (t ) − g¯ (t ) dω dξ∗ dξ dx ds ≡ N0 +

a2 R

t [N11 + N12 + N13 + N14 ] ds

2 0

+

a2 e − R

t [N21 + N22 + N23 + N24 ] ds,

2 0

(23)

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

19

where we set r

q˜ 1 (ξ, ξ∗ , ω, r ) :=

(1 + |ξ |2 ) 2 (1 + |ξ  |2 )

r 2

r

(1 + |ξ∗ |2 ) 2

q˜ 2 (ξ, ξ∗ , ω, r ) :=

,

1 r

(1 + |ξ∗ |2 ) 2

.

rp

Lemma 4.3. For a pair ( p , r ) satisfying p −1 > 3, the following estimates hold:

  

 

p

max  

q˜ i (ξ, ξ∗ , ω, r ) p−1 dω dξ∗  

i =1,2

R3 ×S2+

L∞ ξ

 C.

Proof. (1) First, we write the integral for q˜ 1

 q˜ 1 (ξ, ξ∗ , ω, r )

p p −1



 d ω d ξ∗ =

R3 ×S2

r

+

which is bounded by a constant depending on r when (2) Note that



r

(1 + |ξ  |2 ) 2 (1 + |ξ∗ |2 ) 2

R3 ×S2

+

 p−p 1

r

(1 + |ξ |2 ) 2

rp p −1



p

q˜ 2 (ξ, ξ∗ , ω, r ) p−1 dω dξ∗ = 2π

d ω d ξ∗ ,

> 3 (see [7]).



1 + |ξ∗ |2

− 2( prp−1)

d ξ∗ ,

R3

R3 ×S+ 2

rp p −1

which is bounded uniformly when

> 3. 2

We now estimate N0 , N1i and N2i in the following lemma. Lemma 4.4. Let f and ¯f be mild solutions to (1) and (8) in S (α , β, δ) and 1 < p < 3 and r > 3( p − 1)/ p, then the following estimates hold. For t , s  0 and i = 1, . . . , 4, we have



 p −1

(i) N0   g 0 − g¯ 0  L p  g (t ) − g¯ (t ) L p ,











 



p −1 (ii) N1i  O (1)  g (s) L ∞ ( L p ) +  g¯ (s) L ∞ ( L p )  g (s) − g¯ (s) L p  g (t ) − g¯ (t ) L p , x

ξ

x

ξ

 O(1)   g (t ) − g¯ (t ) pp−1 . (iii) N2i  L 3 (1 + s) Proof. (i) For N0 , we use the Hölder inequality to see



  p −1 | g 0 − g¯ 0 || g − g¯ | p −1 dξ dx   g 0 − g¯ 0 L p  g (t ) − g¯ (t )L p .

R6

(ii) Since the estimates for N1i and N2i are lengthy and straightforward, we leave their estimates in Appendix A. 2 With this lemma we are ready to prove the second main theorem.

20

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

Proof of Theorem 2.2. For the weighted L r -stability of the classical solutions f and ¯f to (1) with r weight function ϕr (ξ ) = (1 + |ξ |2 ) 2 , it is enough to show the L p -stability of g = ϕr f and g¯ = ϕr ¯f . On the other hand, from Lemma 4.4 and (23) we have p

       g (t ) − g¯ (t ) pp   g 0 − g¯ 0 L p  g (t ) − g¯ (t ) pp−1 + e − R  g (t ) − g¯ (t ) pp−1 L L L

t 0

  p −1 + R  g (t ) − g¯ (t )L p

t

O(1) ds (1 + s)3

      O(1)  g (s) L ∞ ( L p ) +  g¯ (s) L ∞ ( L p )  g (s) − g¯ (s) L p ds, x

x

ξ

ξ

0

or equivalently

   g (t ) − g¯ (t )

Lp

t

  g 0 − g¯ 0 L p + e − R

0

t +R

O(1) ds (1 + s)3

      O(1)  g (s) L ∞ ( L p ) +  g¯ (s) L ∞ ( L p )  g (s) − g¯ (s) L p ds. x

x

ξ

ξ

0

By Lemma 4.2 we can see that

   g (t ) − g¯ (t )

Lp

  g 0 − g¯ 0 L p + e

−R

t 0

O(1) ds + R (1 + s)3

t 0

 O(1)   g (s) − g¯ (s) p ds, L 3 (1 + s)

and we apply Lemma 4.1 to obtain

   g (t ) − g¯ (t )

Lp

 O(1) g 0 − g¯ 0 θL p ,

for some θ ∈ (0, 1).

2

Appendix A. The proof of Lemma 4.4 In this part, we present the estimates N1i and N2i appearing in (23). Case A. (N11 ): We use the Hölder inequality to obtain







 

p −1

q˜ 1 g (s) − g¯ (s) g (s) + g¯ (s) ∗− g (t ) − g¯ (t )

N11 =

dω dξ∗ dξ dx

R9 ×S2+

    R3

p p p −1 q˜ 1 g (t ) − g¯ (t ) dω dξ∗ dξ

 p−p 1

R6 ×S2+

  ×



 p

 p

g (s) + g¯ (s) ∗− g (s) − g¯ (s)

 1p d ω d ξ∗ d ξ

R6 ×S2+

The first factor of an integrand can be treated using Lemma 4.3:

dx.

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

 

p p −1

q˜ 1

g (t ) − g¯ (t ) p dω dξ∗ dξ

21

 p−p 1

R6 ×S2+



g (t ) − g¯ (t ) p

=



R3

p p −1

q˜ 1



 p−p 1

d ω d ξ∗ d ξ

R3 ×S2

+

    

 p −1    p−p 1 p p  p p −1  q˜ 1 dω dξ∗  g (t ) − g¯ (t ) dξ ∞ Lξ

R3 ×S2+

R3

  p −1  O(1) g (x, t ) − g¯ (x, t ) L p . ξ

On the other hand, the second factor can be estimated as follows:

 



 p  p g (s) + g¯ (s) ∗− g (s) − g¯ (s) dω dξ∗ dξ

 1p

R6 ×S2+

 



=

p p g (s) + g¯ (s) ∗+ g (s) − g¯ (s) dω dξ∗ dξ

 1p

R6 ×S2+

 



 R3









p p g (s) + g¯ (s) ∗+ dω dξ∗ g (s) − g¯ (s) dξ

 1p

R3 ×S2+

  1p  1p    p g (s) − g¯ (s) p dξ  O(1) g (s) + g¯ (s) ∗+ dω dξ∗ R3

R3

       O(1)  g (s) L ∞ ( L p ) +  g¯ (s) L ∞ ( L p )  g (x, s) − g¯ (x, s) L p . x

x

ξ

ξ

ξ

p −1

Hence we have N11  O (1)( g (s) L ∞ ( L p ) +  g¯ (s) L ∞ ( L p ) ) g (s) − g¯ (s) L p  g (t ) − g¯ (t ) L p . x

x

ξ

ξ

Case B. (N12 ): We use the Hölder inequality to estimate as we did for N11 ,













  p −1 q˜ 1 g (s) − g¯ (s) ∗− g (s) + g¯ (s) g (t ) − g¯ (t ) dω dξ∗ dξ dx

N12 = R9 ×S2+

  

p p −1

q˜ 1

 R3

g (t ) − g¯ (t ) p dω dξ∗ dξ

 p−p 1

R6 ×S2+

  × R6 ×S2+



 p g (s) − g¯ (s)  p dω dξ∗ dξ ∗−

g (s) + g¯ (s)

 1p dx.

22

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

The first factor of integrand is the same as that of N11 . For the second factor, we have

 



 p  p g (s) + g¯ (s) g (s) − g¯ (s) ∗− dω dξ∗ dξ

 1p

R6 ×S2+

 



=

p p g (s) + g¯ (s) g (s) − g¯ (s) ∗− dω dξ∗ dξ

 1p

R6 ×S2+

 



p

g (s) + g¯ (s)

 1p  

g (s) − g¯ (s) p dω dξ∗ ∗−

dξ

R3

 1p

R3 ×S2+

    g (s) + g¯ (s) L ∞ ( L p ) x

 

g (s) − g¯ (s) p dω dξ∗ ∗−

ξ

 1p .

R3 ×S2+

Then we have

  N12  O(1) g (s) + g¯ (s) L ∞ ( L p ) x



ξ

   g (x, t ) − g¯ (x, t ) pp−1

 

Lξ

R3

 1p dx

R3 ×S2

+

    p −1  O(1) g (t ) + g¯ (t )L ∞ ( L p )  g (t ) − g¯ (t ) L p x

g (s) − g¯ (s) p dω dξ∗ ∗−

 

ξ

 1p g (s) − g¯ (s) p dω dξ∗ dx ∗−

R6 ×S2+

      p −1    O(1)  g (s) L ∞ ( L p ) +  g¯ (s) L ∞ ( L p )  g (s) − g¯ (s) L p  g (t ) − g¯ (t )L p . x

x

ξ

ξ

Case C. (N13 ): Note that

  p−p 1    1p p   p p −1 q˜ 2 g (s) + g¯ (s) ∗+ dω dξ∗  O (1) q˜ 2 dω dξ∗ g (s) + g¯ (s) ∗+ dξ∗





R3

R3 ×S2+

R3

   O(1) g (s) + g¯ (s) L ∞ ( L p ) , x

(A.1)

ξ

and we use (A.1) to see











p −1

q˜ 2 g (s) − g¯ (s) g (s) + g¯ (s) ∗+ g (t ) − g¯ (t )

N13 =

dω dξ∗ dξ dx

R9 ×S2+

  





 

q˜ 2 g (s) + g¯ (s) ∗+ dω dξ∗

 R3

R3 ×S2+

R3

 g (s) − g¯ (s) g (t ) − g¯ (t ) p −1 dξ dx

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

   O(1) g (s) + g¯ (s) L ∞ ( L p ) x

 

g (s) − g¯ (s) p dξ

ξ

R3

 1p  

R3

23

g (t ) − g¯ (t ) p dξ

 p−p 1 dx

R3

      p −1    O(1)  g (s) L ∞ ( L p ) +  g¯ (s) L ∞ ( L p )  g (s) − g¯ (s) L p  g (t ) − g¯ (t )L p . x

ξ

ξ

Case D. (N14 ): We now treat N14 with the Hölder inequality









R9 ×S2+

  





q˜ 2 g (s) − g¯ (s) ∗+ dω dξ∗

 R3

 

  

p

p −1 q˜ 2 dω dξ∗



 p−p 1  

R3 ×S2+

 ×











p −1 g (s) + g¯ (s) g (t ) − g¯ (t ) dξ dx

R3

R3 ×S2+

R3





p −1 q˜ 2 g (s) − g¯ (s) ∗+ g (s) + g¯ (s) g (t ) − g¯ (t ) dω dξ∗ dξ dx

N14 =

g (s) − g¯ (s) p dω dξ∗ ∗+

 1p

R3 ×S2+

p g (s) + g¯ (s) dξ

 1p  

R3

g (t ) − g¯ (t ) p dξ

 p−p 1 dx

R3

   O(1) g (s) + g¯ (s) L ∞ ( L p ) x

  

g (s) − g¯ (s) p dω dξ∗ ∗+

ξ

R3

g (t ) − g¯ (t ) p dξ

 p−p 1 dx

R3

R3 ×S2

+

    p −1  O(1) g (s) + g¯ (s) L ∞ ( L p )  g (t ) − g¯ (t )L p x

 1p  

 

 1p g (s) − g¯ (s) p dω dξ∗ dx ∗+

ξ

R6 ×S2+

      p −1    O(1)  g (s) L ∞ ( L p ) +  g¯ (s) L ∞ ( L p )  g (s) − g¯ (s) L p  g (t ) − g¯ (t ) L p . x

x

ξ

ξ

Case E. (N21 and N22 ): First, we use the Hölder inequality to estimate N21





R9 ×S2



=



 

p −1

q˜ 1 e |ξ −ξ∗ | g (s) − g¯ (s) g (s) + g¯ (s) ∗− g (t ) − g¯ (t )

N21 = +

g (t ) − g¯ (t ) p −1

R6

  q˜ 1 e



|ξ −ξ∗ |

dω dξ∗ dξ dx

    g (s) − g¯ (s) g (s) + g¯ (s) ∗− dω dξ∗ dξ dx

R6 ×S2

+

  p −1   g (t ) − g¯ (t ) p

   q˜ 1 e

L

R6

R6 ×S2

+



|ξ −ξ∗ |

   g (s) − g¯ (s) g (s) + g¯ (s) ∗− dω dξ∗ 

 :=N21

p

 1p dξ dx

.



r r Recall that g = (1 + |ξ |2 ) 2 f and g¯ = (1 + |ξ |2 ) 2 ¯f where f and ¯f are in the exponentially decaying functional class, then we have

24

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25



 N21 



1 + |ξ |2

 2r

   e |ξ −ξ∗ | f (s) − ¯f (s) f (s) + ¯f (s) ∗− dω dξ∗

R3 ×S2+



 r  O(1)δ 2 1 + |ξ |2 2 e |ξ |

 2  2  2  2 e |ξ∗ | e −α |x+s(ξ −ξ )| −β|ξ | e −α |x−aω+s(ξ −ξ∗ )| −β|ξ∗ | dω dξ∗

R3 ×S2+



 r  O(1)δ 2 1 + |ξ |2 2 e |ξ |

2 2 2 2 e |ξ∗ | e −α |x| −β|ξ | e −α |x−aω+s(ξ −ξ∗ )| −β|ξ∗ | dω dξ∗ ,

R3 ×S2+

where we use Lemma 2.2 and apply the same calculation as we did in the proof of Proposition 2.1 to see r

 N21  O(1)δ 2

(1 + |ξ |2 ) 2 e |ξ | e −α |x| (1 + s)3

2

−β|ξ |2

.

Finally we can have

  p −1 O(1) N21   g (t ) − g¯ (t ) L p (1 + s)3





1 + |ξ |

2

 rp2

e

p |ξ | −α p |x|2 −

e

βp 2

|ξ |2

 1p dξ dx

R6

  p −1 O(1)   g (t ) − g¯ (t )L p (1 + s)3 and N22 can be treated in the same way. Case F. (N23 and N24 ): Because we can deal N24 similarly to N23 , we only calculate N23 :





p −1   g (s) − g¯ (s) g (s) + g¯ (s)

q˜ 2 e |ξ −ξ∗ | g (t ) − g¯ (t )

N23 = R9 ×S2+





g (t ) − g¯ (t ) p −1

R6

 





∗+

dω dξ∗ dξ dx





q˜ 2 e |ξ −ξ∗ | g (s) − g¯ (s) g (s) + g¯ (s) ∗+ dω dξ∗ dξ dx

R3 ×S2+

  p −1   g (t ) − g¯ (t ) L p

   R6





R3 ×S2+



 as follows We now rewrite N23



R3 ×S2+





p



 :=N23

  p −1 O(1)   g (t ) − g¯ (t ) L p . (1 + s)3

 N23 =



q˜ 2 e |ξ −ξ∗ | g (s) − g¯ (s) g (s) + g¯ (s) ∗+ dω dξ∗



q˜ 2 e |ξ −ξ∗ | g (s) − g¯ (s) g (s) + g¯ (s) ∗+ dω dξ∗

 1p dξ dx

S.-Y. Ha, S.E. Noh / J. Differential Equations 251 (2011) 1–25

 = R3 ×S2+



=



1

(1 + |ξ∗

|2 )r /2



r /2

1 + |ξ |2



25



e |ξ −ξ∗ | g (s) − g¯ (s) g (s) + g¯ (s) ∗+ dω dξ∗







e |ξ −ξ∗ | f (s) − ¯f (s) f (s) + ¯f (s) ∗+ dω dξ∗ .

R3 ×S2+  can be estimated exactly the same as N  . Here we can see that N23 21

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