Global existence for the Boltzmann equation in L r v L ∞ t L ∞ x spaces
aa r X i v : . [ m a t h . A P ] J un GLOBAL EXISTENCE FOR THE BOLTZMANN EQUATION IN L rv L ∞ t L ∞ x SPACES
KOYA NISHIMURAAbstract. We study the Boltzmann equation near a global Maxwellian.We prove the global existence of a unique mild solution with initial datawhich belong to the L rv L ∞ x spaces where r ∈ ( , ∞] by using the excessconservation laws and entropy inequality introduced in [5]. The Boltzmann Equation
Recall that the Boltzmann equation is given by ∂ t F + v · ∇ x F (cid:3) Q ( F , F ) , F ( , x , v ) (cid:3) F ( x , v ) , (1.1)where F ( t , x , v ) is the distribution function for the particles at time t ≥ x ∈ Ω (cid:3) R or T , and velocity v ∈ R . The collision operator isdefined by Q ( F , G )( v ) (cid:3) ∫ R × S du d ω | v − u | γ b ( θ ) [ F ( v ′ ) G ( u ′ ) − F ( v ) G ( u )] . Here the angle θ is defined by cos θ (cid:3) [ v − u ]· ω /| v − u | and B ( θ ) satisfies theangular cutoff assumption 0 ≤ b ( θ ) ≤ C | cos θ | . We assume hard potentials0 ≤ γ ≤
1. The post-collisional velocities satisfy v ′ (cid:3) v + [( u − v ) · ω ] ω, u ′ (cid:3) u − [( u − v ) · ω ] ω, v ′ + u ′ (cid:3) v + u , | v ′ | + | u ′ | (cid:3) | v | + | u | . (1.2)Denoting a normalized global Maxwellian by µ ( v ) (cid:3) e −| v | , µ satisfies(1.1) by (1.2), and so we define the perturbation f ( t , x , v ) to µ as F (cid:3) µ + √ µ f . We consider the Boltzmann equation for the perturbation f : [ ∂ t + v · ∇ x + ν ( v ) − K ] f (cid:3) Γ ( f , f ) , f ( , x , v ) (cid:3) f ( x , v ) . (1.3)Above ν ( v ) (cid:3) Γ loss ( , √ µ ) ≈ ( + | v |) γ is a multiplication operator definedby (3.2) below, and K is a integral operator. (the kernel satisfies (3.11) below.Also, see [2] for its form.) Since Q ( µ, µ ) (cid:3)
0, the remaining nonlinear part Γ (· , ·) is defined as Γ ( g , h ) (cid:3) √ µ Q (√ µ g , √ µ h ) . Mathematics Subject Classification.
Lastly, the mild form of (1.3) is given by f ( t , x , v ) (cid:3) e − ν ( v ) t f ( x − vt , v ) + ∫ t e − ν ( v )( t − s ) (cid:2) K (cid:0) f (cid:1) + Γ (cid:0) f , f (cid:1)(cid:3) ( s , x − v ( t − s ) , v ) ds , (1.4)and its equivalent form is given by (3.9) below.
2. Main Results
Notation.
In this paper, we use the notation L ∞ x (cid:3) L ∞ ( Ω ) and L rv (cid:3) L r ( R v ) ( r ∈ ( , ∞]) . We also write the L ∞ norm on the time interval [ , t ] as || · || L ∞ t . For a function g : [ , ∞) × R x × R v → R , we define the mixed norms (cid:12)(cid:12)(cid:12)(cid:12) g (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t L rv L ∞ x (cid:3) sup s ∈[ , t ] "∫ R du ( sup y ∈ R (cid:12)(cid:12) g ( s , y , u ) (cid:12)(cid:12)) r r , and (cid:12)(cid:12)(cid:12)(cid:12) g (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t L ∞ x (cid:3) "∫ R du ( sup s ∈[ , t ] , y ∈ R (cid:12)(cid:12) g ( s , y , u ) (cid:12)(cid:12)) r r . Similarly for the norm || · || L rv L ∞ x . For r , we denote the conjugate exponent to r by r ′ . We define a weight function w ( v ) (cid:3) + | v | .For a solution to the Boltzmann equation (1.1), we have formally theexcess conservations of mass and energy and the excess entropy inequality: ∬ R × R F ( t , x , v ) − µ ( v ) dv dv (cid:3) ∬ F − µ ≡ M , ∬ R × R | v | (cid:2) F ( t , x , v ) − µ ( v ) (cid:3) dv dv (cid:3) ∬ | v | (cid:2) F − µ (cid:3) ≡ E , ∬ R × R F ( t , x , v ) ln F ( t , x , v ) − µ ( v ) ln µ ( v ) dv dx ≤ ∬ R × R F ln F − µ ln µ dv dx ≡ H . (2.1)The following local and global existence results are valid. Theorem 2.1.
Let r ∈ ( , ∞] and l > max { / r ′ , / r ′ + ( γ + )/ , γ } , and F (cid:3) µ + √ µ f ≥ . For any < M < ∞ , there exist T ⋆ ( M ) > and ǫ > suchthat if (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ x ≤ M / , and sup ≤ t ≤ T ⋆ , x ∈ R ∫ R dv e − | v | (cid:12)(cid:12) f ( x − vt , v ) (cid:12)(cid:12) ≤ ǫ, (2.2) then there is a unique local solution (1.4), f ( t , x , v ) , to (1.3) in [ , T ⋆ ] × Ω × R satisfying (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ T ⋆ L ∞ x ≤ M , LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 3 and F (cid:3) µ + √ µ f ≥ . Moreover, if M , E , and H are finite, then (2.1) holds. Theorem 2.2.
In addition to the assumptions as Theorem 2.1, let l > / r ′ + γ . Forany < M < ∞ , there exist ǫ > and C ( r , l ) > such that if || w l f || L rv L ∞ x ≤ M ,(2.2) and sup t ≥ T ⋆ , x ∈ R ∫ R e − ν ( v ) t (cid:12)(cid:12) f ( x − vt , v ) (cid:12)(cid:12) dv + | M | + | E | + | H | ≤ ǫ, (2.3) then there is a unique global solution (1.4), f ( t , x , v ) , to (1.3) satisfying (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x ≤ C (cid:0) M + M (cid:1) ∀ t > , and moreover F (cid:3) µ + √ µ f ≥ . Remark 2.3. In Theorem 2.1, when r ∈ [ /( − γ ) , ∞] we need not assume sup ≤ t ≤ T ⋆ , x ∈ R ∫ dv e − | v | (cid:12)(cid:12) f ( x − vt , v ) (cid:12)(cid:12) ≪ , (2.4) and the L rv L ∞ t , x norm can be replaced by the L ∞ t L rv L ∞ x norm in both theorems.(2.4) is required only when r ∈ ( , /( − γ )) in Theorem 2.1. In the case,we use Lemma 3.3 to get a decay of the collision term of (3.10) as | v | → ∞ .(Note that we consider the hard potential case.) Recently, the case r (cid:3) ∞ was proved in [1] . In our results, we can takelarge initial data in L rv L ∞ x ( r > ) and we need not take the uniform normwith respect to velocity variable v , but the L ∞ t norm is taken before the L rv norm. In [5], the L ∞ estimate using the excess conservation laws and entropyinequality (2.1) is established. And in [1], to obtain global existence, it wasshown that by a similar argument one can make the L v norm of a solution(1.4) small as in Lemma 4.2. Also for the L ∞ estimate of the collision term,the L v norm is involved as in Lemma 4.1 below. The case r < ∞ is technicallymore complicated to handle than the case r (cid:3) ∞ . For instance, in Lemma4.1, we will need to split the integral domain of the gain term into fourparts, and we change variables several times. The purpose of this paperis to extend the global existence results of [1]. For historical results of theBoltzmann equation, see the article and the references therein.This article is organized as follows. In Section 3 we prove local existence(Theorem 2.1). And in Section 4, we establish a L ∞ t , x L rv estimate (Lemma4.2) and a L rv L ∞ t , x estimate (Lemma 4.4). Global existence (Theorem 2.2) isfollows easily from them.
3. Local Solutions
As usual, we split Γ ( g , h ) (cid:3) Γ gain ( g , h ) − Γ loss ( g , h ) as Γ gain ( g , h )( v ) (cid:3) p µ ( v ) ∫ R × S du d ω | v − u | γ b ( θ ) · (cid:0) √ µ g (cid:1) ( v ′ ) · (cid:0) √ µ h (cid:1) ( u ′ ) , (3.1) LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 4 Γ loss ( g , h )( v ) (cid:3) ∫ R × S du d ω | v − u | γ b ( θ ) p µ ( u ) g ( v ) h ( u ) . (3.2)To begin with, we give a estimate for Γ gain . Lemma 3.1.
Let r ∈ [ /( − γ ) , ∞] and l > / r ′ . For g ( v ) , h ( v ) ≥ , we have (cid:12)(cid:12)(cid:12)(cid:12) w l Γ gain ( g , h ) (cid:12)(cid:12)(cid:12)(cid:12) L rv ≤ C r , l (cid:12)(cid:12)(cid:12)(cid:12) w l g (cid:12)(cid:12)(cid:12)(cid:12) L rv (cid:12)(cid:12)(cid:12)(cid:12) w l h (cid:12)(cid:12)(cid:12)(cid:12) L rv . (3.3) Proof.
We estimate w l Γ gain ( g , h ) as follows. Since w l ( v ) ≤ Cw l ( u ′ ) + Cw l ( v ′ ) by (1.2), w l ( v ) Γ gain ( g , h )( v )≤ C p µ ( v ) ∬ | v − u | γ b ( θ ) · (cid:0) √ µ g (cid:1) ( v ′ ) · (cid:16) w l √ µ h (cid:17) ( u ′ ) du d ω + C p µ ( v ) ∬ | v − u | γ b ( θ ) · (cid:16) w l √ µ g (cid:17) ( v ′ ) · (cid:0) √ µ h (cid:1) ( u ′ ) du d ω. (3.4)As in Proposition 2.1 of [1], it suffices to estimate only the first term (because,one may interchange u ′ and v ′ in the second term. we refer to page 41-42of [2]). By Hölder’s inequality, the first term is bounded by ∬ | v − u | γ b ( θ ) e − | u | w − l ( v ′ ) · (cid:16) w l g (cid:17) ( v ′ ) · (cid:16) w l h (cid:17) ( u ′ ) du d ω ≤ C (cid:20)∬ [| v − u | γ | cos θ |] r ′ e − r ′ | u | w − r ′ l ( v ′ ) du d ω (cid:21) r ′ × (cid:20)∬ (cid:12)(cid:12)(cid:12)(cid:16) w l g (cid:17) ( v ′ ) · (cid:16) w l h (cid:17) ( u ′ ) (cid:12)(cid:12)(cid:12) r du d ω (cid:21) r , (3.5)with the standard modification when r (cid:3) ∞ . By changing u (cid:3) z + v andspliting z || (cid:3) [ z · ω ] ω, z ⊥ (cid:3) z − z || , the integral of the first factor can bebounded by C ∬ (cid:2) | z | γ − | z || | (cid:3) r ′ e − r ′ | z + v | w − r ′ l ( z || + v ) (cid:12)(cid:12) z || (cid:12)(cid:12) − dz ⊥ dz || . (3.6)The further substitution y (cid:3) z || + v and the inequality | z | ≥ p | z || | · | z ⊥ | yieldthat (3.6) is bounded by ∬ (cid:12)(cid:12) y − v (cid:12)(cid:12) γ + r ′ − e − r ′ | y + z ⊥| w − r ′ l ( y ) dz ⊥ dy ≤ C ∫ (cid:12)(cid:12) y − v (cid:12)(cid:12) γ + r ′ − w − r ′ l ( y ) dy ≤ C r , l ( + | v |) γ + r ′ − . (3.7)This is bounded if r ≥ /( − γ ) , so by taking the L r norm of (3.4) and notingthat du dv (cid:3) du ′ dv ′ , we can obtain the lemma. (cid:3) LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 5
Remark 3.2.
When r ∈ ( , /( − γ )) note that (3.7) is not bounded, but asimple modification of the argument in the proof of the lemma shows that for l > / r ′ + ( γ + )/ , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w l − γ + + r ′ Γ gain ( g , h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L rv ≤ C r (cid:12)(cid:12)(cid:12)(cid:12) w l g (cid:12)(cid:12)(cid:12)(cid:12) L rv (cid:12)(cid:12)(cid:12)(cid:12) w l h (cid:12)(cid:12)(cid:12)(cid:12) L rv . (3.8)We now prove Theorem 2.1. To this end, rewrite the mild form of (1.3) asfollows. f ( t , x , v ) (cid:3) e − ∫ t g f ( s , y + vs , v ) ds f ( y , v ) + ∫ t e − ∫ ts g f ( s , y + vs , v ) ds K f ( s , y + vs , v ) ds + ∫ t e − ∫ ts g f ( s , y + vs , v ) ds Γ gain ( f , f )( s , y + vs , v ) ds , (3.9)Here we have used the notation y (cid:3) x − vt , and for a function r ( s , x , v ) , g r ( s , x , v ) (cid:3) ∬ du d ω | v − u | γ b ( θ ) h µ ( u ) + p µ ( u ) r ( s , x , u ) i . Proof of Theorem 2.1.
We use the following iterating sequence ( n ≥ f n + ( t , x , v ) (cid:3) e − ∫ t g f n ( s , y + vs , v ) ds f ( y , v ) + ∫ t e − ∫ ts g f n ( s , y + vs , v ) ds K f n ( s , y + vs , v ) ds + ∫ t e − ∫ ts g f n ( s , y + vs , v ) ds Γ gain ( f n , f n )( s , y + vs , v ) ds . (3.10)We set f n ( , x , v ) (cid:3) f for n ≥
1, and f (cid:3) . It is easily verified as inProposition 2.1 of [1] that µ + √ µ f n ≥ µ + √ µ f ≥
0. First, we consider thecase r ∈ [ /( − γ ) , ∞] and we will show that if sup ≤ t ≤ T ⋆ || w l f n ( t )|| L rv L ∞ x ≤ M then sup ≤ t ≤ T ⋆ || w l f n + ( t )|| L rv L ∞ x ≤ M . We denote by k ( v , u ) the kernel of K where k ( v , u ) satisfies | k ( v , u )| ≤ C | v − u | e − | v | + | u | + C | v − u | − e − | v − u | − [ | v | −| u | ] | v − u | , (3.11)and for l ∈ R , ∫ | k ( v , u )| w l ( v ) w l ( u ) du ≤ C ( + | v |) − . (3.12)For the proof, see Lemma 7 of [4] for instance. (When l < l ≥ | u | ≤ | u − v | + | v | ) From (3.12) and Lemma 3.1 we can obtain (cid:12)(cid:12)(cid:12)(cid:12) w l f n + ( t ) (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ x ≤ (cid:12)(cid:12)(cid:12)(cid:12) w l f n + ( ) (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ x + Ct (cid:12)(cid:12)(cid:12)(cid:12) w l f n (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t L rv L ∞ x + C r t (cid:12)(cid:12)(cid:12)(cid:12) w l f n (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t L rv L ∞ x , LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 6 and hence (cid:12)(cid:12)(cid:12)(cid:12) w l f n + (cid:12)(cid:12)(cid:12)(cid:12) L ∞ T ⋆ L rv L ∞ x ≤ M / + CMT ⋆ + C r M T ⋆ ≤ M when T ⋆ is sufficiently small. As for uniqueness, we take the difference f − h where f and h satisfy (3.9), as follows. w l ( v ) (cid:12)(cid:12) (cid:2) f − h (cid:3) ( t , x , v ) (cid:12)(cid:12) ≤ (cid:26) w l ( v ) (cid:12)(cid:12) f ( y , v ) (cid:12)(cid:12) + ∫ t w l ( v ) (cid:12)(cid:12) K f ( s , y + vs , v ) (cid:12)(cid:12) ds + ∫ t w l ( v ) (cid:12)(cid:12) Γ gain ( f , f )( s , y + vs , v ) (cid:12)(cid:12) ds (cid:27) × ∫ ts (cid:12)(cid:12) (cid:2) g f − g h (cid:3) ( s , y + vs , v ) (cid:12)(cid:12) ds + ∫ t w l ( v ) (cid:12)(cid:12) K (cid:2) f − h (cid:3) ( s , y + vs , v ) (cid:12)(cid:12) ds + ∫ t w l ( v ) (cid:12)(cid:12) Γ gain ( f − h , f )( s , y + vs , v ) (cid:12)(cid:12) ds + ∫ t w l ( v ) (cid:12)(cid:12) Γ gain ( h , f − h )( s , y + vs , v ) (cid:12)(cid:12) ds . (3.13)Here we have used the inequality | e − a − e − b | ≤ | a − b | , ∀ a , b ≥
0. Note that l > γ and ν ( v ) (cid:3) w γ ( v ) . Clearly ∫ ts (cid:12)(cid:12) (cid:2) g f − g h (cid:3) ( s , y + vs , v ) (cid:12)(cid:12) ds ≤ C r w l ( v ) t (cid:12)(cid:12)(cid:12)(cid:12) f − h (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t L rv L ∞ x , so the first term on the right hand side of (3.13) is bounded by C r Mt (cid:12)(cid:12)(cid:12)(cid:12) f − h (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t L rv L ∞ x . By (1.2), the second term from the last of (3.13) is bounded by C ∬ du d ω w γ ( v )[ w ( v ′ ) w ( u ′ )] l e − | u | (cid:16) w l f (cid:17) ( v ′ ) · (cid:16) w l [ f − h ] (cid:17) ( u ′ ) + ∬ du d ω | v − u | γ b ( θ ) e − | u | w − l ( v ′ ) (cid:16) w l f (cid:17) ( v ′ ) · (cid:16) w l [ f − h ] (cid:17) ( u ′ ) . As in the proof of Lemma 3.1, its L ∞ t L rv L ∞ x norm is bounded by C r Mt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w l [ f − h ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ t L rv L ∞ x . Similarly for the last term of (3.13). We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w l (cid:2) f − h (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ T ⋆ L rv L ∞ x ≤ C r MT ⋆ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w l (cid:2) f − h (cid:3)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L ∞ T ⋆ L rv L ∞ x . LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 7
Hence uniqueness follows. Similarly, we can also prove that ( f n ) is a Cauchysequence. Letting n → ∞ we obtain a unique mild solution (3.9), f ( t , x , v ) ,in [ , T ⋆ ] × Ω × R . For the remaining assertions we refer to the proof ofProposition 2.1 of [1].For the other case r ∈ ( , /( − γ )) we replace the L ∞ t L rv L ∞ x norm by L rv L ∞ t , x and use (3.8) and the fact ∫ t e − ν ( v )( t − s ) ν δ ( v ) ds ≤ η + C η t , ≤ δ < , for any η >
0, and Lemma 3.3 below. Taking γδ (cid:3) ( γ + )/ − / r ′ , the L rv L ∞ t , x norm of the last term of (3.10) is bounded by ∫ t ds e − ν ( v )( t − s ) ν δ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) w l − γδ Γ gain ( f n , f n ) (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x ≤ C r η (cid:12)(cid:12)(cid:12)(cid:12) w l f n (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x . The remaining proof is a simple modification of the case r ≥ /( − γ ) . (cid:3) With the same assumptions as Theorem 2.1, we have the following lemmafor the sequence (3.10).
Lemma 3.3.
For any η > , there exists T ⋆ ( η, M ) > such that if sup ≤ t ≤ T ⋆ , x ∈ R ∫ dv e − | v | (cid:12)(cid:12) f ( x − vt , v ) (cid:12)(cid:12) ≤ η / , and sup ≤ t ≤ T ⋆ (cid:12)(cid:12)(cid:12)(cid:12) w l f n − ( t ) (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ x ≤ M , then sup ≤ s ≤ T ⋆ , x ∈ R ∫ du e − | u | (cid:12)(cid:12) f n ( s , y + vs , u ) (cid:12)(cid:12) ≤ η. (3.14) Moreover, we have − ∫ ts ds g f n ( s , y + vs , v ) ≤ − ν ( v )( t − s )/ , (3.15) when η is sufficiently small and ≤ t ≤ T ⋆ . Proof.
From (3.10), we get ∫ e − | u | (cid:12)(cid:12) f n ( s , y + vs , u ) (cid:12)(cid:12) du ≤ ∫ e − | u | (cid:12)(cid:12) f ( y + vs − u s , u ) (cid:12)(cid:12) du + ∫ s ∬ e − | u | | k ( u , u )| · (cid:12)(cid:12)(cid:12)(cid:12) f n − ( s , u ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ x du du ds + ∫ s ∫ e − | u | Γ gain h(cid:12)(cid:12)(cid:12)(cid:12) f n − (cid:12)(cid:12)(cid:12)(cid:12) L ∞ x , (cid:12)(cid:12)(cid:12)(cid:12) f n − (cid:12)(cid:12)(cid:12)(cid:12) L ∞ x i ( s , u ) du ds . LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 8
By (3.12), the second term on the right hand side is bounded by ∫ s ds ∫ du | k ( u , u )| ∫ du (cid:12)(cid:12)(cid:12)(cid:12) f n − ( s , u ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ x ≤ C ∫ s ds ∫ du (cid:12)(cid:12)(cid:12)(cid:12) f n − ( s , u ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ x ≤ C r , l Ms . Using (1.2) and the fact du dv (cid:3) du ′ dv ′ , the last term is bounded by C ∫ s ds ∭ du du d ω e − [ | u ′ | + | u ′ | ] (cid:12)(cid:12)(cid:12)(cid:12) f n − ( s , u ′ ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ x · (cid:12)(cid:12)(cid:12)(cid:12) f n − ( s , u ′ ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ x ≤ C ∫ s ds (cid:20)∫ du ′ (cid:12)(cid:12)(cid:12)(cid:12) f n − ( s , u ′ ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ x (cid:21) ≤ C r , l M s . We have ∫ e − | u | (cid:12)(cid:12) f n ( s , y + vs , u ) (cid:12)(cid:12) du ≤ η / + C r , l Ms + C r , l M s , and hence we have (3.14) if we choose T ⋆ small. Moreover, (3.15) followsfrom this and − g f n ( s , y + vs , v )≤ − ν ( v ) + ∬ | v − u | γ b ( θ ) p µ ( u ) (cid:12)(cid:12) f n ( s , y + vs , u ) (cid:12)(cid:12) du d ω ≤ − ν ( v ) + C ν ( v ) ∫ e − | u | (cid:12)(cid:12) f n ( s , y + vs , u ) (cid:12)(cid:12) du . (cid:3)
4. Global Existence
It is important to bound the nonlinear term by using the L v norm. Lemma 4.1.
Let r ∈ ( , ∞] , l > / r ′ , n > , and g ( v ) ≥ . For any η > wehave (cid:12)(cid:12)(cid:12)(cid:12) w l − γ Γ gain ( g , g ) (cid:12)(cid:12)(cid:12)(cid:12) L rv ≤ C r ,η (cid:12)(cid:12)(cid:12)(cid:12) g (cid:12)(cid:12)(cid:12)(cid:12) nr ′ L v (cid:12)(cid:12)(cid:12)(cid:12) w l g (cid:12)(cid:12)(cid:12)(cid:12) + r + n ′ r ′ L rv + C r , l η Õ p (cid:3) , r (cid:12)(cid:12)(cid:12)(cid:12) w l g (cid:12)(cid:12)(cid:12)(cid:12) + p L rv , (4.1) and (cid:12)(cid:12)(cid:12)(cid:12) w l − γ Γ loss ( g , g ) (cid:12)(cid:12)(cid:12)(cid:12) L rv ≤ C r (cid:12)(cid:12)(cid:12)(cid:12) g (cid:12)(cid:12)(cid:12)(cid:12) L v (cid:12)(cid:12)(cid:12)(cid:12) w l g (cid:12)(cid:12)(cid:12)(cid:12) L rv . (4.2) LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 9
Proof.
The inequality for the loss term is trivial, so we only estimate the gainterm. Using (1.2) and then interchanging v ′ and u ′ as in Lemma 3.1, w l − γ ( v ) Γ gain ( g , g )( v ) ≤ C ∬ (cid:2) w l ( v ′ ) + w l ( u ′ ) (cid:3) e − | u | g ( v ′ ) g ( u ′ ) du d ω ≤ C ∬ g ( v ′ ) · e − | u | · (cid:16) w l g (cid:17) ( u ′ ) du d ω. We split the last integral into four parts. First, for L > C ∬ | u |≥ L g ( v ′ ) · e − | u | · (cid:16) w l g (cid:17) ( u ′ ) du d ω ≤ Ce − L ∬ e − | u | · g ( v ′ ) · (cid:16) w l g (cid:17) ( u ′ ) du d ω ≤ Ce − L (cid:20)∬ (cid:16) w l g (cid:17) r ( v ′ ) · (cid:16) w l g (cid:17) r ( u ′ ) du d ω (cid:21) r , (4.3)so by du dv (cid:3) du ′ dv ′ , the L rv norm of (4.3) is bounded by Ce − L || w l g || L rv .Next, let | v | ≤ L and set k (cid:3) + ( r − )/ n ′ for fixed n >
3. Then we get1 (cid:3) /( n r ′ ) + k / r and C ∬ | u |≤ L g ( v ′ ) · e − | u | · (cid:16) w l g (cid:17) ( u ′ ) du d ω ≤ C (cid:20)∬ | u |≤ L g n ( v ′ ) e − | u | du d ω (cid:21) r ′ (cid:20)∬ | u |≤ L g k ( v ′ ) · (cid:16) w l g (cid:17) r ( u ′ ) du d ω (cid:21) r . (4.4)For the first factor, we use the same change of variables as (3.6) and (3.7).The following integral calculus holds. ∬ | u |≤ L g n ( v ′ ) e − | u | du d ω ≤ C ∬ g n ( v + z || ) e − | v + z | | z || | dz || dz ⊥ ≤ C ∫ | y |≤ L g n ( y ) | y − v | dy ≤ C (cid:20)∫ | y |≤ L g ( y ) dy (cid:21) n (cid:20)∫ | y |≤ L | y − v | n ′ dy (cid:21) n ′ ≤ C r , L (cid:12)(cid:12)(cid:12)(cid:12) g (cid:12)(cid:12)(cid:12)(cid:12) n L v . (4.5) LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 10 By du dv (cid:3) du ′ dv ′ and k < r , the L r ({| v | ≤ L }) norm of the second factorof (4.4) is bounded by C r , L (cid:20)∫ | v ′ |≤ L dv ′ g k ( v ′ ) ∫ | u ′ |≤ L du ′ (cid:16) w l g (cid:17) r ( u ′ ) (cid:21) r ≤ C r , L (cid:12)(cid:12)(cid:12)(cid:12) g (cid:12)(cid:12)(cid:12)(cid:12) kr L rv (cid:12)(cid:12)(cid:12)(cid:12) w l g (cid:12)(cid:12)(cid:12)(cid:12) L rv ≤ C r , L (cid:12)(cid:12)(cid:12)(cid:12) w l g (cid:12)(cid:12)(cid:12)(cid:12) kr + L rv . (4.6)The L r ({| v | ≤ L }) norm of (4.4) is bounded by C r , L || g || nr ′ L v || w l g || + kr L rv . Lastly,when | v | ≥ L , we consider the two cases |( u − v )· ω | ≤ L and |( u − v )· ω | ≥ L .For the former, then | v ′ | (cid:3) | v + [( u − v ) · ω ] ω | ≥ L − L (cid:3) L , so C ∬ |( u − v )· ω |≤ L e − | u | · g ( v ′ ) · (cid:16) w l g (cid:17) ( u ′ ) du d ω ≤ CL − l ∬ e − | u | · (cid:16) w l g (cid:17) ( v ′ ) · (cid:16) w l g (cid:17) ( u ′ ) du d ω. (4.7)As in (4.3), the L r ({| v | ≥ L }) norm of (4.7) is bounded by CL − l || w l g || L rv . Inthe latter, since | z || | ≥ L , as in (4.4) and (4.5) (take n (cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C ∬ |( u − v )· ω |≥ L e − | u | · g ( v ′ ) · (cid:16) w l g (cid:17) ( u ′ ) du d ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L r ({| v |≥ L }) ≤ C r , l L − r ′ (cid:12)(cid:12)(cid:12)(cid:12) w l g (cid:12)(cid:12)(cid:12)(cid:12) + r L rv , and then we have (4.1) by collecting above estimates and choosing L large. (cid:3) For simplicity, we use the notation E (cid:3) [| M | + | E | + | H |] m − for 0 < m < k l ( v , u ) (cid:3) k ( v , u ) ν l ( v )/ ν l ( u ) where k ( v , u ) isthe kernel of the integral operator K . As in [5] or [1], when t − s ≥ κ ( <κ < ) , N >
0, we can obtain ∬ | v |≤ N , | u |≤ N (cid:12)(cid:12) f ( s , x − v ( t − s ) , u ) (cid:12)(cid:12) du dv ≤ C N (cid:0) + κ − (cid:1) E , (4.8)which is the key estimate to global solvability. Recall that n and k weredefined in the proof of Lemma 4.1. Under the assumption of Theorem2.2, from (4.8), the following two lemmas are valid and then Theorem 2.2follows easily (see Proof of Theorem 1.1 in [1]). Lemma 4.2.
For any η > there exists C η ( r , l ) > such that sup s ∈[ T ⋆ , t ] (cid:12)(cid:12)(cid:12)(cid:12) f ( s ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ x L v ≤ ǫ + C r , l η Õ p (cid:3) , (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) p L rv L ∞ t , x + C η E + C η E Õ p (cid:3) r , r / k (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) + p L rv L ∞ t , x , (4.9) LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 11
Remark 4.3.
On the interval [ , T ⋆ ] , Theorem 2.1 yields || f || L ∞ T ⋆ L ∞ x L v ≤ C r , l M . Lemma 4.4.
For any η > there exists C η ( r , l ) > such that (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x ≤ CM + C η E + C r (cid:12)(cid:12)(cid:12)(cid:12) f (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t , x L v (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x + C η (cid:12)(cid:12)(cid:12)(cid:12) f (cid:12)(cid:12)(cid:12)(cid:12) nr ′ L ∞ t , x L v (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) + kr L ∞ t , x L rv + C r , l η Õ p (cid:3) ∞ , , r (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) + p L rv L ∞ t , x . (4.10) Proof of Lemma 4.2.
From (1.4), ∫ (cid:12)(cid:12) f ( t , x , v ) (cid:12)(cid:12) dv ≤ Õ j (cid:3) G j ( t , x ) , where G ( t , x ) (cid:3) ∫ e − ν ( v ) t (cid:12)(cid:12) f ( y , v ) (cid:12)(cid:12) dv , G ( t , x ) (cid:3) ∫ t ds ∫ dv ∫ du e − ν ( v )( t − s ) (cid:12)(cid:12) k ( v , u ) f ( s , y + vs , u ) (cid:12)(cid:12) , G ( t , x ) (cid:3) ∫ t ds ∫ dv ∬ du d ω e − ν ( v )( t − s ) | v − u | γ b ( θ ) p µ ( u )× (cid:12)(cid:12) f ( s , y + vs , u ) f ( s , y + vs , v ) (cid:12)(cid:12) , G ( t , x ) (cid:3) ∫ t ds ∫ dv ∬ du d ω e − ν ( v )( t − s ) | v ′ − u ′ | γ b ( θ ) p µ ( u )× (cid:12)(cid:12) f ( s , y + vs , u ′ ) f ( s , y + vs , v ′ ) (cid:12)(cid:12) . Here we have used the notation y (cid:3) x − vt . Note that | v − u | (cid:3) | v ′ − u ′ | .We further split G j , j (cid:3) , , G j ( t , x ) (cid:3) ∫ tt − κ ∬ + ∫ t − κ ∫ ∫ | u |≥ N + ∫ t − κ ∫ | v |≥ N ∫ | u |≤ N + ∫ t − κ ∫ | v |≤ N ∫ | u |≤ N {· · · } du dv ds ≡ G j ( t , x ) + G j ( t , x ) + G j ( t , x ) + G j ( t , x ) . By assumption G ≤ ǫ. First we will show that for any η >
0, if κ and N − aresufficiently small, then G jk ≤ η Í p (cid:3) , || w l f || p L rv L ∞ t , x for j (cid:3) , , , k (cid:3) , , . To see this for G k , recall (3.12). For G k , k (cid:3) , , v before u . It is not hard to see that G ≤ C r , l κ || w l f || L rv L ∞ t , x . Also, G ≤ ∫ t − κ ds e − ν ( v )( t − s ) ∫ | u |≥ N du (cid:18)∫ dv | k ( v , u )| (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) f ( u ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t , x ≤ CN − ∫ du (cid:12)(cid:12)(cid:12)(cid:12) f ( u ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t , x ≤ C r , l N − (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x . LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 12
Let | v | ≥ N and | u | ≤ N . Then | v − u | ≥ N . Since there is also the case γ (cid:3)
0, we estimate G as follows. G ≤ ∫ | u |≤ N du (cid:18)∫ | v |≥ N dv | k ( v , u )| (cid:19) (cid:18)∫ t − κ ds e − ν ( v )( t − s ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) f ( u ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t , x ≤ C ∫ du (cid:18)∫ dv e − N | k ( v , u )| e | v − u | (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) f ( u ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t , x ≤ Ce − N ∫ du (cid:12)(cid:12)(cid:12)(cid:12) f ( u ) (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x ≤ C r , l e − N (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x , Thus the claim for G k follows by choosing κ and N − small. The terms G k and G k are easy to estimate. Noting du dv (cid:3) du ′ dv ′ and 3 − ( l − γ ) r ′ < Õ k (cid:3) , , G k + G k ≤ ∫ tt − κ ds e − ν ( v )( t − s ) (cid:20)∫ dv w γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) f ( v ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t , x (cid:21) + ∫ t − κ ds e − ν ( v )( t − s ) (cid:20)∫ | v |≥ N dv w γ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) f ( v ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t , x (cid:21) ≤ C r , l κ (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x + CN −( l − γ ) r ′ (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x . Next, we estimate G j , j (cid:3) , , k (cid:3) + ( r − )/ n ′ for fixed n >
3. From the same calculus as (4.5), we have ∫ | v |≤ N dv ∫ | u |≤ N du ∫ d ω (cid:12)(cid:12) f ( s , y + vs , v ′ ) f ( s , y + vs , u ′ ) (cid:12)(cid:12) ≤ (cid:20)∭ | v |≤ N , | u |≤ N (cid:12)(cid:12) f ( s , y + vs , v ′ ) (cid:12)(cid:12) n du dv d ω (cid:21) r ′ × (cid:20)∫ | v ′ |≤ N (cid:12)(cid:12)(cid:12)(cid:12) f ( v ′ ) (cid:12)(cid:12)(cid:12)(cid:12) kL ∞ t , x dv ′ ∫ | u ′ |≤ N (cid:12)(cid:12)(cid:12)(cid:12) f ( u ′ ) (cid:12)(cid:12)(cid:12)(cid:12) rL ∞ t , x du ′ (cid:21) r ≤ C r , N (cid:20)∬ | v |≤ N , | y |≤ N (cid:12)(cid:12) f ( s , y + vs , y ) (cid:12)(cid:12) dy dv (cid:21) nr ′ × (cid:20)∬ | v |≤ N , | y |≤ N | y − v | n ′ dy dv (cid:21) n ′ (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) + kr L rv L ∞ t , x ≤ C r , N ,κ E (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) + kr L rv L ∞ t , x . (4.11) LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 13
Moreover, we easily get ∫ | v |≤ N dv ∫ | u |≤ N du (cid:12)(cid:12) f ( s , y + vs , v ) f ( s , y + vs , u ) (cid:12)(cid:12) ≤ (cid:20)∬ | v |≤ N , | u |≤ N (cid:12)(cid:12) f ( s , y + vs , u ) (cid:12)(cid:12) du dv (cid:21) r ′ × (cid:20)∬ | v |≤ N , | u |≤ N (cid:12)(cid:12) f ( s , y + vs , u ) (cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) f ( v ) (cid:12)(cid:12)(cid:12)(cid:12) rL ∞ t , x du dv (cid:21) r ≤ (cid:20)∬ | v |≤ N , | u |≤ N (cid:12)(cid:12) f ( s , y + vs , u ) (cid:12)(cid:12) du dv (cid:21) r ′ × (cid:20)∫ (cid:12)(cid:12)(cid:12)(cid:12) f ( u ) (cid:12)(cid:12)(cid:12)(cid:12) L ∞ t , x du ∫ (cid:12)(cid:12)(cid:12)(cid:12) f ( v ) (cid:12)(cid:12)(cid:12)(cid:12) rL ∞ t , x dv (cid:21) r ≤ C r , N ,κ E (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) + r L rv L ∞ t , x . (4.12)Hence G + G ≤ C r , N ,κ E (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) + r L rv L ∞ t , x ∫ t − κ ds e − ν ( v )( t − s ) ≤ C r , N ,κ E Õ p (cid:3) r , r / k (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) + p L rv L ∞ t , x . For G , in view of (3.11), we need to approximate k l ( v , u ) by k l , N ( v , u ) smooth with compact support such thatsup | u |≤ N ∫ | v |≤ N dv (cid:12)(cid:12) k l ( v , u ) − k l , N ( v , u ) (cid:12)(cid:12) ≤ N − r ′ . (4.13)We have ∫ | v |≤ N dv ∫ | u |≤ N du (cid:12)(cid:12)(cid:12) k l , N ( v , u ) · (cid:16) ν l f (cid:17) ( s , y + vs , u ) (cid:12)(cid:12)(cid:12) ≤ C l , N ∫ | v |≤ N dv ∫ | u |≤ N du (cid:12)(cid:12) f ( s , y + vs , u ) (cid:12)(cid:12) ≤ C l , N ,κ E , ∫ | v |≤ N , | u |≤ N du dv (cid:12)(cid:12) k l ( v , u ) − k l , N ( v , u ) (cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:16) w l f (cid:17) ( s , y + vs , u ) (cid:12)(cid:12)(cid:12) ≤ C r N − r ′ (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x . Hence G ≤ C l , N ,κ E + C r N − r ′ (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x . LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 14
We obtain the lemma by collecting above estimates and choosing κ and N − small. (cid:3) Proof of Lemma 4.4.
From now on we use the notation y (cid:3) x − v ( t − s ) , y (cid:3) y − u ( s − s ) , and recall that k l ( v , u ) (cid:3) w l ( v ) k ( v , u )/ w l ( u ) . By applying (1.4) to the secondterm on the right hand side of (1.4), we have w l ( v ) f ( t , x , v ) (cid:3) Õ j (cid:3) H j ( t , x , v ) , where H ( t , x , v ) (cid:3) e − ν ( v ) t w l ( v ) f ( x − vt , v ) , H ( t , x , v ) (cid:3) ∫ du k l ( v , u ) ∫ t ds e − ν ( v )( t − s ) × e − ν ( u ) s w l ( u ) f ( y − u s , u ) , H ( t , x , v ) (cid:3) ∬ du du k l ( v , u ) k l ( u , u ) ∫ t ds e − ν ( v )( t − s ) × ∫ s ds e − ν ( u )( s − s ) w l ( u ) f ( s , y , u ) , H ( t , x , v ) (cid:3) ∫ du k l ( v , u ) ∫ t ds e − ν ( v )( t − s ) × ∫ s ds e − ν ( u )( s − s ) w l ( u ) Γ [ f , f ]( s , y , u ) , H ( t , x , v ) (cid:3) ∫ t ds e − ν ( v )( t − s ) w l ( v ) Γ [ f , f ] ( s , x − v ( t − s ) , v ) . Clearly || H || L rv L ∞ t , x and || H || L rv L ∞ t , x are bounded by CM . For H and H we canapply Lemma 4.1, so their L rv L ∞ t , x norm are bounded by the last three termsof (4.10). Thus it remains only to estimate H . We compute the L rv L ∞ t , x normof H by dividing it into four parts. First, by repeating Hölder’s inequality,we can get ∬ du du k ( v , u ) k ( u , u ) w l ( v ) f ( s , y , u )≤ C (cid:20)∫ du | k l ( v , u )| (cid:21) r ′ × (cid:20)∬ du du | k l ( v , u ) k l ( u , u )| (cid:12)(cid:12)(cid:12)(cid:12) w l f ( u ) (cid:12)(cid:12)(cid:12)(cid:12) rL ∞ t , x (cid:21) r , (4.14) LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 15 so the L r ({| v | ≥ L }) norm of H is bounded by C r N − r ′ || w l f || L rv L ∞ t , x . Sincethe first factor of (4.14) is bounded, we also have || H || L rv L ∞ t , x ≤ C (cid:20)∫ dv | k l ( v , u )| ∫ du | k l ( u , u )| ∫ du (cid:12)(cid:12)(cid:12)(cid:12) w l f ( u ) (cid:12)(cid:12)(cid:12)(cid:12) rL ∞ t , x (cid:21) r , (4.15)and hence if either | v | ≤ N , | u | ≥ N or | u | ≤ N , | u | ≥ N then || H || L rv L ∞ t , x ≤ C r N − r || w l f || L rv L ∞ t , x . Lastly, we consider the L r ({| v | ≤ N }) normof the remaining part of H which is given by H ( t , x , v ) (cid:3) ∫ | u |≤ N du ∫ | u |≤ N du k l ( v , u ) k l ( u , u ) ∫ t κ ds e − ν ( v )( t − s ) × ∫ s s − κ ds e − ν ( u )( s − s ) w l ( u ) f ( s , y , u ) + ∫ | u |≤ N du ∫ | u |≤ N du k l ( v , u ) k l ( u , u ) ∫ κ ds e − ν ( v )( t − s ) × ∫ s ds e − ν ( u )( s − s ) w l ( u ) f ( s , y , u ) + ∫ | u |≤ N du ∫ | u |≤ N du k l ( v , u ) k l ( u , u ) ∫ t κ ds e − ν ( v )( t − s ) × ∫ s − κ ds e − ν ( u )( s − s ) w l ( u ) f ( s , y , u ) . (4.16)From (4.15), clearly, the L r ({| v | ≤ N } ; L ∞ t , x ) norms of the first two terms arebounded by C r , N κ || w l f || L rv L ∞ t , x . For the last term, we use (4.8). As before, weapproximate k l by k l , N satisfying (4.13). Then k l ( v , u ) k l ( u , u ) (cid:3) (cid:2) k l ( v , u ) − k l , N ( v , u ) (cid:3) k l ( u , u ) + (cid:2) k l ( u , u ) − k l , N ( u , u ) (cid:3) k l , N ( v , u ) + k l , N ( v , u ) k l , N ( u , u ) , LOBAL EXISTENCE FOR THE BOLTZMANN EQUATION 16 and from this and (4.15), the L r ({| v | ≤ N } ; L ∞ t , x ) norm of the last term of(4.16) is bounded by C r N − rr ′ (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x + C l , N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∫ t κ ds e − ν ( v )( t − s ) ∫ s − κ ds e − ν ( u )( s − s ) × ∫ | u |≤ N du ∫ | u |≤ N du (cid:12)(cid:12) f ( s , y , u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x ≤ C r N − rr ′ (cid:12)(cid:12)(cid:12)(cid:12) w l f (cid:12)(cid:12)(cid:12)(cid:12) L rv L ∞ t , x + C l , N ,κ E . Hence we can obtain the lemma from above estimates. (cid:3)