Operator splitting for dispersion-generalized Benjamin-Ono equations
aa r X i v : . [ m a t h . A P ] J a n OPERATOR SPLITTING FOR DISPERSION-GENERALIZEDBENJAMIN-ONO EQUATIONS
TAKANOBU TOKUMASU
Abstract.
We consider the operator splitting for a class of nonlinear equation,which includes the KdV equation, the Benjamin-Ono equation, and the Burgersequation. We prove a first-order approxomation in ∆ t in the Sobolev space forthe Godunov splitting, and second-order approximation for the Strang splitting. Introduction
In this paper, we consider the operator splitting of ∂ t u + uu x − Ku = 0 , ( x, t ) ∈ R × [0 , T ] ,u ( · ,
0) = u ∈ H s ( R ) (1.1)where K = F − [ k ( ξ ) F ] is a Fourier multiplier, and u and u are R -valued. Through-out this paper, we assume that p ∈ [0 , ∞ ) and k satisfies ℜ ( k ( ξ )) ≤ , k ( − ξ ) = k ( ξ ) , | k ( ξ ) | . h ξ i p , (1.2) | ( ξ + η ) k ( ξ + η ) − ηk ( η ) − ξk ( ξ ) | . | ξ |h η i p + | η |h ξ i p . (1.3)for all ξ , η ∈ R , where h ξ i = (1 + | ξ | ) / . Note that (1.3) is satisfied if k ∈ C ( R ).Here we give some examples. When k ( ξ ) = − iξ , (1.1) is the Korteweg-de Vries(KdV) equation. When k ( ξ ) = iξ | ξ | , (1.1) is the Benjamio-Ono equation. When k ( ξ ) = − ξ , (1.1) is the Burgers equation. The following are the case p is a fractionalnumber. The extended Whitham equation, which is a model for surface water waves(see [9] and [10]), is u t + uu x − Z R e ixξ (1 + β | ξ | ) (cid:16) tanh ξξ (cid:17) iξ b u ( ξ ) dξ = 0 , where β is a measure of surface tension. (see [10]) When β >
0, this satisfies (1.2)and (1.3) with p = 3 /
2. The case β = 0 is the case of no surface tension and calledthe Whitham equation (see [9]), and also the model for purely gravitational waves(see [10]), and this satisfies (1.2) and (1.3) with p = 1 / C [ t ] u ∈ X , where X is some normed space,denotes the solution of the differential equation ∂ t u = C ( u ) at t = t , where u is the initial value. Typical C includes a differential operator in the spatial variables.Here, we assume that C = A + B . In our setting, A = K and B ( u ) = − uu x . In theGodunov operator splitting, the approximate solution is u ( t n ) ≈ (cid:16) Φ A (∆ t ) ◦ Φ B (∆ t ) (cid:17) n u , where n ∈ N , ∆ t ≪
1, and t n = n ∆ t . In the Strang operator splitting, the approxi-mate solution is u ( t n ) ≈ (cid:16) Φ A ( ∆ t ◦ Φ B (∆ t ) ◦ Φ A ( ∆ t (cid:17) n u . Let N = N (∆ t, T ) be the largest integer such that N ∆ t ≤ T for given T > t >
0. Let n ∈ N satisfy 1 ≤ n ≤ N . The Godunov splitting for (1.1) is defined in( x, t, τ ) ∈ R × Π ( n ) G as ∂ t v + vv x = 0 , x ∈ R , ( t, τ ) ∈ Σ ( n ) G, , (1.4) ∂ τ v − Kv = 0 , x ∈ R , ( t, τ ) ∈ Σ ( n ) G, , (1.5) v (0 ,
0) = u ∈ H s ( R ) , (1.6)where Π ( n ) G = Σ ( n ) G, ∪ Σ ( n ) G, andΣ ( n ) G, = n [ l =1 Ω ( l )1 , Σ ( n ) G, = n [ l =1 Ω ( l )2 , and Ω ( n )1 = ( t n − , t n ] × { t n − } , Ω ( n )2 = [ t n − , t n ] × ( t n − , t n ] . The Strang splitting for (1.1) is defined in ( x, t, τ ) ∈ R × Π ( n ) S as ∂ t v + vv x = 0 , x ∈ R , ( t, τ ) ∈ Σ ( n ) S, , (1.7) ∂ τ v − Kv = 0 , x ∈ R , ( t, τ ) ∈ Σ ( n ) S, , (1.8) v (0 ,
0) = u ∈ H s ( R ) , (1.9)where Π ( n ) S = Σ ( n ) S, ∪ Σ ( n ) S, andΣ ( n ) S, = n [ l =1 (cid:16) Ω ( l )1 , ∪ Ω ( l )1 , (cid:17) , Σ ( n ) S, = n [ l =1 (cid:16) Ω ( l )2 , ∪ Ω ( l )2 , (cid:17) . and Ω ( n )1 , = ( t n − , t n − / ] × { t n − } , Ω ( n )1 , = ( t n − / , t n ] × [ t n − / , t n ] , Ω ( n )2 , = [ t n − , t n − / ] × ( t n − , t n − / ] , Ω ( n )2 , = { t n − / } × ( t n − / , t n ] , where t n − / = ( n − / t for n ∈ N . PERATOR SPLITTING FOR DISPERSION-GENERALIZED BENJAMIN-ONO EQUATIONS 3
Since the splitting method requires the existence of the solution for the full equa-tion (1.1), we give some known results. For the KdV equation, Colliander-Keel-Steffilani-Takaoka-Tao proved global-wellposedness (GWP) in H s for s > − / H − / (see [2]). For the Benjamin-Ono equation,Tao proved GWP in H (see [3]) and Ionescu-Kenig proved GWP in L (see [4]). For k ( ξ ) = − iξ | ξ | p − , Guo proved the local-wellposedness (LWP) in H − p for p ∈ [2 , H ( p − / for p ∈ (7 / ,
3] are also proved in [5].On the other hand, we have no results for the the global existence of the solutionfor the full equation (1.1) for general k . Throughout this paper, we assume thatthere exists a solution u ∈ C ([0 , T ] : H s ) of (1.1) and a constant C > t ∈ [0 ,T ] k u ( t ) k H s ( R ) ≤ C . (1.10)The splitting method also requires the solvability for (1.4), (1.5), (1.7), and (1.8).The global existence for the linear equations are obvious. In [6] Kato proved theexistence and the uniqueness of a local solution for a class of nonlinear equations,which includes the nonlinear equations, (1.4) and (1.7). (Theorem 2.1)In this paper, we prove two types of the error estimate. These proofs are basedon the method used in [7]. The first is the first-order approximation in ∆ t for theGodunov splitting, that is the following. Theorem 1.1.
Let
T > and s > / { , p } . Assume that u ∈ C ([0 , T ] : H s ( R )) satisfies (1 . on [0 , T ] and C > satisfies (1 . . Then there exists ∆ t =∆ t ( C , s, p, T ) > such that for all ∆ t ∈ [0 , ∆ t ] , there exists a unique solution v ∈ C (Π ( N ) G : H s ) of (1 . – (1 . . In addition, there exists C = C ( C , s, p, T ) > such that sup t ∈ [0 ,N ∆ t ] k v ( t, t ) − u ( t ) k H s − max { ,p } ≤ C ∆ t. Remark . Theorem 1.1 is also true for the Cauchy problem of ∂ t v − Kv = 0 , x ∈ R , ( t, τ ) ∈ Σ ( n ) G, ,∂ τ v + vv x = 0 , x ∈ R , ( t, τ ) ∈ Σ ( n ) G, ,v (0 ,
0) = u ∈ H s ( R ) . The proof is the same as that of Theorem 1.1.The second is the second-order approximation in ∆ t for the Strang splitting, thatis the following. TAKANOBU TOKUMASU
Theorem 1.3.
Let
T > and s > / { , p } . Assume that u ∈ C ([0 , T ] : H s ( R )) satisfies (1 . on [0 , T ] and C > satisfies (1 . . Then there exists ∆ t =∆ t ( C , s, p, T ) > such that for all ∆ t ∈ [0 , ∆ t ] , there exists a unique solution v ∈ C (Π ( N ) S : H s ) of the Cauchy problem (1 . – (1 . . In addition, there exists C = C ( C , s, p, T ) > such that sup t ∈ [0 ,N ∆ t ] k v ( t, t ) − u ( t ) k H s − { ,p } ≤ C (∆ t ) . Remark . Theorem 1.3 is also true for the Cauchy problem of ∂ t v − Kv = 0 , x ∈ R , ( t, τ ) ∈ Σ ( n ) S, ,∂ τ v + vv x = 0 , x ∈ R , ( t, τ ) ∈ Σ ( n ) S, ,v (0 ,
0) = u ∈ H s ( R ) . The proof is the same as that of Theorem 1.3.Here we mention the privious results for the splitting method. Holden-Karlsen-Risebro-Tao proved the first-order approximation in ∆ t for the KdV equation in H s − when u ∈ H s and s ≥ H s − when u ∈ H s and s ≥
17 is an odd integer. Our result is a generalization of these resultsbecause we can apply Theorem 1.1 for the KdV equation for s > / s > /
2, respectively. Holden-Karlsen-Risebro provedthe first-order approximation in ∆ t in H s − p and the second-order approximation in H s − p +1 for the case k is a polynomial, where s is sufficiently large, u ∈ H s , and p ≥ k (see [8]). Dutta-Holden-Koley-Risebro proved the first-orderapproximation in ∆ t in L for the Benjamin-Ono equation for u ∈ H / and thesecond-order approximation in L for the Benjamin-Ono equation for u ∈ H / in[11]. As far as the author know, no previous results of the splitting method exist forthe extended Whitham equation mentioned above. On the other hand, our resultsinclude that for the extended Whitham equation, because our theorems work for p ≥ PERATOR SPLITTING FOR DISPERSION-GENERALIZED BENJAMIN-ONO EQUATIONS 5 preliminaries First, we mention the solvability of the inviscid Burgers equation ( ∂ t v + vv x = 0 , x ∈ R , t ∈ [0 , T ′ ] , (2.1) v (0) = v ∈ H s ( R ) . (2.2) Theorem 2.1.
Let s > / . Then, there exists T ′ = T ′ ( s, k v k H s ( R ) ) > such thatfor all T ≤ T ′ there exists a unique solution v ∈ C ([0 , T ] : H s ( R )) ∩ C ([0 , T ] : H s − ( R )) of (2 . and (2 . defined on t ∈ [0 , T ] . We have Theorem 2.1 from Theorem II in [6] by Kato.
Corollary 2.2.
Let σ > / , τ ≥ , v ( τ , τ ) ∈ H σ ( R ) , and M > satisfy k v ( τ , τ ) k H σ ≤ M . Then, there exists ∆ t B = ∆ t B ( σ, M ) > such that for all ∆ t ≤ ∆ t B there exists a unique solution v ∈ C ([ τ , τ + ∆ t ] : H σ ( R )) ∩ C ([ τ , τ +∆ t ] : H σ − ( R )) of ( ∂ t v + vv x = 0 , x ∈ R , ( t, τ ) ∈ ( τ , τ + ∆ t ] × { τ } , (2.3) ∂ τ v − Kv = 0 , x ∈ R , ( t, τ ) ∈ [ τ , τ + ∆ t ] × ( τ , τ + ∆ t ] . (2.4)We can solve (2.4) as v ( t, τ, x ) = F − ξ [ e k ( ξ )( τ − σ ) F x [ v ( t, σ, x )]]. Therefore we haveCorollary 2.2 from Theorem 2.1. Lemma 2.3.
Let s ≥ σ > / . Then there exists C = C ( s ) > such that for all f , g ∈ H s ( R ) k ∂ x h ∂ x i s ( f g ) − ( ∂ x h ∂ x i s f ) g − f ( ∂ x h ∂ x i s g ) k L ≤ C ( k f k H s k g k H σ + k f k H σ k g k H s ) , where h ∂ x i s = F − ξ h ξ i s F x .Proof. We put h ( ξ ) = ξ h ξ i s . First, we prove | h ( ξ ) − h ( ξ − ξ ) − h ( ξ ) | ≤ C ( | ξ |h ξ − ξ i s + | ξ − ξ |h ξ i s ) . (2.5)By symmetry, we only need to prove the case | ξ | ≤ | ξ − ξ | . We have | h ′ ( ξ ) | = | s + 1) ξ |h ξ i s − ≤ C h ξ i s . By the mean-value theorem, we have | h ( ξ ) − h ( ξ − ξ ) | = | ξ || h ′ ( ξ − θξ ) | ≤ C | ξ |h ξ − θξ i s ≤ C | ξ |h ξ − ξ i s , where θ ∈ (0 , | ξ | ≤ | ξ − ξ | , we have | h ( ξ ) | ≤ | ξ |h ξ − ξ i s . Thus wehave (2.5). Next, we prove Lemma 2.3. By the Sobolev inequality, the Plancherel TAKANOBU TOKUMASU equality, and (2.5), we have
L.H.S. ≤ C (cid:13)(cid:13)(cid:13) Z R { h ( ξ ) − h ( ξ − ξ ) − h ( ξ ) } b f ( ξ − ξ ) b g ( ξ ) dξ (cid:13)(cid:13)(cid:13) L ≤ C (cid:13)(cid:13)(cid:13) Z R {| ξ |h ξ − ξ i s + | ξ − ξ |h ξ i s }| b f ( ξ − ξ ) || b g ( ξ ) | dξ (cid:13)(cid:13)(cid:13) L ≤ C ( kh ξ i s | b f |k L k| ξ || b g |k L + kh ξ i s | b g |k L k| ξ || b f |k L ) ≤ C ( k f k H s k g k H σ + k g k H s k f k H σ ) . (cid:3) Proposition 2.4.
Let s ≥ σ > / . Assume that f and g are R -valued functions. ( A ) Then there exists C = C ( s ) > such that for all f , g ∈ H s ( R ) |h f, ( f g ) x i H s | ≤ C k f k H s k g k H s +1 . (2.6)( B ) Then there exists C = C ( s, σ ) > such that for all f ∈ H s ( R ) |h f, f f x i H s | ≤ C k f k H s k f k H σ . (2.7) Proof.
First, we show (A). By the definition of inner product of H s ( R ), we have h f, ( f g ) x i H s = hh ∂ x i s f, ( ∂ x h ∂ x i s f ) g i L + hh ∂ x i s f, f ( ∂ x h ∂ x i s g ) i L + hh ∂ x i s f, ∂ x h ∂ x i s ( f g ) − ( ∂ x h ∂ x i s f ) g − f ( ∂ x h ∂ x i s g ) i L . (2.8)For the first term, by integration by parts and the Sobolev inequality, we have |hh ∂ x i s f, ( ∂ x h ∂ x i s f ) g i L | = | − hh ∂ x i s f, ( h ∂ x i s f )( ∂ x g ) i L |≤ C k f k H s k g k H σ . (2.9)For the second term, by the Sobolev inequality, we have |hh ∂ x i s f, f ( ∂ x h ∂ x i s g ) i L | ≤ C k f k H s k f k H σ − k g k H s +1 . By Lemma 2.3, the third term in (2.8) is bounded by k f k H s k ∂ x h ∂ x i s ( f g ) − ( ∂ x h ∂ x i s f ) g − f ( ∂ x h ∂ x i s g ) k L ≤ C k f k H s ( k f k H s k g k H σ + k f k H σ k g k H s ) . (2.10)Since s ≥ σ , we have the desired result. Next, we prove (B). We put g = f in (2.8).Then the second term in (2.8) is equal to the first term. Therefore, we have thedesired result by (2.9) and (2.10). (cid:3) The following lemma is so called bootstrap lemma, which follows from the conti-nuity of v and the connectivity of Π ( n ) . PERATOR SPLITTING FOR DISPERSION-GENERALIZED BENJAMIN-ONO EQUATIONS 7
Lemma 2.5. (bootstrap lemma) Let σ > / , ∆ t > , n ∈ N , and Π ( n ) = Π ( n ) G or Π ( n ) S . Assume that v ∈ C (Π ( n ) : H σ ) and C > satisfy the following two conditions. ( A ) k v (0 , k H σ ≤ C . ( B ) sup ( t,τ ) ∈ Π ( n ) k v ( t, τ ) k H σ ≤ C / holds if sup ( t,τ ) ∈ Π ( n ) k v ( t, τ ) k H σ ≤ C .Then we have sup ( t,τ ) ∈ Π ( n ) k v ( t, τ ) k H σ ≤ C / . Lemma 2.6.
Let ∆ t > , T > , n ∈ N such that n ∆ t ≤ T , Π ( n ) = Π ( n ) G (resp. Π ( n ) S ) and s ≥ σ > / . Assume that v ∈ C (Π ( n ) : H σ ) satisfy (1 . – (1 . (resp. (1 . – (1 . ) on Π ( n ) . Assume that C > and v satisfy sup ( t,τ ) ∈ Π ( n ) k v ( t, τ ) k H σ ≤ C .Then there exists C ′ = C ′ ( k u k H s , C , s, σ, T ) > such that sup ( t,τ ) ∈ Π ( n ) k v ( t, τ ) k H s ≤ C ′ . (2.11) Proof.
We prove the case Π ( n ) = Π ( n ) G . From (1.2) and (1.5), k v ( t, τ ) k H s is monoton-ically decresing with respect to τ in Ω ( n )2 . So we only need to prove the boundnessof k v ( t, t n − ) k H s in Ω ( n )1 . By (1.4), (2.7), and sup ( t,τ ) ∈ Π ( n ) G k v ( t, τ ) k H σ ≤ C , we have ddt k v ( t, t n − ) k H s = − h v ( t, t n − ) , v ( t, t n − ) v x ( t, t n − ) i H s ≤ C k v ( t, t n − ) k H s k v ( t, t n − ) k H σ ≤ CC k v ( t, t n − ) k H s . (2.12)We have k v ( t, t n − ) k H s ≤ k v ( t n − , t n − ) k H s e CC ( t − t n − ) by applying the Gronwallinequality to (2.12). Therefore it follows that k v ( t, τ ) k H s ≤ k v ( t n − , t n − ) k H s e CC ( t − t n − ) ≤ k v ( t n − , t n − ) k H s e CC ( t n − − t n − ) e CC ( t − t n − ) ≤ · · · ≤ k v (0 , k H s e CC t ≤ k u k H s e CC T . (2.13)Similar arguments apply to the case Π ( n ) = Π ( n ) S . (cid:3) Estimate for the Godunov splitting
The main estimate in this section is Proposition 3.1 below.
Proposition 3.1.
Let ∆ t > , T > , n ∈ N such that n ∆ t ≤ T , and s = s − max { , p } > / . Assume that v ∈ C (Π ( n ) G : H s ) and C ′ > satisfy (1 . – (1 . and (2 . . Then there exists C = C ( C , C ′ , s, s , T ) > such that sup t ∈ [0 ,n ∆ t ] k v ( t, t ) − u ( t ) k H s ≤ C ∆ t. TAKANOBU TOKUMASU
Proposition 3.1 follows from Lemmas 3.2 and 3.3 below.
Lemma 3.2.
Let F = v t + vv x and F ( t ) = F ( t, t ) . Under the same assumptions ofProposition 3.1, there exists C = C ( C , C ′ , s, s , T ) > such that k v ( t, t ) − u ( t ) k H s ≤ C Z t k F ( t ′ ) k H s dt ′ (3.1) for all t ∈ [0 , n ∆ t ] .Proof. Let w ( t ) = v ( t, t ) − u ( t ). By (1.1), (1.4), and the definition of F , we have ∂∂t w ( t, x ) = v t ( t, τ, x ) | τ = t + v τ ( t, τ, x ) | τ = t − u t ( t, x )= − ww x − ( uw ) x + Kw + v t + vv x = − ww x − ( uw ) x + Kw + F (3.2)In view of (3.2), we call F the forcing term. Then we have ddt k w ( t ) k H s = 2 h w, − ww x − ( uw ) x + Kw + F i H s . (3.3)Note that h w, Kw i H s ≤ ddt k w ( t ) k H s ≤ C {k w k H s ( k w k H s + k u k H s ) + k F k H s }≤ C ( k w k H s + k F k H s ) . (3.4)Here we used k w k H s ≤ k u k H s + k v k H s ≤ C + C ′ and k u k H s ≤ C . Applying theGronwall inequality and w (0) = 0 to (3.4), and we have k w ( t ) k H s ≤ C Z t k F ( σ ) k H s dσ. (cid:3) Lemma 3.3.
Let F = v t + vv x and F ( t ) = F ( t, t ) . Under the same assumptions ofProposition 3.1, there exists C = C ( k u k H s , C ′ , s, s , T ) > such that sup t ∈ [0 ,n ∆ t ] k F ( t ) k H s ≤ C ∆ t. (3.5) Proof.
By (1.5) and the definition of F , the forcing term F satisfies ∂ τ F ( t, τ ) − KF ( t, τ ) = X ( t, τ ) (3.6)in ( t, τ ) ∈ Π ( n ) G , where X ( t, τ ) = X ( t, τ ) + X ( t, τ ) and X ( t, τ ) and X ( t, τ ) aredefined as below. X ( t, τ ) = − n K ( v ) x − vKv x o , X ( t, τ ) = − v x Kv PERATOR SPLITTING FOR DISPERSION-GENERALIZED BENJAMIN-ONO EQUATIONS 9
By (3.6), we get ∂ τ k F ( t, τ ) k H s = 2 h F, KF i H s + 2 h F, X i H s . (3.7)The first term in (3.7) is equal to or less than 0 because of (1.2). For the secondterm, by (1.3), the Sovolev inequality, and Lemma 2.6, we have k X k H s = 12 kh ξ i s Z R { ξk ( ξ ) − ξ k ( ξ ) − ( ξ − ξ ) k ( ξ − ξ ) } b v ( t, τ, ξ − ξ ) b v ( t, τ, ξ ) dξ k L ≤ C ( kh ξ i s | ξ || b v |k L kh ξ i p | b v |k L + kh ξ i s h ξ i p | b v |k L k| ξ || b v |k L ) ≤ C ( k v k H s k v k H s p + k v k H s p k v k H s ) ≤ C, (3.8)where we used s = s + max { , p } and s > / . s and s written above, by the Sobolev inequality,we have k X k H s ≤ C k v k H s k v k H s p ≤ C . Therefore, we have ∂ τ k F ( t, τ ) k H s ≤ C. (3.9)Take k ∈ N such that ( t, τ ) ∈ [ t k , t k +1 ] . Since F ( t, t k ) = 0, k F ( t, τ ) k H s = k F ( t, τ ) k H s − k F ( t, t k ) k H s = Z τt k ∂ σ k F ( t, σ ) k H s dσ ≤ C | τ − t k | ≤ C ∆ t. (cid:3) As a corollary of Proposition 3.1, we have the following.
Corollary 3.4.
Let
T > and s = s − max { , p } > / . Then there exist ∆ t ∗ =∆ t ∗ ( C , s, s , T ) > and C ∗ = C ∗ ( C , s, s , T ) > such that, for all ∆ t ≤ ∆ t ∗ , n ∈ N satisfying n ∆ t ≤ T , and v ∈ C (Π ( n ) G : H s ) satisfying (1 . – (1 . on Π ( n ) G , itfollows that v ∈ C (Π ( n ) G : H s ) and sup ( t,τ ) ∈ Π ( n ) G k v ( t, τ ) k H s ≤ C , (3.10)sup t ∈ [0 ,n ∆ t ] k v ( t, t ) − u ( t ) k H s ≤ C ∗ ∆ t. (3.11) Proof.
First, we prove (3.10). For that purpose, we only need to prove (A) and (B)of Lemma 2.5 with C = 4 C . Obviously, (A) holds by (1.10). Next, we prove (B).Assume that sup ( t,τ ) ∈ Π ( n ) G k v ( t, τ ) k H s ≤ C . By Lemma 2.6 with C = 4 C , thereexists C = C ( C , s, s , T ) > ( t,τ ) ∈ Π ( n ) G k v ( t, τ ) k H s ≤ C . By s + p ≤ s , (1.2), and (1.5), we have k v ( t, τ ) − v ( t, t ) k H s ≤ Z τt k ∂ σ v ( t, σ ) k H s dσ ≤ | t − τ | sup ( t,τ ) ∈ Π ( n ) G k v ( t, τ ) k H s p ≤ C ∆ t. (3.12)Then, by Proposition 3.1, it follows that for all ∆ t ≤ ∆ t ∗ , k v ( t, τ ) k H / ǫ ≤ k v ( t, τ ) − v ( t, t ) k H s + k v ( t, t ) − u ( t ) k H s + k u ( t ) k H s ≤ C ∆ t + C ≤ C . Here we take ∆ t ∗ such that C ∆ t ∗ = C . Thus, we obtain (B). Therefore, we have(3.10) by Lemma 2.5.By applying Lemma 2.6 to (3.10), it is also proved that v ∈ C (Π ( n ) G : H s ) and thereexists C = C ( C , s, s , T ) > ( t,τ ) ∈ Π ( n ) G k v ( t, τ ) k H s ≤ C . Therefore, byProposition 3.1, we have (3.11). (cid:3) Finally we prove Theorem 1.1.
Proof.
Let ∆ t = min { ∆ t ∗ , ∆ t B ( s , C ) } , where ∆ t B = ∆ t B ( σ, M ) is defined inCorollary 2.2. Note that ∆ t = ∆ t ( C , s, s , T ).We put conditions ( A ) n and ( B ) n for n ∈ N satisfying 1 ≤ n ≤ N as below.( A ) n : For any ∆ t ≤ ∆ t , there exists a unique solution v ∈ C (Π ( n ) G : H s ( R )) whichsatisfies (1 . . B ) n : For any ∆ t ≤ ∆ t , the solution v ∈ C (Π ( n ) G : H s ( R )) of (1.4)–(1.6) satisfies v ∈ C (Π ( n ) G : H s ( R )), (3.10), and (3.11).The proof is by induction on n . Obviously, ( A ) and ( B ) are true. Let l ∈ N suchthat 1 ≤ l ≤ N −
1. We assume ( A ) l and ( B ) l , and prove ( A ) l +1 and ( B ) l +1 . First,we prove that ( A ) l +1 holds. Since ∆ t ≤ ∆ t ≤ ∆ t B ∗ , we have ( A ) l +1 by (3.10) with n = l and Corollary 2.2. Next, we prove ( B ) l +1 from ( A ) l +1 , but this has alreadybeen proved as Corollary 3.4.By induction on n , we have Theorem 1.1 for C = C ( C , C ′ | C =2 C , s, s , T ), where C is the constant in Proposition 3.1. (cid:3) Estimate for the Strang splitting
In this section, we prove Theorem 1.3. We put w ( t ) = v ( t, t ) − u ( t ), Λ ( n )1 = ∪ nl =1 (Ω ( l )1 , ∪ Ω ( l )2 , ), Λ ( n )2 = ∪ nl =1 (Ω ( l )2 , ∪ Ω ( l )1 , ), and tilde is a time-shift operator, that is e f ( t, x ) = f ( t + ∆ t/ , x ). PERATOR SPLITTING FOR DISPERSION-GENERALIZED BENJAMIN-ONO EQUATIONS 11
The main proposition in this section is Proposition 4.1 below. We obtain Theorem1.3 in the same manner as in Section 3 if we use Proposition 4.1 instead of Proposition3.1. Therefore, we only need to prove Proposition 4.1.
Proposition 4.1.
Let ∆ t > , T > , n ∈ N such that n ∆ t ≤ T , and s = s − { , p } > / . Assume that there exists a unique solution v ∈ C (Π ( n ) S : H s ) of (1 . – (1 . and a constant C ′ > satisfies (2 . . Then, there exists C = C ( C , C ′ , s, s , T ) > such that sup t ∈ [0 ,t n ] k w ( t ) k H s ≤ C (∆ t ) . (4.1)For the proof of Proposition 4.1, we only need to provesup t ∈ [0 ,t n − / ] k w ( t ) + e w ( t ) k H s ≤ C ′ (∆ t ) (4.2)sup t ∈ [0 ,t n − / ] k w ( t ) − e w ( t ) k H s ≤ C ′ (∆ t ) (4.3)instead of (4.1), sincesup t ∈ [0 ,t n ] k w ( t ) k H s ≤ sup t ∈ [0 ,t n − / ] k w ( t ) k H s + sup t ∈ [0 ,t n − / ] k e w ( t ) k H s ≤ sup t ∈ [0 ,t n − / ] k w ( t ) + e w ( t ) k H s + sup t ∈ [0 ,t n − / ] k w ( t ) − e w ( t ) k H s . First, we prepare some notations to prove (4.2). We put F ( t, τ ) = v t ( t, τ ) + v ( t, τ ) v x ( t, τ ) , ( t, τ ) ∈ Λ ( n )1 , , ( t, τ ) ∈ Λ ( n )2 , (4.4) G ( t, τ ) = , ( t, τ ) ∈ Λ ( n )1 ,v τ ( t, τ ) − Kv ( t, τ ) , ( t, τ ) ∈ Λ ( n )2 . (4.5)We also define the total forcing term H in Π ( n ) S as H ( t, τ ) = F ( t, τ ) + G ( t, τ ) and H ( t ) = H ( t, t ). By (1 . . .
4) for the case ( t, t ) ∈ Λ ( n )1 and (1 . . .
5) for the case ( t, t ) ∈ Λ ( n )2 , w ′ is written in t ∈ [0 , t n ] as w ′ = − ww x − ( uw ) x + Kw + H ( t ) . (4.6)For simplicity, we put z ( t ) = w ( t ) + e w ( t ) in t ∈ [0 , t n − / ].Next, we prepare some lemmas to estimate k z ( t ) k H s . Lemma 4.2.
Let H ( t, τ ) = F ( t, τ ) + G ( t, τ ) and H ( t ) = H ( t, t ) , where F satisfies (4.4) and G satisfies (4.5) . Under the same assumption of Proposition 4.1, there exists C = C ( C , C ′ , s, s , T ) > such that for all t ∈ [0 , t n − / ] , k z ( t ) k H s ≤ k z (0) k H s e Ct + Ct { sup t ∈ [0 ,t n − / ] k H ( t ) + e H ( t ) k H s + ( sup t ∈ [0 ,t n ] k w k H s ) + sup t ∈ [0 ,t n ] k w k H s sup t ∈ [0 ,t n − / ] k e u − u k H s } . (4.7) Proof.
By (4.6), it follows that for t ∈ [0 , t n − / ], e w ′ = e H ( t ) + K e w − ( e u e w ) x − e w e w x . (4.8)By (4 .
6) and (4 . z ′ in t ∈ [0 , t n − / ] as below. z ′ = H ( t ) + e H ( t ) − (cid:16) z + uz (cid:17) x + Kz − n e w ( e u − u ) + w e w o x . (4.9)Then it follows that ddt k z ( t ) k H s = D z, H ( t ) + e H ( t ) − (cid:16) z + uz (cid:17) x + Kz − n e w ( e u − u ) + w e w o x E H s . Note that h z, Kz i H s ≤ t ∈ [0 , t n − / ], ddt k z ( t ) k H s ≤ C {k z k H s + k H ( t ) + e H ( t ) k H s + k w k H s k e w k H s + k e w k H s k e u − u k H s } . (4.10)Here we used the following inequality. k z k H s ≤ k w k H s + k e w k H s ≤ k u k H s + k v k H s + k e u k H s + k e v k H s ≤ C. Applying the Gronwall inequality to (4.10), we have (4.7). (cid:3)
Lemma 4.3.
Let X = −{ K ( v ) x / − v x Kv − vKv x } . Under the same assumptionof Proposition 4.1 , there exists C = C ( k u k H s , C ′ , s, s , T ) > such that k X ( t ) − X ( t ) k H s ≤ C | t − t | (4.11) for all t , t ∈ [0 , t n ] .Remark . Since F = v t + vv x , G = v τ − Kv , and v satisfies (1.7) and (1.8), wehave F τ − KF = X , ( t, τ ) ∈ Λ ( n )1 , { G t + ( vG ) x } = − X , ( t, τ ) ∈ Λ ( n )2 . Before proving Lemma 4.3, we estimate v t and v τ . PERATOR SPLITTING FOR DISPERSION-GENERALIZED BENJAMIN-ONO EQUATIONS 13
Lemma 4.5.
Let j = 1 , , and / < σ ≤ s − j max { , p } for each j . Under thesame assumption of Proposition 4.1, there exists C = C ( k u k H s , C ′ , s, s , T ) > such that k ( ∂ t ) j v ( t, τ ) k H σ ≤ j Y m =0 sup ( t,τ ) ∈ Π ( n ) S k v ( t, τ ) k H σ + m , (4.12) k ( ∂ τ ) j v ( t, τ ) k H σ ≤ C sup ( t,τ ) ∈ Π ( n ) S k v ( t, τ ) k H σ + jp . (4.13) Proof.
First, we prove (4.12) for the case ( t, τ ) ∈ Λ ( n )1 with j = 1. Since (1.2) and(1.8), it follows that for ( t, τ ) ∈ Λ ( n )1 ∩ Σ ( n ) S, and 3 / < σ ≤ s , ∂ τ k v t ( t, τ ) k H σ = 2 h v t , Kv t i H σ ≤ . (4.14)By (1.7), it follows that for ( t, τ ) ∈ Λ ( n )1 , l ∈ N such that t ∈ [ t l − , t l − / ], and3 / < σ ≤ s − k v t ( t, τ ) k H σ ≤ k v t ( t, t l − ) k H σ ≤ k v ( t, t l − ) k H σ k v ( t, t l − ) k H σ +1 ≤ Y j =0 sup ( t,τ ) ∈ Π ( n ) S k v ( t, τ ) k H σ + j . Next, we prove (4.12) for the case ( t, τ ) ∈ Λ ( n )1 with j = 2 ,
3. By induction argument,we have that ∂ t ( vv x ) becomes the ( p + 1)-th polynomial with p derivatives. By (1.2),we have ∂ τ k ( ∂ t ) j v ( t, τ ) k H σ ≤
0. Therefore, the same argument as the case j = 1works and we have (4.12).Next, we prove (4.12) for the case ( t, τ ) ∈ Λ ( n )2 with j = 1. By (1.8), it followsthat for ( t, τ ) ∈ Λ ( n )2 and 3 / < σ ≤ s − k v t ( t, τ ) k H σ ≤ k v ( t, τ ) k H σ k v ( t, τ ) k H σ +1 ≤ Y j =0 sup ( t,τ ) ∈ Π ( n ) S k v ( t, τ ) k H σ + j . Then, we have (4.12). (4.12) for the case ( t, τ ) ∈ Λ ( n )2 with j = 2 , ∂ t ( vv x ) becomes the ( p + 1)-th polynomial with p derivatives.Next, we prove (4.13) for the case ( t, τ ) ∈ Λ ( n )2 with j = 1. By (1.7) and Proposi-tion 2.4 (A), it follows that for ( t, τ ) ∈ Λ ( n )2 ∩ Σ ( n ) S, and 3 / < σ ≤ s − ∂ t k v τ ( t, τ ) k H σ = 2 h v τ , ( vv x ) τ i H σ ≤ C k v τ k H σ k v k H σ +1 . By the Gronwall inequality, (1.2), and (1.8), it follows that for ( t, τ ) ∈ Λ ( n )2 , l ∈ N such that τ ∈ [ t l − / , t l ], and 3 / < σ ≤ s − max { , p } , k v τ ( t, τ ) k H σ ≤ e CC ′ ( t − t l − / ) k v τ ( t l − / , τ ) k H σ ≤ e CC ′ ∆ t k v ( t l − / , τ ) k H σ + p ≤ e CC ′ ∆ t sup ( t,τ ) ∈ Π ( n ) S k v ( t, τ ) k H σ + p . Next, we prove (4.13) for the case ( t, τ ) ∈ Λ ( n )2 with j = 2 ,
3. In the same man-ner as the case j = 1, by the Gronwall inequality, we have k ( ∂ τ ) j v ( t, τ ) k H σ ≤ e CC ′ ( t − t l − / ) k ( ∂ τ ) j v ( t l − / , τ ) k H σ . Thus, by (1.8), we have (4.13). Finally, we prove(4.13) for the case ( t, τ ) ∈ Λ ( n )1 with j = 1 , ,
3. By (1.2) and (1.8), it follows thatfor ( t, τ ) ∈ Λ ( n )1 and 3 / < σ ≤ s − j , k ( ∂ τ ) j v ( t, τ ) k H σ ≤ k v ( t, τ ) k H σ + jp ≤ sup ( t,τ ) ∈ Π ( n ) S k v ( t, τ ) k H σ + jp . Therefore, we have (4.13). (cid:3)
Next, we prove Lemma 4.3.
Proof.
For the proof of Lemma 4.3, we only need to prove the boundness of X t and X τ in H s . We have the boundness for ( t, τ ) ∈ Λ ( n )2 in the same manner as for( t, τ ) ∈ Λ ( n )1 , so we only give the proof for the case ( t, τ ) ∈ Λ ( n )1 . First, we prove theboundness of X t in H s . By the definition of X , we have X t = ( Kv t ) v x + ( Kv x ) v t + v xt ( Kv ) + ( Kv xt ) v − K ( v xt v ) − K ( v t v x ) . Since s + 2 + p ≤ s , we have k X t k H s ≤ C k v k H s p k v k H s p ≤ C. (4.15)Next, we prove for the boundness of X τ in H s . Since s + 1 + 2 p ≤ s , (1.8), andRemark 4.4, in the same manner as (4.15), we have k X τ k H s ≤ k ( K v ) v x k H s + 2 k ( Kv )( Kv x ) k H s + k v ( K v x ) k H s + k K ( v ( Kv x )) k H s + k K (( Kv ) v x ) k H s ≤ C. (cid:3) Next, we estimate H + e H = F + G + e F + e G . In view of e F = G = 0 in Λ ( n )1 , F = e G = 0 in Λ ( n − , and Remark 4.4, it is natural to estimate e F + G and F + e G . Lemma 4.6.
Let l ∈ N , F satisfy (4.4) , and G satisfy (4.5) . Under the sameassumptions of Proposition 4.1, there exists C = C ( k u k H s , C ′ , s, s , T ) > suchthat (cid:13)(cid:13)(cid:13) F ( t ) + G ( t + ∆ t (cid:13)(cid:13)(cid:13) H s ≤ C (∆ t ) for all t ∈ [ t l − , t l − / ] ⊂ [0 , t n − / ] , and (cid:13)(cid:13)(cid:13) F ( t ) + G ( t − ∆ t (cid:13)(cid:13)(cid:13) H s ≤ C (∆ t ) for all t ∈ [ t l − , t l − / ] ⊂ [ t , t n − / ] . PERATOR SPLITTING FOR DISPERSION-GENERALIZED BENJAMIN-ONO EQUATIONS 15
Proof.
Let Φ = F tt + 2 F tτ + F ττ and Ψ = G tt + 2 G tτ + G ττ . By Taylor expansion at t = t l − , we have F ( t ) + G ( t ± ∆ t { F ( t l − ) + G ( t l − ± ) } + ( t − t l − ) { ( F t + F τ )( t l − ) + ( G t + G τ )( t l − ± ) } + ( t − t l − ) Z { Φ( θ ( t − t l − ) + t l − ) + Ψ( θ ( t ± ∆ t − t l − ± ) + t l − ± ) } dθ. Since F = 0 in Λ ( n )2 ∪ Σ ( n ) S, because of (1.7) and (4.4), and G = 0 in Λ ( n )1 ∪ Σ ( n ) S, becauseof (1.8) and (4.5), we have F ( t l − ) = G ( t l − ± / ) = 0 and F t ( t l − ) = G τ ( t l − ± / ) =0. By Remark 4.4 and F ( t l − ) = G ( t l − ± / ) = 0, we have F τ ( t l − ) + G t ( t l − ± ) = X ( t l − ) − X ( t l − ± ). Then, we have the second order estimate in ∆ t of the secondterm in H s space by Lemma 4.3. k Φ k H s + k Ψ k H s ≤ C is proved by (1.7)–(1.9), Proposition 2.4, (2.11), (4.12) , and (4.13). Therefore it follows that for t ∈ [ t l − , t l − / ], k F ( t ) + G ( t ± ∆ t k H s ≤ | t − t l − |k X ( t l − ± ) − X ( t l − ) k H s + C | t − t l − | ≤ C (∆ t ) . (4.16) (cid:3) Lemma 4.7.
Let s = s − { , p } > / . Assume that u ∈ C ([0 , t / ] : H s ) satisfies (1 . on [0 , t / ] and v ∈ C (Λ (1)1 : H s ) satisfies (1 . – (1 . on Λ (1)1 . Assumethat a constant C ′ > satisfies sup ( t,τ ) ∈ Λ (1)1 k v ( t, τ ) k H s ≤ C ′ . Then, there exists C = C (sup [0 ,t / ] k u ( t ) k H s , C ′ , s, s ) > such that k w ( t ) k H s ≤ C (∆ t ) for all t ∈ [0 , t / ] .Proof. We use the Taylor expantion of w ( t ) at t = 0. That is w ( t ) = w (0) + t (cid:0) ∂ t v ( t, τ ) | t = τ =0 + ∂ τ v ( t, τ ) | t = τ =0 − u ′ (0) (cid:1) + t Z { ∂ tt v ( σt, σt ) + 2 ∂ tτ v ( σt, σt ) + ∂ ττ v ( σt, σt ) } dσ. The term of the 0th order of t is 0 because of w (0) = v (0 , − u (0) and u (0) = v (0 , . . . u (0) = v (0 ,
0) to the term of the 1st order of t , wehave ∂ t v ( t, τ ) | t = τ =0 + ∂ τ v ( t, τ ) | t = τ =0 − u ′ (0)= − v (0 , v x (0 ,
0) + Kv (0 ,
0) + u (0) u x (0) − Ku (0) = 0 . By (4.12) and (1.8), it follows that k ∂ tt v ( σt, σt ) + 2 ∂ tτ v ( σt, σt ) + ∂ ττ v ( σt, σt ) k H s ≤ k ∂ tt v ( σt, σt ) k H s + 2 k K∂ t v ( σt, σt ) k H s + k K v ( σt, σt ) k H s ≤ Y j =0 max ( t,τ ) ∈ Λ (1)1 k v ( t, τ ) k H s j + Y j =0 max ( t,τ ) ∈ Λ (1)1 k v ( t, τ ) k H s p + j + max ( t,τ ) ∈ Λ (1)1 k v ( t, τ ) k H s p ≤ C (4.17)Therefore, k w ( t ) k H s ≤ Ct . Since t ≤ (∆ t ) /
2, we have the desired result. (cid:3)
Next, we prove (4.2) by Lemmas 4.2, 4.7, Theorem 1.1, and Remark 1.2.
Proof.
First, we prove three inequalities k z (0) k H s ≤ C (∆ t ) , k e u − u k H s ≤ C ∆ t, k w k H s ≤ C ∆ t. (4.18)We have the first inequality in (4 .
18) by applying Lemma 4.7 for t = (∆ t ) /
2. Notethat z (0) = w (0) + e w (0) = w ( ∆ t ). To prove the second one, we use (1 . . u ( t ) − u ( t ) = R ∆ t u ′ ( t + σ ) dσ . The third one is already proved as Theorem 1.1and Remark 1.2.Finally, we prove (4.2). For ( t, t ) ∈ Λ ( n )1 , since e F = G = 0, we have k H + e H k H s = k F + e G k H s ≤ C (∆ t ) by Lemma 4.6. For ( t, t ) ∈ Λ ( n − (not Λ ( n )2 ), since F = e G = 0,we have k H + e H k H s = k e F + G k H s ≤ C (∆ t ) by Lemma 4.6. Thus, it follows thatfor ( t, t ) ∈ Λ ( n )1 ∪ Λ ( n − , k H ( t ) + e H ( t ) k H s ≤ C (∆ t ) . (4.19)Since ( t, t ) ∈ Λ ( n )1 ∪ Λ ( n − is equivalent to t ∈ [0 , t n − / ], (4.2) follows by applying(4.18) and (4.19) to Lemma 4.2. (cid:3) Next, we prove (4.3). We use the following lemma to prove (4.3) later.
Lemma 4.8.
Let l ∈ N and assumptions of Proposition 4.1 hold. ( A ) Then there exists C = C ( C , C ′ , s, s , T ) > such that for all t ∈ [ t l − , t l − / ] ⊂ [0 , t n ] , k w ( t ) − w ( t l − ) k H s ≤ C (∆ t ) . ( B ) Then there exists C = C ( C , C ′ , s, s , T ) > such that for all t ∈ [ t l − / , t l ] ⊂ [0 , t n ] , k w ( t ) − w ( t l − / ) k H s ≤ C (∆ t ) . PERATOR SPLITTING FOR DISPERSION-GENERALIZED BENJAMIN-ONO EQUATIONS 17
Proof.
We have (B) in the same manner as (A), so we only prove (A). Let V ( t, x )satisfy ∂ t V − V V x + KV = 0 , ( x, t ) ∈ R × [ t l − , T ] ,V ( · , t l − ) = v ( · , t l − , t l − ) ∈ H s ( R ) . (4.20)Note that the first equations in (1.1) and (4.20) are the same, and we may also apply(1.10) to (4.20). First, we decompose w ( t ) − w ( t l − ) as w ( t ) − w ( t l − )= { v ( t ) − V ( t ) } + { V ( t ) − v ( t l − , t l − ) } − { u ( t ) − u ( t l − ) } . (4.21)We have the second-order approximation in ∆ t from the first term in (4.21) byLemma 4.7. The second and the third terms are { V ( t ) − v ( t l − , t l − ) } − { u ( t ) − u ( t l − ) } = Z tt l − { V t ( t ′ ) − u t ( t ′ ) } dt ′ = Z tt l − { V ( t ′ ) V x ( t ′ ) − KV ( t ′ ) − u ( t ′ ) u x ( t ′ ) + Ku ( t ′ ) } dt ′ = Z tt l − { V ( t ′ )( V ( t ′ ) − u ( t ′ )) x + ( V ( t ′ ) − u ( t ′ )) u x ( t ′ ) − K ( V − u )( t ′ ) } dt ′ . Let s ′ = s + max { , p } (= s − { , p } ). Since k V k H s ≤ C and k u x k H s ≤ C ,we have (cid:13)(cid:13)(cid:13) { V ( t ) − v ( t l − , t l − ) } − { u ( t ) − u ( t l − ) } (cid:13)(cid:13)(cid:13) H s ≤ C Z tt l − k V ( t ′ ) − u ( t ′ ) k H s ′ dt ′ ≤ C ∆ t sup t ∈ [ t l − ,t l − / ] k V ( t ) − u ( t ) k H s ′ . k V ( t ) − u ( t ) k H s ′ is estimated from (1.1), (1.10), (4.20), Theorem 1.1, and Remark1.2 as k V ( t ) − u ( t ) k H s ′ ≤ k V ( t ) − v ( t l − , t l − ) k H s ′ + k w ( t l − ) k H s ′ + k u ( t l − ) − u ( t ) k H s ′ ≤ Z tt l − k V ′ ( t ′ ) k H s ′ dt ′ + C ∆ t + Z tt l − k u ′ ( t ′ ) k H s ′ dt ′ ≤ C ∆ t. (4.22)Thus, we have the second-order approximation in ∆ t from the second and the thirdterms in (4.21). Therefore we have Lemma 4.8 (A). (cid:3) Finally, we prove (4.3).
Proof.
Let l ∈ N . By Lemma 4.8, it follows that for t ∈ [ t l − , t l − / ] ⊂ [0 , t n − / ], k e w ( t ) − w ( t ) k H s ≤ k w ( t + ∆ t/ − w ( t l − / ) k H s + k w ( t l − / ) − w ( t l − ) k H s + k w ( t l − ) − w ( t ) k H s ≤ C (∆ t ) , and for t ∈ [ t l − / , t l ] ⊂ [0 , t n − / ], k e w ( t ) − w ( t ) k H s ≤ k w ( t + ∆ t/ − w ( t l ) k H s + k w ( t l ) − w ( t l − / ) k H s + k w ( t l − / ) − w ( t ) k H s ≤ C (∆ t ) . (cid:3) Acknowledgements.
The author would like to thank his advisor, Kotaro Tsugawa,for suggesting the problem and for all the guidance and encourarement along theway
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