The IVP for the Benjamin-Ono equation in weighted Sobolev spaces
aa r X i v : . [ m a t h . A P ] S e p THE IVP FOR THE BENJAMIN-ONO EQUATION INWEIGHTED SOBOLEV SPACES
GERM ´AN FONSECA AND GUSTAVO PONCE
Abstract.
We study the initial value problem associated to the Benjamin-Ono equation. The aim is to establish persistence properties of the solutionflow in the weighted Sobolev spaces Z s,r = H s ( R ) ∩ L ( | x | r dx ), s ∈ R , s ≥ s ≥ r . We also prove some unique continuation properties of the solutionflow in these spaces. In particular, these continuation principles demonstratethat our persistence properties are sharp. Introduction
This work is concerned with the initial value problem (IVP) for the Benjamin-Ono (BO) equation(1.1) ( ∂ t u + H ∂ x u + u∂ x u = 0 , t, x ∈ R ,u ( x,
0) = u ( x ) , where H denotes the Hilbert transform(1.2) H f ( x ) = 1 π p.v.( 1 x ∗ f )( x )= 1 π lim ǫ ↓ Z | y |≥ ǫ f ( x − y ) y dy = − i (sgn( ξ ) b f ( ξ )) ∨ ( x ) . The BO equation was deduced by Benjamin [3] and Ono [29] as a model forlong internal gravity waves in deep stratified fluids. It was also shown that it is acompletely integrable system (see [2], [7] and references therein).Several works have been devoted to the problem of finding the minimal regularity,measured in the Sobolev scale H s ( R ) = (cid:0) − ∂ x (cid:1) − s/ L ( R ), which guarantees thatthe IVP (1.1) is locally or globally wellposed (LWP and GWP, resp.), i.e. existenceand uniqueness hold in a space embedded in C ([0 , T ] : H s ( R )), with the map data-solution from H s ( R ) to C ([0 , T ] : H s ( R )) being locally continuous. Let us recallthem : in [32] s > s > /
2, in [31] s ≥ /
2, in [24] s > /
4, in [21] s > /
8, in [35] s ≥
1, in [5] s > /
4, and finally in [16] s ≥ Mathematics Subject Classification.
Primary: 35B05. Secondary: 35B60.
Key words and phrases.
Benjamin-Ono equation, weighted Sobolev spaces.
Real valued solutions of the IVP (1.1) satisfy infinitely many conservation laws(time invariant quantities), the first three are the following:(1.3) I ( u ) = Z ∞−∞ u ( x, t ) dx, I ( u ) = Z ∞−∞ u ( x, t ) dx,I ( u ) = Z ∞−∞ ( | D / x u | − u x, t ) dx, where D x = H ∂ x .Roughly, for k ≥ k -conservation law I k provides an a priori estimate of the L -norm of the derivatives of order ( k − / k D ( k − / x u ( t ) k .This allows one to deduce GWP from LWP results.For existence of solutions with non-decaying at infinity initial data we refer to[19] and [12].In the BO equation the dispersive effect is described by a non-local operatorand is significantly weaker than that exhibited by the Korteweg-de Vries (KdV)equation ∂ t u + ∂ x u + u∂ x u = 0 . Indeed, it was proven in [26] that for any s ∈ R the map data-solution from H s ( R )to C ([0 , T ] : H s ( R )) is not locally C , and in [25] that it is not locally uniformlycontinuous. This implies that no LWP results can be obtained by an argumentbased only on a contraction method. This is certainly not the case of the KdV (see[23]).Our interest here is to study real valued solutions of the IVP (1.1) in weightedSobolev spaces(1.4) Z s,r = H s ( R ) ∩ L ( | x | r dx ) , s, r ∈ R , and decay properties of solutions of the equation (1.1). In this direction R. Iorio[17] proved the following results: Theorem A. ([17]) (i) The IVP (1.1) is GWP in Z , .(ii) If b u (0) = 0 , then the IVP (1.1) is GWP in ˙ Z , .(iii) If u ( x, t ) is a solution of the IVP (1.1) such that u ∈ C ([0 , T ] : Z , ) forarbitrary T > , then u ( x, t ) ≡ . Above we have introduced the notation(1.5) ˙ Z s,r = { f ∈ H s ( R ) ∩ L ( | x | r dx ) : b f (0) = 0 } , s, r ∈ R . Notice that the conservation law I in (1.3) tells us that the property b u (0) = 0 ispreserved by the solution flow.We observe that the linear part of the equation in (1.1) L = ∂ t + H ∂ x commuteswith the operator Γ = x − t H ∂ x , i.e.[ L ; Γ] = L Γ − Γ L = 0 . In fact, one can deduce (see [17]) that for a solution v ( x, t ) of the associated linearproblem(1.6) v ( x, t ) = U ( t ) v ( x ) = e − it H ∂ x v ( x ) = ( e − itξ | ξ | b v ) ∨ ( x ) , HE BENJAMIN-ONO EQUATION 3 to satisfy that v ( · , t ) ∈ L ( | x | k dx ) , t ∈ [0 , T ], one needs v ∈ Z k,k , k ∈ Z + for k = 1 , Z ∞−∞ x j v ( x ) dx = 0 , j = 0 , , ..., k − , if k ≥ . Also one notices that the traveling wave φ c ( x + t ) , c > φ ( x ) = −
41 + x , φ c ( x + t ) = c φ ( c ( x + ct )) , has very mild decay at infinity. In this case, the traveling wave is negative andtravels to the left. To get a positive traveling wave moving to the right one needsto consider the equation(1.7) ∂ t v − H ∂ x v + v∂ x v = 0 , t, x ∈ R , and observes that if u ( x, t ) is a solution of (1.1) then v ( x, t ) = − u ( x, − t ) , satisfies equation (1.7). In particular, (1.7) has the traveling wave solution v ( x, t ) = ψ c ( x − t ) = cψ ( c ( x − ct )) , c > ψ ( x ) = − φ ( x ) . In [18] R. Iorio strengthened his unique continuation result in Z , found in [17](Theorem A, part (iii)) by proving: Theorem B. ([18])
Let u ∈ C ([0 , T ] : H ( R )) be a solution of the IVP (1.1) . Ifthere exist three different times t , t , t ∈ [0 , T ] such that (1.8) u ( · , t j ) ∈ Z , , j = 1 , , , then u ( x, t ) ≡ . Our goal in this work is to extend the results in Theorem A and Theorem B frominteger values to the continuum optimal range of indices ( s, r ). Our main resultsare the following:
Theorem 1. (i) Let s ≥ , r ∈ [0 , s ] , and r < / . If u ∈ Z s,r , then the solution u ( x, t ) of the IVP (1.1) satisfies that u ∈ C ([0 , ∞ ) : Z s,r ) .(ii) For s > / ( s ≥ / ), r ∈ [0 , s ] , and r < / the IVP (1.1) is LWP (GWPresp.) in Z s,r .(iii) If r ∈ [5 / , / and r ≤ s , then the IVP (1.1) is GWP in ˙ Z s,r . Theorem 2.
Let u ∈ C ([0 , T ] : Z , ) be a solution of the IVP (1.1) . If there existtwo different times t , t ∈ [0 , T ] such that (1.9) u ( · , t j ) ∈ Z / , / , j = 1 , , then b u (0) = 0 , ( so u ( · , t ) ∈ ˙ Z / , / ) . Theorem 3.
Let u ∈ C ([0 , T ] : ˙ Z , ) be a solution of the IVP (1.1) . If there existthree different times t , t , t ∈ [0 , T ] such that (1.10) u ( · , t j ) ∈ Z / , / , j = 1 , , , then u ( x, t ) ≡ . Remarks : (a) Theorem 2 shows that the condition b u (0) = 0 is necessary tohave persistence property of the solution in Z s, / , with s ≥ /
2, so in that regardTheorem 1 parts (i)-(ii) are sharp. Theorem 3 affirms that there is an upper limitof the spacial L -decay rate of the solution (i.e. | x | / u ( · , t ) / ∈ L ∞ ([0 , T ] : L ( R )), GERM´AN FONSECA AND GUSTAVO PONCE for any
T >
0) regardless of the decay and regularity of the non-zero initial data u . In particular, Theorem 3 shows that Theorem 1 part (iii) is sharp.(b) In part (ii) of Theorem 1 we shall use that in that case the solution u ( x, t )satisfies ∂ x u ∈ L ([0 , T ] : L ∞ ( R )) , (see [21], [24], and [31]) to establish that the map data-solution is locally continuousfrom Z s,r into C ([0 , T ] : Z s,r ).(c) The condition in Theorem 3 involving three times seems to be technical andmay be reduced to two different times as that in Theorem 2 . We recall that uniquecontinuation principles for the nonlinear Schr¨odinger equation and the generalizedKorteweg-de Vries equation have been established in [10] and [11] resp. underassumptions on the solutions at two different times. Following the idea in [18] onefinds from the equation (1.1) that(1.11) ddt Z ∞−∞ xu ( x, t ) dx = 12 k u ( t ) k = 12 k u k , so the first momentum of a non-null solution of the BO equation is strictly increas-ing. On the other hand, using the integral equation version of the BO equation fromthe hypotheses one can deduce that the first momentum must vanish somewhere inthe time intervals ( t , t ) and ( t , t ). This implies that u ( x, t ) ≡ u ∈ C ([0 , T ] : H s ( R )) of (1.1) one has that ∃ t ∈ [0 , T ] such that u ( x, t ) ∈ H s ′ ( R ) , s ′ > s , then u ∈ C ([0 , T ] : H s ′ ( R )). So weshall mainly consider the most interesting case s = r in (1.4).(e) Consider the IVP for generalized Benjamin-Ono (gBO) equation(1.12) ( ∂ t u + H ∂ x u ± u k ∂ x u = 0 , t, x ∈ R , k ∈ Z + ,u ( x,
0) = u ( x ) , with u a real valued function. In this case the best LWP available results are : for k = 2 , s ≥ / k = 3 , s > /
34 (see [36]), and for k ≥ , s ≥ / − /k (see [36]). So for any power k = 1 , , ... with focussing (+) or defocusing ( − ) non-linearity the IVP (1.12) is LWP in H ( R ). So the local results in Theorems 1 andTheorem 2 and their proofs extend to the IVP (1.12) with possible different values s = s ( k ) for the minimal regularity required. This is also the case for Theorem 3when the power k in (1.12) is odd in the focusing and defocusing regime.(f) In [20] the number 7 / H . Among them we shall use the A p condition introduced in [27], (see Definition 1). It was proven in [15] that this isa necessary and sufficient condition for the Hilbert transform H to be bounded in L p ( w ( x ) dx ) (see [15], ), i.e. w ∈ A p , < p < ∞ if and only if(1.13) ( Z ∞−∞ |H f | p w ( x ) dx ) /p ≤ c ∗ ( Z ∞−∞ | f | p w ( x ) dx ) /p , (see Theorem 4).In order to justify some of our arguments in the proofs we need some furthercontinuity properties of the Hilbert transform. More precisely, our proof requiresthe constant c ∗ in (1.13) to depend only on c ( w ) the constant describing the A p condition (see (2.2)) and on p . In [30] precise bounds for the constant c ∗ in (2.3) HE BENJAMIN-ONO EQUATION 5 were given which are sharp in the case p = 2 and sufficient for our purpose (seeTheorem 5).It will be essential in our arguments that some commutator operators involvingthe Hilbert transform H are of “order zero”. More precisely, we shall use thefollowing estimate: ∀ p ∈ (1 , ∞ ) , l, m ∈ Z + ∪ { } , l + m ≥ ∃ c = c ( p ; l ; m ) > k ∂ lx [ H ; a ] ∂ mx f k p ≤ c k ∂ l + mx a k ∞ k f k p . In the case l + m = 1, (1.14) is Calder´on’s first commutator estimate [6]. In thecase l + m ≥
2, (1.14) was proved in [8].The rest of this paper is organized as follows: section 2 contains some preliminaryestimates to be utilized in the coming sections. Theorem 1 will be proven in section3. Finally, the proofs of Theorem 2 and Theorem 3 will be given in sections 4 and5, respectively. 2.
Preliminary Estimates
We shall use the following notations:(2.1) k f k p = ( Z ∞∞ | f ( x ) | p dx ) /p , ≤ p < ∞ , k f k ∞ = sup x ∈ R | f ( x ) | , k f k s, = k (1 − ∂ x ) s/ f k , s ∈ R . Let us first recall the definition of the A p condition. We shall restrict here tothe cases p ∈ (1 , ∞ ) and the 1-dimensional case R (see [27]). Definition 1.
A non-negative function w ∈ L loc ( R ) satisfies the A p inequality with1 < p < ∞ if(2.2) sup Q interval (cid:18) | Q | Z Q w (cid:19) (cid:18) | Q | Z Q w − p ′ (cid:19) p − = c ( w ) < ∞ , where 1 /p + 1 /p ′ = 1. Theorem 4. ([15])
The condition (2.2) is necessary and sufficient for the bound-edness of the Hilbert transform H in L p ( w ( x ) dx ) , i.e. (2.3) ( Z ∞−∞ |H f | p w ( x ) dx ) /p ≤ c ∗ ( Z ∞−∞ | f | p w ( x ) dx ) /p . In the case p = 2, a previous characterization of w in (2.3) was found in [14](for further references and comments we refer to [9], [13], and [34]). However, eventhough we will be mainly concerned with the case p = 2, the characterization (2.3)will be the one used in our proof. In particular, one has that in R (2.4) | x | α ∈ A p ⇔ α ∈ ( − , p − . In order to justify some of the arguments in the proof of Theorem 1 we needsome further continuity properties of the Hilbert transform. More precisely, ourproof requires the constant c ∗ in (2.3) to depend only on c ( w ) in (2.2) and on p (infact, this is only needed for the case p = 2). Theorem 5. ([30])
For p ∈ [2 , ∞ ) the inequality (2.3) holds with c ∗ ≤ c ( p ) c ( w ) ,with c ( p ) depending only on p and c ( w ) as in (2.2) . Moreover, for p = 2 thisestimate is sharp. GERM´AN FONSECA AND GUSTAVO PONCE
Next, we define the truncated weights w N ( x ) using the notation h x i = (1+ x ) / as(2.5) w N ( x ) = ( h x i if | x | ≤ N ,2 N if | x | ≥ N,w N ( x ) are smooth and non-decreasing in | x | with w ′ N ( x ) ≤ x ≥ Proposition 1.
For any θ ∈ ( − , and any N ∈ Z + , w θN ( x ) satisfies the A inequality (2.2) . Moreover, the Hilbert transform H is bounded in L ( w θN ( x ) dx ) with a constant depending on θ but independent of N ∈ Z + . The proof of Proposition 1 follows by combining the fact that for a fixed θ ∈ ( − ,
1) the family of weights w θN ( x ) , N ∈ Z + satisfies the A inequality in (2.2)with a constant c independent of N , and Theorem 5.Next, we have the following generalization of Calder´on commutator estimates [6]found in [8] and already commented in the introduction: Theorem 6.
For any p ∈ (1 , ∞ ) and l, m ∈ Z + ∪ { } , l + m ≥ there exists c = c ( p ; l ; m ) > such that (2.6) k ∂ lx [ H ; a ] ∂ mx f k p ≤ c k ∂ l + mx a k ∞ k f k p . We shall also use the pointwise identities[ H ; x ] ∂ x f = [ H ; x ] ∂ x f = 0 , and more generally [ H ; x ] f = 0 if and only if Z f dx = 0 . We recall the following characterization of the L ps ( R n ) = (1 − ∆) − s/ L p ( R n ) spacesgiven in [33]. Theorem 7.
Let b ∈ (0 , and n/ ( n + 2 b ) < p < ∞ . Then f ∈ L pb ( R n ) if andonly if (2.7) ( a ) f ∈ L p ( R n ) , ( b ) D b f ( x ) = ( Z R n | f ( x ) − f ( y ) | | x − y | n +2 b dy ) / ∈ L p ( R n ) , with (2.8) k f k b,p ≡ k (1 − ∆) b/ f k p = k J b f k p ≃ k f k p + k D b f k p ≃ k f k p + kD b f k p . Above we have used the notation: for s ∈ R D s = ( − ∆) s/ with D s = ( H ∂ x ) s , if n = 1 . For the proof of this theorem we refer the reader to [33]. One sees that from(2.7) for p = 2 and b ∈ (0 ,
1) one has(2.9) kD b ( f g ) k ≤ k f D b g k + k g D b f k . We shall use this estimate in the proof of Theorem 3. As applications of Theorem7 we have the following estimate:
HE BENJAMIN-ONO EQUATION 7
Proposition 2.
Let b ∈ (0 , . For any t > D b ( e − itx | x | ) ≤ c ( | t | b/ + | t | b | x | b ) . For the proof of Proposition 2 we refer to [28].As a further direct consequence of Theorem 7 we deduce the following result tobe used in the proof of Theorem 3.
Proposition 3.
Let p ∈ (1 , ∞ ) . If f ∈ L p ( R ) such that there exists x ∈ R for which f ( x +0 ) , f ( x − ) are defined and f ( x +0 ) = f ( x − ) , then for any δ > , D /p f / ∈ L p loc ( B ( x , δ )) and consequently f / ∈ L p /p ( R ) . Also as consequence of the estimate (2.9) one has the following interpolationinequality.
Lemma 1.
Let a, b > . Assume that J a f = (1 − ∆) a/ f ∈ L ( R ) and h x i b f = (1 + | x | ) b/ f ∈ L ( R ) . Then for any θ ∈ (0 , k J θa ( h x i (1 − θ ) b f ) k ≤ c kh x i b f k − θ k J a f k θ . Moreover, the inequality (2.11) is still valid with w N ( x ) in (2.5) instead of h x i witha constant c independent of N .Proof. It will suffice to consider the case : a = 1 + α, α ∈ (0 , ρ ( x )a function equal to h x i or equal to w N ( x ) as in (2.5) and consider the function F ( z ) = e ( z − Z ∞−∞ J az ( ρ b (1 − z ) f ( x )) g ( x ) dx with g ∈ L ( R n ) with k g k = 1, which is continuous in { z = η + iy : 0 ≤ η ≤ } and analytic in its interior. Moreover, | F (0 + iy ) | ≤ e − ( y +1) k ρ b f k , and since | ρ ′ /ρ | + | ρ ′′ /ρ | ≤ c (independent of N ) combining (2.7) and (2.9) one has | F (1 + iy ) | ≤ e − y k J a ( ρ iby f ) k ≤ e − y ( k ρ iby f k + k D α ∂ x ( ρ iby f ) k ) ≤ e − y ( k f k + k D α ( ρ iby ∂ x f ) k + | by | k D α ( ρ iby − ρ ′ f ) k ) ≤ e − y ( k f k + kD α ( ρ iby ∂ x f ) k + | by | kD α ( ρ iby − ρ ′ f ) k ) ≤ e − y ( k f k + kD α ( ρ iby ) ∂ x f k + k ( ρ iby ) D α ∂ x f k + | by | kD α ( ρ iby − ρ ′ ) f k + | by | k ( ρ iby − ρ ′ ) D α f k ) ≤ c α e − y (1 + | yb | )( k f k + kD α f k + k ∂ x f k + kD α ∂ x f k ) ≤ c α e − y (1 + | yb | ) k J α f k = c α e − y (1 + | yb | ) k J a f k , using that for α ∈ (0 , kD α h k ∞ ≤ c α ( k h k ∞ + k ∂ x h k ∞ ) . Therefore, the three lines theorem yields the desired result. (cid:3)
We shall also employ the following simple estimate.
Proposition 4. If f ∈ L ( R ) and φ ∈ H ( R ) , then (2.12) k [ D / ; φ ] f k ≤ c k φ k , k f k . GERM´AN FONSECA AND GUSTAVO PONCE
Finally, to complete this section we recall the result obtained in [31] concerningregularity properties of the solutions of the IVP (1.1) with data u ∈ H s ( R ) , s ≥ /
2. This will be used in the proof of Theorem 3.
Theorem 8.
For any u ∈ H s ( R ) with s ≥ / the IVP (1.1) has a unique globalsolution u ∈ C ([0 , T ] : H s ( R )) such that for any T > J s +1 / u ∈ l ∞ k ( L ([ k, k + 1] × [0 , T ])) , Ju ∈ l k ( L ∞ ([ k, k + 1] × [0 , T ])) and J s − / ∂ x u ∈ L ([0 , T ] : L ∞ ( R )) . Proof of Theorem 1
We consider several cases :Case 1: s = 1 and r = θ ∈ (0 , w θN u (see (2.5)) with 0 < θ ≤ R to obtain(3.1) 12 ddt Z ( w θN u ) dx + Z w θN H ∂ x u w θN u dx + Z w θN u ∂ x u dx = 0 . To handle the second term on the left hand side (l.h.s.) of (3.1) we write w θN H ∂ x u = [ w θN ; H ] ∂ x u + H ( w θN ∂ x u )= A + H ∂ x ( w θN u ) − H ( ∂ x w θN ∂ x u ) − H ( ∂ x w θN u )= A + A + A + A . We observe that by Theorem 6 and our assumption on θ ∈ (0 ,
1] the terms A , A are bounded by the L -norm of the solution u and A is bounded by the H -normof the solution with constants independent of N , thus they are bounded uniformlyon N ∈ Z + by M = sup t ∈ [0 ,T ] k u ( t ) k , . We insert the term A in (3.1) and use integration by parts, to get that Z H ∂ x ( w θN u ) w θN udx = 0 . Finally, using integration by parts, we bound the nonlinear term (the third termon the l.h.s.) in (3.1) as(3.2) | Z w θN u ∂ x u dx | ≤ c k u k ∞ k u k k w θN u k ≤ c k u k , k w θN u k . Inserting this information in (3.1) we get ddt k w θN u ( t ) k ≤ cM, with c independent of N, which tells us thatsup t ∈ [0 ,T ] k w θN u ( t ) k ≤ c kh x i θ u k e T M , with c independent of N, which yields the result u ∈ L ∞ ([0 , T ] : L ( | x | θ )) for any T > u ∈ C ([0 , T ] : L ( | x | θ )) one considers the sequence( w θN u ) N ∈ Z + ⊆ C ([0 , T ] : L ( R )) , HE BENJAMIN-ONO EQUATION 9 and reapply the above argument to find that it is a Cauchy sequence.Finally, we point out that the use of the differential equation in (1.1) can bejustified by the locally continuous dependence of the solution upon the data from H s ( R ) to C ([0 , T ] : H s ( R )).Case 2: s ∈ (1 ,
2] and r = s . Part (i) in Theorem 1.We multiply the differential equation by w θN u (see (2.5)) with 0 ≤ θ ≤ R to obtain(3.3) 12 ddt Z ( w θN u ) dx + Z w θN H ∂ x u w θN u dx + Z w θN u ∂ x u dx = 0 . To control the second term on the l.h.s. of (3.3) we write w θN H ∂ x u = [ w θN ; H ] ∂ x u + H ( w θN ∂ x u )= B + H ∂ x ( w θN u ) − H ( ∂ x w θN ∂ x u ) − H ( ∂ x w θN u )= B + B + B + B . We observe that by Theorem 6 and our assumption θ ∈ (0 ,
1] the terms B , B arebounded by the L -norm of the solution. Inserting the term B on the (3.3) andusing integration by parts one finds that its contribution is null. So it remains tocontrol B = − H ( ∂ x w θN ∂ x u ). Since | ∂ x w θN | = | (1 + θ ) w θN ∂ x w N | ≤ c w θN , c independent of N, one has(3.4) k B k ≤ c k w θN ∂ x u k ≤ c k ∂ x ( w θN u ) k + c k ∂ x w θN u k ≤ c k ∂ x ( w θN u ) k + c k u k . Then by the interpolation inequality in (2.11) it follows that(3.5) k ∂ x ( w θN u ) k ≤ k J ( w θN u ) k ≤ c k w θN u k θ/ (1+ θ )2 k J θ u k / (1+ θ )2 , with a constant c independent of N . So by Young’s inequality in (3.5) and (3.4)the term B is appropriately bounded. Finally, for the last term on the l.h.s. of(3.3) we write(3.6) | Z w θN u ∂ x u dx | ≤ c k u k ∞ k w θN u k ≤ c k u k , k w θN u k , with c independent of N .So inserting the above information in (3.3) we obtain the result.Case 3: s ∈ (9 / ,
2] and r = s . Part (ii) in Theorem 1.In this case it remains to establish the continuous dependence of the solution C ([0 , T ] : Z s,r ) upon the data in Z s,r . We are considering the most interesting case s = r ∈ (9 / , u, v ∈ C ([0 , T ] : Z s,s ) are two solutions of the BOequation in (1.1) corresponding to data u , v respectively. Hence,(3.7) ∂ t ( u − v ) + H ∂ x ( u − v ) + ∂ x u ( u − v ) + v∂ x ( u − v ) = 0 . We will reapply the argument used in the previous case. However, we notice thatthe nonlinear term in (3.7) is different than that in (3.3). So we recall the result in[21] which affirms that for s > / ∂ x u, ∂ x v ∈ L ([0 , T ] : L ∞ x ( R )) , and use integration by parts to obtain that(3.9) | Z w θN ( ∂ x u ( u − v ) + v ( u − v ) ∂ x ( u − v )) dx |≤ c ( k ∂ x u ( t ) k ∞ + k ∂ x v ( t ) k ∞ + k v ( t ) k ∞ ) k w θN ( u − v ) k . Hence, combining the argument in the previous section, the estimates (3.9) and(3.8), and the continuous dependence of the solution in C ([0 , T ] : H s ( R )) upon thedata in H s ( R ) the desired result follows.Case 4: s = r ∈ (2 , / s ≥ r ∈ (0 , r = 2 + θ, θ ∈ (0 , / x w θN u (see (2.5)) and integrate on R to obtain(3.10) 12 ddt Z ( w θN xu ) dx + Z w θN x H ∂ x u w θN x u dx + Z x w θN u ∂ x u dx = 0 . From our previous proofs it is clear that we just need to handle the second term onthe l.h.s. of (3.10). First we write the identity(3.11) x H ∂ x u = H ( x∂ x u ) = H ( ∂ x ( xu )) − H ∂ x u = E + E . To bound the contribution of the term E inserted in (3.10) we shall use that w θN with θ ∈ (0 , /
2) satisfies the A inequality uniformly in N (see Proposition 1)so(3.12) k w θN E k = 2 k w θN H ∂ x u k ≤ c k w θN H ∂ x u k + c k w θN x H ∂ x u k ≤ c k w θN ∂ x u k + c k w θN H ( x∂ x u ) k ≤ c k w θN ∂ x u k + c k w θN x ∂ x u k = F + F . Now using complex interpolation one gets (see Lemma 1)(3.13) k w θN ∂ x u k ≤ k ∂ x ( w θN u ) k + k ∂ x w θN u k ≤ k ∂ x ( w θN u ) | + c k u k ≤ c || J ( w θN u ) k + c k u k ≤ c k J u k / kh x i θ u k / + c k u k , which has been bounded in the previous cases. So it remains to bound the term(3.14) F = k w θN x ∂ x u k , which will be considered later.Inserting the term E in (3.11) into (3.10) one obtains the term(3.15) G = Z w θN H ∂ x ( xu ) w θN x u dx. As before we write(3.16) w θN H ∂ x ( xu ) = − [ H ; w θN ] ∂ x ( xu ) + H ( w θN ∂ x ( xu ))= K + H ( ∂ x ( w θN xu )) − H ( ∂ x w θN ∂ x ( xu )) − H ( ∂ x w θN ( xu ))= K + K + K + K . Thus, by Theorem 6 and the results in the previous cases the contribution of K , K in (3.15) is bounded. Also inserting the term K in (3.15) one has by HE BENJAMIN-ONO EQUATION 11 integration by parts that its contribution is null. So in (3.16) it only remains toconsider the contribution from K in (3.15). But using that(3.17) k K k = kH ( ∂ x w θN ∂ x ( xu )) k = k ∂ x w θN ∂ x ( xu ) k ≤ k ∂ x w θN u k + k ∂ x w θN x ∂ x u k ≤ c ( k w θN u k + k w θN x ∂ x u k ) = R + R , since R was previously bounded, it remains to estimate R which is equal to theterm F in (3.14). To estimate this term we use the BO equation in (1.1) to obtainthe new equation(3.18) ∂ t ( x ∂ x u ) + H ∂ x ( x ∂ x u ) − H ∂ x u + x ∂ x ( u∂ x u ) = 0 . The differential equation (3.18) multiplied by w θN x ∂ x u leads to the identity(3.19) 12 ddt Z ( w θN x∂ x u ) dx + Z w θN H ∂ x ( x∂ x u ) w θN ( x∂ x u ) dx − Z w θN H ∂ x u w θN x∂ x u dx + Z w θN x ∂ x ( u∂ x u ) w θN x∂ x u dx = 0 . Sobolev inequality and integration by parts lead to(3.20) | Z w θN x ∂ x ( u∂ x u ) w θN x∂ x u dx |≤ c k u k , k w θN x∂ x u k ( k w θN x∂ x u k + k w θN u k ) , and since(3.21) w θN H ∂ x ( x∂ x u ) = − [ H ; w θN ] ∂ x ( x∂ x u ) + H ( w θN ∂ x ( x∂ x u ))= V + H ∂ x ( w θN x∂ x u ) − H ( ∂ x w θN ∂ x ( x∂ x u )) − H ( ∂ x ( w θN ) x ∂ x u )= V + V + V + V , Theorem 6, the previous results, and interpolation allow to bound the L -norm ofthe terms V and V . As before by integration by parts the contribution of the term V in (3.19) is null. So it just remains to consider the term V in (3.21). In fact, V = − H ( ∂ x w θN ∂ x u ) − H ( ∂ x w θN ( x∂ x u )) = V , + V , , so one just needs to handle the term V , . Using that | ∂ x w θN x | ≤ c w θN , c independent of N, it suffices to consider(3.22) k w θN ∂ x u k ≤ c k J ( w θN u ) k + c k u k , + c k w θN u k , with c independent on N . So we just need to consider the first term on the r.h.s.of the inequality (3.22). Using interpolation it follows that(3.23) k J ( w θN u ) k ≤ c k J θ u k / (2+ θ )2 k w θN u k θ/ (2+ θ ) s . We notice that the first term on the r.h.s. of (3.23) is bounded and the secondone is bounded by the one we were estimating in (3.10). Therefore, (3.10) and(3.19) yield closed differential inequalities for k xw θN u k and k w θN x ∂ x u k , andconsequently the desired result.Case 5: s = r ∈ [5 / , / First, by differentiating the BO equation in (1.1) one gets ∂ t ( ∂ x u ) + H ∂ x ( ∂ x u ) + u∂ x ( ∂ x u ) + ∂ x u ∂ x u = 0 , so by reapplying the argument in the previous cases it follows that(3.24) sup t ∈ [0 ,T ] kh x i s − ∂ x u ( t ) k ≤ M, with M depending on k u k s, , kh x i s u k , and T .Next, we multiply the BO equation in (1.1) by x w ˜ θN with ˜ θ ∈ [1 / , /
2) to get(3.25) ∂ t x w ˜ θN u + x w ˜ θN H ∂ x u + x w ˜ θN u∂ x u = 0 , so a familiar argument leads to(3.26) 12 ddt Z ( x w ˜ θN u ) dx + Z x w ˜ θN H ∂ x u x w ˜ θN u dx + Z x w ˜ θN u∂ x ux w ˜ θN u dx = 0 . Using the identity x H ∂ x u = H ∂ x ( x u ) + 4 H ∂ x ( xu ) + H u, the linear dispersive part of (3.25) (the second term on the l.h.s. of (3.25)) can bewritten(3.27) w ˜ θN x H ∂ x u = w ˜ θN H ∂ x ( x u ) + 4 w ˜ θN H ∂ x ( xu ) + w ˜ θN H u = Q + Q + Q . Since Z ∞−∞ u ( x ) dx = Z ∞−∞ u ( x, t ) dx = 0 , then H ( xu ) = xHu, for ˜ θ ∈ [1 / ,
1] one has k Q k = k w ˜ θN H u k ≤ k (1 + | x | ) H u k ≤ k u k + k x u k , and for ˜ θ ∈ (1 , /
2) using Proposition 1 k Q k = k w ˜ θN H u k ≤ k (1 + | x | ) w ˜ θ − N H u k ≤ k w ˜ θ − N u k + k w ˜ θ − N x u k , so in both cases by the previous results Q in (3.27) is bounded in L .To control Q we first consider the case ˜ θ ∈ [1 / ,
1] and use Calder´on commutatortheorem to get k Q k = 4 k w ˜ θN H ∂ x ( xu ) k ≤ c ( k [ H ; w ˜ θN ] ∂ x ( xu ) k + kH ( w ˜ θN ∂ x ( xu )) k ) ≤ c ( k xu k + k w ˜ θN x∂ x u k + k w ˜ θN u k ) . Thus, in the case ˜ θ ∈ [1 / , L -norm of Q For the case ˜ θ = 1 + θ, θ ∈ (0 , /
2) we combine Proposition 1 and the hypothesison the mean value of u to deduce that k Q k = 4 k w ˜ θN H ∂ x ( xu ) k ≤ c ( k w θN x H ∂ x ( xu ) k + k w θN H ∂ x ( xu ) k ) ≤ c ( k w θN H ( x ∂ x ( xu )) k + k w θN ∂ x ( xu ) k ) ≤ c ( k w θN x ∂ x ( xu ) k + k w θN ∂ x ( xu ) k ) . Hence, (3.24) yields the appropriate bound on the L -norm of Q .Finally, we turn to the contribution of the term Q when inserted in (3.27).Thus, we write w ˜ θN H ∂ x ( x u ) = − [ H ; w ˜ θN ] ∂ x ( x u ) + H ( w ˜ θN ∂ x ( x u ))= V + H ( ∂ x ( w ˜ θN x u )) − H ( ∂ x w ˜ θN ∂ x ( x u )) − H ( ∂ x w ˜ θN ( x u ))= V + V + V + V . From the previous cases it follows that the L -norm of the terms V , V arebounded. By integration by parts, the contribution of the term V is null. So itjust remains to consider V = − H ( ∂ x w ˜ θN ∂ x ( x u )) in L , but ∂ x w ˜ θN ∂ x ( x u ) = ∂ x w ˜ θN ( x ∂ x u + 2 x u ) = V , + V , . Since ˜ θ ∈ (1 / , / k V , k ≤ c kh x i u k which has been found to be bounded in the previous cases. Now since | ∂ x w ˜ θN x | ≤ h x i θ , it follows that k V , k ≤ kh x i θ ∂ x u k , so (3.24) gives the bound. Gathering the above information one completes the proofof Theorem 1. 4. Proof of Theorem 2
Without loss of generality we assume that t = 0 < t .Since u ( t ) ∈ Z , , we have that u ∈ C ([0 , T ] : H / ∩ L ( | x | − dx )).Let us denote by U ( t ) u = ( e − it | ξ | ξ c u ) ∨ the solution of the IVP for the linearequation associated to the BO equation with datum u . Therefore, the solution tothe IVP (1.1) can be represented by Duhamel’s formula(4.1) u ( t ) = U ( t ) u − Z t U ( t − t ′ ) u ( t ′ ) ∂ x u ( t ′ ) dt ′ . From Plancherel’s equality we have that for any t , | x | / U ( t ) u ∈ L ( R ) if andonly if D / ξ ∂ ξ ( e − it | ξ | ξ b u ) ∈ L ( R ) and since(4.2) ∂ ξ ( e − it | ξ | ξ b u ) = − e − it | ξ | ξ (4 t ξ b u + 2 it sgn( ξ ) b u + 4 it | ξ | ∂ ξ b u − ∂ ξ b u ) , we show that with the hyphotesis on the initial data, all terms in Duhamel’s formulafor our solution u except the one involving sgn( ξ ), arising from the linear part in(4.2), have the appropriate decay at a later time. The argument in our proof requireslocalizing near the origin in Fourier frequencies by a function χ ∈ C ∞ , supp χ ⊆ ( − ǫ, ǫ ) and χ = 1 on ( − ǫ/ , ǫ/ Let us start with the computation for the linear part in (4.1) by introducing acommutator as follows χD / ξ ∂ ξ ( e − it | ξ | ξ b u ) = [ χ ; D / ξ ] ∂ ξ ( e − it | ξ | ξ b u ) + D / ξ ( χ∂ ξ ( e − it | ξ | ξ b u ))= A + B. (4.3)From Proposition 4 and identity (4.2) we have that k A k = k [ χ ; D / ξ ] ∂ ξ ( e − it | ξ | ξ b u ) k ≤ c k ∂ ξ ( e − it | ξ | ξ b u ) k ≤ c ( t k ξ b u k + t k sgn( ξ ) b u k + t k| ξ | ∂ ξ b u k + k ∂ ξ b u k ) ≤ c ( t k ∂ u k + t k u k + t k ∂ x ( xu ) k + k x u k ) , (4.4)which are all finite since u ∈ Z , .On the other hand, B = D / ξ ( χ∂ ξ ( e − it | ξ | ξ b u ))= 4 D / ξ ( χe − it | ξ | ξ t ξ b u ) + 2 iD / ξ ( χe − it | ξ | ξ t sgn( ξ ) b u )+ 4 iD / ξ ( χe − it | ξ | ξ t | ξ | b u ) − D / ξ ( χe − it | ξ | ξ ∂ ξ b u )= B + B + B + B . (4.5)Next, we shall estimate B in L ( R ). From Theorem 7, Proposition 2, and thefractional product rule type inequality (2.10) we get that k B k ≤ c ( k χe − it | ξ | ξ ∂ ξ b u k + kD / ξ ( χe − it | ξ | ξ ∂ ξ b u ) k ) ≤ c ( k ∂ ξ b u k + kD / ξ ( e − it | ξ | ξ ) χ∂ ξ b u k + k e − it | ξ | ξ D / ξ ( χ∂ ξ b u ) k ≤ c ( k x u k + k ( t / + t / | ξ | / ) χ∂ ξ b u k + kD / ξ ( χ∂ ξ b u ) k ) ≤ c ( T ) ( k x u k + kD / ξ ( χ ) k k ∂ ξ b u k ∞ + k χ k ∞ kD / ξ ( ∂ ξ b u ) k ) ≤ c ( T ) kh x i / u k . (4.6)Estimates for B and B in L ( R ) are easily obtained in a similar manner in-volving lower decay and regularity of the initial data. On the other hand for theanalysis of B we introduce ˜ χ ∈ C ∞ ( R ) such that ˜ χ ≡ supp ( χ ) . Then we canexpress this term as D / ξ ( χe − it | ξ | ξ t sgn( ξ ) b u ) = tD / ξ ( e − it | ξ | ξ ˜ χχ sgn( ξ ) b u )= t ([ e − it | ξ | ξ ˜ χ, D / ξ ] χ sgn( ξ ) b u + e − it | ξ | ξ ˜ χD / ξ ( χ sgn( ξ ) b u ))= t ( S + S ) . (4.7)Proposition 4 can be applied to bound S in L ( R ) as k S k ≤ k [ e − it | ξ | ξ ˜ χ, D / ξ ] χ sgn( ξ ) b u k ≤ c k χ sgn( ξ ) b u k ≤ c k u k . (4.8) HE BENJAMIN-ONO EQUATION 15
Therefore, once we show that the integral part in Duhamel‘s formula (4.1) liesin L ( | x | dx ), we will be able to conclude that S , ˜ χD / ξ χ sgn( ξ ) b u , D / ξ ( ˜ χχ sgn( ξ ) b u ) ∈ L ( R ) , then from Proposition 3 it will follow that b u (0) = 0, and from the conservationlaw I in (1.3), this would necessarily imply that b u (0) = R u ( x, t ) dx = 0.As we just mentioned above, in order to complete the proof, we consider theintegral part in Duhamel’s formula. We localize again with the help of χ ∈ C ∞ ( R )so that the integral in equation (4.1) after weights and a commutator reads now inFourier space as(4.9) Z t ([ χ ; D / ξ ]( e − i ( t − t ′ ) | ξ | ξ (4( t − t ′ ) ξ b z + 2 i ( t − t ′ )sgn( ξ ) b z + 4 i ( t − t ′ ) | ξ | ∂ ξ b z − ∂ ξ b z ))+ D / ξ ( χ ( e − i ( t − t ′ ) | ξ | ξ (4( t − t ′ ) ξ b z + 2 i ( t − t ′ )sgn( ξ ) b z + 4 i ( t − t ′ ) | ξ | ∂ ξ b z − ∂ ξ b z )))) dt ′ = A + A + A + A + B + B + B + B where b z = [ ∂ x u = i ξ b u ∗ b u .We limit our attention to the terms in (4.9) involving the highest order derivativesof u , i.e. A and B , and remark that the others can be treated in a similar wayby using that the function b z vanishes at ξ = 0.Combining Proposition 4, Holder’s inequality and Theorem 8 one has that kA k L ∞ T L x ≤ c k ( t − t ′ ) ξ ξ b u ∗ b u k L T L x ≤ c ( T ) k ∂ x ( uu ) k L T L x ≤ c ( T ) ( k u∂ x u k L T L x + k ∂ x u∂ x u k L T L x ) ≤ c ( T ) ( k u k l k L ∞ T L ∞ x ( Q Tk ) k ∂ x u k l ∞ k L T L x ( Q Tk ) + k ∂ x u k L ∞ T L ∞ x k ∂ x u k L ∞ T L x ) ≤ c ( T )( k u k l k L ∞ T L ∞ x ( Q Tk ) k ∂ x u k l ∞ k L T L x ( Q Tk ) + k u k L ∞ T H ) , (4.10)where Q Tk = [ k, k + 1] × [0 , T ].For B we obtain from Theorem 7(4.11) kB k L ∞ T L x ≤ c Z T k D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ξ ξ b u ∗ b u ) k dt ≤ c ( k e − i ( t − t ′ ) | ξ | ξ χ ξ ξ b u ∗ b u k L T L x + kD / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ξ ξ b u ∗ b u ) k L T L x )= Y + Y . These terms can be handled as follows(4.12) Y ≤ c k b u ∗ b u k L T L x ≤ c kk u k ∞ k u k k L T ≤ cT sup [0 ,T ] k u ( t ) k , , and using Proposition 2, (2.9), (2.10), and (4.12)(4.13) Y = kD / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ξ ξ b u ∗ b u ) k L T L x ≤ c kD / ξ ( e − i ( t − t ′ ) | ξ | ξ ) χ ξ ξ b u ∗ b u k L T L x + c kD / ξ ( χ ξ ξ b u ∗ b u ) k L T L x ) ≤ c k ( t / + t / | ξ | / ) χ ξ ξ b u ∗ b u k L T L x + c kkD / ξ ( χ ξ ) k ∞ k b u ∗ b u k k L T + c kk χ ξ k ∞ kD / ξ ( b u ∗ b u ) k k L T ≤ c ( T ) k b u ∗ b u k L T L x + c kD / ξ ( b u ∗ b u k L T L x ≤ c ( T ) sup [0 ,T ] k u ( t ) k , + c ( T ) k| x | / u k L ∞ T L x sup [0 ,T ] k u ( t ) k , . Hence the terms in (4.9) are all bounded, so by applying the argument afterinequality (4.8) we complete the proof.5.
Proof of Theorem 3
From the previous results and the hypothesis we have that for any ǫ > u ∈ C ([0 , T ] : ˙ Z / , / − ǫ ) and u ( · , t j ) ∈ H / ( R ) , j = 1 , , . Hence, b u ∈ C ([0 , T ] : H / − ǫ ( R ) ∩ L ( | ξ | dξ )) and b u ( · , t j ) ∈ L ( | ξ | dξ ) , j = 1 , , . for any ǫ >
0. Thus, in particular it follows that(5.1) b u ∗ b u ∈ C ([0 , T ] : H ( R ) ∩ L ( | ξ | dξ )) . Let us assume that t = 0 < t < t . An explicit computation shows that F ( t, ξ, b u ) = ∂ ξ ( e − it | ξ | ξ b u )= e − it | ξ | ξ (8 it ξ b u − t ξ b u − t ξ ∂ ξ b u − it sgn( ξ ) ∂ ξ b u − it | ξ | ∂ ξ b u − itδ b u + ∂ ξ b u ) , (5.2)where we observe that since the initial data u has zero mean value the terminvolving the Dirac function in (5.2) vanishes. Hence in order to prove our theorem,via Plancherel’s Theorem and Duhamel‘s formula (4.1), it is enough to show thatthe assumption that(5.3) D / ξ F ( t, ξ, b u ) − Z t D / ξ F ( t − t ′ , ξ, b z ( t ′ )) dt ′ , lies in L ( R ) for times t = 0 < t < t , where b z = [ ∂ x u = i ξ b u ∗ b u , leads to acontradiction. Let us show that the first term in equation (5.3) which arises fromthe linear part in Duhamel’s formula persists in L We proceed as in the proof of Theorem 2 and localize one more time by intro-ducing χ ∈ C ∞ , supp χ ⊆ ( − ǫ, ǫ ) and χ = 1 on ( − ǫ/ , ǫ/
2) so that χ D / ξ ∂ ξ ( e − it | ξ | ξ b u ) = [ χ ; D / ξ ] ∂ ξ ( e − it | ξ | ξ b u ) + D / ξ ( χ∂ ξ ( e − it | ξ | ξ b u ))= e A + e B. (5.4) HE BENJAMIN-ONO EQUATION 17
As for the first term, e A , from Proposition 4, this is bounded in L ( R ) by k ∂ ξ ( e − it | ξ | ξ b u ) k , which is finite as can easily be observed from its explicit rep-resentation in (5.2), the assumption on the initial data u , and the quite similarcomputation already performed in (4.4), therefore we omit the details.On the other hand, for e B , we notice that e B = D / ξ ( χ∂ ξ ( e − it | ξ | ξ b u ))= 8 iD / ξ ( χe − it | ξ | ξ t | ξ | b u ) − D / ξ ( χe − it | ξ | ξ t ξ b u ) − D / ξ ( χe − it | ξ | ξ t ξ ∂ ξ b u ) − iD / ξ ( χe − it | ξ | ξ t sgn( ξ ) ∂ ξ b u ) − iD / ξ ( χe − it | ξ | ξ t | ξ | ∂ ξ b u ) + D / ξ ( χe − it | ξ | ξ ∂ ξ b u )= e B + e B + e B + e B + e B + e B . (5.5)Notice that from the remark made after the identity (5.2) e B does not appear,and that e B and e B are the terms involving the highest regularity and decay ofthe initial data. Therefore we show in detail their L estimates along with theargument to exploit a nice cancellation property of e B , and a term arising in theintegral part in Duhamel’s formula (4.1) .For e B we obtain from Theorem 7, fractional product rule type estimate (2.9) ,(2.10), and Holder’s inequality that k e B k ≤ c ( k χe − it | ξ | ξ t ξ b u k + kD / ξ ( χe − it | ξ | ξ t | ξ | b u ) k ) ≤ c t ( k u k + kD / ξ ( e − it | ξ | ξ ) χ | ξ | b u k + k e − it | ξ | ξ D / ξ ( χ | ξ | b u ) k ) ≤ c t ( k u k + k ( t / + t / | ξ | / ) χ | ξ | b u k + kD / ξ ( χ | ξ | b u ) k ) ≤ c ( T ) ( k u k + kD / ξ ( χξ ) k ∞ k b u k + k χξ k ∞ kD / ξ b u k ) ≤ c ( T ) ( k u k + k| x | / u k ) , (5.6)and similarly k e B k ≤ c ( k χe − it | ξ | ξ ∂ ξ b u k + kD / ξ ( χe − it | ξ | ξ ∂ ξ b u ) k ) ≤ c ( k ∂ ξ b u k + kD / ξ ( e − it | ξ | ξ ) χ∂ ξ b u k + k e − it | ξ | ξ D / ξ ( χ∂ ξ b u ) k ≤ c ( k x u k + k ( t / + t / | ξ | / ) χ∂ ξ b u k + kD / ξ ( χ∂ ξ b u ) k ) ≤ c ( t ) ( k x u k + kD / ξ ( χ ) k ∞ k ∂ ξ b u k + k χ k L ∞ kD / ξ ( ∂ ξ b u ) k ) ≤ c ( T ) ( k x u k + k D / ξ ( ∂ ξ b u ) k ) ≤ c ( T ) kh x i / u k . (5.7) Now, let us go over the integral part that can be written in Fourier space andwith the help of a commutator as Z t ([ χ ; D / ξ ]( e − i ( t − t ′ ) | ξ | ξ (8 i ( t − t ′ ) ξ b z − t − t ′ ) ξ b z − t − t ′ ) ξ ∂ ξ b z − i ( t − t ′ ) | ξ | ∂ ξ b z − i ( t − t ′ ) δ b z + ∂ ξ b z ))+ D / ξ ( χ (8 i ( t − t ′ ) ξ b z − t − t ′ ) ξ b z − t − t ′ ) ξ ∂ ξ b z − i ( t − t ′ ) | ξ | ∂ ξ b z − i ( t − t ′ ) δ b z + ∂ ξ b z ))) dt = e A + e A + e A + e A + e A + e A + e B + e B + e B + e B + e B + e B + e C , (5.8)where(5.9) e C = − i Z t D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ( t − t ′ )sgn( ξ ) ∂ ξ b z ) dt ′ , and b z = [ ∂ x u = i ξ b u ∗ b u .Notice that e A , e B vanish since u∂ x u has zero mean value and for e A , e A , e A , e A , e A , e B , e B , e B , e B and e B the estimates in L ( R ) are essentially the same for theircounterparts in equation (4.9), in the proof of Theorem 2, so we omit the details oftheir estimates.Therefore from the assumption that u , u ( t ) ∈ ˙ Z , , equation (5.8), and theestimates above, we conclude that R = − iD / ξ ( e − it | ξ | ξ χ t sgn( ξ ) ∂ ξ b u ) − e C = − iD / ξ ( e − it | ξ | ξ χ t sgn( ξ ) ∂ ξ b u )+ 6 i Z t D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ( t − t ′ )sgn( ξ ) ∂ ξ ( iξ b u ∗ b u )) dt ′ , (5.10)is a function in L ( R ) at time t = t . But R = 6 i Z t ( D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ( t − t ′ )sgn( ξ )( ∂ ξ ( iξ b u ∗ b u ) − ∂ ξ ( iξ b u ∗ b u )(0))) dt ′ − iD / ξ ( e − it | ξ | ξ χ t sgn( ξ )( ∂ ξ b u − ∂ ξ b u (0))) − iD / ξ ( e − it | ξ | ξ χ t sgn( ξ ) ∂ ξ b u (0))+ 6 i Z t D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ( t − t ′ )sgn( ξ )( ∂ ξ ( iξ b u ∗ b u (0))) dt ′ = R + R + R + R . (5.11)We shall show that R and R are L ( R ) functions. This will imply that ( R + R )( t ) is also an L ( R ) function.For R we observe that from (5.1) e − i ( t − t ′ ) | ξ | ξ χ ( ξ )sgn( ξ )( ∂ ξ ( iξ b u ∗ b u )( ξ, t ′ ) − ∂ ξ ( iξ b u ∗ b u )(0 , t ′ )))is a Lipschitz function with compact support in the ξ variable. Therefore, usingTheorem 7 one sees that R ( t ) ∈ L ( R ). A similar argument shows that R ( t ) ∈ L ( R ). Therefore, we have that ( R + R )( t ) ∈ L ( R ). HE BENJAMIN-ONO EQUATION 19
On the other hand ∂ ξ ( iξ b u ∗ b u )(0) = \ − ixu∂ x u (0) = − i Z xu∂ x u dx = i k u k , and from the Benjamin-Ono equation we have(5.12) ddt Z xudx + Z x∂ x H udx + Z xu∂ x udx = 0 , which implies that(5.13) ddt Z xudx = − Z xu∂ x udx = 12 k u k , and hence ∂ ξ ( iξ b u ∗ b u )(0) = i ddt Z xudx. Substituting this into R gives us after integration by parts R = − Z t D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ( t − t ′ )sgn( ξ )( ddt ′ Z xudx )) dt ′ = − D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ( t − t ′ )sgn( ξ ) Z xudx | t ′ = tt ′ =0 )+ 6 Z t D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ( i | ξ | ξ ( t − t ′ ) − ξ )( Z xudx )) dt ′ = 6 D / ξ ( e − it | ξ | ξ χ t sgn( ξ ) Z xu ( x ) dx )+ 6 i Z t D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ ( t − t ′ ) | ξ | ξ sgn( ξ )( Z xudx )) dt ′ − Z t D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ sgn( ξ )( Z xudx )) dt ′ . (5.14)We observe that the second term after the last equality in (5.14) belongs to L ( R )and the first cancels out with R since ∂ ξ b u (0) = − i d xu (0) = − i Z xu ( x ) dx, (5.15)and therefore(5.16) R = − D / ξ ( e − it | ξ | ξ χ t sgn( ξ ) Z xu ( x ) dx ) . So the argument above implies that(5.17) − Z t D / ξ ( e − i ( t − t ′ ) | ξ | ξ χ sgn( ξ )( Z xu ( x, t ′ ) dx )) dt ′ is in L ( R ) at time t = t , and from Theorem 7 this is equivalent to have that(5.18) D / ξ ( χ ( ξ ) sgn( ξ ) Z t e − i ( t − t ′ ) | ξ | ξ ( Z xu ( x, t ′ ) dx ) dt ′ ) ∈ L ( R ) , which from Proposition 3 (choosing the support ( − ǫ, ǫ ) of χ sufficiently small)implies that R t ( R xu ( x, t ′ ) dx ) dt ′ = 0 and consequently R xu ( x, t ) dx must be zeroat some time in (0 , t ). We reapply the same argument to conclude that R xu ( x, t ) dx is again zero at some other time in ( t , t ). Finally, the identity (1.11) completesthe proof of the theorem. ACKNOWLEDGMENT : The authors would like to thank J. Duoandikoetxeafor fruitful conversations concerning this paper. This work was done while G. F.was visiting the Department of Mathematics at the University of California-SantaBarbara whose hospitality he gratefully acknowledges. G.P. was supported by NSFgrant DMS-0800967.
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E-mail address : [email protected] (G. Ponce) Department of Mathematics, University of California, Santa Barbara,CA 93106, USA.
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