Global well-posedness of the KP-I initial-value problem in the energy space
aa r X i v : . [ m a t h . A P ] M a y GLOBAL WELL-POSEDNESS OF THE KP-I INITIAL-VALUEPROBLEM IN THE ENERGY SPACE
A. D. IONESCU, C. E. KENIG, AND D. TATARU
Abstract.
We prove that the KP-I initial-value problem ( ∂ t u + ∂ x u − ∂ − x ∂ y u + ∂ x ( u /
2) = 0 on R x,y × R t ; u (0) = φ, is globally well-posed in the energy space E ( R ) = { φ : R → R : k φ k E ( R ) ≈ k φ k L + k ∂ x φ k L + k ∂ − x ∂ y φ k L < ∞} . Contents
1. Introduction 12. Notation and definitions 53. Global linear, bilinear and energy estimates 94. Proof of the main theorem 125. L bilinear estimates 176. Energy estimates 237. Dyadic bilinear estimates, I 298. Dyadic bilinear estimates, II 33References 351. Introduction
In this paper we consider the KP-I initial-value problem ( ∂ t u + ∂ x u − ∂ − x ∂ y u + ∂ x ( u /
2) = 0; u (0) = φ, (1.1)on R x,y × R t . The KP-I equation and the KP-II equation, in which the sign ofthe term ∂ − x ∂ y u in (1.1) is + instead of − , arise in physical contexts as modelsfor the propagation of dispersive long waves with weak transverse effects. The first author was supported in part by an NSF grant and a Packard Fellowship. Thesecond author was supported in part by an NSF grant. The third author was supported in partby an NSF grant.
The KP-II equation is well understood from the point of view of well-posedness:the KP-II initial-value problem is globally well-posed for suitable data in L , onboth R and T = S × S , see [4], as well as in some spaces larger than L , see[18] and the references therein.On the other hand, it has been shown in [14] that the KP-I initial-value problemis badly behaved with respect to Picard iterative methods in standard Sobolevspaces, since the flow map fails to be real-analytic at the origin in these spaces. On the positive side, it is known that the KP-I initial value problem is globallywell-posed in the “second” energy spaces on both R (see [11], and also [15] and[16]) and T (see [7]), as well as locally well-posed in larger spaces. These globalwell-posedness results rely on refined energy methods. In this paper we show thatthe KP-I initial-value problem is globally well-posed in the natural energy spaceof the equation.Let ξ , µ and τ denote the Fourier variables with respect to x , y and t respec-tively. For σ = 1 , , . . . we define the Banach spaces E σ = E σ ( R ), E σ = { φ : R → R : k φ k E σ = k b φ ( ξ, µ ) · p ( ξ, µ )(1 + | ξ | ) σ k L ξ,µ < ∞} , (1.2)where b φ denotes the Fourier transform of φ and p ( ξ, µ ) = 1 + | µ || ξ | + | ξ | . (1.3)Clearly, p ( ξ, µ )(1 + | ξ | ) σ = (1 + | ξ | ) σ + | µ/ξ | · (1 + | ξ | ) σ − . Let E ∞ = ∞ \ σ =1 E σ with the induced metric. We recall the KP-I conservation laws (see, for example,[15] for formal justifications): if t < t ∈ R u ∈ C ([ t , t ] : E ∞ ) is a solution ofthe equation ∂ t u + ∂ x u − ∂ − x ∂ y u + ∂ x ( u /
2) = 0 on R × ( t , t ) then e E ( u ( t )) = e E ( u ( t )) and e E ( u ( t )) = e E ( u ( t )) , (1.4)where, for any φ ∈ E , e E ( φ ) = Z R φ dxdy, (1.5)and e E ( φ ) = Z R ( ∂ x φ ) dxdy + Z R ( ∂ − x ∂ y φ ) dxdy − Z R φ dxdy. (1.6) Picard iterative methods can be applied, however, to produce local in time solutions forsmall low-regularity data in suitably weighted spaces, see [6].
LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 3
Consequently, if t ∈ [ t , t ] and k u ( t ) k E ≤ t ∈ [ t ,t ] k u ( t ) k E . k u ( t ) k E . (1.7)Our main theorem concerns global well-posedness of the KP-I initial-value prob-lem in the energy space E . Theorem 1.1. (a) Assume φ ∈ E ∞ . Then there is a unique global solution u = S ∞ ( φ ) ∈ C ( R : E ∞ ) of the initial-value problem (1.1) . In addition, for any T ∈ [0 , ∞ ) and any σ ∈{ , , . . . } sup | t |≤ T k S ∞ ( φ )( t ) k E σ ≤ C ( T, σ, k φ k E σ ) . (1.8) (b) Assume T ∈ R + . Then the mapping S ∞ T = [ − T,T ] ( t ) · S ∞ : E ∞ → C ([ − T, T ] : E ∞ ) extends uniquely to a continuous mapping S T : E → C ([ − T, T ] : E ) , and e E j ( u ( t )) = e E j ( φ ) for any t ∈ [ − T, T ] and j ∈ { , } . We remark that the global existence of smooth solutions stated in Theorem 1.1(a) is also new. The earlier global existence theorems of [11], [15], and [16] relyof the conservation of the second energy, which requires the stronger momentumcondition ∂ − x ∂ y φ ∈ L . Also, a simple additional argument shows that (1.8) canbe improved to sup | t |≤ T k S ∞ ( φ )( t ) k E σ ≤ C ( T, σ, k φ k E ) · k φ k E σ . We discuss now some of the ingredients in the proof of Theorem 1.1. One mighttry a direct perturbative approach (which goes back to work on the KdV equationin [9], [3], [10], and nonlinear wave equations in [12]), based on the properties ofsolutions to the linear equation ( ∂ t u + ∂ x u − ∂ − x ∂ y u = f ; u (0) = φ. (1.9)For some suitable spaces F ( T ) and N ( T ) one would like to prove a linear boundfor solutions to (1.9) on R × [ − T, T ], T ∈ (0 , k u k F ( T ) . k φ k E + k f k N ( T ) , (1.10)together with a matching nonlinear estimate k − ∂ x ( u / k N ( T ) . k u k F ( T ) . (1.11) A. D. IONESCU, C. E. KENIG, AND D. TATARU
Due to [14], it is known however that the inequalities (1.10), (1.11) cannot holdfor any choice of the spaces F ( T ) and N ( T ); this forces us to approach theproblem in a less perturbative way.To prove Theorem 1.1 (a) we define instead the normed spaces F ( T ), N ( T ),and the semi-normed space B ( T ) and show that if u is a smooth solution of (1.1)on R × [ − T, T ], T ∈ (0 , k u k F ( T ) . k u k B ( T ) + k − ∂ x ( u / k N ( T ) ; k − ∂ x ( u / k N ( T ) . k u k F ( T ) ; k u k B ( T ) . k φ k E + k u k F ( T ) . (1.12)The inequalities (1.12) and a simple continuity argument still suffice to control k u k F ( T ) , provided that k φ k E ≪ L and in ˙ H − x . To exploit thesesymmetries, we define the normed spaces F , N , and the semi-normed space B ,and prove a second set of linear, bilinear, and energy estimates, similar to (1.12).Then we adapt the Bona-Smith method [2] to prove the continuity of the flow inthe space E .We explain now our strategy to define the main normed and semi-normedspaces. Ideally, one would like to use standard X s,b - type structures (as in [3],[10]) for the spaces F ( T ) and N ( T ). For such spaces, however, the bilinearestimate k ∂ x ( uv ) k N ( T ) . k u k F ( T ) k v k F ( T ) cannot hold even for solutions u, v of the linear homogeneous equation. This bilinear estimate is only possible ifwe weaken significantly the contributions of the components of the functions u and v of high frequency and low modulation. To achieve this we still use X s,b -type structures for the spaces F ( T ) and N ( T ), but only on small, frequencydependent time intervals. A similar method was used recently in [5] and [13] toprove a-priori bounds for the 1-d cubic nonlinear Schr¨odinger equation in negativeSobolev spaces.The second step is to define k u k B ( T ) sufficiently large to be able to still provethe linear estimate k u k F ( T ) . k u k B ( T ) + k − ∂ x ( u / k N ( T ) . Finally, we usefrequency-localized energy estimates and the symmetries of the equation (1.1) toprove the energy estimate k u k B ( T ) . k φ k E + k u k F ( T ) . These symmetries allowus to trade high frequencies for low frequencies in trilinear forms, improving the The two main symmetries used at this stage are the fact that the solution u is real-valuedand the precise form of the nonlinearity − ∂ x ( u / LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 5 timescale from frequency dependent time intervals (as guaranteed by the bilinearestimates) to frequency independent time intervals.A new twist arises in the proof of part (b) of Theorem 1.1. The symmetriesof the difference equation are not as good as the symmetries of the nonlinearequation, which causes difficulties in the proofs of suitable energy estimates. Thelow frequency part of the solution turns out to be particularly harmful in thedifference equation. To avoid this difficulty we define the normed spaces F , N ,and the semi-normed space B , which have a special low-frequency structure.The rest of the paper is organized as follows: in section 2 we summarize mostof the notation, define the main normed spaces, and prove some of their basicproperties. In section 3 we state our main global linear, bilinear, and energy esti-mates. The proof of the bilinear estimate Proposition 3.3 depends on the dyadicbilinear estimates proved in sections 7 and 8; the energy estimates Proposition3.4 and Proposition 3.6 are proved in section 6. In section 4 we prove the maintheorem, using the linear, bilinear, and energy estimates of section 3. In section5, which is self-contained, we prove the bilinear L estimates in Corollary 5.3;these L estimates are the main building blocks in all the dyadic estimates insections 6, 7, and 8. The key technical ingredient is the scale-invariant estimatein Lemma 5.1 (a), which is also used in [6]. In section 6 we prove the energyestimates Proposition 3.4 and Proposition 3.6. Finally, in sections 7 and 8 weprove the dyadic bilinear estimates used in Proposition 3.3.2. Notation and definitions
Let Z + = Z ∩ [0 , ∞ ). For k ∈ Z let I k = { ξ : | ξ | ∈ [(3 / · k , (3 / · k ] } and e I k = { ξ : | ξ | ∈ [2 k − , k +1 ] } . Let η : R → [0 ,
1] denote an even smooth function supported in [ − / , /
5] andequal to 1 in [ − / , / k ∈ Z ∩ [1 , ∞ ) let η k ( ξ ) = η ( ξ/ k ) − η ( ξ/ k − ).For k ∈ Z + let η ≤ k = P kk ′ =0 η k ′ . For k ∈ Z let χ k ( ξ ) = η ( ξ/ k ) − η ( ξ/ k − ). For( ξ, µ ) ∈ R \ { } × R let ω ( ξ, µ ) = ξ + µ /ξ. (2.1)For k ∈ Z we define the dyadic X s,b -type normed spaces X k = X k ( R ), X k = { f ∈ L ( R ) : f is supported in e I k × R × R and k f k X k = ∞ X j =0 j/ k η j ( τ − ω ( ξ, µ )) · f k L < ∞} . (2.2)We use an l Besov-type norm with respect to modulations. Structures of thistype were introduced, for instance, in [19], and are useful in order to prevent highfrequency losses in bilinear and trilinear estimates.
A. D. IONESCU, C. E. KENIG, AND D. TATARU
The definition shows easily that if k ∈ Z and f k ∈ X k then (cid:13)(cid:13)(cid:13)(cid:13)Z R | f k ( ξ, µ, τ ′ ) | dτ ′ (cid:13)(cid:13)(cid:13)(cid:13) L ξ,µ . k f k k X k . (2.3)Moreover, if k ∈ Z , l ∈ Z + , and f k ∈ X k then ∞ X j = l +1 j/ (cid:13)(cid:13)(cid:13)(cid:13) η j ( τ − ω ( ξ, µ )) · Z R | f k ( ξ, µ, τ ′ ) | · − l (1 + 2 − l | τ − τ ′ | ) − dτ ′ (cid:13)(cid:13)(cid:13)(cid:13) L + 2 l/ (cid:13)(cid:13)(cid:13)(cid:13) η ≤ l ( τ − ω ( ξ, µ )) · Z R | f k ( ξ, µ, τ ′ ) | · − l (1 + 2 − l | τ − τ ′ | ) − dτ ′ (cid:13)(cid:13)(cid:13)(cid:13) L . k f k k X k . (2.4)In particular, if k ∈ Z , l ∈ Z + , t ∈ R , f k ∈ X k , and γ ∈ S ( R ), then kF [ γ (2 l ( t − t )) · F − ( f k )] k X k . k f k k X k . (2.5)For k ∈ Z let k + = max( k, P k denote the operator on L ( R )defined by the Fourier multiplier ( ξ, µ, τ ) → I k ( ξ ). By a slight abuse of notation,we also let P k denote the operator on L ( R ) defined by the Fourier multiplier( ξ, µ ) → I k ( ξ ). For l ∈ Z let P ≤ l = X k ≤ l P k , P ≥ l = X k ≥ l P k . With p as in (1.3), for k ∈ Z define the frequency localized initial data spaces E k = { φ : R → R : F ( φ ) = e I k ( ξ ) F ( φ ) and k φ k E k = k b φ · p ( ξ, µ ) k L ξ,µ < ∞} , (2.6)and E k = { φ : R → R : F ( φ ) = e I k ( ξ ) F ( φ ) and k φ k E k = k b φ k L ξ,µ < ∞} . (2.7)The corresponding frequency localized energy spaces for the solutions are C ( R : E k ) = { u k ∈ C ( R : E k ) : u k is supported in R × [ − , } C ( R : E k ) = { u k ∈ C ( R : E k ) : u k is supported in R × [ − , } . At frequency 2 k we will use the X s,b structure given by the X k norm, uniformlyon the 2 − k + time scale. For k ∈ Z we define the normed spaces F k = { u k ∈ C ( R : E k ) : k u k k F k = sup t k ∈ R k p ( ξ, µ ) · F [ u k · η (2 k + ( t − t k ))] k X k < ∞} , (2.8)and F k = { u k ∈ C ( R : E k ) : k u k k F k = sup t k ∈ R kF [ u k · η (2 k + ( t − t k ))] k X k < ∞} . (2.9) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 7
For k ∈ Z we define the normed spaces N k = C ( R : E k ) and N k = C ( R : E k ) (asvector spaces), which are used to measure the frequency 2 k part of the nonlinearterm, with norms k f k k N k = sup t k ∈ R k p ( ξ, µ )( τ − ω ( ξ, µ ) + i k + ) − · F [ f k · η (2 k + ( t − t k ))] k X k , (2.10)and k u k k N k = sup t k ∈ R k ( τ − ω ( ξ, µ ) + i k + ) − · F [ f k · η (2 k + ( t − t k ))] k X k . (2.11)The bounds we obtain for solutions of the KP-I equation are on a fixed timeinterval, while the above function spaces are not. To remedy this, for any time T ∈ (0 ,
1] we define the normed spaces F k ( T ) = { u k ∈ C ([ − T, T ] : E k ) : k u k k F k ( T ) = inf e u k = u k in R × [ − T,T ] k e u k k F k < ∞} ; N k ( T ) = { f k ∈ C ([ − T, T ] : E k ) : k u k k N k ( T ) = inf e f k = f k in R × [ − T,T ] k e u k k N k < ∞} , (2.12)where the infimum is taken over all extensions e u k ∈ C ( R : E k ) of u k . Similarlywe define the normed spaces F k ( T ) = { u k ∈ C ([ − T, T ] : E k ) : k u k k F k ( T ) = inf e u k = u k in R × [ − T,T ] k e u k k F k < ∞} ; N k ( T ) = { f k ∈ C ([ − T, T ] : E k ) : k f k k N k ( T ) = inf e f k = f k in R × [ − T,T ] k e f k k N k < ∞} , (2.13)where the infimum is taken over all extensions e u k ∈ C ( R : E k ) of u k .So far we have defined the dyadic function spaces where we measure the solution u to (1.1) and the nonlinearity. We assemble these in a straightforward mannerusing a Littlewood-Paley decomposition to obtain the global function spaces forthe solutions. In what follows we let σ ∈ Z + and T ∈ (0 , E σ = { φ : R → R : k φ k E σ = k b φ · p ( ξ, µ )(1 + | ξ | ) σ k L ξ,µ < ∞} , (2.14)and E σ = { φ : R → R : k φ k E σ = k b φ · (1 + | ξ | − / )(1 + | ξ | ) σ k L ξ,µ < ∞} . (2.15)Their intersections are denoted by E ∞ = ∞ \ σ =1 E σ , E ∞ = ∞ \ σ =1 E σ . For u ∈ C ([ − T, T ] : E ∞ ), respectively u ∈ C ([ − T, T ] : E ∞ ), we define k u k B σ ( T ) = k P ≤ ( u (0)) k E σ + X k ≥ sup t k ∈ [ − T,T ] σk k P k ( u ( t k )) k E k , (2.16) A. D. IONESCU, C. E. KENIG, AND D. TATARU and k u k B σ ( T ) = k P ≤ ( u (0)) k E + X k ≥ sup t k ∈ [ − T,T ] σk k P k ( u ( t k )) k E k . (2.17)Notice that the E σ and B σ ( T ) norms, which are used for the difference equation,have the added factor (1 + | ξ | − / ). This gives extra decay at low frequencies,and is essential in our analysis.Finally, the X s,b - type control of the solutions, respectively the nonlinearity isachieved using the normed spaces F σ ( T ) = { u ∈ C ([ − T, T ] : E ∞ ) : k u k F σ ( T ) = X k ∈ Z σk + k P k ( u ) k F k ( T ) < ∞} , N σ ( T ) = { u ∈ C ([ − T, T ] : E ∞ ) : k u k N σ ( T ) = X k ∈ Z σk + k P k ( u ) k N k ( T ) < ∞} . (2.18)For the difference equation we use the normed spaces F ( T ) = { u ∈ C ([ − T, T ] : E ∞ ) : k u k F ( T ) = X k ∈ Z (1 + 2 − k/ ) k P k ( u ) k F k ( T ) < ∞} , N ( T ) = { u ∈ C ([ − T, T ] : E ∞ ) : k u k N ( T ) = X k ∈ Z (1 + 2 − k/ ) k P k ( u ) k N k ( T ) < ∞} . (2.19)For any k ∈ Z we define the set S k of k-acceptable time multiplication factors S k = { m k : R → R : k m k k S k = X j =0 − jk + k ∂ j m k k L ∞ < ∞} . (2.20)Direct estimates using the definitions and (2.4) show that for any σ ∈ Z + and T ∈ (0 , (cid:13)(cid:13)P k ∈ Z m k ( t ) · P k ( u ) (cid:13)(cid:13) F σ ( T ) . (sup k ∈ Z k m k k S k ) · k u k F σ ( T ) ; (cid:13)(cid:13)P k ∈ Z m k ( t ) · P k ( u ) (cid:13)(cid:13) N σ ( T ) . (sup k ∈ Z k m k k S k ) · k u k N σ ( T ) ; (cid:13)(cid:13)P k ∈ Z m k ( t ) · P k ( u ) (cid:13)(cid:13) B σ ( T ) . (sup k ∈ Z k m k k S k ) · k u k B σ ( T ) , (2.21)and (cid:13)(cid:13)P k ∈ Z m k ( t ) · P k ( u ) (cid:13)(cid:13) F ( T ) . (sup k ∈ Z k m k k S k ) · k u k F ( T ) ; (cid:13)(cid:13)P k ∈ Z m k ( t ) · P k ( u ) (cid:13)(cid:13) N ( T ) . (sup k ∈ Z k m k k S k ) · k u k N ( T ) ; (cid:13)(cid:13)P k ∈ Z m k ( t ) · P k ( u ) (cid:13)(cid:13) B ( T ) . (sup k ∈ Z k m k k S k ) · k u k B ( T ) . (2.22) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 9 Global linear, bilinear and energy estimates
In this section we state our main linear, bilinear and energy estimates. Weshow first that F σ ( T ) ֒ → C ([ − T, T ] : E σ ) for σ ∈ Z + , T ∈ (0 , Lemma 3.1. If σ ∈ Z + , T ∈ (0 , , and u ∈ F σ ( T ) , then sup t ∈ [ − T,T ] k u ( t ) k E σ . k u k F σ ( T ) . (3.1) Proof of Lemma 3.1.
In view of the definitions, it suffices to prove that if k ∈ Z , t k ∈ [ − , e u k ∈ F k then k p ( ξ, µ ) · F [ e u k ( t k )] k L ξ,µ . k p ( ξ, µ ) · F [ e u k · η (2 k + ( t − t k ))] k X k . (3.2)Let f k = F [ e u k · η (2 k + ( t − t k ))], so F [ e u k ( t k )]( ξ, µ ) = c Z R f k ( ξ, µ, τ ) e it k τ dτ. Thus, using (2.3), k p ( ξ, µ ) · F [ e u k ( t k )] k L ξ,µ . k p ( ξ, µ ) · f k k L ξ,µ L τ . k p ( ξ, µ ) · f k k X k , which gives (3.2). (cid:3) We prove now a linear estimate.
Proposition 3.2.
Assume T ∈ (0 , , u, v ∈ C ([ − T, T ) : E ∞ ) and ∂ t u + ∂ x u − ∂ − x ∂ y u = v on R × ( − T, T ) . (3.3) (a) Then, for any σ ∈ Z + , k u k F σ ( T ) . k u k B σ ( T ) + k v k N σ ( T ) . (3.4) (b) Assume, in addition, that u (0) ∈ E ∞ and v ∈ N ( T ) . Then u ∈ F ( T ) and k u k F ( T ) . k u k B ( T ) + k v k N ( T ) . (3.5) Proof of Proposition 3.2.
In view of the definitions, it suffices to prove that if k ∈ Z and u, v ∈ C ([ − T, T ] : E k ) solve (3.3), then k P k ( u ) k F k ( T ) . k P k ( u (0)) k E k + k v k N k ( T ) if k ≤ k P k ( u ) k F k ( T ) . sup t k ∈ [ − T,T ] k P k ( u ( t k )) k E k + k v k N k ( T ) if k ≥ . (3.6)Let e v ∈ C ( R : E k ) denote an extension of v such that k e v k N k ≤ C k v k N k ( T ) . Using(2.21), we may assume that e v is supported in R × [ − T − − k + − , T + 2 − k + − ], k ∈ Z . For t ∈ R let W ( t ) denote the solution at time t of the free KP-I evolution, i.e. the operator on L ( R ) defined by the Fourier multiplier ( ξ, µ ) → e itω ( ξ,µ ) .For t ≥ T we define e u ( t ) = η (2 k + +5 ( t − T )) h W ( t − T ) P k ( u ( T )) + Z tT W ( t − s )( P k ( e v ( s ))) ds i . For t ≤ − T we define e u ( t ) = η (2 k + +5 ( t + T )) h W ( t + T ) P k ( u ( − T )) + Z t − T W ( t − s )( P k ( e v ( s ))) ds i . With e u ( t ) = u ( t ) for t ∈ [ − T, T ], it is clear that e u ∈ C ( R : E k ) is an extensionof u . Also, using (2.21), k u k F k ( T ) . sup t k ∈ [ − T,T ] k p ( ξ, µ ) · F [ e u · η (2 k + ( t − t k ))] k X k , where the supremum is taken over t k ∈ [ − T, T ].In view of the definitions and (2.21), it suffices to prove that if k ∈ Z , φ k ∈ E k ,and v k ∈ N k then k p ( ξ, µ ) · F [ u k · η (2 k + t )] k X k . k p ( ξ, µ ) · c φ k k L ξ,µ + k p ( ξ, µ )( τ − ω ( ξ, µ ) + i k + ) − · F ( v k ) k X k , (3.7)where u k ( t ) = W ( t )( φ k ) + Z t W ( t − s )( v k ( s )) ds. (3.8)It follows from (3.8) that F [ u k · η (2 k + t )]( ξ, µ, τ ) = b φ k ( ξ, µ ) · − k + b η (2 − k + ( τ − ω ( ξ, µ )))+ C Z R F ( v k )( ξ, µ, τ ′ ) · − k + b η (2 − k + ( τ − τ ′ )) − − k + b η (2 − k + ( τ − ω ( ξ, µ ))) τ ′ − ω ( ξ, µ ) dτ ′ . We observe now that (cid:12)(cid:12)(cid:12) − k + b η (2 − k + ( τ − τ ′ )) − − k + b η (2 − k + ( τ − ω ( ξ, µ ))) τ ′ − ω ( ξ, µ ) · ( τ − ω ( ξ, µ ) + i k + ) (cid:12)(cid:12)(cid:12) . − k + (1 + 2 − k + | τ − τ ′ | ) − + 2 − k + (1 + 2 − k + | τ − ω ( ξ, µ ) | ) − , and the bound (3.7) follows from (2.3) and (2.4). (cid:3) We continue with our main bilinear estimates.
Proposition 3.3. a) If σ ∈ { , , } , T ∈ [0 , , and u, v ∈ F σ ( T ) then k ∂ x ( uv ) k N σ ( T ) . k u k F σ ( T ) · k v k F ( T ) + k u k F ( T ) · k v k F σ ( T ) . (3.9) b) If T ∈ (0 , , u ∈ F ( T ) and v ∈ F ( T ) then k ∂ x ( uv ) k N ( T ) . k u k F ( T ) · k v k F ( T ) . (3.10) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 11
Proof of Proposition 3.3.
We fix extensions e u, e v ∈ C ( R : E ∞ ) of u, v such that k P k ( e u ) k F k ≤ k P k ( u ) k F k ( T ) and k P k ( e v ) k F k ≤ k P k ( v ) k F k ( T ) , k ∈ Z . Let e u k = P k ( e u ) and e v k = P k ( e v ), k ∈ Z . It follows from Lemma 7.1, Lemma 7.2, andLemma 7.5 that ( k + k P k ( ∂ x ( e u k e v k )) k N k . −| k | / k k e u k k F k · k k e v k k F k if k ≤ k and | k − k | ≤ . (3.11)Also, it follows from Lemma 7.3, Lemma 7.4, and Lemma 7.5 that ( k + k P k ( ∂ x ( e u k e v k )) k N k . − max( | k | , | k | , | k | ) / · k k e u k k F k · k k e v k k F k if | k − k | ≤ k ≤ min( k , k ) − . (3.12)The bound (3.9) follows from (3.11) and (3.12).Consider now part (b) of the proposition. We fix extensions e u ∈ C ( R : E ∞ ) of u and e v ∈ C ( R : E ∞ ) of v such that k P k ( e u ) k F k ≤ k P k ( u ) k F k ( T ) and k P k ( e v ) k F k ≤ k P k ( v ) k F k ( T ) , k ∈ Z . Let e u k = P k ( e u ) and e v k = P k ( e v ), k ∈ Z . It follows fromLemma 8.1, Lemma 8.2, and Lemma 8.4 that(1 + 2 − k/ ) k P k ( ∂ x ( e u k e v k )) k N k . − max( | k | , | k | , | k | ) / (1 + 2 − k / ) k e u k k F k · (1 + 2 k ) k v k k F k if | k − k | ≥
5. It follows from Lemma 8.1, Lemma 8.2, and Lemma 8.3 that(1 + 2 − k/ ) k P k ( ∂ x ( e u k e v k )) k N k . −| k | / (1 + 2 − k / ) k e u k k F k · (1 + 2 k ) k v k k F k if | k − k | ≤
4. The proposition follows. (cid:3)
The last ingredients in the proof of Theorem 1.1 are energy estimates. For part(a) we need the following:
Proposition 3.4.
Assume that T ∈ (0 , and u ∈ C ([ − T, T ] : E ∞ ) is a solutionof the initial value problem ( ∂ t u + ∂ x u − ∂ − x ∂ y u + ∂ x ( u /
2) = 0 on R × [ − T, T ]; u (0) = φ. (3.13) Then, for σ ∈ { , , } we have k u k B σ ( T ) . k φ k E σ + k u k F ( T ) · k u k F σ ( T ) . (3.14)The linearized equation lacks the full set of symmetries of the nonlinear equa-tion. Consequently, we have good estimates for it only at the L level: Proposition 3.5.
Assume T ∈ (0 , , u ∈ F ( T ) ∩ F ( T ) , v ∈ F ( T ) and ( ∂ t u + ∂ x u − ∂ − x ∂ y u + ∂ x ( uv ) = 0 on R × ( − T, T ); u (0) = φ, (3.15) Then k u k B ( T ) . k φ k E + k v k F ( T ) · k u k F ( T ) . (3.16)Finally, to estimate differences of solutions in F we need to differentiate thedifference equation. To get bounds for this equation we need a stronger versionof Proposition 3.5. Proposition 3.6.
Assume T ∈ (0 , , u ∈ F ( T ) , u = P ≥− ( u ) , v ∈ F ( T ) , w , w , w ∈ F ( T ) ∩ F ( T ) , w ′ , w ′ , w ′ ∈ F ( T ) , h ∈ F , h = P ≤− ( h ) , and ∂ t u + ∂ x u − ∂ − x ∂ y u = P ≥− ( v · ∂ x u ) + P m =1 P ≥− ( w m · w ′ m ) + P ≥− ( h ); u (0) = φ, (3.17) on R × ( − T, T ) . Then k u k B ( T ) . k φ k E + k v k F ( T ) · k u k F ( T ) + X m =1 k u k F ( T ) k w m k F ( T ) k w ′ m k F ( T ) . (3.18)We observe that Proposition 3.5 follows from Proposition 3.6: let u ′ = P ≥− u and observe that, using (3.15) and the definitions ( k u k B ( T ) . k u ′ k B ( T ) + C k φ k E ; ∂ x ( uv ) = v · ∂ x u ′ + P ≥− v · ∂ x P ≤− u + u · ∂ x v + P ≤− v · ∂ x P ≤− u. We prove Proposition 3.4 and Proposition 3.6 in section 6.4.
Proof of the main theorem
In this section we use the linear, bilinear and energy estimates in the previoussection to prove Theorem 1.1. Our starting point is a well-posedness result formore regular solutions:
Proposition 4.1.
Assume φ ∈ E ∞ . Then there is T = T ( k φ k E ) ∈ (0 , and aunique solution u = S ∞ T ( φ ) ∈ C ([ − T, T ] : E ∞ ) of the initial value problem ( ∂ t u + ∂ x u − ∂ − x ∂ y u + ∂ x ( u /
2) = 0 on R × ( − T, T ); u (0) = φ. (4.1) In addition, for any σ ≥ t ∈ [ − T,T ] k u ( t ) k E σ ≤ C ( σ, k φ k E σ , sup t ∈ [ − T,T ] k u ( t ) k E ) . LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 13
Proposition 4.1 follows by standard energy estimates (see [8]), since k φ k L ∞ + k ∂ x φ k L ∞ . k φ k E . To prove Theorem 1.1 (a), by scaling we may assume that k φ k E ≤ ε ≪ . (4.2)The uniqueness part of Theorem 1.1 (a) follows from Proposition 4.1. For globalexistence, in view of the conservation laws (1.4), we only need to construct thesolution on the time interval [ − , T ∈ (0 ,
1] and u ∈ C ([ − T, T ] : E ∞ ) is a solution of (4.1) with k φ k E ≤ ε ≪ t ∈ [ − T,T ] k u ( t ) k E . k φ k E . (4.3)We first use a continuity argument to establish an F bound on u in the interval[ − T, T ]. By taking σ = 1, it follows from Proposition 3.2 (a), Proposition 3.3(a), and Proposition 3.4 that for any T ′ ∈ [0 , T ] we have k u k F ( T ′ ) . k u k B ( T ′ ) + k ∂ x ( u ) k N ( T ′ ) ; k ∂ x ( u ) k N ( T ′ ) . k u k F ( T ′ ) ; k u k B ( T ′ ) . k φ k E + k u k F ( T ′ ) . (4.4)We denote X ( T ′ ) = k u k B ( T ′ ) + k ∂ x ( u ) k N ( T ′ ) and eliminate k u k F ( T ′ ) to obtain X ( T ′ ) . k φ k E + X ( T ′ ) + X ( T ′ ) . (4.5)Assuming that X ( T ′ ) is continuous and satisfieslim T ′ → X ( T ′ ) . k φ k E (4.6)if ε is sufficiently small, we would conclude that X ( T ′ ) . k φ k E . Using (4.4), k u k F ( T ) . k φ k E , (4.7)To obtain (4.6) and the continuity of X ( T ′ ) we first observe that for u ∈ C ([ − T, T ] : E ∞ ) the mapping T ′ → k u k B ( T ′ ) is increasing and continuous on theinterval [ − T, T ] and lim T ′ → k u k B ( T ′ ) . k φ k E . The similar properties of k ∂ x ( u ) k N ( T ′ ) are obtained by applying the followinglemma to v = ∂ x ( u ). Lemma 4.2.
Assume T ∈ (0 , and v ∈ C ([ − T, T ] : E ∞ ) . Then the mapping T ′ → k v k N ( T ′ ) is increasing and continuous on the interval [0 , T ] and lim T ′ → k v k N ( T ′ ) = 0 . (4.8) Proof of Lemma 4.2.
In view of the definitions and (2.21), k v k N ( T ′ ) . k p ( ξ, µ ) · F ( v · [ − T ′ ,T ′ ] ( t )) k L . T ′ / sup t ∈ [ − T ′ ,T ′ ] k v ( t ) k E . (4.9)The limit in (4.8) follows. It remains to prove the continuity of the mapping T ′ → k v k N ( T ′ ) at some point T ′ ∈ (0 , T ]. We fix ε >
0. Let D r ( v )( x, y, t ) = v ( x, y, t/r ), r ∈ [1 / , k v k N ( T ′ ) − k D T ′ /T ′ ( v ) k N ( T ′ ) . sup t ∈ [ − T ′ ,T ′ ] k v ( t ) − D T ′ /T ′ ( v )( t ) k E ≤ ε, for any T ′ ∈ (0 , T ] sufficiently close to T ′ . Also, using the definitionslim r → k D r ( v ) k N ( rT ′ ) = k v k N ( T ′ ) , which completes the proof of the lemma. (cid:3) To prove (4.3) we combine again Proposition 3.2 (a), Proposition 3.3 (a), andProposition 3.4 with σ = 2 , k u k F σ ( T ) . k u k B σ ( T ) + k ∂ x ( u ) k N σ ( T ) ; k ∂ x ( u ) k N σ ( T ) . k u k F ( T ) · k u k F σ ( T ) ; k u k B σ ( T ) . k φ k E σ + k u k F ( T ) · k u k F σ . (4.10)Using (4.2) and (4.7), it follows that k u k F σ ( T ) . k φ k E σ for σ ∈ { , } . (4.11)Then the inequality (4.3) follows from Lemma 3.1.We prove now Theorem 1.1 (b). Assume φ ∈ E is fixed, { φ n : n ∈ Z + } ⊆ E ∞ and lim n →∞ φ n = φ in E . It suffices to prove that the sequence S ∞ T ( φ n ) ∈ C ([ − T, T ] : E ∞ ) is a Cauchysequence in C ([ − T, T ] : E ). By scaling, we may assume k φ k E ≤ ε ≪ k φ n k E ≤ ε ≪ n ∈ Z + . (4.12)Using the conservation laws (1.4) it suffices to prove that for any δ > M δ such thatsup t ∈ [ − , k S ∞ ( φ m )( t ) − S ∞ ( φ n )( t ) k E ≤ δ for any m, n ≥ M δ . (4.13)For K ∈ Z + let φ Kn = P ≤ K φ n . We show first that for any K ∈ Z + there is M δ,K such thatsup t ∈ [ − , k S ∞ ( φ Km )( t ) − S ∞ ( φ Kn )( t ) k E ≤ δ for any m, n ≥ M δ,K . (4.14) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 15
Using Theorem 1.1 (a)sup t ∈ [ − , k S ∞ ( φ Kn ) k E ≤ C ( K ) for any n ∈ Z + . Standard energy estimates for the difference equation show thatsup t ∈ [ − , k S ∞ ( φ Km )( t ) − S ∞ ( φ Kn )( t ) k E ≤ C ′ ( K ) · k φ m − φ n k E , and (4.14) follows.We show now that for any δ > K ∈ Z + and M δ such thatsup t ∈ [ − , k S ∞ ( φ n )( t ) − S ∞ ( φ Kn )( t ) k E ≤ δ for any n ≥ M δ . (4.15)The main bound (4.13) would follow from (4.14) and (4.15). To prove (4.15)we need to estimate differences of solutions. We summarize our main result inProposition 4.3 below. The bound (4.15) follows from (4.17) and Lemma 3.1:sup t ∈ [ − , k S ∞ ( φ n )( t ) − S ∞ ( φ Kn )( t ) k E . k S ∞ ( φ n )( t ) − S ∞ ( φ Kn )( t ) k F (1) . k φ n − φ Kn k E + C k φ Kn k E k φ n − φ Kn k E . k φ − φ n k E + k φ − P ≤ K φ k E . Proposition 4.3.
Let u , u ∈ F (1) be solutions to (1.1) with initial data φ , φ ∈ E ∞ satisfying k φ k E + k φ k E ≤ ε ≪ , φ − φ ∈ E . Then k u − u k F (1) . k φ − φ k E , (4.16) and k u − u k F (1) . k φ − φ k E + k φ k E k φ − φ k E (4.17) Proof of Proposition 4.3.
The difference v = u − u solves the equation ( ∂ t v + ∂ x v − ∂ − x ∂ y v = − ∂ x [ v ( u + u ) / v (0) = φ = φ − φ . (4.18)By (4.7) we can bound k u k F (1) + k u k F (1) . ε . (4.19)By Proposition 3.2 (b) , Proposition 3.3 (b), and Proposition 3.5 we obtain k v k F (1) . k v k B (1) + k ∂ x [ v ( u + u ) / k N (1) ; k ∂ x [ v ( u + u ) / k N (1) . k v k F (1) ( k u k F (1) + k u k F (1) ); k v k B (1) . k φ k E + k v k F (1) ( k u k F (1) + k u k F (1) ) . Clearly, ∂ x [ v ( u + u ) / ∈ N (1) since ∂ x [ v ( u + u ) / ∈ C ([ − ,
1] : E ). Combining this with (4.19) we obtain the estimate (4.16).To prove (4.17) we first use Proposition 3.2 (a) and Proposition 3.3 (a) toobtain ( k v k F (1) . k v k B (1) + k ∂ x [ v ( u + u ) / k N (1) ; k ∂ x [ v ( u + u ) / k N (1) . k v k F (1) · ( k u k F (1) + k u k F (1) ) . Since k P ≤ ( v ) k B (1) = k P ≤ ( φ ) k E , it follows from (4.19) that k v k F (1) . k P ≥ ( v ) k B (1) + k φ k E (4.20)By (4.16) and (4.11), for (4.17) it remains to prove the estimate k P ≥ ( v ) k B (1) . k φ k E + k u k F (1) · k v k F (1) ( k P ≥ ( v ) k B (1) + k φ k E ) . (4.21)In view of the definitions, k P ≥ ( v ) k B (1) ≈ k P ≥ ( ∂ x v ) k B (1) + k P ≥ ( ∂ − x ∂ y v ) k B (1) . (4.22)Using (4.18) we write the equation for U = P ≥− ( ∂ x v ) in the form ( ∂ t U + ∂ x U − ∂ − x ∂ y U = P ≥− ( − u · ∂ x U ) + P ≥− ( G ); U (0) = P ≥− ( ∂ x φ ) , where G = − P ≥− ( u ) · ∂ x P ≤− ( v ) − P ≤− ( u ) · ∂ x P ≤− ( v ) − ∂ x v · ∂ x ( u + u ) − v · ∂ x u . It follows from Proposition 3.6 (with h = − P ≤− ( u ) · ∂ x P ≤− ( v )) that k P ≥− ( ∂ x v ) k B (1) . k ∂ x φ k E + k u k F (1) k P ≥− ( ∂ x v ) k F (1) + k P ≥− ( ∂ x v ) k F (1) · k ∂ x v k F (1) · h k ∂ x u k F (1) + k ∂ x u k F (1) i + k P ≥− ( ∂ x v ) k F (1) · k v k F (1) · k ∂ x u k F (1) . Using (4.19) and (4.20), it follows that k P ≥− ( ∂ x v ) k B (1) . k φ k E + ε ( k P ≥ ( v ) k B (1) + k φ k E )+ ( k P ≥ ( v ) k B (1) + k φ k E ) · k v k F (1) · k u k F (1) . (4.23)Using (4.18) we write the equation for U = P ≥− ( ∂ − x ∂ y v ) in the form ( ∂ t U + ∂ x U − ∂ − x ∂ y U = P ≥− ( − u · ∂ x U ) + P ≥− G ; U (0) = P ≥− ( ∂ − x ∂ y φ ) , where G = − P ≥− ( u ) · ∂ x P ≤− ( ∂ − x ∂ y v ) − P ≤− ( u ) · ∂ x P ≤− ( ∂ − x ∂ y v ) − v · ∂ y u . LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 17
It follows from Proposition 3.6 (with h = − P ≤− ( u ) · ∂ x P ≤− ( ∂ − x ∂ y v )) that k P ≥− ( ∂ − x ∂ y v ) k B (1) . k P ≥− ( ∂ − x ∂ y φ ) k E + k u k F (1) k v k F (1) + k P ≥− ( ∂ − x ∂ y v ) k F (1) · k v k F (1) · k ∂ x ( ∂ − x ∂ y u ) k F (1) . Using (4.19) and (4.20), it follows that k P ≥− ( ∂ − x ∂ y v ) k B (1) . k φ k E + ε ( k P ≥ ( v ) k B (1) + k φ k E )+ ( k P ≥ ( v ) k B (1) + k φ k E ) · k v k F (1) · k u k F (1) . (4.24)We add up (4.23) and (4.24) and use (4.22). The bound (4.21) follows. (cid:3) L bilinear estimates For k ∈ Z and l, j ∈ R let D k,l,j = { ( ξ, µ, τ ) : ξ ∈ e I k , | µ | ≤ l , | τ − ω ( ξ, µ ) | ≤ j } , and let D k, ∞ ,j = ∪ l ∈ Z D k,l,j . Lemma 5.1. (a) Assume k , k , k ∈ Z , j , j , j ∈ Z + , and f i : R → R + are L functions supported in D k i , ∞ ,j i , i = 1 , , . If max( j , j , j ) ≤ k + k + k −
20 (5.1) then Z R ( f ∗ f ) · f . ( j + j + j ) / · − ( k + k + k ) / · k f k L k f k L k f k L . (5.2) (b)Assume k , k , k ∈ Z , l , l , l ∈ Z , j , j , j ∈ Z + , and f i : R → R + are L functions supported in D k i ,l i ,j i , i = 1 , , . Then Z R ( f ∗ f ) · f . [min( k ,k ,k )+min( l ,l ,l )+min( j ,j ,j )] / · k f k L k f k L k f k L . (5.3) Proof of Lemma 5.1.
Part (b) follows easily from the Minkowski inequality. Part(a) is proved also in [6]; we reproduce the proof here for the sake of completeness.We observe that Z R ( f ∗ f ) · f = Z R ( e f ∗ f ) · f = Z R ( e f ∗ f ) · f , (5.4)where e f i ( ξ, µ, τ ) = f i ( − ξ, − µ, − τ ), i = 1 ,
2. In view of the symmetry of (5.2) wemay assume j = max( j , j , j ) . (5.5) We define f i ( ξ, µ, θ ) = f i ( ξ, µ, θ + ω ( ξ, µ )), i = 1 , , k f i k L = k f i k L . Werewrite the left-hand side of (5.2) in the form Z R f ( ξ , µ , θ ) · f ( ξ , µ , θ ) × f ( ξ + ξ , µ + µ , θ + θ + Ω(( ξ , µ ) , ( ξ , µ ))) dξ dξ dµ dµ dθ dθ , (5.6)whereΩ(( ξ , µ ) , ( ξ , µ )) = − ω ( ξ + ξ , µ + µ ) + ω ( ξ , µ ) + ω ( ξ , µ )= − ξ ξ ξ + ξ h ( √ ξ + √ ξ ) − (cid:16) µ ξ − µ ξ (cid:17) i . (5.7)The functions f i are supported in the sets { ξ, µ, θ ) : ξ ∈ e I k i , µ ∈ R , | θ | ≤ j i } .We will prove that if g i : R → R + are L functions supported in e I k i × R , i = 1 ,
2, and g : R → R + is an L function supported in e I k × R × [ − j , j ], j ≤ k + k + k −
15, then Z R g ( ξ , µ ) · g ( ξ , µ ) · g ( ξ + ξ , µ + µ , Ω(( ξ , µ ) , ( ξ , µ ))) dξ dξ dµ dµ . j/ · − ( k + k + k ) / · k g k L k g k L k g k L . (5.8)This suffices for (5.2), in view of (5.5) and (5.6).To prove (5.8), we observe first that we may assume that the integral in theleft-hand side of (5.8) is taken over the set R ++ = { ( ξ , µ , ξ , µ ) : ξ + ξ ≥ µ /ξ − µ /ξ ≥ } . Using the restriction j ≤ k + k + k −
15 and (5.7), we may assume also that theintegral in the left-hand side of (5.8) is taken over the set e R ++ = { ( ξ , µ , ξ , µ ) ∈ R ++ : |√ ξ + ξ ) | − | µ /ξ − µ /ξ | ≤ − | ξ + ξ |} . To summarize, it suffices to prove that Z e R ++ g ( ξ , µ ) · g ( ξ , µ ) · g ( ξ + ξ , µ + µ , Ω(( ξ , µ ) , ( ξ , µ ))) dξ dξ dµ dµ . j/ · − ( k + k + k ) / · k g k L k g k L k g k L . (5.9)We make the changes of variables µ = √ ξ + β ξ and µ = −√ ξ + β ξ , There are four identical integrals of this type.
LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 19 with dµ dµ = ξ ξ dβ dβ . The left-hand side of (5.9) is bounded by C k + k Z S g ( ξ , √ ξ + β ξ ) · g ( ξ , −√ ξ + β ξ ) × g ( ξ + ξ , √ ξ − √ ξ + β ξ + β ξ , e Ω(( ξ , β ) , ( ξ , β ))) dξ dξ dβ dβ , (5.10)where S = { ( ξ , β , ξ , β ) : ξ + ξ ≥ | β − β | ≤ − ( ξ + ξ ) } , (5.11)and e Ω(( ξ , β ) , ( ξ , β )) = ξ ξ ( β − β ) (cid:16) √ β − β ξ + ξ (cid:17) . (5.12)We define the functions h i : R → R + supported in e I k i × R , i = 1 , ( h ( ξ , β ) = 2 k / · g ( ξ , √ ξ + β ξ ); h ( ξ , β ) = 2 k / · g ( ξ , −√ ξ + β ξ ) , with k h i k L ≈ k g i k L . Thus, for (5.8) it suffices to prove that2 ( k + k ) / Z S h ( ξ , β ) · h ( ξ , β ) × g ( ξ + ξ , √ ξ − √ ξ + β ξ + β ξ , e Ω(( ξ , β ) , ( ξ , β ))) dξ dξ dβ dβ . j/ · − ( k + k + k ) / · k h k L k h k L k g k L . (5.13)To prove (5.13), we may assume without loss of generality that k ≤ k . (5.14)We make the change of variables β = β + β . In view of (5.11), (5.12), andthe restriction on the support of g , we may assume | β | ≤ j − k − k +4 . Thus, theintegral in the left-hand side of (5.13) is equal to2 ( k + k ) / Z e S h ( ξ , β + β ) · h ( ξ , β ) · [ − , ( β/ j − k − k +4 ) × g ( ξ + ξ , A ( ξ , ξ , β ) + β ( ξ + ξ ) , B ( ξ , ξ , β )) dξ dξ dβdβ , (5.15)where e S = { ( ξ , ξ , β, β ) ∈ R : ξ + ξ ≥ | β | ≤ − ( ξ + ξ ) } , and ( A ( ξ , ξ , β ) = √ ξ − √ ξ + βξ ; B ( ξ , ξ , β ) = ξ ξ β · (2 √ β/ ( ξ + ξ )) . (5.16)Let j ′ = j − k − k + 4 and decompose, for i = 1 , h i ( ξ ′ , β ′ ) = X m ∈ Z h i ( ξ ′ , β ′ ) · [0 , ( β ′ / j ′ − m ) = X m ∈ Z h mi ( ξ ′ , β ′ ) . The expression in (5.15) is dominated by C ( k + k ) / X | m − m ′ |≤ Z e S h m ( ξ , β + β ) · h m ′ ( ξ , β ) × g ( ξ + ξ , A ( ξ , ξ , β ) + β ( ξ + ξ ) , B ( ξ , ξ , β )) dξ dξ dβdβ . (5.17)Also, for i = 1 , k h i k L = (cid:2) X m ∈ Z k h mi k L (cid:3) . Thus, to prove (5.13), we may assume h = h m and h = h m ′ for some fixed m, m ′ ∈ Z with | m − m ′ | ≤
4. To summarize, it suffices to prove that if F i : R → [0 , ∞ ) are L functions supported in e I k i × R , g is as before, and m ∈ Z then2 ( k + k ) / Z e S F ( ξ , β + β ) · F ( ξ , β ) · [ m − ,m +1] ( β / j ′ ) × g ( ξ + ξ , A ( ξ , ξ , β ) + β ( ξ + ξ ) , B ( ξ , ξ , β )) dξ dξ dβdβ . j/ · − ( k + k + k ) / · k F k L k F k L k g k L . (5.18)To prove (5.18) we use the Minkowski inequality in the variables ( ξ , ξ , β ): with S ′ = { ( ξ , ξ , β ) ∈ R : ξ i ∈ e I k i , ξ + ξ ≥ , | β | ≤ − ( ξ + ξ ) } , the left-hand side of (5.18) is dominated by C ( k + k ) / Z R [ m − ,m +1] ( β / j ′ ) · (cid:16) Z S ′ | F ( ξ , β + β ) · F ( ξ , β ) | dξ dξ dβ (cid:17) / × (cid:16) Z S ′ | g ( ξ + ξ , A ( ξ , ξ , β ) + β ( ξ + ξ ) , B ( ξ , ξ , β )) | dξ dξ dβ (cid:17) / dβ . (5.19)For (5.18), it is easy to see that it suffices to prove that (cid:16) Z S ′ | g ( ξ + ξ , A ( ξ , ξ , β ) + β ( ξ + ξ ) , B ( ξ , ξ , β )) | dξ dξ dβ (cid:17) / . − ( k + k + k ) / k g k L . (5.20)for any β ∈ R . Indeed, assuming (5.20), we can bound the expression in (5.19)by C ( k + k ) / Z R [ m − ,m +1] ( β / j ′ ) · k F k L k F ( ., β ) k L ξ · − ( k + k + k ) / k g k L dβ , which suffices since 2 j ′ / ( k + k ) / ≈ j/ .Finally, to prove (5.20), we may assume first that β = 0. We examine (5.16)and make the change of variable β = √ ξ + ξ ) · ν . The left-hand side of (5.20) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 21 is dominated by C (cid:16) k Z S ′′ | g ( ξ + ξ , √ ξ + ξ )( ξ − ξ + νξ ) , ξ ξ ( ξ + ξ ) ν (2+ ν )) | dξ dξ dν (cid:17) / , (5.21)where S ′′ = { ( ξ , ξ , ν ) ∈ R : ξ i ∈ e I k i , | ν | ≤ − } . We define the function h ( ξ, x, y ) = 2 k · | g ( ξ, √ ξ · x, ξ · y ) | , so k h k| L ≈ k g k L . The expression in (5.21) is dominated by C − k/ (cid:16) Z S ′′ | h ( ξ + ξ , ξ − ξ + νξ , ξ ξ · ν (2 + ν )) | dξ dξ dν (cid:17) / . Therefore, it remains to prove that Z S ′′ | h ( ξ + ξ , ξ − ξ + νξ , ξ ξ · ν (2 + ν )) | dξ dξ dν . − ( k + k ) k h k L for any function h ∈ L ( R ). This is clear since the absolute value of the deter-minant of the change of variables ( ξ , ξ , ν ) → [ ξ + ξ , ξ − ξ + νξ , ξ ξ · ν (2 + ν )]is equal to (2 + ν ) | ξ | · | ξ (2 + ν ) + ξ ν | ≈ k + k , see (5.14) and the definition ofthe set S ′′ . (cid:3) Lemma 5.2.
Assume k , k , k ∈ Z , j , j , j ∈ Z + , and f i : R → R + are L functions supported in D k i , ∞ ,j i , i = 1 , , . Then Z R ( f ∗ f ) · f . ( j + j + j ) / · − max( j ,j ,j ) / · k f k L k f k L k f k L . (5.22) Proof.
Using the symmetry (5.4), we may assume j = max( j , j , j ). Then Z R ( f ∗ f ) · f . k f k L · k f ∗ f k L . k f k L kF − ( f ) k L kF − ( f ) k L . We use the scale-invariant Strichartz estimate of [1]: (cid:13)(cid:13)(cid:13)(cid:13)Z R φ ( ξ, µ ) e ix · ξ e iy · µ e it · ω ( ξ,µ ) dξdµ (cid:13)(cid:13)(cid:13)(cid:13) L x,y,t . k φ k L , (5.23)for any φ ∈ L ( R ). With f i , i = 1 ,
2, defined as in the proof of Lemma 5.1, weestimate (cid:13)(cid:13)(cid:13)(cid:13)Z R f i ( ξ, µ, τ ) · e ix · ξ e iy · µ e it · τ dξdµdτ (cid:13)(cid:13)(cid:13)(cid:13) L x,y,t = (cid:13)(cid:13)(cid:13)(cid:13)Z R f i ( ξ, µ, θ ) · e it · θ · e ix · ξ e iy · µ e it · ω ( ξ,µ ) dξdµdθ (cid:13)(cid:13)(cid:13)(cid:13) L x,y,t . j i / k f i ( ξ, µ, θ ) k L , which gives (5.22). (cid:3) As a consequence of Lemma 5.1 and Lemma 5.2, we have the following L bilinear estimates. Corollary 5.3. (a) Assume k , k , k ∈ Z , j , j , j ∈ Z + , and f i : R → R + are L functions supported in D k i , ∞ ,j i , i = 1 , . Then k D k, ∞ ,j · ( f ∗ f ) k L . ( j + j + j ) / (2 max( j ,j ,j ) + 2 k + k + k ) − / · k f k L k f k L . (5.24) (b) Assume k , k , k ∈ Z , j , j , j ∈ Z + , and f i : R → R + are L functionssupported in D k i , ∞ ,j i , i = 1 , . If k ≤ then k D k, ∞ ,j · ( f ∗ f ) k L . [ k +min( k ,k ,k )+min( j ,j ,j )] / ·k p ( ξ , µ ) · f k L k f k L . (5.25) If k ≥ − then k D k, ∞ ,j · ( f ∗ f ) k L . [2 k +min( k ,k ,k )+min( j ,j ,j )] / ·k p ( ξ , µ ) · f k L k f k L . (5.26) Proof of Corollary 5.3.
Part (a) follows from (5.2) and (5.22). For part (b), recall(see (1.3)) that p ( ξ, µ ) = 1 + | µ | / ( | ξ | + | ξ | ). To prove (5.25) we decompose f = f ,k + X l = k +1 f ,l = f · η ( µ / k ) + ∞ X l = k +1 f · χ l ( µ ) . Using (5.3), the left-hand side of (5.25) is dominated by ∞ X l = k k D k, ∞ ,j · ( f ,l ∗ f ) k L . [min( k ,k ,k )+min( j ,j ,j )] / k f k L ∞ X l = k l / k f ,l k L . [min( k ,k ,k )+min( j ,j ,j )] / k f k L · k / k p ( ξ , µ ) · f k L , as desired. To prove (5.26) we decompose f = f , k + X l =2 k +1 f ,l = f · η ( µ / k ) + ∞ X l =2 k +1 f · χ l ( µ ) . Using (5.3), the left-hand side of (5.25) is dominated by ∞ X l =2 k k D k, ∞ ,j · ( f ,l ∗ f ) k L . [min( k ,k ,k )+min( j ,j ,j )] / k f k L ∞ X l =2 k l / k f ,l k L . [min( k ,k ,k )+min( j ,j ,j )] / k f k L · k k p ( ξ , µ ) · f k L , as desired. (cid:3) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 23 Energy estimates
In this section we prove the energy estimates in Proposition 3.4 and Proposi-tion 3.6. To prove dyadic energy estimates we introduce a new Littlewood-Paleydecomposition with smooth symbols. With χ k ( ξ ) = η ( ξ/ k ) − η ( ξ/ k − ) , k ∈ Z , let e P k denote the operator on L ( R ) defined by the Fourier multiplier ( ξ, µ, τ ) → χ k ( ξ ). By a slight abuse of notation, we also let e P k denote the operator on L ( R )defined by the Fourier multiplier ( ξ, µ ) → χ k ( ξ ). For l ∈ Z let e P ≤ l = X k ≤ l e P k , e P ≥ l = X k ≥ l e P k . Assume that, for some k ∈ Z and u, v ∈ C ([ − T, T ] : E k ) ( ∂ t u + ∂ x u − ∂ − x ∂ y u = v on R × ( − T, T ); u (0) = φ. (6.1)We multiply by u and integrate to conclude thatsup | t k |≤ T k u ( t k ) k L ≤ k φ k L + sup | t k |≤ T (cid:12)(cid:12)(cid:12)(cid:12)Z R × [0 ,t k ] u · v dxdydt (cid:12)(cid:12)(cid:12)(cid:12) . (6.2)To prove Proposition 3.4 and Proposition 3.6 we need to replace v by thecorresponding bilinear expressions. Thus we need to estimate integrals of trilinearforms. However, instead of direct estimates we seek to take advantage of thespecial form of the nonlinearities. This allows us to place the derivative in thenonlinearity on the lowest frequency factor. We summarize the main dyadicestimates we need in Lemma 6.1 below. Lemma 6.1. (a) Assume T ∈ (0 , , k , k , k ∈ Z with max { k , k , k } ≥ ,and u i ∈ F k i ( T ) , i = 1 , , . Assume in addition that u i ∈ F k i ( T ) for some i ∈ { , , } . Then (cid:12)(cid:12)(cid:12)(cid:12)Z R × [0 ,T ] u u u dxdydt (cid:12)(cid:12)(cid:12)(cid:12) . − min( k ,k ,k ) / Y i =1 k u k i k F ki ( T ) . (6.3) (b) Assume T ∈ (0 , , k ∈ Z + , k ≤ k − , u ∈ F ( T ) , and v ∈ F k ( T ) . Then (cid:12)(cid:12)(cid:12)(cid:12)Z R × [0 ,T ] e P k ( u ) e P k ( ∂ x u · e P k ( v )) dxdydt (cid:12)(cid:12)(cid:12)(cid:12) . k / k v k F k ( T ) X | k ′ − k |≤ k e P k ′ ( u ) k F k ′ ( T ) . (6.4) Proof of Lemma 6.1.
For part (a), we may assume that k ≤ k ≤ k . In orderfor the integral to be nontrivial we must also have | k − k | ≤
4. The integralin the left-hand side of (6.3) converges absolutely, since one of the factors is in F k ( T ), thus bounded. We fix extensions e u i ∈ F k i such that k e u i k F ki ≤ k u i k F ki ( T ) , i = 1 , ,
3. Let γ : R → [0 ,
1] denote a smooth function supported in [ − ,
1] withthe property that X n ∈ Z γ ( x − n ) ≡ , x ∈ R . The left-hand side of (6.3) is dominated by C X | n |≤ C k (cid:12)(cid:12)(cid:12) Z R × R ( γ (2 k t − n ) [0 ,T ] ( t ) e u ) × ( γ (2 k t − n ) [0 ,T ] ( t ) e u ) · ( γ (2 k t − n ) [0 ,T ] ( t ) e u ) dxdydt (cid:12)(cid:12)(cid:12) . (6.5)To estimate the integrals in (6.5) we observe that, in view of (5.24), if k , k , k ∈ Z , f k i ∈ X k i , i = 1 , ,
3, and | m | ≤ (cid:12)(cid:12)(cid:12) Z R Z R m ( ξ, ξ ) · f ( − ξ, − µ, − τ ) f ( ξ − ξ , µ − µ , τ − τ ) × f ( ξ , µ , τ ) dξdµdτ dξ dµ dτ (cid:12)(cid:12)(cid:12) . (1 + 2 k + k + k ) − / Π , (6.6)where Π = || f || X k || f || X k || f || X k . In addition, as in (2.5), if I ⊆ R is aninterval, k ∈ Z , f k ∈ X k , and f Ik = F ( I ( t ) · F − ( f k )) thensup j ∈ Z + j/ k η j ( τ − ω ( ξ, µ )) · f Ik k L . k f k k X k . Thus, using (5.24) again, if k , k , k ∈ Z , f k i ∈ X k i , i = 1 , , I i ⊆ R , i = 1 , , | m | ≤ (cid:12)(cid:12)(cid:12) Z R Z R m ( ξ, ξ ) · f I ( − ξ, − µ, − τ ) f I ( ξ − ξ , µ − µ , τ − τ ) f I ( ξ , µ , τ ) dξdµdτ dξ dµ dτ (cid:12)(cid:12)(cid:12) . (1 + 2 k + k + k ) − / max(1 , k , k , k ) Π . (6.7)We apply now the bound (6.7) up to 4 times (for the integers n for which γ (2 k t − n ) [0 ,T ] ( t ) = γ (2 k t − n )) and the bound (6.6) about 2 k times to boundthe sum in (6.5) by the right-hand side of (6.3) (using also (2.5)). This completesthe proof of part (a).For part (b), we observe first that the expression in the left-hand side of (6.4)is dominated by C (cid:12)(cid:12)(cid:12)(cid:12)Z R × [0 ,T ] e P k ( u ) · e P k ( ∂ x u ) · e P k ( v ) dxdydt (cid:12)(cid:12)(cid:12)(cid:12) + C (cid:12)(cid:12)(cid:12)(cid:12)Z R × [0 ,T ] e P k ( u ) · [ e P k ( ∂ x u · e P k ( v )) − e P k ( ∂ x u ) · e P k ( v )] dxdydt (cid:12)(cid:12)(cid:12)(cid:12) . (6.8) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 25
We integrate by parts and use (6.3) to conclude that (cid:12)(cid:12)(cid:12)(cid:12)Z R × [0 ,T ] e P k ( u ) · e P k ( ∂ x u ) · e P k ( v ) dxdydt (cid:12)(cid:12)(cid:12)(cid:12) . k / k e P k ( v ) k F k ( T ) · k e P k ( u ) k F k ( T ) , (6.9)which suffices for (6.4).To control the term in the second line of (6.8) we fix extensions e u of u and e v of v and use the formula F [ e P k ( e P k ( e v ) · ∂ x e u ) − e P k ( e v ) · e P k ( ∂ x e u )]( ξ, µ, τ )= C Z R F ( e P k ( ∂ x e v ))( ξ , µ , τ ) · F ( e u )( ξ − ξ , µ − µ , τ − τ ) · m ( ξ, ξ ) dξ dµ dτ , (6.10)where | m ( ξ, ξ ) | = (cid:12)(cid:12)(cid:12) ( ξ − ξ )( χ k ( ξ ) − χ k ( ξ − ξ )) ξ (cid:12)(cid:12)(cid:12) . X | k ′ − k |≤ χ k ′ ( ξ − ξ ) . The bound (6.4) follows by decomposing the integral in the second line of (6.8)into at most C k integrals over time-intervals of length ≈ − k (as in (6.5)),and using the formula (6.10) and the bounds (6.6) and (6.7) to bound theseintegrals. (cid:3) We prove now Proposition 3.4 and Proposition 3.6.
Proof of Proposition 3.4.
Recall that u solves the initial-value problem ( ∂ t u + ∂ x u − ∂ − x ∂ y u + ∂ x ( u /
2) = 0 on R × [ − T, T ]; u (0) = φ. (6.11)We observe that k u k B σ ( T ) − k P ≤ ( φ ) k E σ . X k ≥ sup t k ∈ [ − T,T ] (2 σk k e P k ( u ( t k )) k L + 2 (2 σ − k || e P k ( ∂ − x ∂ y u ( t k )) || L ) . (6.12)Therefore it suffices to prove that for σ ∈ { , , } X k ≥ sup t k ∈ [ − T,T ] σk k e P k ( u ( t k )) k L + X k ≥ sup t k ∈ [ − T,T ] (2 σ − k || e P k ( ∂ − x ∂ y u ( t k )) || L ) . k φ k E σ + k u k F ( T ) · k u k F σ ( T ) . (6.13)We show first that X k ≥ sup t k ∈ [ − T,T ] σk k e P k ( u ( t k )) k L − σk k e P k ( φ ) k L . k u k F ( T ) · k u k F σ ( T ) . (6.14) For k ∈ Z + we use (6.2) and the equation (6.11) to estimate the increment2 σk k e P k ( u ( t k )) k L − σk k e P k ( φ ) k L . σk (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) e P k ( u · ∂ x u ) dxdydt (cid:12)(cid:12)(cid:12) . (6.15)The right-hand side of (6.15) is dominated by C σk X k ≤ k − (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · e P k ( e P k ( u ) · ∂ x u ) dxdydt (cid:12)(cid:12)(cid:12) + C σk X k ≥ k − ,k ∈ Z (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · e P k ( u ) · ∂ x e P k ( u ) dxdydt (cid:12)(cid:12)(cid:12) . (6.16)Using (6.4), the sum in the first line of (6.16) is dominated by C k u k F ( T ) · X | k ′ − k |≤ σk ′ k e P k ′ ( u ) k F k ′ ( T ) . Using (6.3), the sum in the second line of (6.16) is dominated by C σk X | k − k |≤ ,k ≤ k +10 k / k e P k ( u ) k F k ( T ) k e P k ( u ) k F k ( T ) k e P k ( u ) k F k ( T ) + C σk X k ≥ k +10 , | k − k |≤ k − k/ k e P k ( u ) k F k ( T ) k e P k ( u ) k F k ( T ) k e P k ( u ) k F k ( T ) . k u k F ( T ) · X | k ′ − k |≤ σk ′ k e P k ′ ( u ) k F k ′ ( T ) + 2 k/ k e P k ( u ) k F k ( T ) · k u k F σ ( T ) . The bound (6.14) follows.We show now that X k ≥ sup t k ∈ [ − T,T ] (2 σ − k k e P k ( ∂ − x ∂ y u ( t k )) k L − (2 σ − k k e P k ( ∂ − x ∂ y φ ) k L . k u k F ( T ) · k u k F σ ( T ) . (6.17)For k ∈ Z + and t k ∈ [ − T, T ] we use (6.2) and the the equation (6.11) to estimatethe increment2 (2 σ − k k e P k ( ∂ − x ∂ y u ( t k )) k L − (2 σ − k k e P k ( ∂ − x ∂ y φ ) k L . (2 σ − k (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( ∂ − x ∂ y u ) e P k ( u · ∂ y u ) dxdydt (cid:12)(cid:12)(cid:12) . (6.18) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 27
The right-hand side of (6.18) is dominated by C (2 σ − k X k ≤ k − (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( v ) · e P k ( e P k ( u ) · ∂ x v ) dxdydt (cid:12)(cid:12)(cid:12) + C (2 σ − k X k ≥ k − ,k ∈ Z (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( v ) · e P k ( u ) · ∂ x e P k ( v ) dxdydt (cid:12)(cid:12)(cid:12) , (6.19)where v = ∂ − x ∂ y u . Using (6.4), the sum in the first line of (6.19) is dominatedby C k u k F ( T ) · X | k ′ − k |≤ (2 σ − k ′ k e P k ′ ( ∂ − x ∂ y u ) k F k ′ ( T ) . Using (6.3), the sum in the second line of (6.19) is dominated by C (2 σ − k X | k − k |≤ ,k ≤ k +10 k / k e P k ( v ) k F k ( T ) k e P k ( u ) k F k ( T ) k e P k ( v ) k F k ( T ) + C (2 σ − k X k ≥ k +10 , | k − k |≤ k − k/ k e P k ( v ) k F k ( T ) k e P k ( u ) k F k ( T ) k e P k ( v ) k F k ( T ) . k u k F ( T ) · X | k ′ − k |≤ σk ′ k e P k ′ ( u ) k F k ′ ( T ) + C k/ k e P k ( u ) k F k ( T ) · k u k F σ ( T ) . The bound (6.17) follows, which completes the proof of Proposition 3.4. (cid:3)
Proof of Proposition 3.6.
Recall that u = P ≥− ( u ) solves the equation ∂ t u + ∂ x u − ∂ − x ∂ y u = P ≥− ( v · ∂ x u ) + P m =1 P ≥− ( w m · w ′ m ) + P ≥− ( h ); u (0) = φ, (6.20)on R × ( − T, T ). It suffices to prove that X k ≥ sup t k ∈ [ − T,T ] (cid:0) k e P k ( u ( t k )) k L − k e P k ( φ ) k L (cid:1) . k v k F ( T ) · k u k F ( T ) + X m =1 k u k F ( T ) k w m k F ( T ) k w ′ m k F ( T ) . (6.21) Using (6.2) and the equation (6.20), for k ≥ k e P k ( u ( t k )) k L −k e P k ( φ ) k L . (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · e P k ( e P ≤ k − ( v ) · ∂ x u ) dxdydt (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · ∂ x u · e P ≥ k − ( v ) dxdydt (cid:12)(cid:12)(cid:12) + X m =1 (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · w m · w ′ m dxdydt (cid:12)(cid:12)(cid:12) . (6.22)We observe that the term P ≥− ( h ) plays no role in the proof of (6.21) (this termis needed, however, to prove the bounds (4.23) and (4.24)).Using (6.4), X k ≥ (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · e P k ( e P ≤ k − ( v ) · ∂ x u ) dxdydt (cid:12)(cid:12)(cid:12) . k v k F ( T ) k u k F . Using (6.3), X k ≥ (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · ∂ x u · e P ≥ k − ( v ) dxdydt (cid:12)(cid:12)(cid:12) . X k ≥ X k ≥ k − X k ≤ k +20 (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · ∂ x e P k ( u ) · e P k ( v ) dxdydt (cid:12)(cid:12)(cid:12) . X k ≥ X k ≥ k − X k ≤ k +20 k − min( k ,k ) / · k e P k ( u ) k F k ( T ) k e P k ( u ) k F k ( T ) k e P k ( v ) k F k ( T ) . k v k F ( T ) k u k F . Using (6.3), with k med = k + k + k − min( k, k , k ) − max( k, k , k ), k max =max( k, k , k ) X k ≥ (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · w m · w ′ m dxdydt (cid:12)(cid:12)(cid:12) . X k,k ,k ∈ Z (cid:12)(cid:12)(cid:12) Z R × [0 ,t k ] e P k ( u ) · e P k ( w m ) · e P k ( w ′ m ) dxdydt (cid:12)(cid:12)(cid:12) . X | k med − k max |≤ − min( k,k ,k ) / k e P k ( u ) k F k ( T ) k e P k ( w m ) k F k ( T ) k e P k ( w ′ m ) k F k ( T ) . k u k F ( T ) k w m k F ( T ) k w ′ m k F ( T ) . This last inequality uses the fact that, for any v ∈ F ( T ), X k ∈ Z − k/ k e P k ( v ) k F k ( T ) . k v k F ( T ) , LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 29 which is the main reason for the low-frequency condition on functions in F .The main bound (6.21) follows, which completes the proof of the proposition. (cid:3) Dyadic bilinear estimates, I
In this section we prove several dyadic bounds which are used in the proof ofProposition 3.3 (a). We estimate first Low × High → High interactions.
Lemma 7.1.
Assume k, k , k ∈ Z , k ≤ k , k ≤ , k ≥ , | k − k | ≤ , u k ∈ F k and v k ∈ F k . Then k P k ( ∂ x ( u k v k )) k N k . k · k u k k F k · k v k k F k . (7.1) Proof of Lemma 7.1.
Using the definitions and (2.21), the left-hand side of (7.1)is dominated by C sup t k ∈ R k p ( ξ, µ )( τ − ω ( ξ, µ ) + i k ) − · k I k ( ξ ) ·F [ u k · η (2 k ( t − t k ))] ∗ F [ v k · η (2 k ( t − t k ))] k X k . Let f k = F [ u k · η (2 k ( t − t k ))] and f k = F [ v k · η (2 k ( t − t k ))]. Using the bounds(2.4) and (2.21), it suffices to prove that if j , j ≥ k , and f k i ,j i : R → R + aresupported in D k i , ∞ ,j i , i = 1 ,
2, then2 k X j ≥ k − j/ k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . k · j / k p ( ξ , µ ) · f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (7.2)Since j, j , j ≥ k it suffices to prove the L product estimate k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . k · min( j ,j ) / k p ( ξ , µ ) · f k ,j k L · k p ( ξ , µ ) · f k ,j k L . (7.3)Using the obvious bound p ( ξ, µ ) . | ξ || ξ | − p ( ξ , µ ) + p ( ξ , µ ) , (7.4)this is a consequence the estimates k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . k / k min( j ,j ) / k f k ,j k L k p ( ξ , µ ) f k ,j k L , and k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . k · min( j ,j ) / k p ( ξ , µ ) f k ,j k L · k f k ,j k L , which follow from (5.26) and (5.25) repectively. (cid:3) Lemma 7.2.
Assume k, k , k ∈ Z , k ≤ k , k ≥ , k ≥ , | k − k | ≤ , u k ∈ F k , and v k ∈ F k . Then k P k ( ∂ x ( u k v k )) k N k . (1 + k )2 − k / · k u k k F k · k v k k F k . (7.5) Proof of Lemma 7.2.
As in the proof of Lemma 7.1, using the definitions and thebounds (2.4) and (2.21), it suffices to prove that if j , j ≥ k and f k i ,j i : R → R + are supported in D k i , ∞ ,j i , i = 1 ,
2, then2 k X j ≥ k − j/ k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . (1 + k )2 − k / · j / k p ( ξ , µ ) · f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (7.6)Since j, j , j ≥ k , the large modulations j ≥ k + 4 k in the output are controlledby the L product estimate k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . k / · min( j ,j ) / k p ( ξ , µ ) · f k ,j k L · k p ( ξ , µ ) · f k ,j k L . (7.7)In this case we have p ( ξ, µ ) . | ξ | | ξ | − p ( ξ , µ ) + p ( ξ , µ ) . (7.8)Hence (7.7) is a consequence of the estimates k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . k / k min( j ,j ) / k f k ,j k L k p ( ξ , µ ) f k ,j k L and k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . k / min( j ,j ) / k p ( ξ , µ ) f k ,j k L · k f k ,j k L , which follow from (5.26).It remains to estimate the small modulations k ≤ j ≤ k + 4 k in the output.There are about 1 + k possible values for j , therefore we need to prove that k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . − k / − k · ( j + j + j ) / k p ( ξ , µ ) · f k ,j k L · k p ( ξ , µ ) · f k ,j k L . (7.9)We observe that (cid:18) µ ξ − µξ (cid:19) . | ξ | + | ξ || ξ | − | Ω(( ξ , µ ) , ( ξ , µ )) | which leads to | µ || ξ | . | µ || ξ | + | ξ | + | ξ | | ξ | − | Ω(( ξ , µ ) , ( ξ , µ )) | / , (7.10)therefore p ( ξ, µ ) . p ( ξ , µ ) + 2 k / − k max( j ,j ,j ) / . (7.11) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 31
We eliminate the expression p ( ξ, µ ) on the left using (7.11), neglecting the re-maining p ( ξ, µ ) factors on the right. Then it suffices to show that k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . − k / − k ( j + j + j ) / k f k ,j k L k f k ,j k L , and k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . − k + k ( j + j + j ) / − max( j ,j ,j ) / k f k ,j k L k f k ,j k L . Both these estimates follow from (5.24). (cid:3)
We estimate now High × High → Low interactions. Let γ : R → [0 ,
1] denote asmooth function supported in [ − ,
1] with the property that P m ∈ Z γ ( x − m ) ≡ x ∈ R . Lemma 7.3.
Assume k, k , k ∈ Z + , | k − k | ≤ , k ≤ min( k , k ) − , u k ∈ F k , and v k ∈ F k . Then k P k ( ∂ x ( u k v k )) k N k . k k − k/ · k u k k F k · k v k k F k . (7.12) Proof of Lemma 7.3.
Using the definitions and (2.21), the left-hand side of (7.12)is dominated by C sup t k ∈ R (cid:13)(cid:13)(cid:13) p ( ξ, µ )( τ − ω ( ξ, µ ) + i k ) − · k I k ( ξ ) · X | m |≤ C k − k F [ u k η (2 k ( t − t k )) γ (2 k ( t − t k ) − m )] ∗F [ v k η (2 k ( t − t k )) γ (2 k ( t − t k ) − m )] (cid:13)(cid:13)(cid:13) X k Using the definitions and the bounds (2.4) and (2.21), it suffices to prove that if j , j ≥ k , and f k i ,j i : R → R + are supported in D k i , ∞ ,j i , i = 1 ,
2, then2 k k − k X j ≥ k − j/ k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . k k − k/ · j / k p ( ξ , µ ) · f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (7.13)Due to the rough estimate p ( ξ, µ ) . k − k ( p ( ξ , µ ) + p ( ξ , µ )) (7.14)the bound (7.13) follows from (5.24) in the region for j/ ≥ k − k/
2. Thereforeit remains to prove that2 − j/ k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . − k/ · j / k p ( ξ , µ ) · f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (7.15)We now seek to improve (7.14). We observe that (cid:18) µ ξ − µξ (cid:19) . | ξ | + | ξ | − | Ω(( ξ , µ ) , ( ξ , µ )) | , which leads to | µ || ξ | . | µ || ξ | + | ξ | + | ξ | − / | Ω(( ξ , µ ) , ( ξ , µ )) | , (7.16)therefore p ( ξ, µ ) . k − k p ( ξ , µ ) + 2 − k/ max( j ,j ,j ) / Thus (7.15) follows from the bounds k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . − k − k/ ( j + j + j ) / k f k ,j k L k f k ,j k L , (7.17)and k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . ( j + j + j ) / − max( j,j ,j ) / k f k ,j k L k f k ,j k L , (7.18)both of which are consequences of (5.24). (cid:3) Lemma 7.4.
Assume that k , k ∈ Z + , | k − k | ≤ , k ∈ Z ∩ ( −∞ , , k ≤ min( k , k ) − , u k ∈ F k , and v k ∈ F k . Then k P k ( ∂ x ( u k v k )) k N k . ( k − k )2 k + k/ · k u k k F k · k v k k F k . (7.19) Proof of Lemma 7.4.
As in the proof of Lemma 7.3, using the definitions and thebounds (2.4) and (2.21), it suffices to prove that if j , j ≥ k , and f k i ,j i : R → R + are supported in D k i , ∞ ,j i , i = 1 ,
2, then2 k k X j ≥ − j/ k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . ( k − k )2 k + k/ · j / k p ( ξ , µ ) · f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (7.20)Instead of (7.14) we now have p ( ξ, µ ) . k − k ( p ( ξ , µ ) + p ( ξ , µ )) (7.21)which shows that the bound (7.13) follows from (5.24) for j/ ≥ k − k/ − j/ k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . − k/ · j / k p ( ξ , µ ) · f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (7.22)We still have (7.16), but now this leads to p ( ξ, µ ) . k p ( ξ , µ ) + 2 − k/ max( j ,j ,j ) / Then (7.22) reduces to (7.17) and (7.18), which follow as before from (5.24). (cid:3)
Finally, we estimate low-frequency interactions.
Lemma 7.5.
Assume k, k , k ∈ ( −∞ , ∩ Z , u k ∈ F k , and v k ∈ F k . Then k P k ( ∂ x ( u k v k )) k N k . ( k + k + k ) / · k u k k F k · k v k k F k . (7.23) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 33
Proof of Lemma 7.5.
As in the proof of Lemma 7.1, using the definitions and thebounds (2.4) and (2.21), it suffices to prove that if j , j ∈ Z + , and f k i ,j i : R → R + are supported in D k i , ∞ ,j i , i = 1 ,
2, then2 k X j ≥ − j/ k D k, ∞ ,j · p ( ξ, µ ) · ( f k ,j ∗ f k ,j ) k L . ( k + k + k ) / · j / k p ( ξ , µ ) · f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (7.24)We may assume that k ≤ k (which forces k ≤ k + 4). We use the simple bound p ( ξ, µ ) . k − k ( p ( ξ , µ ) + p ( ξ , µ )) , and (5.25). The bound (7.24) follows. (cid:3) Dyadic bilinear estimates, II
In this section we prove several dyadic bounds which are used in the proof ofProposition 3.3 (b). We estimate first low-frequency interactions.
Lemma 8.1.
Assume k, k , k ∈ Z ∩ ( −∞ , , u k ∈ F k , and v k ∈ F k . Then k P k ( ∂ x ( u k v k )) k N k . k/ k / · k u k k F k · k v k k F k . (8.1) Proof of Lemma 8.1.
Using the definitions and the bounds (2.4) and (2.21), itsuffices to prove that if j , j ∈ Z + , and f k i ,j i : R → R + are supported in D k i , ∞ j i , i = 1 ,
2, then 2 k X j ≥ − j/ k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . k/ k / · j / k f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (8.2)This is a direct consequence of (5.25). (cid:3) We estimate now High × High → Low interactions.
Lemma 8.2.
Assume k, k , k ∈ Z , k , k ≥ max( k − , , u k ∈ F k , and v k ∈ F k . Then k P k ( ∂ x ( u k v k )) k N k . (3 k − | k | ) / · k u k k F k · k v k k F k . (8.3) Proof of Lemma 8.2.
As in the proof of Lemma 7.3, using the definitions and thebounds (2.4) and (2.21), it suffices to prove that if j , j ≥ k , and f k i ,j i : R → R + are supported in D k i , ∞ ,j i , i = 1 ,
2, then2 k k − k + X j ≥ − j/ k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . (3 k − | k | ) / · j / k f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (8.4)Assume first that k ≥ . Then, using (5.24), the left-hand side of (8.4) is dominated by C k k − (2 k + k ) / j / k f k ,j k L · j / k f k ,j k L , which suffices for (8.4). Assume now that k ≤ k + 2 k ≥ . (8.5)Then, using (5.24), the left-hand side of (8.4) is dominated by C k + k k − (2 k + k ) / j / k f k ,j k L · j / k f k ,j k L , which suffices for (8.4) in view of (8.5). Finally, assume that k + 2 k ≤ C k + k j / k f k ,j k L · j / k f k ,j k L , which suffices for (8.4) in view of (8.6). (cid:3) Finally, we estimate Low × High → High interactions.
Lemma 8.3.
Assume k, k , k ∈ Z , k ≥ , k ≤ k − , | k − k | ≤ , u k ∈ F k ,and v k ∈ F k . Then k P k ( ∂ x ( u k v k )) k N k . k · k u k k F k · k v k k F k if k ≤ , (8.7) and k P k ( ∂ x ( u k v k )) k N k . k − k / · k u k k F k · k v k k F k if k ≥ . (8.8) Proof of Lemma 8.3.
As in the proof of Lemma 7.1, using the definitions and thebounds (2.4) and (2.21), it suffices to prove that if j , j ≥ k , and f k i ,j i : R → R + are supported in D k i , ∞ ,j i , i = 1 ,
2, then2 k X j ≥ k − j/ k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . c ( k ) · j / k f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (8.9)where c ( k ) = 2 k for k ≤
0, respectively c ( k ) = k − k / for k > k ≤
0. Since j, j , j ≥ k , the above bound is a directconsequence of (5.25).If k ≥ j ≥ k + 4 k is obtained directly from(5.26). Therefore it remains to prove that k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . − k − k / · ( j + j + j ) / k f k ,j k L · k p ( ξ , µ ) · f k ,j k L . (8.10)This follows from (5.24). (cid:3) LOBAL WELL-POSEDNESS OF THE KP-I EQUATION 35
Lemma 8.4.
Assume k, k , k ∈ Z , k ≥ , k ≤ k − , | k − k | ≤ , u k ∈ F k ,and v k ∈ F k . Then k P k ( ∂ x ( u k v k )) k N k . − k / k · k u k k F k · k v k k F k . (8.11) Proof of Lemma 8.4.
As in the proof of Lemma 7.1, using the definitions and thebounds (2.4) and (2.21), it suffices to prove that if j , j ≥
0, and f k i ,j i : R → R + are supported in D k i , ∞ ,j i , i = 1 ,
2, then2 k X j ≥ − j/ k D k, ∞ ,j · ( f k ,j ∗ f k ,j ) k L . − k / k · j / k f k ,j k L · j / k p ( ξ , µ ) · f k ,j k L . (8.12)Using (5.24), the left-hand side of (8.12) is dominated by2 k k · − (2 k + k ) / k f k ,j k L · k f k ,j k L , which suffices. (cid:3) References [1] M. Ben-Artzi and J.-C. Saut, Uniform decay estimates for a class of oscillatory integralsand applications, Differential Integral Equations (1999), 137–145.[2] J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation,Philos. Trans. Roy. Soc. London Ser. A (1975), 555–601.[3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and ap-plications to nonlinear evolution equations II. The KdV-equation, Geom. Funct. Anal. (1993), 209–262.[4] J. Bourgain, On the Cauchy problem for the Kadomstev–Petviashvili Equation, Geom.Funct. Anal. (1993), 315–341.[5] M. Christ, J. Colliander, and T. Tao, A priori bounds and weak solutions for the nonlinearSchr¨odinger equation in Sobolev spaces of negative order, Preprint (2006).[6] J. Colliander, A. D. Ionescu, C. E. Kenig, and G. Staffilani, Weighted low-regularity solu-tions of the KP-I initial-value problem, Preprint (2007).[7] A. D. Ionescu and C. E. Kenig, Local and global well-posedness of periodic KP-I equations,Preprint (2005).[8] R. J. Iorio and W. V. L. Nunes, On equations of KP-type, Proc. Roy. Soc. Edinburgh Sect.A (1998), 725–743.[9] C. E. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the gener-alized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. (1993), 527–620.[10] C. E. Kenig, G. Ponce, and L. Vega, The Cauchy problem for the Korteweg-de Vriesequation in Sobolev spaces of negative indices. Duke Math. J. (1993), 1–21.[11] C. E. Kenig, On the local and global well-posedness theory for the KP-I equation, Ann.Inst. H. Poincar´e Anal. Non Lin´eaire (2004), 827–838.[12] S. Klainerman and M. Machedon, Space-time estimates for null forms and the local exis-tence theorem, Comm. Pure Appl. Math. (1993), 1221–1268.[13] H. Koch and D. Tataru, A-priori bounds for the 1-d cubic NLS in negative Sobolev spaces,Preprint (2006). [14] L. Molinet, J.-C. Saut, and N. Tzvetkov, Well-posedness and ill-posedness results for theKadomtsev-Petviashvili-I equation, Duke Math. J. (2002), 353–384.[15] L. Molinet, J.-C. Saut, and N. Tzvetkov, Global well-posedness for the KP-I equation,Math. Ann. (2002), 255–275.[16] L. Molinet, J.-C. Saut, and N. Tzvetkov, Correction: Global well-posedness for the KP-Iequation, Math. Ann. (2004), 707–710.[17] J.-C. Saut, Remarks on the generalized Kadomstev-Petviashvili equations, Indiana Univ.Math. J. (1993), 1011–1026.[18] H. Takaoka and N. Tzvetkov, On the local regularity of the Kadomtsev-Petviashvili-IIequation, Int. Math. Res. Not. (2001), 77–114.[19] D. Tataru, Local and global results for wave maps I, Comm. Partial Differential Equations (1998), 1781–1793. University of Wisconsin–Madison
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