Well-posedness of the Fifth Order Kadomtsev-Petviashvili I Equation in Anisotropic Sobolev Spaces with Nonnegative Indices
aa r X i v : . [ m a t h . A P ] J a n Well-posedness of the Fifth OrderKadomtsev-Petviashvili I Equation inAnisotropic Sobolev Spaces withNonnegative Indices ∗ Junfeng Li
School of Mathematical SciencesLaboratory of Math and Complex Systems, Ministry of EducationBeijing Normal University, Beijing 100875, P. R. ChinaEmail: junfl[email protected]
Jie Xiao
Department of Mathematics and StatisticsMemorial University of Newfoundland, St John’s, NL AIC 5S7, CanadaEmail: [email protected]
Abstract
In this paper we establish the local and global well-posedness of the real valuedfifth order Kadomstev-Petviashvili I equation in the anisotropic Sobolev spaceswith nonnegative indices. In particular, our local well-posedness improves Saut-Tzvetkov’s one and our global well-posedness gives an affirmative answer to Saut-Tzvetkov’s L -data conjecture. Key Words:
Fifth KP-I equation, anisotropic Sobolev space, Bourgain space, dyadicdecomposed Strichartz estimate, smoothing effect.
In their J. Math. Pures Appl. (2000) paper on the initial value problem (IVP) of thereal valued fifth order Kadomtsev-Petviashvili I (KP-I) equation (for ( α, t, x, y ) ∈ R ): (cid:26) ∂ t u + α∂ x u + ∂ x u + ∂ − x ∂ y u + u∂ x u = 0 ,u (0 , x, y ) = φ ( x, y ) , (1)J.C. Saut and N. Tzvzetkov obtained the following result (cf. [16, Theorems 1 & 2]): ∗ This project was completed when the first-named author visited Memorial University of Newfound-land under the financial support from the NNSF of China No.10626008 as well as the second-namedauthor’s NSERC (Canada) grant and Dean of Science (MUN, Canada) Start-up fund. aut-Tzvzetkov’s Theorem (i) The IVP (1) is locally well-posed for initial data φ satisfying k φ k L ( R ) + (cid:13)(cid:13) |− i∂ x | s φ (cid:13)(cid:13) L ( R ) + (cid:13)(cid:13) |− i∂ y | k φ (cid:13)(cid:13) L ( R ) < ∞ with s − , k ≥
0; ˆ φ ( ξ, η ) | ξ | ∈ S ′ ( R ) . (2)(ii) The IVP (1) is globaly well-posed for initial data φ satisfying k φ k L ( R ) < ∞ ; 12 Z R | ∂ x φ | + α Z R | ∂ x φ | + 12 Z R | ∂ − x ∂ y φ | − Z R φ < ∞ . (3) Here and henceforth, | − i∂ x | s and | − i∂ y | s are defined via the Fourier transform: \ | − i∂ x | s φ ( ξ, η ) = | ξ | s ˆ φ ( ξ, η ) and \ | − i∂ y | s φ ( ξ, η ) = | η | s ˆ φ ( ξ, η ) . Since they simultaneously found in [16, Theorem 3] that the condition k φ k L ( R ) < ∞ ; | ξ | − ˆ φ ( ξ, η ) ∈ S ′ ( R ) (4)ensures the gobal well-posedness for the real valued fifth order Kadomtsev-PetviashviliII (KP-II) equation (for ( α, t, x, y ) ∈ R ): (cid:26) ∂ t u + α∂ x u − ∂ x u + ∂ − x ∂ y u + u∂ x u = 0 ,u (0 , x, y ) = φ ( x, y ) , (5)they made immediately a conjecture in [16, Remarks, p. 310] which is now reformulatedin the following form: Saut-Tzvzetkov’s L -data Conjecture The IVP (1) is globally well-posed for initialdata φ satisfying (4) .In the above and below, as “local well-posedness” we refer to finding a Banach space( X, k · k X ) – when the initial data φ ∈ X there exists a time T depending on k φ k X such that (1) has a unique solution u in C ([ − T, T ]; X ) ∩ Y (where Y is one of the Bour-gain spaces defined in Section 2) and u depends continuously on φ (in some reasonabletopology). If this existing time T can be extended to the positive infinity, then “localwell-posedness” is said to be “global well-posedness”. Of course, the choice of a Banachspace relies upon the boundedness of the fundamental solution to the correspondinghomogenous equation or the conservation law for equation itself.In our current paper, we settle this conjecture through improving the above-citedSaut-Tzvzetkov’s theorem. More precisely, we have the following: Theorem 1.1
The IVP (1) is not only locally but also globally well-posed for initial data φ satisfying φ ∈ H s ,s ( R ) with s , s ≥ | ξ | − ˆ φ ( ξ, η ) ∈ S ′ ( R ) . (6)Here and henceafter, the symbol H s ,s ( R ) = n f ∈ S ′ ( R ) : k f k H s ,s ( R ) = (cid:13)(cid:13) (1 + | ξ | ) s (1 + | η | ) s ˆ f ( ξ, η ) (cid:13)(cid:13) L ( R ) < ∞ o stands for the anisotropic Sobolev space with nonnegative indices s , s ∈ [0 , ∞ ). Obvi-ously, if s = s = 0 then H s ,s ( R ) = L ( R ) and hence (6) goes back to (4) which maybe regarded as the appropriate constraint on the initial data φ deriving the global well-posednedness of the IVP for the fifth order KP-I equation. And yet the understandingof Theorem 1.1 is not deep enough without making three more observations below:2 Observation 1
The classification of the fifth order KP equations is determined bythe dispersive function: ω ( ξ, µ ) = ± ξ − αξ + µ ξ , (7)where the signs ± in (7) produce the fifth order KP-I and KP-II equations respectively.The forthcoming estimates play an important role in the analysis of the fifth order KPequations – for the fifth order KP-I equation, we have | ξ | > | α | ⇒ |∇ ω ( ξ, µ ) | = (cid:12)(cid:12)(cid:12)(cid:16) ξ + 3 αξ − µ ξ , µξ (cid:17)(cid:12)(cid:12)(cid:12) & | ξ | ; (8)and for the fifth order KP-II equation, we have | ξ | > | α | ⇒ |∇ ω ( ξ, µ ) | = (cid:12)(cid:12)(cid:12)(cid:16) ξ + 3 αξ + µ ξ , µξ (cid:17)(cid:12)(cid:12)(cid:12) & | ξ | . (9)By (9), we can get more smooth effect estimates than by (8). These imply that wecan get a well-posedness (in other words, a lower regularity) for the fifth order KP-IIequation better than that for the fifth order KP-I equation. Another crucial concept isthe resonance function: R ( ξ , ξ , µ , µ )= ω ( ξ + ξ , µ + µ ) − ω ( ξ , µ ) − ω ( ξ , µ )= ξ ξ ( ξ + ξ ) (cid:18) ( ξ + ξ ) h ξ + ξ ξ + ξ ) − α i ∓ (cid:16) µ ξ − µ ξ (cid:17) (cid:19) . (10)Evidently, the fifth order KP-II equation (corresponding to “+” in (10)) always enjoys | R ( ξ , ξ , µ , µ ) | & (cid:0) max {| ξ | , | ξ | , | ξ + ξ |} (cid:1) min {| ξ | , | ξ | , | ξ + ξ |} . (11)Nevertheless, this last inequality (11) is no longer true for the fifth order KP-I equation.In the foregoing and following the notation A . B (i.e., B & A ) means: there existsa constant C > A and B such that A ≤ CB.
In addition, if there existtwo positive constants c and C such that 10 − < c < C < and cA ≤ B ≤ CB thenthe notation A ∼ B will be used. • Observation 2
Perhaps it worths pointing out that the well-posedness of the fifthorder KP-II equation is relatively easier to establish but also its result is much betterthan that of the fifth order KP-I equation. Although the study of the well-posedness forthe fifth order KP-II equation (without the third order partial derivative term) usuallyfocuses on the critical cases (which means s + 2 s = − H s ,s ( R ) with s > − , s ≥ L ( R ). On the other hand, in [8] Isaza-L´opez-Mej´ıa established the local well-posedness for H s ,s ( R ) with s > − , s ≥ H s ,s ( R ) with s > − , s ≥
0. More recently, Hadac [4] alsogained the same local well-posedness in a broader context. Meanwhile in the fifth orderKP-I equation case, the attention is mainly paid on those spaces possessing conservationlaw such as L ( R ) and the energy space E ( R ) = n f ∈ L ( R ) : (cid:13)(cid:13) (1 + | ξ | + | ξ | − | µ | ) ˆ f ( ξ, µ ) (cid:13)(cid:13) L ( R ) < ∞ o .
3o obtain the local well-posedness of KP-I in E ( R ), in [16], besides the above-mentionedresults Saut and Tzvekov also got the local well-posedness in ˜ H s,k ( R ) with s − , k ≥ H s,k ( R ) = n f ∈ L ( R ) : (cid:13)(cid:13) (1 + | ξ | s + | ξ | − | η | k ) ˆ f ( ξ, η ) (cid:13)(cid:13) L ( R ) < ∞ o . For the energy case ˜ H , ( R ) = E ( R ), they obtained the global well-posedness of (1).In [5], Ionescu and Kenig got the global well-posedness for the fifth order periodic KP-Iequation (without the third order dispersive term) in the standard energy space E ( R ).Recently, in [3] Chen-Li-Miao obtained the local well-posedness in E s ( R ) = n f ∈ L ( R ) : k (1 + | ξ | + | ξ | − | µ | ) s ˆ f ( ξ, µ ) k L ( R ) < ∞ o , < s ≤ . • Observation 3
The well-posedness for the IVP of the third order KP equations in R : (cid:26) ∂ t u ∓ ∂ x u + ∂ − x ∂ y u + u∂ x u = 0 ,u (0 , x, y ) = φ ( x, y ) , (12)in which the sign ∓ give the third order KP-I and KP-II equations respectively, is an im-portant background material of the investigation of the well-posedness for the fifth orderKP equations. Molinet, Saut and Tzvetkov showed in [13, 14] that, for the third orderKP-I equation one cannot obtain the local well-posedness in any type of nonisotropic L -based Sobolev space or in the energy space using Picard’s iteration – see also [12]; whileI´orio and Nunes [7] applied a compactness method to deduce the local well-posednes forthe third KP-I equation with data being in the normal Sobolev space H s ( R ) , s > n f ∈ L ( R ) : k f k L ( R ) + k ∂ − x ∂ y f k L ( R ) + k ∂ x f k L ( R ) + k ∂ − x ∂ y f k L ( R ) < ∞ o . As far as we know, the best well-posed result on the third order KP-I equation is due toIonescu, Kenig and Tataru [6] which gives the global well-posedness for the third orderKP-I equation in the energy space n f ∈ L ( R ) : k f k L R + k ∂ − x ∂ y f k L ( R ) + k ∂ x f k L ( R ) < ∞ o . Relatively speaking, the results on the third order KP-II equation are nearly perfect.In [2], Bourgain proved the global well-posedness of the third order KP-II equation in L ( R ) – the assertion was then extended by Takaoka and Tzvetkov [18] and Isaza-Mej´ıa[9] from L ( R ) to H s ,s ( R ) with s > − , s ≥
0. In [17], Takaoka obtained the localwell-posedness for the third order KP-II equation in H s ,s ( R ) with s > − , s = 0under an additional low frequency condition | − i∂ x | − + ε φ ∈ L ( R ) which was removedsuccessfully in Hadac’s recent paper [4]. These results are very close to the critical index s + 2 s = − which follows from the scaling argument.The rest of this paper is devoted to an argument for Theorem 1.1. In Section 2 wecollect some useful and basically known linear estimates for the fifth order KP-I equation.In Section 3 we present the necessary and crucial bilinear estimates in order to set up thelocal (and hence global) well-posedness – this part is partially motivated by [16] though– the main difference between their treatment and ours is how to dispose the “high-highinteraction” – their method exhausts no geometric structure of the resonant set of thefifth order KP-I equation while ours does fairly enough. In Section 4 we complete theargument through applying the facts verified in Sections 2 and 3 and Picard’s iterationprinciple to the integral equation corresponding to (1).4 Linear Estimates
We begin with the IVP of linear fifth order KP-I equation: (cid:26) ∂ t u + α∂ x u + ∂ x u + ∂ − x ∂ y u = 0 ,u (0 , x, y ) = φ ( x, y ) . (13)By the Fourier transform c ( · ), the solution of (13) can be defined as u ( t )( x, y ) = (cid:0) S ( t ) φ (cid:1) ( x, y ) = Z R e i ( xξ + yµ + tω ( ξ,µ )) b φ ( ξ, µ ) dξdµ. By Duhamel’s formula, (1) can be reduced to the integral representation below: u ( t ) = S ( t ) φ − Z t S ( t − t ′ ) ∂ x ( u ( t ′ )) dt ′ . (14)So, in order to get the locall well-posedenss we will apply a Picard fixed point argumentin a suitable function space to the following integral equation: u ( t ) = ψ ( t ) S ( t ) φ − ψ T ( t )2 Z t S ( t − t ′ ) ∂ x ( u ( t ′ )) dt ′ , (15)where t belongs to R , ψ is a time cut-off function satisfying ψ ∈ C ∞ ( R ); supp ψ ⊂ [ − , ψ = 1 on [ − , , and ψ T ( · ) represents ψ ( · /T ) for a given time T ∈ (0 , s , s ≥ b ∈ R the notation X s ,s b is used as theBourgain space with norm: k u k X s ,s b = k < τ − ω ( ξ, µ ) > b < ξ > s < µ > s ˆ u ( τ, ξ, µ ) k L ( R ) , where < · > stands for (1 + | · | ) ∼ | · | . Furthermore, for an interval I ⊂ R thelocalized Bourgain space X s ,s b ( I ) can be defined via requiring k u k X s ,s b ( I ) = inf w ∈ X s ,s b (cid:8) k w k X s ,s b : w ( t ) = u ( t ) on interval I (cid:9) . The following two results are known.
Proposition 2.1 [16] If T ∈ (0 , ∞ ); s , s ≥ − < b ′ ≤ ≤ b ≤ b ′ + 1 , then k ψS ( t ) φ k X s ,s b . k φ k H s ,s ( R ) . (16) (cid:13)(cid:13)(cid:13) ψ ( t/T ) Z t S ( t − t ′ ) h ( t ′ ) dt ′ (cid:13)(cid:13)(cid:13) X s ,s b . T − b + b ′ k h k X s ,s b ′ . (17) for any k h k X s ,s b ′ < ∞ . Proposition 2.2 [1] If r ∈ [2 , ∞ ) , then there exists a constant c > independent of T ∈ (0 , such that (cid:13)(cid:13) | − i∂ x | − r (cid:0) S ( t ) φ (cid:1) ( x, y ) (cid:13)(cid:13) L rr − T L r ( R ) ≤ c k φ k L ( R ) , (18)5here k f k L rr − T L r ( R ) = Z T − T (cid:18)Z R | f ( x, y, t ) | r dxdy (cid:19) r − dt ! r − r . To reach our bilinear inequalities in Section 3, we will use ( · ) ∨ for the inverse Fouriertransform, and take the dyadic decomposed Strichartz estimates below into account. Proposition 2.3
Let η be a bump function with compact support in [ − , ⊂ R and η = 1 on ( − , ⊂ R . For each integer j ≥ set η j ( x ) = η (2 − j x ) − η (2 − j x ) , η ( x ) = η ( x ) , η j ( ξ, µ, τ ) = η j ( τ − ω ( ξ, µ )) , and f j ( ξ, µ, τ ) = ( η j ( ξ, µ, τ ) | ˆ f | ( ξ, µ, τ )) ∨ for any given f ∈ L ( R ) . Then for given r ∈ [2 , ∞ ) and any T ∈ (0 , we have (cid:13)(cid:13) | − i∂ x | − r f j (cid:13)(cid:13) L rr − T L r ( R ) . j k f j k L ( R ) . (19) In particular, (cid:13)(cid:13) | − i∂ x | f j (cid:13)(cid:13) L T L ( R ) . j k f j k L ( R ) . (20) Proof : Note first that f j ( x, y, t ) = Z R e i ( xξ + yµ + tτ ) | ˆ f | η j ( ξ, µ, τ ) dξdµdτ. So, changing variables and using c f λ ( ξ, µ ) = | ˆ f | ( ξ, µ, λ + ω ) we can write f j ( x, y, t ) = Z R e i ( xξ + yµ + t ( λ + ω )) | ˆ f | ( ξ, µ, λ + ω ) η j ( λ ) dξdµdλ = Z R e itλ η j ( λ ) h Z R e i ( xξ + yµ + tω ) | ˆ f | ( ξ, µ, λ + ω ) dξdµ i dλ = Z R e itλ η j ( λ ) S ( t ) f λ ( x, y ) dλ. Now the estimate (19) follows from Minkowski’s inequality, the Strichartz estimate (18)and the Cauchy-Schwarz inequality.The following well-known elementary inequalities are also useful – see for example[16, Proposition 2.2].
Proposition 2.4
Let γ > . Then Z R dt< t > γ < t − a > γ . < a > − γ (21) and Z R dt< t > γ | t − a | . < a > − (22) hold for any a ∈ R Bilinear Estimates
Although there were many works on the so-called bilinear estimates, we have found thatthe Kenig-Ponce-Vega’s bilinear estimation approach introduced in [11] is quite suitablefor our purpose. With the convention: when a ∈ R the number a ± equals a ± ǫ forarbitrarily small number ǫ >
0, we can state our bilinear estimate as follows.
Theorem 3.1 If s , s ≥ and functions u, v have compact time support on [ − T, T ] with < T < , then k ∂ x ( uv ) k X s ,s −
12 + . k u k X s ,s
212 + k v k X s ,s
212 + . (23) Proof
In what follows, we derive (23) using the duality; that is, we are required todominate the integral Z A ∗ | ξ | < ξ > s < µ > s < τ − ω ( ξ, µ ) > − g ( ξ, µ, τ ) | ˆ u | ( ξ , µ , τ ) | ˆ v | ( ξ , µ , τ ) dξ dµ dτ dξ dµ dτ , (24)where g ≥ k g k L ( R ) ≤ A ∗ = (cid:8) ( ξ , µ , τ , ξ , µ , τ ) ∈ R : ξ + ξ = ξ, µ + µ = µ, τ + τ = τ (cid:9) . Let σ = τ − ω ( ξ, µ ); σ = τ − ω ( ξ , µ ); σ = τ − σ ( ξ , µ ) . Define two functions below: f ( ξ , µ , τ ) = < ξ > s < µ > s < σ > + | ˆ u ( ξ , µ , τ ) | and f ( ξ , µ , τ ) = < ξ > s < µ > s < σ > + | ˆ v ( ξ , µ , τ ) | . Then we need to bound the integral Z A ∗ K ( ξ , µ , τ , ξ , µ , τ ) g ( ξ, µ, τ ) f ( ξ , µ , τ ) f ( ξ , µ , τ ) dξ dµ dτ dξ dµ dτ (25)from above by using a constant multiple of k f k L ( R ) k f k L ( R ) . Here K ( ξ , µ , τ , ξ , µ , τ ) = (cid:18) | ξ + ξ | < σ > − < σ > + < σ > + (cid:19) × (cid:18) < ξ + ξ > s < ξ > s < ξ > s (cid:19) (cid:18) < µ + µ > s < µ > s < µ > s (cid:19) . It is clear that for s , s ≥ K ( ξ , µ , τ , ξ , µ , τ ) . | ξ + ξ | < σ > − < σ > + < σ > + . Keeping a further assumption | ξ | ≥ | ξ | (which follows from symmetry) in mind, weare about to fully control the integral in (25) through handling two situations. • Situation 1 – Low Frequency | ξ + ξ | . max { , | α |} . ◦ High+High → Low | ξ | , | ξ | & max { , | α |} . We first deduce a dyadic decompo-sition. Employing η j in Proposition 2.3, we have P j ≥ η j = 1, and consequently (25)can be bounded from above by a constant multiple of X j ≥ − j ( − ) Z A ∗ η j ( σ ) g ( ξ, µ, τ ) (cid:18) f ( ξ , µ , τ ) < σ > + (cid:19) (cid:18) f ( ξ , µ , τ ) < σ > + (cid:19) dξ dµ dτ dξ dµ dτ . (26)7e may assume that for each natural number j , G j ( x, y, t ) = F − (cid:16) η j ( σ ) g ( ξ, µ, τ ) (cid:17) ( x, y, t ) , has support compact in the interval [ − T, T ] whenever it acts as a time-dependent func-tion, where F − also denotes the inverse Fourier transform. In fact, if we consider thefollowing functions generated by F − : F l ( x, y, t ) = F − f l ( ξ l , µ l , τ k ) < σ l > + ! ( x, y, t ) for l = 1 , , then the integral in (26) can be written as an L inner product h G j , F F i . Since u and v (acting as time-dependent functions) have compact support in [ − T, T ], so does F F . Asa result, the inner product h G j , F F i can be restricted on the interval [ − T, T ], namely,we may assume that G j has the same compact support (with respect to time) as F F ’s.Now, an application of (20) yields that the sum in (26) is bounded by a constant multipleof X j ≥ − j ( − ) h G j , F F i . X j,j ,j ≥ (cid:16) − j ( − ) − j ( +) − j ( +) × (cid:13)(cid:13) | − i∂ x | ( η j ( σ ) f ) ∨ (cid:13)(cid:13) L T L ( R ) (cid:13)(cid:13) | − i∂ x | ( η j ( σ ) f ) ∨ (cid:13)(cid:13) L T L ( R ) k η j ( σ ) g k L ( R ) (cid:17) . X j,j ,j ≥ (cid:16) − j ( − ) − j [( +) − ] − j [( +) − ] × k η j ( σ ) f k L ( R ) k η j ( σ ) f k L ( R ) k η j ( σ ) g k L ( R ) (cid:17) . k f k L ( R ) k f k L ( R ) . ◦ Low+Low → Low | ξ | , | ξ | . max { , | α |} . Via changing variables and using theCauchy-Schwarz inequality we can bound (25) with Z K ll (cid:18)Z | f ( ξ , µ , τ ) f ( ξ − ξ , µ − µ , τ − τ ) | dτ dξ dµ (cid:19) g ( ξ, µ, τ ) dξdµdτ, where K ll = | ξ | < σ > − (cid:18)Z dτ dξ dµ < τ − ω ( ξ , µ ) > < τ − τ − ω ( ξ − ξ , µ − µ ) > (cid:19) . We need only to control K ll using a constant independent of ξ, µ, τ . By (21) we have K ll . | ξ | < σ > − (cid:18)Z dξ dµ < τ − ω ( ξ, µ ) − ω ( ξ − ξ , µ − µ ) > (cid:19) . An elementary computation with the change of variables: ν = τ − ω ( ξ, µ ) − ω ( ξ − ξ , µ − µ )shows (cid:12)(cid:12)(cid:12) dνdµ (cid:12)(cid:12)(cid:12) & | ξ | | σ + ξξ ( ξ − ξ )(5 ξ − ξξ + 5 ξ − α ) − ν | K ll . | ξ | < σ > − Z dξ dν< ν > | σ + ξξ ( ξ − ξ )(5 ξ − ξξ + 5 ξ − α ) − ν | ! . By (22) we further get K ll . Z | ξ | . max { , | α |} dξ < σ + ξξ ( ξ − ξ )(5 ξ − ξξ + 5 ξ − α ) > ! . . • Situation 2 – High Frequency | ξ + ξ | & max { , | α |} . ◦ High+Low → High | ξ | . max { , | α |} . | ξ | ∼ | ξ | . As above, we apply theCauchy-Schwarz inequality to bound the integral in (25) from above with a constantmultiple of Z K hl (cid:18)Z | f ( ξ , µ , τ ) f ( ξ − ξ , µ − µ , τ − τ ) | dτ dξ dµ (cid:19) g ( ξ, µ, τ ) dξdµdτ where K hl = | ξ | < σ > − (cid:18)Z dτ dξ dµ < τ − ω ( ξ , µ ) > < τ − τ − ω ( ξ − ξ , µ − µ ) > (cid:19) , but also we have the following estimate K hl . | ξ | < σ > − (cid:18)Z dξ dµ < τ − ω ( ξ, µ ) − ω ( ξ − ξ , µ − µ ) > (cid:19) . Under the change of variables κ = ξξ ( ξ − ξ )(5 ξ − ξξ + 5 ξ − α ); ν = τ − ω ( ξ, µ ) − ω ( ξ − ξ , µ − µ )the Jacobian determinant J enjoys J . | κ | | ξ | | σ + κ − ν | ( | ξ | − | κ | ) . As a by-product of the last inequality and (22), we obtain K hl . | ξ | < σ > − Z | κ | dκdν | σ + κ − ν | ( | ξ | − | κ | ) < ν > ! . | ξ | < σ > − Z | κ | dκ< σ + κ > ( | ξ | − | κ | ) ! . Since | ξ − ξ | . max { , | α |} , we have | κ | . | ξ | , whence getting K hl . | ξ | < σ > − Z | κ | . | ξ | dκ< σ + κ > ! . . ◦ High+High → High | ξ | , | ξ | & max { , | α |} . Since | ξ | ≥ | ξ | , we have | ξ | & | ξ + ξ | . Under this circumstance, we will deal with two cases in the sequel.9 Case (i) max {| σ | , | σ |} & | ξ | . Decomposing the integral according to | ξ | ∼ m where m = 1 , , · · · , we can run the dyadic decomposition: | σ | ∼ j , | σ | ∼ j , | σ | ∼ j for j, j , j = 0 , , , .... If | σ | ≥ | σ | ≥ | ξ | , then an application of (20) yields that the integral in (25) is boundedfrom above by a constant multiple of X m ≥ X j ≥ m X j ,j ≥ (cid:16) m − j ( − ) − j ( +) − j ( +) k η j ( σ ) g k L ( R ) × (cid:13)(cid:13) | − i∂ x | (cid:0) η m ( ξ ) η j ( σ ) f (cid:1) ∨ (cid:13)(cid:13) L T L ( R ) (cid:13)(cid:13) | − i∂ x | (cid:0) η j ( σ ) f (cid:1) ∨ (cid:13)(cid:13) L T L ( R ) (cid:17) . X m ≥ X j ≥ m X j ,j ≥ (cid:16) − j ( − ) m − j [( +) − ] − j [( +) − ] × k η j ( σ ) f k L ( R ) k η j ( σ ) f k L ( R ) k η j ( σ ) g k L ( R ) (cid:17) . k f k L ( R ) k f k L ( R ) . If | σ | ≥ | σ | ≥ | ξ | , then a further use of (20) derives that the integral in (25) is boundedfrom above by a constant multiple of X m ≥ X j ≥ m,j ≥ X j ≥ (cid:16) m − j ( − ) − j ( +) − j ( +) k η j ( σ ) f k L ( R ) × (cid:13)(cid:13) | − i∂ x | ( η m ( ξ ) η j ( σ ) f ) ∨ (cid:13)(cid:13) L T L ( R ) (cid:13)(cid:13) | − i∂ x | ( η j ( σ ) g ) ∨ (cid:13)(cid:13) L T L ( R ) (cid:17) . X m ≥ X j ≥ m X j ,j ≥ (cid:16) − j ( +) m − j [( +) − ] − j [( − ) − ] × k η j ( σ ) f k L ( R ) k η j ( σ ) f k L ( R ) k η j ( σ ) g k L ( R ) (cid:17) . k f k L ( R ) k f k L ( R ) . ⋄ Case (ii) max {| σ | , | σ |} . | ξ | . In this case, we need to consider the size ofthe resonance function even more carefully. This consideration will be done via splittingthe estimate into two pieces according to the size of resonance function. ⊲ Subcase (i) max {| σ | , | σ | , | σ |} & | ξ | . This means that the resonant interac-tion does not happen and consequently | σ | & | ξ | . The dyadic decomposition and (20)are applied to deduce that the integral in (25) is bounded from above by a constantmultiple of X m ≥ X j ≥ m X m ≥ j,j ≥ (cid:16) m − j ( − ) − j ( +) − j ( +) k η m ( ξ ) η j ( σ ) f k L ( R ) × (cid:13)(cid:13) | − i∂ x | ( η j ( σ ) f ) ∨ (cid:13)(cid:13) L T L ( R ) (cid:13)(cid:13) | − i∂ x | ( η j ( σ ) g ) ∨ (cid:13)(cid:13) L T L ( R ) (cid:17) . X m ≥ X j ≥ m X m ≥ j,j ≥ (cid:16) m − j ( +) − j [( − ) − ] − j [( +) − ] × k η j ( σ ) f k L ( R ) k η j ( σ ) f k L ( R ) k η j ( σ ) g k L ( R ) (cid:17) . k f k L ( R ) k f k L ( R ) .⊲ Subcase (ii) max {| σ | , | σ | , | σ |} . | ξ | . This means that the resonant interac-tion does happen. By the definition of the resonant function we have (cid:12)(cid:12)(cid:12) µ ξ − µ ξ (cid:12)(cid:12)(cid:12) > − | ξ + ξ | | ξ + ξ ξ + ξ ) − α | . θ = τ − ω ( ξ , µ ); θ = τ − ω ( ξ , µ ) , and A j,j ,j be the image of the following subset of A ∗ (cid:8) | ξ | ≥ | ξ | & max { , | α |} ; | σ | ∼ j , | σ | ∼ j , | σ | ∼ j ; max {| σ | , | σ | , | σ |} . | ξ | } under the transformation: ( ξ , µ , τ , ξ , µ , τ ) ( ξ , µ , θ , ξ , µ , θ ). If in addition f j = η j ( σ ) f ( ξ , µ , τ ); f j = η j ( σ ) f ( ξ , µ , τ ) , then the integral in (25) is controlled from above by a constant multiple of X j> X j ,j ≥ (cid:16) − j ( − ) − j ( +) − j ( +) × Z A j,j ,j h | ξ | g (cid:0) ξ, µ, θ + ω ( ξ , µ ) + θ + ω ( ξ + µ ) (cid:1) × η j (cid:0) θ + θ + ω ( ξ , µ ) + ω ( ξ + µ ) − ω ( ξ + ξ , µ + µ ) (cid:1) × f j (cid:0) ξ , µ , θ + ω ( ξ , µ ) (cid:1) f j (cid:0) ξ , µ , θ + ω ( ξ , µ ) (cid:1)i dξ dµ dξ dµ dθ dθ (cid:17) . (27)To get the desired estimate, we are led to dominate the following sum for each fixednatural number j : X j ,j ≥ (cid:16) − j ( +) − j ( +) × Z A j,j ,j h | ξ | g (cid:0) ξ, µ, θ + ω ( ξ , µ ) + θ + ω ( ξ + µ ) (cid:1) × η j (cid:0) θ + θ + ω ( ξ , µ ) + ω ( ξ + µ ) − ω ( ξ + ξ , µ + µ ) (cid:1) × f j ( ξ , µ , θ + ω ( ξ , µ )) f j ( ξ , µ , θ + ω ( ξ , µ )) i dξ dµ dξ dµ dθ dθ (cid:17) . (28)This will be accomplished via considering two more settings. ⋆ Subsubcase (i) (cid:12)(cid:12)(cid:12) ξ − ξ ) − α ( ξ − ξ ) − h(cid:16) µ ξ (cid:17) − (cid:16) µ ξ (cid:17) i(cid:12)(cid:12)(cid:12) > j . Under this circumstance, we change the variables u = ξ + ξ v = µ + µ w = θ + ω ( ξ , µ ) + θ + ω ( ξ + µ ) µ = µ , (29)and then obtain its Jacobian determinant J µ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ − αξ − µ ξ ξ − αξ − µ ξ µ ξ µ ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 5( ξ − ξ ) − α ( ξ − ξ ) − h(cid:16) µ ξ (cid:17) − (cid:16) µ ξ (cid:17) i . (30)11uppose now A (1) j,j ,j is the image of the subset of all points ( ξ , µ , θ , ξ , µ , θ ) ∈ A j,j ,j obeying the just-assumed Subsubcase (i) condition under the transformation (29). Thenit is not hard to deduce that | J µ | & j and so that the sum in (28) is . X j ,j ≥ − j ( +) − j ( +) Z A (1) j,j ,j | u | g ( u, v, w ) | J µ | H ( u, v, w, µ , θ , θ ) dudvdwdµ dθ dθ , (31)where H ( u, v, w, µ , θ , θ ) is just η j f j f j with respect to the transformation (29). Forthe fixed variables: θ , θ , ξ , ξ , µ , we calculate the set length, denoted by ∆ µ , wherethe free variable µ can range. More precisely, if f ( µ ) = θ + θ − ξ ξ ( ξ + ξ ) (cid:18) ( ξ + ξ ) h ξ + ξ ξ + ξ ) − α i − (cid:16) µ ξ − µξ (cid:17) (cid:19) , then | f ′ ( µ ) | > | ξ | & | u | , and hence ∆ µ . j | u | − follows from | θ + θ + ω ( ξ , µ ) + ω ( ξ + µ ) − ω ( ξ + ξ , µ + µ ) | = (cid:12)(cid:12)(cid:12)(cid:12) θ + θ − ξ ξ ( ξ + ξ ) (cid:18) ( ξ + ξ ) h ξ + ξ ξ + ξ ) − α i − (cid:16) µ ξ − µ ξ (cid:17) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∼ j . By the Cauchy-Schwarz inequality and the inverse change of variables we have Z A (1) j,j ,j | u | g ( u, v, w ) | J µ | − H ( u, v, w, µ , θ , θ ) dudvdwdµ dθ dθ . j Z | u | g ( u, v, w ) (cid:18)Z | J µ | − H ( u, v, w, µ , θ , θ ) dµ (cid:19) dudvdwdθ dθ . j k g k L ( R ) Z (cid:18)Z | J µ | − H ( u, v, w, µ , θ , θ ) dudvdwdµ (cid:19) dθ dθ . k g k L ( R ) Z (cid:16) Z Y i =1 , f j i ( ξ i , µ i , θ i + ω ( ξ i , µ i )) dξ dµ dξ dµ (cid:17) dθ dθ . j j k g k L ( R ) k f k L ( R ) k f k L ( R ) . It follows from (28) that the sum in (27) is . k f k L ( R ) k f k L ( R ) .⋆ Subsubcase (ii) (cid:12)(cid:12)(cid:12) ξ − ξ ) − α ( ξ − ξ ) − h(cid:16) µ ξ (cid:17) − (cid:16) µ ξ (cid:17) i(cid:12)(cid:12)(cid:12) ≤ j . In this setting, the change of variables taken in Subsubcase (i) does not work becausethe determinant of the Jacobian may be zero. So, we cannot help finding a new changeof variables. Before doing this, we notice that the size | ξ | ∼ m (for m ≥
0) can be usedbut also the integral in (25) may be rewritten as12 j ,m ≥ X m>j,j ≥ (cid:16) − j ( − ) − j ( +) − j ( +) m × Z A j,j ,j h g ( ξ, µ, θ + ω ( ξ , µ ) + θ + ω ( ξ + µ )) × η j (cid:0) θ + θ + ω ( ξ , µ ) + ω ( ξ + µ ) − ω ( ξ + ξ , µ + µ ) (cid:1) × f m,j (cid:0) ξ , µ , θ + ω ( ξ , µ ) (cid:1) f j (cid:0) ξ , µ , θ + ω ( ξ , µ ) (cid:1)i dξ dµ dξ dµ dθ dθ (cid:17) , (32)where f m,j = η m ( ξ ) η j ( σ ) f ( ξ , µ , τ ) . Now, we choose the following transformation: u = ξ + ξ v = µ + µ w = θ + ω ( ξ , µ ) + θ + ω ( ξ + µ ) ξ = ξ , (33)and moreover assume that A (2) j,j ,j is the image under (33) of the set of those points( ξ , µ , θ , ξ , µ , θ ) ∈ A j,j ,j satisfying the just-given Subsubcase (ii) condition. Acalculation yields that the associated Jacobian determinant of the last transformation(33) is J ξ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ − αξ − µ ξ ξ − αξ − µ ξ µ ξ µ ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 2 (cid:16) µ ξ − µ ξ (cid:17) . (34)From this formula it follows that | J ξ | & | ξ | . Next, we fix θ , θ , ξ , µ , µ , and estimatethe interval length ∆ ξ of the free variable ξ . Putting h ( ξ ) = 5( ξ − ξ ) − α ( ξ − ξ ) − h(cid:16) µ ξ (cid:17) − (cid:16) µ ξ (cid:17) i , (35)we compute h ′ ( ξ ) = 20 ξ − αξ + 2( µ /ξ ) ξ − . (36)Since now h ′ ( ξ ) has the same sign as ξ ’s, we conclude | h ′ ( ξ ) | & | ξ | , thereby finding∆ ξ . j − m . Consequently, the sum in (32) is . X j ,m ≥ X m>j,j ≥ j ( − +) m Z A (2) j,j ,j g ( u, v, w ) | J ξ | H ( u, v, w, ξ , θ , θ ) dudvdwdξ dθ dθ , (37)13here H ( u, v, w, ξ , θ , θ ) equals η j f m,j f j under the change of variables (33). Notethat by the Cauchy-Schwarz inequality Z A (2) j,j ,j g ( u, v, w ) | J ξ | H ( u, v, w, ξ , θ , θ ) dudvdwdξ dθ dθ . − m j Z g ( u, v, w ) (cid:18)Z | J ξ | − H ( u, v, w, ξ , θ , θ ) dξ (cid:19) dudvdwdθ dθ . − m j k g k L ( R ) Z (cid:18)Z | J ξ | − H ( u, v, w, ξ , θ , θ ) dudvdwdξ (cid:19) dθ dθ . − m j k g k L ( R ) Z (cid:18)Z | J ξ | − H ( u, v, w, ξ , θ , θ ) dudvdwdξ (cid:19) dθ dθ . − m j k g k L ( R ) Z (cid:16) Z Y l =1 , f l ( ξ l , µ l , θ l + ω ( ξ l , µ l )) dξ l dµ l (cid:17) dθ dθ . − m j j j k g k L ( R ) k f k L ( R ) k f k L ( R ) . Thus the sum in (32) is . X m,j ≥ X m>j ,j ≥ (cid:16) − j ( − ) − m j − j (( +) − ) j (( +) − ) ×k g k L ( R ) k f k L ( R ) k f k L ( R ) (cid:17) . k f k L ( R ) k f k L ( R ) . • Local well-posedness.
Consider the integral equation associated with (1) u ( t ) = ψ ( t ) S ( t ) φ − ψ T ( t )2 Z t S ( t − t ′ ) ∂ x (cid:0) u ( t ′ ) (cid:1) dt ′ , (38)where 0 < T <
1, and ψ T ( t ) is the bump function defined in Section 2. It is clear that asolution to (38) is a fixed point of the nonlinear operator L ( u ) = ψ ( t ) S ( t ) φ − ψ T ( t )2 Z t S ( t − t ′ ) ∂ x (cid:0) u ( t ′ ) (cid:1) dt ′ . (39)Therefore we are required to verify that L is a contractive mapping from the followingclosed set to itself B a = n u ∈ X s ,s b : k u k X s ,s b ≤ a = 4 c k φ k H s ,s ( R ) , − < b o . (40)Here and hereafter c > σ > k L ( u ) k X s ,s b ≤ c k φ k H s ,s ( R ) + cT σ k u k X s ,s b . (41)Next, since ∂ x ( u ) − ∂ x ( v ) = ∂ x [( u − v )( u + v )], we similarly get k L ( u ) − L ( v ) k X s ,s b ≤ cT σ k u − v k X s ,s b (cid:16) k u k X s ,s b + k v k X s ,s b (cid:17) . (42)14hoosing T = T ( k φ k H s ,s ( R ) ) such that 8 cT σ k φ k H s ,s ( R ) <
1, we deduce from (41)and (42) that L is strictly contractive on the ball B a . Thus there exists a unique solution u ∈ X s ,s b ([ − T, T ]) ⊆ C (cid:0) [ − T, T ]; H s ,s ( R ) (cid:1) (thanks to b > /
2) to the IVP of the fifthorder KP-I equation. The smoothness of the mapping from H s ,s ( R ) to X s ,s b ([ − T, T ])follows from the fixed point argument. Because the dispersive function ω ( ξ, µ ) is singularat ξ = 0, the requirement | ξ | − ˆ φ ∈ S ′ ( R ) is necessary in order to have a well definedtime derivative of S ( t ) φ . So, the argument for the local well-posedness is complete. • Global well-posedness.
We first handle the global well-posedness of (1) in theanisotropic Sobolev space H s , ( R ) with s ≥
0. Suppose φ ∈ H s , ( R ). Then by localwell-posedness there exists a unique solution u ∈ C (cid:0) [ − T, T ]; H s , ( R ) (cid:1) of (1). We claimthat there exists T , depending on k φ k L ( R ) , such that on the interval [ − T, T ] one hassup | t |≤ T k u ( t ) k H s , ( R ) ≤ c k φ k H s , ( R ) . (43)With the help of this claim and the local well-posedness part of Theorem 1.1 with u ( T )and u ( − T ) being initial values, we can extend the exit time to the positive infinity stepby step in that the exist time T ′ depends only on k u ( T ) k L ( R ) = k u ( − T ) k L ( R ) = k φ k L ( R ) and max (cid:8) k u ( T ) k H s , ( R ) , k u ( − T ) k H s , ( R ) (cid:9) ≤ c k φ k H s , ( R ) . To check the claim, let J s x = ( I − ∂ x ) s / . Then from the definitions of the anisotropicSobolev space and the Bourgain space it follows that k J s x u k L ( R ) = k u k H s , ( R ) and k J s x u k X , b = k u k X s , b . Letting J s x act on both sides of the integral equation (38), we derive J s x u ( t ) = ψ ( t ) S ( t ) J s x φ − ψ T ( t )2 Z t S ( t − t ′ ) J s x ∂ x (cid:0) u ( t ′ ) (cid:1) dt ′ . (44)By Proposition 2.1, we have k ψS ( t ) J s x φ k X , b ≤ c k φ k H s , ( R ) , (45)as well as (cid:13)(cid:13)(cid:13) ψ T ( t ) Z t S ( t − t ′ ) J s x ∂ x (cid:0) u ( t ′ ) (cid:1) dt ′ (cid:13)(cid:13)(cid:13) X , b ≤ cT − b + b ′ k J s x ∂ x (cid:0) u ( t ′ ) (cid:1) k X , b ′ . (46)A slight modification of the argument for the bilinear estimates carried out in Section 3can produce the following bilinear estimate k J s x ∂ x ( u ) k X , −
12 + ≤ c k u k X ,
012 + k J s x u k X ,
012 + . (47)Combining (45), (46) and (47), we get k J s x u k X , b ≤ c k φ k H s , + cT σ k u k X , b k J s x u k X , b . By (41) with s = s = 0, we can choose T = T ( k φ k L ( R ) ) such that cT σ k u k X , b < . Thus by (47), we have k J s x ∂ x ( u ) k X , b ≤ c k φ k H s , ( R ) + 2 − k J s x u k X , b . b > , we obtain the fundamental embedding inequalitysup | t |≤ T k u ( t ) k H s , ( R ) ≤ k J s x u k X , b ≤ c k φ k H s , ( R ) , as well as (43) which verifies the claim.Similarly, the operator J s y = ( I − ∂ y ) s / can act on both side of the integral equation(38). As a result, we get k J s y ∂ x ( u ) k X s , −
12 + ≤ c k u k X s ,
012 + k J s y u k X s ,
012 + , (48)thereby obtaining the following estimate k J s y ∂ x ( u ) k X s , −
12 + ≤ c k φ k H s ,s ( R ) + cT σ k u k X s , b k J s y u k X s , b . By (43), we can also choose a time T so that it depends on k φ k H s , ( R ) and obeys cT σ k u k X s , < . Finally, we arrive atsup | t |≤ T k u ( t ) k H s ,s ( R ) = sup | t |≤ T k J s y u ( t ) k H s , ( R ) ≤ k J s y u k X s ,s b ≤ c k φ k H s ,s ( R ) . Note that the previous constant c >
References [1] M. Ben-Artzi and J. Saut,
Uniform decay estimates for a class of oscillatory integralsand applications.
Diff. Int. Eq. (1999), 137-145.[2] J. Bougain, On the Cauchy problem for the Kadomtsev-Petviashvili equation.
Geom.Funct. Anal. (4) (1993), 315-341.[3] W. Chen, J. Li and C. Miao, On the low regularity of the fifth order Kadomtsev-Petviashvili I equation. arXiv:0709.0103v1 (2007).[4] M. Hadac,
Well-posedness for the Kadomtsev-Petviashili II equation and generalisa-tions.
Trans. Amer. Math. Soc. (to appear). arXiv:math.AP/0611197v1 (2006).[5] A. Ionescu and C. Kenig,
Local and global well-posedness of periodic KP-I equations. in Mathematical Aspects of Nonlinear Dispersive Equations, Princeton UniversityPress (2007), 181-212.[6] A. Ionescu, E. Kenig and D. Tataru,
Global well-posedness of the KP-I initial-valueproblem in the energy space. arXiv:0705.4239v1 (2007).[7] R. I´orio and W. Nunes,
On equations of KP-type.
Proc. Roy. Soc. Edinburgh Sect.A. (1998), 725-743.[8] P. Isaza, J. L´opez and J. Mej´ıa,
Cauchy problem for the fifth order Kadomtsev Petvi-ashvili (KPII) equation.
Communication on Pure and Applied Analysis (2006),887-905. 169] P. Isaza and J. Mej´ıa, Local and global Cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices.
Comm. PartialDiff. Eq. (2001), 1027-1054.[10] C. Kenig, On the local and global well-posedness theory for the KP-I equation.
Ann.I. H. Poincar´e-AN. (2004), 827-838.[11] C. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdVequation,
J. Amer. Math. Soc. (1996), 573-603.[12] H. Koch and N. Tzvetkov, On finite energy solutions of the KP-I equation.
Math. Z. (2008), 55-68.[13] L. Molinet, J. Saut and N. Tzvetkov,
Well-posedness and ill-posedness results for theKadomtsev-Petviashvili-I.
Duke Math. J. (2002), 353-384.[14] L. Molinet, J. Saut and N. Tzvetkov,
Global well-posedness for the KP-I equation.
Math. Ann. (2002), 225-275.[15] J. Saut and N. Tzvetkov,
The Cauchy problem for higher order KP equations.
J.Differential Equations (1999), 196-222.[16] J. Saut and N. Tzvetkov,
The Cauchy problem for the fifth order KP equations,
J.Math. Pures Appl. (2000), 307-338.[17] H. Takaoka, Global well-posedness for the Kadomtsev-Petviashvili II equation.
Dis-crete Contin. Dynam. Systems. (2) (2000), 483-499.[18] H. Takaoka and N. Tzvetkov, On the local regularity of Kadomtsev-Petviashili IIequation.
Internat. Math. Res. Notices.2