Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations

We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space H^{s_1,s_2}(R^2) with s_1>-1/2 and s_2 \geq 0. On the H^{s_1,0}(R^2) scale this result includes the full subcritical range without any additional low frequency assumption on the initial data. More generally, we prove the local in time well-posedness of the Cauchy problem for a dispersion generalised KP II type equation. We also deduce a global well-posedness result for the generalised equation.

We are interested in low regularity well-posedness of (2). By using refined Fourier restriction norm spaces we will prove new bilinear estimates which allow us to apply the contraction mapping principle.
For the fifth order KP-II equation (4) local well-posedness was shown by Saut and Tzvetkov (see [12,13]) for s 1 ≥ − 1 4 and s 2 ≥ 0. (Note that the equation considered in [12,13] is slightly more general than (4) because it also contains the third order term.) Very recently, Isaza, López and Mejía [6] improved the local well-posedness result to s 1 > − 5 4 and s 2 ≥ 0. (These authors also show global well-posedness of (4) for s 1 > − 4 7 and s 2 = 0.) For general α ∈ ( 4 3 , 6] Iório and Nunes [5] showed the local well-posedness for initial values u 0 in the isotropic Sobolev space H s (R 2 ), s > 2, with the additional low frequency condition ∂ −1 x u 0 ∈ H s (R 2 ) using parabolic regularization. Let us note that they consider much more general equations and do not use the dispersive structure of the equation.
For a recent result concerning the so called mass constraint property for solutions of equations of type (2) see [10].
Our main result for equation (1) is the following More generally, we will show the following theorem concerning equation (2) Remark 1.3. If we (formally) apply the operator ∂ −1 x to equation (2) and use Duhamel's formula, equation (2) is (for suitable u) equivalent to the integral equation where U α is the unitary group on H s1,s2 (R 2 ) defined by We define a solution of (2) in X T (for T ≤ 1) to be a solution of the operator equation where Γ T is the bilinear operator on X T defined for smooth u 1 , u 2 by and ψ ∈ C ∞ 0 (R) is a cut-off function with ψ(t) = 1 for |t| ≤ 1 and ψ(t) = 0 for |t| ≥ 2. Furthermore ψ T (t) = ψ(t/T ).
Remark 1.4. In the particular case α = 4 of the fifth order KP II equation Theorem 1.2 shows the local well-posedness of (4) for s 1 > − 5 4 and s 2 ≥ 0. We therefore get a local well-posedness result for the same class of initial data as Isaza, López and Mejía in [6]. Note though that the spaces X T where the local well-posedness result of Theorem 1.2 holds true are different from those used in [6] (see Remark 4.4).
By combining the local well-posedness result of Theorem 1.2 with the conservation of the L 2 -norm which holds for real valued solutions of (7) we get the following global result, where H s1,0 (R 2 ; R) denotes the subspace of all real valued functions in H s1,0 (R 2 ). Theorem 1.6. Let 4 3 < α ≤ 6, s 1 ≥ 0 and T > 0. Then there exists a Banach space X T ֒→ C([−T, T ]; H s1,0 (R 2 ; R)), such that for every u 0 ∈ H s1,0 (R 2 ; R) there is exactly one solution u of equation (2) in X T . Let us fix some notation we use throughout the paper: • For ξ ∈ R let ξ := (1 + |ξ| 2 ) 1 2 . • For u ∈ S ′ (R n ) the Fourier transformation of u in R n is denoted by u or F u.
A partial Fourier transformation with respect to some of the n variables, is denoted for example by F 1 for the Fourier transformation in the first variable, etc. • µ = (τ, ξ, η) ∈ R 3 always denotes the Fourier variable dual to (t, x, y).
• A B means that there is a (harmless) constant C such that A ≤ CB.
• For X and Y Banach spaces X ֒→ Y means that there is a continuous embedding from X into Y . Furthermore C b (R; X) denotes the space of all continuous and bounded functions f : R → X with the sup-norm. The author would like to thank S. Herr and H. Koch for valuable discussions and suggestions on the subject.

Definition of the solution spaces
Definition 2.1. Let us consider the following space of test functions be the completion ofS with respect to the norm (10).
Remark 2.2. At least for b > − 1 2 − σ, we can identify X s1,s2,b σ with the subspace of tempered distributions u on R 3 such that u is a regular distribution and Remark 2.4. The spaces X s1,s2,b σ are modifications of spaces first used by Bourgain [1,2] in the context of the KdV and Schrödinger equations.
We have the following well-known linear estimates Proof. See for example [4].
Proof. For σ = 0 see [4]. For σ = 0 consider the operator I σ defined for u ∈S by is an isometric isomorphism. Therefore we have We have the following well-known embedding result for the X s1,s2,b σ -spaces Proof. For σ = 0 see [4]. But for σ > 0 we have that u X s 1 , Then X T ֒→ C([−T, T ]; H s1,s2 (R 2 )).

Strichartz and refined Strichartz estimates
Exactly as in the case α = 2 (see [11]) we show the following Strichartz' estimates for the solution of the linear equation. For the convenience of the reader we will give the full proof here.
Now by the theorem of dominated convergence again we can take the limit δ 2 → 0+ in this last expression and get Now for ξ = 0 we set ψ(ξ) := e −δ1ξ 2 +i sign( ξ t ) π 4 and φ(ξ) := ξ|ξ| α . Then with our choice of θ we have |φ ′′ (ξ)| ∼ |ξ| 1−2θ and we can use Corollary 2.9 of [9] to see that where the implicit constant does not depend on δ 1 > 0. Therefore we get that and then by continuity also for all u 0 ∈ L 1 (R 2 ). By Plancherel we also have that |D x | iκ U α (t)u 0 L 2 = u 0 L 2 . Now using the interpolation theorem of Stein we get for every 2 ≤ r ≤ ∞ that (13) follows from (14) by well-known methods. (See for example [9].) From this linear version of Strichartz estimates we can deduce the following bilinear version (15) (15) is equivalent to (16). By suitable changes of variables we also get (17) and (18).
For the part of the product u 1 u 2 where the ξ-frequency of the first factor is significantly smaller than the ξ-frequency of the second factor we can improve this bilinear Strichartz estimate. To formulate this improvement let us define for c > 0 the following operator We have the following refined bilinear Strichartz estimate which for the case α = 2 was already implicitly used in [7,[14][15][16][17][18].
For the proof of the theorem we need the following Lemma We then have for every ξ, ξ 1 ∈ R Proof of Lemma 3.4. Suppose first that |ξ min | = |ξ 1 |. Then we have Putting these estimates together we get we see that we also get (22) in the other cases.
which by duality is equivalent to By use of the Cauchy-Schwarz inequality it suffices to show that sup µ I(µ) For fixed µ we now use the change of variables T : Let us also recall the definition of λ 1 and λ 2 Observe that Therefore we have Furthermore we have ∂ ξ1 ν = (α + 1)(|ξ − ξ 1 | α − |ξ 1 | α ). As we only consider the region where |ξ 1 | ≤ 1 3 |ξ − ξ 1 |, i. e. |ξ 1 | = |ξ min | and |ξ − ξ 1 | ∼ |ξ max | we have by (22) that |ν| ∼ |ξ 1 ||ξ − ξ 1 | α . We also have |∂ ξ1 ν| |ξ − ξ 1 | α in this region. Therefore we have that Let us notice that it is possible to divide the region of integration into a finite number of open subsets U i such that T is an injective C 1 -function in U i with nonvanishing Jacobian. As we are in the KP-II-case both terms on the right hand side of (24) have the same sign which implies that |ν| ≤ |λ 1 +λ 2 −λ|. So performing the change of variables and using the following elementary inequality , a = 0 we get 1.
Remark 3.5. In fact we get (23) also without the cut-off function χ |ξ1|≤ 1 3 |ξ−ξ1| , as in the region where |ξ 1 | > 1 3 |ξ−ξ 1 | we have that and so the estimate in this region follows from the bilinear Strichartz estimate (16). Like in the case of the bilinear Strichartz estimates we also get the following dual versions of estimate (23) (without the cut-off function) by an appropriate change of variables

The main bilinear estimate
In the following formulation and proof of the crucial bilinear estimate needed to prove Theorem 1.2 we will only consider the case s 2 = 0 (and write s for s 1 ) to simplify the presentation. Note that the case s 2 > 0 follows from this special case, as in the general case we only get an extra term η s 2 η1 s 2 η−η1 s 2 in the integral inequalities we have to prove (see (32)). But this term is always bounded above for s 2 ≥ 0. such that and set b : We then have Remark 4.2. The spaces X andX defined in Theorem 4.1 are built by taking sums and intersections of the Bourgain type spaces of Section 2. Therefore it is easy to see that they also satisfy the linear estimates of Propositions 2.5 and 2.6, i. e.
Remark 4.3. The sum structure of the spaces X andX is the essential ingredient to use the additional weight ( ξ |ξ| ) σ , which is incorporated in the definition of X 2 and X 2 (see (10)), to lower the x-regularity s in the bilinear estimate without imposing a low frequency condition on the initial data. Therefore, in the case α = 2 of the Kadomtsev-Petviashvili II equation we are able to show the local well-posedness for all s > − 1 2 without a low frequency condition on the initial data whereas the counterexamples in [16] show that it is not possible to get the bilinear estimate for − 1 2 < s < − 1 3 and σ = 0. Remark 4.4. In the case α = 4 of the fifth order KP II equation it is possible to get the bilinear estimate (28) in the spaces X = X s,b 0 andX = X s,b ′ 0 (i. e. choosing b 1 = 0 and σ = 0) as is shown in [6]. More generally, this is true for all α > 5 2 which can be seen by refining the estimate of Lemma 4.8 by an additional dyadic decomposition and interpolation argument as used in [16], pp. 89-92.
To prove Theorem 4.1, we will split the nonlinear term ∂ x (u 1 u 2 ) into various pieces and give estimates in appropriate X s,b σ -spaces for each of these pieces (see ). We will then combine these estimates to prove (28). First of all, with P c defined as in (19), we can write As the main bilinear estimate (28) is symmetric in u 1 and u 2 , it suffices to prove it only for ∂ x P 1 (u 1 , u 2 ). This expression can be decomposed further into The operators Q ij are defined by where Let us explain what the meaning of the regions Ξ 1 and Ξ 2 is. In Ξ 1 we have that 2 ≤ 2|ξ − ξ 1 | ≤ 3|ξ| ≤ 4|ξ − ξ 1 |, i. e. ξ and ξ − ξ 1 are comparable in size and are both bounded away from zero, whereas ξ 1 is the smallest of the frequencies dual to the x-variable. In Ξ 2 we have that ξ 1 and ξ − ξ 1 are comparable in size and are both bounded away from zero, whereas ξ may be small here. For each of the operators Q ij we will now show estimates of the form By definition (10) of the X s,b σ -norm and setting f l (µ) := |ξ| −σ ξ s l +σ λ b l u l (µ) this is equivalent to Using duality this estimate is equivalent to The main ingredients we use in the proof of these estimates are the bilinear Strichartz estimates of corollary 3.2 and Theorem 3.3 and a use of the "resonance identity" (24). We already noted that the two terms on the right hand side of (24) have the same sign. Therefore we have where for the last inequality we used (22). Lemma 4.5. We have that provided that b > 1 2 , α ≤ 6 and s ∈ R. Proof. We have to prove that | A00 k 00 (µ 1 , µ) On A 00 we have that |ξ 1 | ≤ |ξ − ξ 1 | ≤ 1 and therefore also |ξ| ≤ 2|ξ − ξ 1 | ≤ 2, so that k 00 (µ 1 , µ) where the last inequality follows from α ≤ 6. Therefore (34) follows from the refined bilinear Strichartz estimate (23). Lemma 4.6. We have that Proof. We have to prove that We show that k 10 is bounded in A 10 , then the lemma follows from the refined bilinear Strichartz estimate (23). In region A 10 we have 1 ≤ |ξ − ξ 1 | ∼ |ξ| ∼ ξ and |λ| = |λ max | |ξ 1 ||ξ| α , so using (36) we get k 10 Because of (36) we have 1 2 1 where the last inequality follows from (37). Lemma 4.7. We have that (36) and (37) hold. Proof. We have to show that . Now the boundedness of k 12 on A 12 follows exactly like the boundedness of k 10 in Lemma 4.6. Then (38) follows from the refined bilinear Strichartz estimate (25).
Lemma 4.8. We have that the boundedness of k 22 follows in exactly the same way as the boundedness of k 21 in Lemma 4.10.
We are now in a position to prove Theorem 4.1.
Proof of Theorem 4.1. With our definitions of b, b ′ , b 1 and σ we have that b > 1 2 , b − b ′ < 1, (36), (37), (41), (42), (44) and (45) hold. We noticed before that because of the symmetry of (28) in u 1 and u 2 , it suffices to show (28) for ∂ x P 1 (u 1 , u 2 ) instead of ∂ x (u 1 u 2 ) where P 1 is the operator defined in (19). We now decompose ∂ x P 1 (u 1 , u 2 ) further as in (31). Therefore we have to show for every Q ij that Let us notice that by the definition of the spaceX we have that u X ≤ u X s,0 , so that it suffices to control Q ij in one of these norms. The norm in X is given by But by the definition of the Bourgain spaces (10) and because of b ′ ≤ 0 and σ ≥ 0 we have u X s,b 0 ≤ u X1 and u X s,b 0 ≤ u X2 which means that we have the continuous embedding X ֒→ X s,b 0 . Therefore (49) follows from which actually holds for all of the Q ij except Q 11 . Let us prove this first. From Lemma 4.5 it follows that In the same way, from Lemmas 4.6, 4.7, 4.9, 4.10 and 4.11 it follows for all of the remaining Q ij except Q 11 . So it remains to consider Q 11 . Now let us decompose u 1 ∈ X as u 1 = v 1 + w 1 with v 1 ∈ X 1 and w 1 ∈ X 2 . For v 1 we have because of (39) of Lemma 4.8 For w 1 we have because of (40) of Lemma 4.8 So putting these two estimates together we have Now taking the infimum over every decomposition of u 1 of the form u 1 = v 1 + w 1 with v 1 ∈ X 1 and w 1 ∈ X 2 we finally get (49) for Q 11 , which finishes the proof.
For the proof of Theorem 1.6 we need the following refined version of (28) Corollary 4.12. Let 4 3 < α ≤ 6 and s > max(1 − 3 4 α, 1 4 − 3 8 α). Let X andX be defined as in Theorem 4.1 We then have for every ρ > 0 But this follows exactly as (28), as the operators J ρ x in (51) only give an additional bounded term ( ξ ξ−ξ1 ) ρ in the dual formulations of the estimates proved in Lemmas 4.5-4.11.

5.
Proof of theorems 1.2 and 1.6 As the methods of proof used here are all well known, we only give Sketch of proof of Theorem 1.2. As explained at the beginning of Section 4 it suffices to consider the case s 2 = 0. Let s = s 1 . For T ≤ 1 and u 1 , u 2 ∈S we define the bilinear operator Γ T by (8). Let X andX be defined as in Theorem 4.1 and set δ : Then by (30) and Theorem 4.1 we have Therefore we can extend Γ T to a continuous, bilinear operator Γ T : X ×X → X. As Γ T (u 1 , u 2 ) |[−T,T ] only depends on u i|[−T,T ] (i = 1, 2), Γ T also defines a continuous, bilinear operator Γ T : X T × X T → X T . Furthermore by (29) we have ψU α (·)u 0 XT ≤ ψU α (·)u 0 X u 0 H s,0 (R 2 ) for u 0 ∈ H s,0 (R 2 ). So if we define (52) Φ T (u, u 0 ) := ψU α (·)u 0 − Γ T (u, u), u ∈ X T , u 0 ∈ H s,0 (R 2 ) we have for u 0 ∈ B R := {u 0 ∈ H s,0 (R 2 ) | u 0 H s,0 < R} and u, v ∈Ā r := {u ∈ X T | u XT ≤ r} that (53) Φ T (u, u 0 ) XT ≤ C( u 0 H s,0 (R 2 ) + T δ u 2 XT ) ≤ CR + CT δ r 2 with some constant C which does not depend on R, r and T and So given R > 0 we choose r = 2CR and T = min{1, (8C 2 R) − 1 δ }. Then, for fixed u 0 ∈ B R , by (53) Φ T (·, u 0 ) mapsĀ r intoĀ r and by (54) Φ T (u 0 , ·) is a contraction. By the Banach fixed point theorem there is exactly one fixed point of Φ T inĀ r . Now by a well known argument the uniqueness of the solution u follows also in X T . Furthermore it is easy to see that the mapping Λ T : X T × B R → X T , Λ T (u, u 0 ) := Φ T (u, u 0 ) − u is analytic. Therefore a standard use of the implicit function theorem yields the analyticity of the flow map F R : u 0 → u.
Sketch of proof of Theorem 1.6. Let u 0 ∈ H s,0 (R 2 ; R) be real valued and let X be defined as in Theorem 4.1. Let T 0 be the supremum of all T ∈ (0, 1] such that there exists a unique u ∈ X T with Φ T (u, u 0 ) = u. We will prove that T 0 = 1. By Theorem 1.2 we see that T 0 > 0. Let T ∈ (0, T 0 ). Let X 0 be defined as X in Theorem 4.1, but with s = 0. We obviously have u X = J s u X 0 . By (29), (30) and Corollary 4.12 we have we have by (53) that u X 0 T ≤ 2C u 0 L 2 and therefore J s u X 0 T ≤ CR + 1 2 J s u X 0 T . It follows that sup |t|≤T u(t) H s,0 ≤C J s u X 0 T ≤ 2CCR. As this upper bound does not depend on T and applying Theorem 1.2 with u(T ) and u(−T ) as initial values, we see that we can extend the solution beyond the interval [−T 0 , T 0 ]. This contradicts the choice of T 0 . Therefore we have T 0 ≥ min(1, (8C 2 u 0 L 2 ) − 1 δ ). This implies that the length of the maximal interval of existence does only depend on u 0 L 2 . But the L 2 -norm of real valued solutions u of (7) is conserved, i. e. u(±T ) L 2 = u 0 L 2 , so if we had T 0 < 1, we could extend the solution beyond the interval [−T 0 , T 0 ] which contradicts the choice of T 0 .