Global well-posedness for the Kawahara equation with low regularity data

We consider the global well-posedness for the Cauchy probelem of the Kawahara equation which is one of the fifth order KdV type equations. We first establish the local well-posedness in a more suitable function space for the global well-posedness by a variant of the Fourier restriction norm method. Next, we extend local solutions globally in time by the I-method. In this paper, we apply the I-method to the modified Bourgain space.


Introduction
We consider the global well-posedness (GWP) for the Cauchy problem of the Kawahara equation which is one of fifth order KdV type equations.
Following an idea of Bejenaru and Tao [1] and Kishimoto and Tsugawa [17], we improved the previous results to have LWP in H s for s ≥ −2 in [10]. On the other hand, when s < −2, we obtained ill-posedness in the following sense.
where u N is the solution to (1.1) with initial data φ N obtained in Theorem 1.1.
For the proof of Theorem 1.1, see [10]. This proof is based on Bejenaru and Tao's work [1]. These theorems imply that the critical exponent s is equal to −2. Next, we extend the local solution obtained above globally in time by the I-method. Yan and Li [20] proved GWP in H s for s > −63/58, following the argument of [5]. Chen and Guo [3] used the argument of [7] to show GWP in H s for s ≥ −7/4. We apply the I-method under the weaker regularity condition on data to obtain the global solution of (1.1) for s ≥ −38/21. But the function space used in [10] is not applicable for the proof of the global existence. In this paper, we adjoint the function space for the global well-posedness and so we reproduce the proof of LWP in the adjusted function space (see Section 3). The main result in this paper is the following. If u solves (1.1), then u λ satisfies the following equation.    ∂ t u λ − ∂ 5 x u λ + βλ −2 ∂ 3 x u λ + ∂ x (u 2 λ ) = 0, (t, x) ∈ [0, λ 5 T ] × R, u λ (0, ·) = u 0,λ (·) ∈ H s (R).
Therefore we can assume smallness of initial data when s > −7/2. So we solves (1.2) for sufficiently small data.
We first recall our local well-posedness result. The main idea is how to define the function space to construct local solutions. Here the Bourgian space X s,b plays an important role when s is small. The Bourgain space X s,b is introduced by Bougain [2] and equipped with the norm, where p λ (ξ) = ξ 5 + βλ −2 ξ 3 and u is the Fourier transform of u. The key is to establish the bilinear estimate of the nonlinearity ∂ x (u 2 ) as follows; where Λ b is the Fourier multiplier defined as Λ b := F −1 τ,ξ τ − p λ (ξ) b F t,x for b ∈ R. Combining (1.3) and some linear estimates, the Fourier restriction norm method works to obtain LWP. Chen, Li, Miao and Wu [4] proved (1.3) in X s,1/2+ε with 0 < ε ≪ 1 when s > −7/4. This result was improved to s = −7/4 by Chen and Guo [3]. However, (1.3) fails for any b ∈ R when s < −7/4. To overcome this difficulty, we modify the Bourgain space X s,b to control strong nonlinear interactions and establish (1.3) for s < −7/4. The idea of the modification of X s,b was introduced by Bejenaru and Tao [1]. Remark that there is no general framework for modifying X s,b . This is one of the most difficult points in our study. Following the similar argument to [17], we obtained (1.3) in the critical case s = −2 in [10]. We now mention how to modify the Bourgain space. From the counterexamples of (1.3) (see appendix in [10]), we find the domain in which strong nonlinear interactions appear and make a suitable modification in this region. We first divide R 2 into three parts as follows; βλ −2 |ξ| 3 and |ξ| ≥ 1 , βλ −2 |ξ| 3 and |ξ| ≥ 1 , From the counterexamples of (1.3), the necessary conditions are Remark that these conditions only come from the high × high → low interaction which means that a low frequency band is generated from an interaction between high and high frequencies bands. So the way to modify the Bourgain space is not strictly restricted. From (1.4), we have to take b ≤ 3/10 in D 3 when s = −2.
Following the above argument, the function space Z s is equipped with the norm, where P Ω is the Fourier projection onto a set Ω and u Y s := ξ s u L 2 (2,1) is the Besov type Bourgain space defined by the norm, where χ Ω is the characteristic function of a set Ω and A j , B k are dyadic decompositions as follows; for j, k ≥ 0. Using the function space Z s , we obtain (1.3) for s ≥ −2. Then the standard argument of the Fourier restriction norm method works to have LWP in Here Z s (I), for a time interval I, was defined by the norm, For the details of the proof, see [10]. Note that the function space constructed in [10] is slightly different from Z s but essentially same as this one.
Next, we extend the local solution obtained above globally in time. But we have no conservation laws when s is negative. To avoid this difficulty, we apply the I-method exploited by Colliander, Keel, Staffilani, Takaoka and Tao [5], [6]. The main idea is to use a modified energy defined for less regular functions, which is not conserved. We now define the modified energy E I (u). The operator I : H s → L 2 is the Fourier multiplier satisfying Here m(ξ) is a smooth and monotone function such that for s < 0 and large N. The functional E (2) If we can control the growth of the modified energy E (2) I (u) in time, this allows to iterate the local theory to continue the solution to any time T . In the I-method, the key estimate is the almost conservation law which implies that the increment of the modified energy is sufficiently small for small time interval and large N. Yan and Li [20] proved the almost conservation law for E   I (u) in order to remove some oscillations in that functional. They proved GWP for the KdV equation in H s when s > −3/4 by using this functional. Kishimoto [16] slightly but essentially modified the Bourgain space to establish GWP for s ≥ −3/4 (see also [9]). Chen and Guo [3] defined the modified energy E Note that we obtain the same result as above if this function space is replaced by X s,1/2 (2,1) . We encounter some difficulty to establish the almost conservation law because the Kawahara equation has less symmetries than the KdV equation. Our purpose is to establish both (1.3) and the almost conservation law when s < −7/4.
In fact, the use of X s,1/2 (2,1) enables us to show that the almost conservation law for E (2,1) breaks down for s < −7/4. Namely, we would not construct the local solution by the iteration argument. From the necessary condition (1.4), we need to take b 1 < 1/2 when s < −7/4 when the norm in D 3 is defined as · X s,b 1 (2,1) . In the case b 1 ≥ 1/2, we can recover two derivatives by the smoothing effects (see [13]). On the other hand, when b 1 < 1/2, it is hard to establish the almost conservation law for s < −7/4 since the smoothing effect is weaker. Using the function space Z s defined above, we obtain (1.3) for s ≥ −2 but the almost conservation law for s ≥ −6/5 since b 1 = 3/10 in the function space Z s . So we simultaneously need to control strong nonlinear interactions which come from (1.3) and the almost conservation law. To overcome this difficulty, we establish the improved bilinear estimate which is sharp in some sense. Roughly speaking, this estimate implies that we can gain 4b 1 derivatives by using X 0,b 1 (2,1) . Following the improved bilinear estimate and the Sobolev inequality, we obtain the almost conservation law for s ≥ −4b 1 . From this and the necessary condition (1.4), the minimum of s is −38/21 with b 1 = 19/42. Following the above argument, we define the function space W s as follows; where s 2 = 25/168 and b 2 = 79/168 which come from the necessary condition (1.5).
By using the function space W s , we obtain both (1.3) and the almost conservation law for −38/21 ≤ s < 0. Remark we need another approach to prove the bilinear estimate because the function space used in [10] is different from the adjusted function space W s for the global well-posedness. Then we give the proof of the bilinear estimate in W s , following Bejenaru and Tao [1]. For the details, see Proposition 3.2 in Section 3. Moreover, we note that the difference between the almost conserved quantity E   Moreover c+ means c + ε, while c− means c − ε, where ε > 0 is enough small. The rest of this paper is planned as follows. In Section 2, we prepare the lemmas to prove the main results. In Section 3, we give the proof of the bilinear estimate and show LWP by the Fourier restriction norm method. In Section 4, we apply the I-method to modified Bougain space to show GWP.
Acknowledgment. The author would like to appreciate his supervisor Professor Yoshio Tsutsumi for many helpful conversation and encouragement and thank Professor Kotaro Tsugawa and Professor Nobu Kishimoto for helpful comments.

Preliminaries
In this section, we prepare important lemmas to show the main estimates. When we use the variables (τ, ξ), (τ 1 , ξ 1 ) and (τ 2 , ξ 2 ), we always assume the relation as follows.
For a normed space X and a set Ω, is the characteristic function of Ω. For dyadic numbers N, M ≥ 1, A N and B M are dyadic decompositions defined as for dyadic numbers N, M ≥ 1.
The next two lemmas play a crucial role to establish the bilinear estimate. . (2.1) Moreover, then we have .
Assume that v is supported on A N for N ≥ 1 and u is an arbitrary .
We put a one parameter semigroup U λ (t) defined by From the definition, W s ([0, T ]) has the following property.
which implies the following linear estimates.
For the proofs of the above propositions, see [1].

Local well-posedness
In this section, we establish LWP in W s ([0, T ]) for s ≥ −38/21. The bilinear estimate in W s is stated as follows.
The proof of the bilinear estimate in [10] is based on the argument of Kenig, Ponce and Vega [15]. But this method is not applicable in the proof of the above bilinear estimate because the function space W s is the Besov type space. Then we use the similar argument to Bejenaru and Tao [1]. Note that W s has the L 2 ξ -property, namely, for dyadic numbers N. From this, we can reduce (3.1) to the following.
Proposition 3.2. Assume that u and v are restricted to A N 1 and A N 2 for dyadic numbers N 1 , N 2 ≥ 1. Then we have for a dyadic number N 0 ≥ 1 in the following six cases.
(i) At least two of N 0 , N 1 , N 2 are less than some universal constant and By using Hölder's and Young's inequalities and Lemmas 2.1 and 2.2, we obtain the above proposition.
Proof of Proposition 3.2. We first see the properties of the function space W s . From the definition, u X s,19/42 . From the Schwarz inequality, .
Estimate for (i). In this case, all N 0 , N 1 and N 2 1. We use the Young inequality to obtain which is bounded by u X s,0 v Y s from the Hölder inequality. In the other cases, we often use the following algebraic relation.
Estimate for (ii). In this case, u * v is supported on D 3 and M max |ξ|N 4 1 . In the case |ξ| N −4 1 , we use the Hölder inequality to obtain (Ia) Consider the case u is supported on D 2 . From the algebraic relation (3.3), In the former case, we use Young's inequality to have from the Hölder inequality. In the latter case, we may assume that v is supported on D 2 and |τ − p λ (ξ)| N 5 1 . Combining the Hölder inequality and the Young inequality, we obtain In the same manner as above, we obtain the desired estimate in the case v is supported on D 2 . Following the above estimates, we only prove (3.2) when both u and v are restricted to D 1 and |τ − p λ (ξ)| N 5 1 . (Ib) Consider the case M max = |τ − p λ (ξ)|. We may assume |τ − p λ (ξ)| ∼ |ξ|N 4 1 . Firstly, we deal with the case N . Secondly, we consider the case N −4 1 , we use Hölder's and Young's inequalities to have . Finally, we estimate the Y s norm of Λ −1 ∂ x (uv). We use Hölder's inequality to have The case M max = |τ 2 − p λ (ξ 2 )| is identical to the above case.
In the former case, we use the Hölder inequality and the Young inequality to have In the latter case, we may assume that |τ − p λ (ξ)| N 5 1 from the above estimate and v is supported on D 2 . Combining the Hölder inequality and the Young inequality, we have N s+1 The case v is supported on D 2 is identical to the above case. Therefore we only consider the case both u and v are restricted to D 1 .
Next, we estimate the norm Y s of Λ −1 ∂ x (uv). Hölder's and Young's inequalities imply that (IIc) Consider the case M max = |τ 1 − p λ (ξ 1 )|. From the above estimate, we may assume |τ − p λ (ξ)| N 5 1 . We use (2.4) with K ∼ N 1 to have , which is an appropriate bound. In the same manner as above, we obtain the desired estimate in the case M max = |τ 2 − p λ (ξ 2 )| by symmetry.
(IIIb) Consider the case M max = |τ − p λ (ξ)|. We use Young's inequality to have which implies the desired estimate from Hölder's inequality.
In the former case, u is supported on D 2 . We use the Young inequality to have From (IVa), we only prove (3.2) in the latter case. In this case, we may assume that u is supported on D 2 and |τ − p λ (ξ)| N 5 0 . We use the Young inequality to obtain (IVc) Consider the case u is supported on D 2 . We use (2.4) with K ∼ N 0 to have .
In the case M max = |τ 1 −p λ (ξ 1 )|, we immediately obtain the desired estimate because Therefore we may assume that u is restricted to D 1 .
We use (2.3) with b = b ′ = 7/16 to obtain , which shows the required estimate.
Combining Propositions 2.3, 2.4 and 3.1, the iteration argument works to construct a local-in-time solution. Here B r (X ) for a Banach space X is defined as B r (X ) := u ∈ X ; u X ≤ r .
We obtain the local well-posedness for (1.1) in the following sense.
For the details of the proof, see [10]. Remark that the solution u obtained above satisfies u W s ([0,T ]) ≤ C u 0 H s for some constant C > 0.

Global well-posedness
In this section, we extend the local-in-time solution obtained in Proposition 3.3 to global one by the I-method. We use the modified energy E (4) I (u) by adding two suitable correction terms to E (2) I (u)(t) = Iu(t) 2 L 2 , which is introduced by Colliander, Keel, Staffilani, Takaoka and Tao [7]. They established the almost conservation law for this functional to obtain GWP for the KdV equation when s > −3/4. Before the definition of the modified energies, we state some notations. Let M : R k → C.
We now define new modified energies by adding some correction terms to the original modified energy E (2) I (u). Following u is real valued and m is even, we have We denote We compute the time derivative of E I (u)(t) to obtain d dt E (2) where ξ ij := ξ i + ξ j for i = j. Note that the quadratic term vanishes because a 2 = 0 and b 2 = 0. So the time derivative of E I (u) has the cubic form as follows; d dt E (2) We add a correction term Λ 3 (σ 3 ) to the modified energy E where the symmetric function σ 3 is determined later. Similar to above, the time derivative of E We choose σ 3 = −M 3 /(a 3 + βλ −2 b 3 ) to cancel the cubic terms. Therefore we have d dt E (3) In the same manner, we define the third modified energy as Then, Our almost conservation law for the modified energy E This estimate implies that the growth of the modified energy E We prepare some lemmas to prove the above proposition. Chen and Guo [3] used the mean value theorem to obtain the upper bound of M 4 as follows.

Compared to the KdV equation, the Kawahara equation has less symmetries. So
it is hard to obtain this upper bound. Next we recall some well-known estimates for the evolution operator e t(∂ 5 x −λ −2 β∂ 3 x ) as follows. , For the proof, see [13] and [14]. Combining Lemmas 4.2 and 4.3, Chen and Guo [3] showed the almost conservation law for E I (u) when s ≥ −7/4. In fact, the use of X s,1/2 (2,1) enables us to gain two derivatives by the smoothing effect. From this, we obtain the almost conservation law for s ≥ −37/20 by using this function space. But (1.3) in X s,1/2 (2,1) fails for s < −7/4. When we use the function space W s , two derivatives cannot be recovered by the smoothing effects. So it is difficult to obtain the almost conservation law when s < −7/4. To overcome this difficulty, we establish the improved bilinear estimate which is L 4 t,x type Strichartz estimate.
We prove these estimates by using the Hölder inequality, the Sobolev inequality and the improved bilinear estimate (4.6). Firstly, we show (4.8).
(Ia) Consider the case u N 1 is restricted to D 2 . The Hölder inequality and the Sobolev inequality imply that which is an appropriate bound where s 2 = 25/168 and b 2 = 79/168. So we only prove (4.8) in the case u N 1 is restricted to D 1 .
(Ib) Consider the case N 1 ∼ N 5 N. We combine (4.3)-(4.5) to obtain the L 4 t,x estimate, u N L 4 t,x N −3/8 u N X 0,3/8 (2,1) , for N ≫ 1. We use this estimate and the Hölder inequality to obtain , which implies the desired estimate. So we may assume N 1 ≫ N 5 . Then we use (4.6) with b = 19/42 to have , which is an appropriate bound.
(IIa) Consider the case u N 2 is supported on D 2 . We use the Hölder inequality and the Sobolev inequality to obtain which implies the desired estimate. So we only estimate (4.9) in the case u N 2 is supported on D 1 .
(IIb) Consider the case N 2 ∼ N 4 N. If there exists at least one of i = 2, 3, 4 such that u N i is supported on D 2 , then we immediately obtain the desired estimate following the above estimate. Therefore we may assume that u N i is restricted to D 1 for all i = 2, 3, 4. In this case, we use (4.3), (4.4) and the Hölder inequality to obtain .
(IIc) Consider the case N 2 ≫ N 4 . We first deal with the case N 4 ≤ 1. We use Hölder's and Sobolev's inequalities to have which is bounded by from (4.6) with b = 19/42. Next, we prove (4.9) when N 4 ≥ 1. We first estimate in the case u N 4 is supported on D 2 . The Hölder inequality and the Sobolev inequality imply that which is bounded by from (4.6) with b = b 2 = 79/168. Next, we consider the case u N 4 is supported on D 1 . We use Hölder's and Sobolev's inequalities and (4.6) with b=1/2 to have , which is an appropriate bound.
Remark. We add a suitable correction term to E  I (u), we probably obtain the almost conservation law in the same regularity s = −38/12. This is a reason why we do not expect to gain more 4b 1 derivatives by smoothing effects when the norm in D 3 is defined as · X s,b 1 (2,1) and any derivatives from M 5 bounds.
Next, we estimate the difference between the modified energies E   for any t 0 ∈ R.
We call this estimate the fixed point difference. Compared to the argument of [7], we need to estimate more sharply in order to obtain the above estimate when −2 ≤ s < −7/4 Proof. We may assume that u is non-negative. From the definition of the modified energy, it suffices to show Note that the mean value theorem shows the following M 3 bounds as follows; where |ξ 1 | ≥ |ξ 2 | ≥ |ξ 3 |. From M 3 bounds and M 4 bounds, (4.10) is reduced to the following estimates.
We use the Hölder inequality and the Young inequality to have which shows the desired estimate. Next, we deal with the case ξ * 4 = |ξ 4 |. It suffices to show that The Sobolev inequality and the Hölder inequality inequality imply that Combining Propositions 4.1 and 4.5, we can find a constant C 1 > 0 such that sup −N −5s ≤t≤N −5s when 0 > s ≥ −38/21. For the details, see [7]. Following