Well-posedness for the supercritical gKdV equation

In this paper we consider the supercritical generalized Korteweg-de Vries equation $\partial_t\psi + \partial_{xxx}\psi + \partial_x(|\psi|^{p-1}\psi) = 0$, where $5\leq p\in\R$. We prove a local well-posedness result in the homogeneous Besov space $\dot B^{s_p,2}_{\infty}(\mathbb{R})$, where $s_p=\frac12-\frac{2}{p-1}$ is the scaling critical index. In particular local well-posedness in the smaller inhomogeneous Sobolev space $H^{s_p}(\mathbb{R})$ can be proved similarly. As a byproduct a global well-posedness result for small initial data is also obtained.

Well-posedness results of the Cauchy problem (1) (with p ≥ 2) has been studied by many authors in recent years. We want to give a brief overview of the best known well-posedness results. The fundamental work on this topic was done by Kenig, Ponce and Vega [6,7] in 1993 and 1996. They proved local and small data global well-posedness for the sub-critical cases p ∈ {2, 3, 4} in H s (R) for certain s. For the KdV equation (p = 2) they proved well-posedness for s > − 3 4 . In the limiting case s = − 3 4 existence of solutions has been obtained by Christ, Colliander, and Tao [2]. Kenig, Ponce and Vega also proved well-posedness of the mKdV equation (p = 3) for s ≥ 1 4 , and of the quartic gKdV equation (p = 4) for s ≥ 1 12 . So far the scaling space H sp with s p = 1 2 − 2 p−1 was not reached for the sub-critical cases. That changed in 2007, when Tao [12] proved local well-posedness (and global well-posedness for small data) of the quartic KdV equation in the scaling critical inhomogeneous Sobolev spaceḢ − 1 6 . In 2012 Koch and Marzuola [8] simplified and strengthened Tao's well-posedness result in the Besov spaceḂ  [11] extended the well-posedness result in the supercritical cases to the homogeneous Besov spaceḂ sp,2 ∞ (R) with integer p. To our knowledge well-posedness results for non-integer p ≥ 5 were not obtained so far. We present a unified proof of well-posedness in the homogeneous Besov spaceḂ sp,2 ∞ (R) for all 5 ≤ p ∈ R. In this paper we pick up techniques of Koch and Marzuola [8] to prove local (and small data global) well-posedness for the supercritical gKdV equation, i.e. (1) with 5 ≤ p ∈ R. The well-posedness is proved in the in the homogeneous Besov spaceḂ Here, u λ denotes the Littlewood-Paley decomposition of u at frequency λ that is defined in Section 2.
In the following, let v be a solution to the Airy equation with same initial data For the quartic gKdV equation Koch and Marzuola [8] proved the following local well-posedness result: Moreover, the function w (and hence ψ) depends analytically on the initial data.
From this local well-posedness result they even obtained global wellposedness for small data ψ 0 , since one easily proves by Strichartz' estimates and the definition of the spaces that In the sequel, we are going to prove the analogue statement in the supercritical case, i.e. for (1) with 5 ≤ p ∈ R: then there exists an unique solution ψ = v + w to (1) with Moreover, the solution map is Lipschitz continuous.
Using the same arguments as Koch and Marzuola, we obtain global well-posedness for small initial data ψ 0 as well: Corollary 1.3. Let 5 ≤ p ∈ R, s p = 1 2 − 2 p−1 and δ 0 (1) be the δ 0 of Theorem 1.2, which depends on r 0 , evaluated at r 0 = 1. Let κ 0 and κ 1 be the constants from Lemma 3.1 and Lemma 2.12, respectively. Then there exists ε 0 > 0 such that for Moreover, the solution map is Lipschitz continuous.
The main ingredient of the proof of Theorem 1.2 is a multi-linear estimate that gives bounds on the Duhamel term of the nonlinearity. A crucial tool to get these estimates are the recently introduced U p and V p spaces. The rest of the proof is a standard fixed point argument to get existence and uniqueness. However, due to the non-integer exponents, this argument gets a bit more delicate.
Remark 1. The analogue local and global well-posedness in the inhomogeneous Sobolev space H sp follows along these lines. Note that the function spaces and the summation has to be modified. Throughout this paper, we will use mixed Lebesgue spaces L p t L q x which are defined via the norm and with obvious modifications for p = ∞. If p = q, then we write L p t,x for brevity. Moreover, we want to mention that we write A B, if there is a harmless constant c > 0 such that A ≤ cB.
This paper is organized as follows: In Section 2 we give a brief introduction to the function spaces used in this paper. Section 3 provides some basic linear and bilinear estimates. Multi-linear estimates to control the Duhamel term of the nonlinearity are proved in Section 4. Theorem 1.2 and the global well-posedness result is proved in Section 5.
Acknowledgments This paper is an extension of the diploma thesis of the author. The author wishes to thank the thesis advisor Herbert Koch and Sebastian Herr for helpful comments while working on this result.

Function spaces
Crucial tools to prove this well-posedness results are the function spaces U p , which have been introduced in the context of dispersive PDEs by Tataru and Koch-Tataru [9,10] as well as the closely related spaces of bounded p-Variation V p due to Wiener [13]. The following exposition of the U p and V p spaces may be found in [5]. We refer the reader to this paper for detailed definitions and proofs.
We consider functions taking values in L 2 = L 2 (R d , R), but in the general part of this section one may replace L 2 by an arbitrary Hilbert space. Let Z be the set of finite partitions −∞ < t 0 < t 1 < . . . < t K ≤ ∞.
k=0 φ k p L 2 = 1 and φ 0 = 0, we call the function a : R → L 2 given by a U p -atom. Furthermore, we define the atomic space Two useful statements about U p are collected in the following (i) We define V p as the normed space of all functions v : R → L 2 such that lim t→±∞ v(t) exists and for which the norm There is a unique number B(u, v) with the property that for all ε > 0 there exists t ∈ Z such that for every t ′ ⊂ t it holds

and the associated bilinear form
is an isometric isomorphism.
Corollary 2.7. For 1 < p < ∞, u ∈ U p and for v ∈ V p the following estimates hold true Following Bourgain's strategy for the Fourier restriction spaces we adapt the U p and V p space to the gKdV equation. Definition 2.9. Define the Airy group S : C(R, L 2 ) → C(R, L 2 ) as By this bilinear map, we obtain similar duality statements as in Corollary 2.7.
We pic up the homogeneous spaceẊ s that was defined in [8].
Definition 2.11. For s ∈ R, we define the real-valued homogeneous spaceẊ s using the norm Furthermore, we denote byẊ s T the functions on the time space set (0, T ) × R.
The following estimate follows directly from the definition of the spaces U p KdV and V p KdV . Lemma 2.12. Let v be a solution to the Airy equation then, for s ∈ R, there exists κ 1 > 0 such that the following estimate holds true v Ẋs

Linear and bilinear estimates
The following Lemma is based on [6] and may be found in [8, formula (3.2) and (7.7)].
In particular, for λ ∈ 1.01 Z we have Proof. This estimate follows directly from Bernstein's inequality and the energy estimate.
The next Corollary immediately follows from interpolating the L 6 t,x -Strichartz estimate and the L ∞ t,x estimate.
KdV , λ ∈ 1.01 Z and q ≥ 6, then we have and if p ≥ 5, then we even have for all q > 2(p − 1) The following bilinear estimate is based on a bilinear estimate of Grünrock [4] and can be found in [8, formula (7.8)].
KdV and let λ, µ ∈ 1.01 Z such that λ ≥ 1.1µ. Then If in addition p ≥ 5, then for all q > p−1 Proof. The first inequality follows by interpolating the bilinear estimate (Lemma 3.4) and the L ∞ t,x estimate (Lemma 3.2) as well as Proposition 2.4. As a consequence the second inequality simply follows from a Littlewood-Paley decomposition of v ≤µ . Note that in (6) q is choosen such that the exponent of µ is larger than zero.

Multi-linear estimates
Lemma 4.1. Let 5 ≤ p ∈ R and λ 2 ≤ . . . ≤ λ 5 ∈ 1.01 Z , µ ∈ 1.01 Z and 1.1λ 5 > µ. There exists r > 0 independent of T such that for given v i , u ∈Ẋ sp T , i = 0, . . . , 5, we have for some small ε > δ > 0 Moreover, we may even replace one factor v i Ẋ sp Proof. In order to prove this multi-linear estimate, we distinguish two cases. First, we consider the case when all frequencies λ i are comparable, i.e. λ 5 ≤ 1.1λ 2 . In the second case, we consider the situation if the frequency λ 5 is much greater than λ 2 , i.e. 1.1λ 2 < λ 5 . In this situation, we can make use of the strong bilinear estimate.

Corollary 3.3 allows to estimate
Furthermore, since q 2 > 2 the bilinear estimate gives The product of the λ i can be estimated by λ . Note that the exponent of λ 2 is bigger than zero for all p ≥ 5.
If we assume that the frequency µ is much greater than all other frequencies, then we can even prove the following Lemma.

By Corollary 3.3 and the definition ofẊ
Finally, the bilinear estimate provides Note that we may change the role of v 3,λ 3 and v 4,λ 4 in the calculation above and hence can also estimate v 3,λ 3 in L 6 t,x .

Proof of the theorem
In this section we are going to prove Theorem 1.2. The solution ψ = v + w of (1) is constructed by studying the following equation where v is a solution to the Airy equation (2).
If w solves (7) and W Ẋ sp T ≤ α, then there exists some c > 0 such that for Proof. For τ ∈ R and λ ∈ 1.01 Z we set F τ λ (u) = u <λ + τ u λ . Using that we define for τ = (τ 1 , . . . , τ n ) ∈ R n and λ = (λ 1 , . . . , λ n ) ∈ 1.01 Z n F τ λ (u) = F τ 1 λ 1 • · · · • F τn λn (u). One easily proves, that for τ = (τ 1 , . . . , τ n ) ∈ [0, 1] n , λ = (λ 1 , . . . , λ n ) ∈ (1.01 Z ) n and µ ∈ 1.01 Z we have Furthermore, one trivially verifies . Set f p (x) = |x| p−1 x, then by the telescoping series we have By a standard trick, using the fundamental theorem of calculus we get In the sequel we use a more compact notation and write u τ λ := F τ λ (u). Reapplying this method three times, we get for λ i = (λ i , . . . , λ 5 ) and τ i = (τ i , . . . , τ 5 ), i = 2, . . . , 5, That is in our context Let w be a solution to (7). By Lemma 2.12 it suffices to show By duality (cf. Corollary 2.7), it suffices to show that for each µ ∈ 1.01 Z we have If we apply the calculation above once, we can rewrite the modulus of the integral as First, we consider the sum S 1 . We integrate by parts, apply the calculation above to v τ 5 λ 5 + W τ 5 λ 5 p−1 and hence have to bound Note that since the spaces V 2 KdV are based on L 2 , the operator ∂x µ is bounded. We expand the factor v τ 5 λ 5 + W τ 5 λ 5 λ 4 . For v τ 5 λ 5 ,λ 4 we apply Lemma 4.1 and keep this factor in L 6 t,x . For W τ 5 λ 5 ,λ 4 we apply Lemma 4.1 and estimate all terms inẊ sp T . Hence, after summing over the frequencies, we obtain that S 1 is less than By the properties of F τ λ (·), we may estimate this by Using the bounds given in Theorem 1.2 and α ≤ κ 1 r 0 we may estimate this by for some c > 0. Since δ 0 ≤ α and α ≤ 1 2crκ p 1 r p−1 0 we obtain which implies the desired estimate w Ẋ sp T ≤ α for S 1 . Now, we consider S 2 . Note that S 2 = 0 if p = 5, since the frequencies do not sum up to zero. Hence we may assume p > 5 in the following. In order to estimate S 2 , we decompose Differentiating this term with respect to x yields We estimate all these terms exactly as for S 1 , but using Lemma 4.2 instead of Lemma 4.1. Note that the additional factor λ i on each term ensures that the summation over the frequencies converges. Note also that the operator ∂x λ i does not play a role, since the V 2 KdV spaces are based on L 2 . By the same argument as before, we can bound S 2 by α κ 1 as before.
Proof of Theorem 1.2. In order to prove Theorem 1.2, we use a fixed point argument to show existence and uniqueness. Let Furthermore, let be two iteration steps. Note that Lemma 5.1 ensures that w 1 Ẋ sp T < α as well. We have to show that there exists q ∈ (0, 1) such that By Lemma 2.12 and duality, it suffices to replace the left hand side by For brevity we define ω 0 = v + w 0 and ω 1 = v + w 1 . Similar as in the proof of Lemma 5.1 we may write Again, we split the sum into two parts, such that First, we consider S 1 . Similar to the previous Lemma, we integrate by parts such that the derivative turns into a factor µ, and we decompose The integrand may be written as Using the fundamental theorem of calculus, we can further manipulate the last term S 5 to get We split S 1 into two terms by expanding ω 0 τ 5 λ 5 ,λ 4 : For the first term we estimate v τ 5 λ 5 ,λ 4 in L 6 t,x , and for the second term we estimate all factors inẊ sp T . Hence, by Lemma 4.1 µ 1+sp u µ V 2 KdV λ 2 ≤...≤λ 5 λ 5 : 1.1λ 5 ≤µ Analogously, we expand either ω 0 All in all, we obtain Now, we may choose α (and hence δ 0 ) small enough such that crκ 1 (r 0 κ 1 ) p−2 (δ 0 + α) < 1 2 , which gives the desired estimate for S 1 . Now, we consider Note that if p = 5 then S 2 = 0 by the same argument as in Lemma 5.1. Hence, we may assume p > 5. We decompose If we differentiate S 1 with respect to x, then we are able to apply Lemma 4.2 and since we obtain an additional factor λ i , we get κ 1 µ sp u µ V 2 KdV λ 3 ≤λ 4 ≤λ 4 λ 5 : 1.1λ 5 ≤µ We can treat S 2 and S 3 analogously. Now, we consider S 4 . Applying the fundamental theorem of calculus, we obtain for Ω(τ ) = ω 0 Differentiating this term with respect to x yields a sum of 5 terms, each of which can be estimated using Lemma 4.2 as above. All in all we obtain κ 1 µ sp u µ V 2 KdV S 2 ≤ crκ 1 (r 0 κ 1 ) p−2 (δ 0 + α) w 0 − w 1 Ẋ sp T . By possibly chooser α smaller again, we have crκ 1 (r 0 κ 1 ) p−2 (δ 0 + α) < 1 2 .
Thus, for small enough α, we have a contraction and Banach fixedpoint theorem gives existence and uniqueness.
The following proof of Corollary 1.3 is an observation of Koch and Marzuola in [8, p. 175-176].
Since this estimate holds true for all 0 < T ≤ ∞, we may apply Theorem 1.2 with T = ∞ to obtain global existence.