Local well-posedness for the periodic higher order KdV type equations

Hirayama, Hiroyuki

doi:10.1007/s00030-011-0147-9

Cited by 29 publications

(29 citation statements)

References 25 publications

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“…They established the bilinear estimate in X s, 1 2 , s ≥ − 1 2 to prove the local well-posedness in H − 1 2 (T). This result was improved by Hirayama [27]. He improved the bilinear estimate established in [23] in H s (T) level, s ≥ − m−1 4 to show the local well-posedness H − m−1 4 , and this estimate was shown to be sharp in the standard X s,b .…”

Section: )mentioning

confidence: 86%

Well-posedness issues on the periodic modified Kawahara equation

Kwak

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on T), which is well-known as a model of capillary-gravity waves in an infinitely long canal over a flat bottom in a long wave regime [26]. We show in this paper some well-posedness results, mainly the global well-posedness in L 2 (T). The proof basically relies on the idea introduced in Takaoka-Tsutsumi's works [69,60], which weakens the non-trivial resonance in the cubic interactions (a kind of smoothing effect) for the local result, and the global well-posedness result immediately follows from L 2 conservation law. An immediate application of Takaoka-Tsutsumi's idea is available only in H s (T), s > 0, due to the lack of L 4 -Strichartz estimate for arbitrary L 2 data, a slight modification, thus, is needed to attain the local well-posedness in L 2 (T). This is the first low regularity (global) well-posedness result for the periodic modified Kwahara equation, as far as we know. A direct interpolation argument ensures the unconditional uniqueness in H s (T), s > 1 2 , and as a byproduct, we show the weak ill-posedness below H 1 2 (T), in the sense that the flow map fails to be uniformly continuous.2010 Mathematics Subject Classification. Primary 35Q53, 76B15, Secondary 35G25. 1 2 (T)). However, we do not take their argument for the proof of Theorem 1.5 in order to avoid abusing the normal form mechanism.Remark 1.6. Theorems 1.4 and 1.5 will be improved in the forthcoming work by developing the argument inspired by, for instance, Okamoto [64] and Molinet, Pilod and Vento [57], respectively. 3 The function space X s,b T is the time localization of the standard X s,b introduced by Bourgain [6]

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Section: )mentioning

confidence: 86%

Well-posedness issues on the periodic modified Kawahara equation

Kwak

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…We first restate several lemmas in Section 2 by modifying those adapted to µ-periodic setting. Lemma A.2 (Hirayama [9]). Let j ∈ N and µ ≥ 1.…”

Section: Nonsqueezing Property When J ≥mentioning

confidence: 99%

“…The local and global well-posedness of (1.1) were widely studied. For the local well-posedness result, Gorsky and Himonas [6] firstly proved this problem for s ≥ − 1 2 and Hirayama [9] improved for s ≥ − j 2 . Both works are based on the standard Fourier restriction norm method.…”

Section: Introductionmentioning

confidence: 99%

“…We encountered difficulties in the algebraic factorization of resonant functions. In order to overcome this issue, we use another argument (see Lemma 2.1 below) comparing with Hirayama's proof (Lemma 2.2 in [9]). Remark that s = − j 2 is sharp in the sense that the bilinear estimate in X s, 1 2 space fails for s < − j 2 (see Theorem 1.4. in [9]).…”

Section: Introductionmentioning

confidence: 99%

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Global well-posedness and nonsqueezing property for the higher-order KdV-type flow

Hong

Kwak

2016

Journal of Mathematical Analysis and Applications

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Abstract. In this paper, we prove that the periodic higher-order KdV-type equationis globally well-posed in H s for s ≥ − j 2 , j ≥ 3. The proof is based on "I-method" introduced by Colliander et al. [4]. We also prove the nonsqueezing property of the periodic higherorder KdV-type equation. The proof relies on Gromov's nonsqueezing theorem for the finite dimensional Hamiltonian system and an approximation argument for the solution flow. More precisely, after taking the frequency truncation to the solution flow, we apply the nonsqueezing theorem. By using the approximation argument, we extend this result to the infinite dimensional system. This argument was introduced by Kuksin [14] and made concretely by Bourgain [2] for the 1D cubic NLS flow, and Colliander et al. [5] for the KdV flow. One of our observations is that the higher-order KdV-type equation has the better modulation effect from the non-resonant interaction than that the KdV equation has. Hence, unlike the work of Colliander et al. [5], we can get the nonsqueezing property for the solution flow without the Miura transform.

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“…became the famous KdV equation, therefor people often call (1.5) as high-order KdV equation [1,2]. More information on higher-order dispersive equations can be found in [3].…”

Section: Introductionmentioning

confidence: 99%

Control and Stabilization of High-Order KdV Equation Posed on the Periodic Domain

Meng¹

2018

JPDE

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In this paper, we study exact controllability and feedback stabilization for the distributed parameter control system described by high-order KdV equation posed on a periodic domain T with an internal control acting on an arbitrary small nonempty subdomain ω of T. On one hand, we show that the distributed parameter control system is locally exactly controllable with the help of Bourgain smoothing effect; on the other hand, we prove that the feedback system is locally exponentially stable with an arbitrarily large decay rate when Slemrod's feedback input is chosen.

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Local well-posedness for the periodic higher order KdV type equations

Cited by 29 publications

References 25 publications

Well-posedness issues on the periodic modified Kawahara equation

Well-posedness issues on the periodic modified Kawahara equation

Global well-posedness and nonsqueezing property for the higher-order KdV-type flow

Control and Stabilization of High-Order KdV Equation Posed on the Periodic Domain

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