Global well-posedness for the Kawahara equation with low regularity

Kato, Tadakatsu

doi:10.3934/cpaa.2013.12.1321

Cited by 26 publications

(15 citation statements)

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“…For the Kawahara equation ((1.2) with p = 2), we refer to [18,17,74,13,32,12,37,39,64] for the well-and ill-posedness results. As the best result in the sense of the low regularity Cauchy problem, Kato [37,39] proved the local well-posedness for s ≥ −2 by modifying X s,b space, the global well-posedness for s > − 38 21 and the ill-posedness for s < −2 in the sense that the flow map is discontinuous at zero. Recently, Okamoto [64] observed the norm inflation with general initial data, which implies that the flow map of the Kawahara equation is discontinuous everywhere in H s (R) with s < −2.…”

Section: )mentioning

confidence: 99%

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: 99%

Well-posedness issues on the periodic modified Kawahara equation

Kwak

2020

Ann. Inst. H. Poincaré Anal. Non Linéaire

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“…The well-posedness of pure initial value problem for (1.1) in R with a j = 0, j = 2, • • • , n, i.e., the fifth-order KdV equation, has been discussed in [8,9]. The controllability problems of (1.1) for this case using nonhomogeneous terms in the equation or boundary conditions have been studied in [32,33].…”

Section: πmentioning

confidence: 99%

A Korteweg–de Vries type of fifth-order equations on a finite domain with point dissipation

Gao

Sun

2016

Journal of Mathematical Analysis and Applications

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The paper discusses initial value problem of a Korteweg-de Vries type of fifth-order equation w t + w xxx − w xxxxx − n X j=1 a j w j w x = 0 , w(x, 0) = w 0 (x) posed on a periodic domain x ∈ [0, 2π] with periodic boundary conditions w ix (0, t) = w ix (2π, t), i = 0, 2, 3, 4 and an L 2-stabilizing feedback control law w x (2π, t) = αw x (0, t) + (1 − α)w xxx (0, t) where n is a fixed positive integer, a j , j = 1, 2, • • • , n, α are real constants, and |α| < 1. It is shown that for w 0 (x) ∈ H 1 α (0, 2π) with the boundary conditions described above, the problem is locally well-posed for w ∈ C([0, T ]; H 1 α (0, 2π)) with a conserved volume of w, [w] = R 2π 0 w(x, t)dx. Moreover, the solution with small initial condition exists globally and approaches to [w 0 (x)]/(2π) as t → +∞.

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“…Moreover, Kato 7 proved that the periodic Kawahara Equation () is ill‐posed in

H^{s} false(𝕋 false)

for any

s < \frac{- 3}{2}

in the sense that the data to the solution map is not C 3 . This critical exponent for the periodic Kwahara equation is quite different to that of the nonperiodic Kawahara equation, considering that Equation () is proved to be locally well‐posedness in

H^{s} false(ℝ false)

with s ≥ −2 and ill‐posed in

H^{s} false(ℝ false)

for any s < −2 (see previous studies 8‐15 ). However, the results on the fifth‐order KdV equations with the nonlinear dispersive term () are less advanced.…”

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confidence: 98%

Local well‐posedness for a type of periodic fifth‐order Korteweg–de Vries equations without nonlinear dispersive term

Zhou

2020

Math Methods in App Sciences

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We consider the Cauchy problem of the fifth‐order Korteweg–de Vries (KdV) equations without nonlinear dispersive term ∂tu−∂x5u+b0u∂xu+b1∂xfalse(∂xufalse)2=0,0.30emfalse(t,xfalse)∈ℝ×𝕋. Recently, Kappeler‐Molnar (2018) proved that the fifth‐order KdV equation with nonlinear dispersive term and Hamiltonian structure is globally well‐posed in Hsfalse(𝕋false) with s ≥ 0. Without the nonlinear dispersive term, Equation () is not integrable, and Kappeler–Molnar's approach is not valid. Using the idea of modifying Bourgain space, we prove that Equation () is locally well‐posed in Hsfalse(𝕋false) with s≥58.

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Global well-posedness for the Kawahara equation with low regularity

Cited by 26 publications

References 20 publications

Well-posedness issues on the periodic modified Kawahara equation

Well-posedness issues on the periodic modified Kawahara equation

A Korteweg–de Vries type of fifth-order equations on a finite domain with point dissipation

Local well‐posedness for a type of periodic fifth‐order Korteweg–de Vries equations without nonlinear dispersive term

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