Norm inflation for the generalized Boussinesq and Kawahara equations

Okamoto, Mamoru

doi:10.1016/j.na.2017.03.011

Cited by 22 publications

(27 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The authors of [8,33] informed us that their proofs of norm inflation results followed the argument in the first version of this article. We also remark that an estimate proved in the first version (Lemma 3.6 below) was employed later in [31,19,37]. 6 In [37] non gauge-invariant nonlinearities were first treated in a general setting.…”

Section: Introductionmentioning

confidence: 99%

“…We also remark that an estimate proved in the first version (Lemma 3.6 below) was employed later in [31,19,37]. 6 In [37] non gauge-invariant nonlinearities were first treated in a general setting. In fact Theorem 1.1 follows as a corollary of [37, Proposition 2.5 and Corollary 2.10].…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of the present article is to apply this method to NLS with general nonlinearities. In the last few years the method has been used to a wide range of equations; see for instance [30,31,19,8,37]. 5 In [33,37], norm inflation based at general initial data was proved for NLS and some other equations.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A remark on norm inflation for nonlinear Schrödinger equations

Kishimoto¹

2019

Communications on Pure &Amp; Applied Analysis

View full text Add to dashboard Cite

We consider semilinear Schrödinger equations with nonlinearity that is a polynomial in the unknown function and its complex conjugate, on R d or on the torus. Norm inflation (ill-posedness) of the associated initial value problem is proved in Sobolev spaces of negative indices. To this end, we apply the argument of Iwabuchi and Ogawa (2012), who treated quadratic nonlinearities. This method can be applied whether the spatial domain is non-periodic or periodic and whether the nonlinearity is gauge/scale-invariant or not.2010 Mathematics Subject Classification. 35Q55, 35B30.1 Proposition A.1 below for the definition). Our argument, which evaluates each term in the power series expansion of the solution directly, is different from the aforementioned works. Note that, for smooth nonlinearities, Theorem 1.1 covers all the remaining cases in the range s < min{s c (d, p), 0} and extends the result to the (partially) periodic setting as well as to the case of general nonlinearities with complex coefficients. Moreover, our argument also gives another proof of the results in [6, 7] on NI s with infinite loss of regularity; see Proposition A.1 for the precise statement.The one-dimensional cubic equation with nonlinearity ±|u| 2 u has been attracting particular attention due to its various physical backgrounds and complete integrability. Note also that this is the only L 2 -subcritical case among smooth and gauge-invariant nonlinearities. In spite of the L 2 subcriticality, the equation becomes unstable below L 2 due to the Galilean invariance, both in R and in T. In fact, the initial value problem was shown to be globally well-posed in L 2 [39, 3], whereas it was shown in [23,9] for R and in [4,9] for T that the solution map fails to be uniformly continuous below L 2 . Ill-posedness below L 2 (T) was established in the periodic case by the lack of continuity of the solution map [11,32] and by the non-existence of solutions [17]. Nevertheless, one can show a priori bound in some Sobolev spaces below L 2 [27,12,28,17], which prevents norm inflation. Recent results in [29,24] finally gave a priori bound on H s for s > − 1 2 , both in R and in T. We remark that NI s at s = − 1 2 shown in Theorem 1.1 ensures the optimality of these results. 3 In [24, Theorem 4.7], Killip, Vişan and Zhang also derived a priori bound of the solutions in the norm which is logarithmically stronger than the critical H − 1 2 . Motivated by this result, in addition to Theorem 1.1 (ii) we also show norm inflation for the one-dimensional cubic equation in some "logarithmically subcritical" spaces; see Proposition B.3 below.Since the work of Kenig, Ponce, and Vega [22], non gauge-invariant nonlinearities have also been intensively studied. In [2], Bejenaru and Tao proposed an abstract framework for proving ill-posedness in the sense of discontinuity of the solution map. They considered the quadratic NLS (1.2) on R with nonlinearity u 2 and obtained a complete dichotomy of Sobolev index s into locally well-posed (s ≥ −1) and illposed (s < −1) in the sense ...

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A remark on norm inflation for nonlinear Schrödinger equations

Kishimoto¹

2019

Communications on Pure &Amp; Applied Analysis

View full text Add to dashboard Cite

show abstract

“…We will prove Theorem 1.1 under the massless case m = M = 0 and norm inflation at zero ψ 0 = φ 0 = φ 1 = 0. This is sufficient for the general case in Theorem 1.1 from the argument in [21].…”

Section: Proof Of Theorem 11mentioning

confidence: 98%

Sharp ill-posedness of the Dirac–Klein–Gordon system in one dimension

Machihara

Okamoto

2020

Nonlinear Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…The Cauchy problems for the Kawahara and modified Kawahara equations posed on R have been extensively studied. 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.…”

Section: )mentioning

confidence: 99%

Well-posedness issues on the periodic modified Kawahara equation

Kwak

2020

Ann. Inst. H. Poincaré C Anal. Non Linéaire

View full text Add to dashboard Cite

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]

show abstract

Norm inflation for the generalized Boussinesq and Kawahara equations

Cited by 22 publications

References 25 publications

A remark on norm inflation for nonlinear Schrödinger equations

A remark on norm inflation for nonlinear Schrödinger equations

Sharp ill-posedness of the Dirac–Klein–Gordon system in one dimension

Well-posedness issues on the periodic modified Kawahara equation

Contact Info

Product

Resources

About