Rough solutions of the fifth-order KdV equations

Guo, Zihua; Kwak, Chulkwang; Kwon, Soonsik

doi:10.1016/j.jfa.2013.08.010

Cited by 38 publications

(59 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Around the time when we completed this work, we learned that Guo, Kwak and Kwon [11] had also worked on the same problem and obtained the same results as ours (in Theorem 1.1 and Proposition 6.2). They also used the short-time X s,b method.…”

Section: On the Other Hand It Was Provedsupporting

confidence: 55%

Well-posedness for the fifth-order KdV equation in the energy space

Kenig

Pilod

2014

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Section: On the Other Hand It Was Provedsupporting

confidence: 55%

Well-posedness for the fifth-order KdV equation in the energy space

Kenig

Pilod

2014

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Recently various new results have been obtained -see [12,13,23,24] and references therein. In particular, in [13,23], wellposedness results were established for initial data in H s (R) with s 2 whereas in [12,23], such results were obtained for initial data in certain classes of Fourier Lebesgue spaces. Since it is believed that for such equations stronger wellposedness results can be obtained on the line than in the periodic setup (cf.…”

Section: Remark 110mentioning

confidence: 99%

On the wellposedness of the KdV/KdV2 equations and their frequency maps

Kappeler¹,

Molnar²

2018

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

View full text Add to dashboard Cite

In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on H s the KdV2 equation is C 0 -wellposed if s 0 and illposed (in a strong sense) if s < 0.

show abstract

“…This method has been very effective in establishing a priori bounds on solutions in low regularities (yielding even uniqueness in some cases), in particular, where a solution map is known to fail to be locally uniformly continuous. See [13,24,28,21,29,22,25].…”

Section: 2mentioning

confidence: 99%

Non-Existence of Solutions for the Periodic Cubic NLS below ${L}^{{2}}$

Guo

2016

Int Math Res Notices

Self Cite

137

View full text Add to dashboard Cite

Abstract. We prove non-existence of solutions for the cubic nonlinear Schrödinger equation (NLS) on the circle if initial data belong to H s (T) \ L 2 (T) for some s ∈ (− 1 8, 0). The proof is based on establishing an a priori bound on solutions to a renormalized cubic NLS in negative Sobolev spaces via the short-time Fourier restriction norm method. where u is a complex-valued function. The Cauchy problem (1.1) has been studied extensively from both theoretical and applied points of view. It is known to be one of the simplest partial differential equations (PDEs) with complete integrability [1,2,19]. In the following, however, we only discuss analytical aspects of (1.1) without using the complete integrable structure of the equation. It is well known that (1.1) enjoys several symmetries. In the following, we concentrate on the scaling symmetry and the Galilean symmetry. The scaling symmetry states that if u(x, t) is a solution to (1.1) on R with initial condition u 0 , then u λ (x, t) = λ −1 u(λ −1 x, λ −2 t) is also a solution to (1.1) with the λ-scaled initial condition u λ 0 (x) = λ −1 u 0 (λ −1 x). Associated to the scaling symmetry, there is a scaling-critical Sobolev index s c such that the homogeneousḢ sc -norm is invariant under the dilation symmetry. It is commonly conjectured that a PDE is ill-posed in H s for s < s c . In the case of the one-dimensional cubic NLS, the scaling-critical Sobolev index is s c = − 1 While there is no dilation symmetry in the periodic setting, the heuristics provided by the scaling argument plays an important role even in the periodic setting. For example, a norm inflation result for (1.1) on T analogous to [12] also holds for s < s c = − x u 0 (x). This symmetry also holds on the circle for β ∈ 2Z. Note that the L 2 -norm is invariant under the Galilean symmetry.2 This induces another critical regularity s ∞ crit = 0. Indeed, there is a strong dichotomy in the behavior of solutions for s ≥ 0 and s < 0.Bourgain [3] introduced the so-called Fourier restriction norm method via the X s,bspaces (see (3.1) below) and proved local well-posedness of (1.1) in L 2 (T). Thanks to the L 2 -conservation, this result was immediately extended to global well-posedness in L 2 (T). On the other hand, (1.1) is known to be ill-posed below L 2 (T). Burq-Gérard-Tzvetkov [5] showed that the solution map Φ t : u 0 ∈ H s → u(t) ∈ H s is not locally uniformly continuous if s < 0. See also [10]. Moreover, Christ-Colliander-Tao [11] and Molinet [31] proved that the solution map is indeed discontinuous if s < 0. Our main result in this paper states an even stronger form of ill-posedness holds true for (1.1) in negative Sobolev spaces. Theorem 1.1 (Non-existence of solutions for the cubic NLS). Let s ∈ (− 1 8 , 0) andNote that the condition (ii) is a natural condition to impose since it would follow from the continuity of the solution map :, which is one of the essential components in the well-posedness theory of evolution equations. On the one hand, the previous ill-posedness results [5,10,11,12,31] ...

show abstract

Rough solutions of the fifth-order KdV equations

Cited by 38 publications

References 26 publications

Well-posedness for the fifth-order KdV equation in the energy space

Well-posedness for the fifth-order KdV equation in the energy space

On the wellposedness of the KdV/KdV2 equations and their frequency maps

Non-Existence of Solutions for the Periodic Cubic NLS below ${L}^{{2}}$

Contact Info

Product

Resources

About