Sharp asymptotics for the minimal mass blow up solution of the critical gKdV equation

Combet, Vianney; Martel, Yvan

doi:10.1016/j.bulsci.2017.01.001

Cited by 5 publications

(17 citation statements)

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“…The result (1.13) for (gKdV) is thus more precise. In fact, for (gKdV), the minimal mass blow up is quite well understood, at least close to the blow up time: in addition to (1.13), sharp asymptotics, both in time (as t ↓ 0) and in space (as x → ±∞) were derived in [18], for any level of derivative of S KdV . Importantly, S KdV is also known to be global for t > 0 and to be the unique minimal mass solution of (gKdV), up to the symmetries of the equation (scaling, translations and sign change), see [42].…”

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confidence: 91%

“…A main difficulty in the present paper is to combine such local norms and nonlocal operators. A hint of the specificity of KdV-type blow up is given by the asymptotics found in [18], showing the existence of a fixed tail for S KdV (t). See also Remark 3.1 for more details.…”

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confidence: 96%

“…For the mass critical generalized Korteweg-de Vries equation (gKdV) u t + (u xx + u 5 ) x = 0, x ∈ R, t ∈ R, (1.12) (the energy space for (1.12) is H 1 (R)), an existence result similar to Theorem 1.1 was proved by Martel, Merle and Raphaël [42], and then sharpened by Combet and Martel [18]. More precisely, let Q KdV ∈ H 1 (R), Q KdV > 0 be the ground state for (1.12), i.e.…”

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confidence: 97%

“…the unique even solution of Q ′′ KdV + Q 5 KdV = Q KdV . It follows from [42] and [18] that there exists a solution S KdV on (0, +∞) such that S KdV (t) L 2 = Q KdV L 2 and…”

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confidence: 99%

“…In particular, Krieger, Lenzmann and Raphaël [32] addressed the case of the half-wave equation in one space dimension, which also involves the nonlocal operator D 1 , and requires the use of commutator estimates. However, as pointed out in [42,18], the minimal mass blow up for KdV-type equations is specific, in some sense less compact that for NLS or wave-type equations, and requires the use of local norms, instead of global norms. A main difficulty in the present paper is to combine such local norms and nonlocal operators.…”

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confidence: 99%

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Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation

Martel¹,

Pilod²

2016

Math. Ann.

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Abstract. We construct a minimal mass blow up solution of the modified Benjamin-Ono equation (mBO) utwhich is a standard mass critical dispersive model. Let Q ∈ H 1 2 , Q > 0, be the unique ground state solution of D 1 Q + Q = Q 3 , constructed using variational arguments by Weinstein (Comm. PDE, 12 (1987), J. Diff. Eq., 69 (1987)) and Albert, Bona and Saut (Proc. Royal London Soc., 453 (1997)), and whose uniqueness was recently proved by Frank and Lenzmann (Acta Math., 210 (2013)).We show the existence of a solution S of (mBO) satisfying S L 2 = Q L 2 andwhereThis existence result is analogous to the one obtained by Martel, Merle and Raphaël (J. Eur. Math. Soc., 17 (2015)) for the mass critical generalized Korteweg-de Vries equation. However, in contrast with the (gKdV) equation, for which the blow up problem is now wellunderstood in a neighborhood of the ground state, S is the first example of blow up solution for (mBO). The proof involves the construction of a blow up profile, energy estimates as well as refined localization arguments, developed in the context of Benjamin-Ono type equations by Kenig, Martel and Robbiano (Ann. Inst. H. Poincaré, Anal. Non Lin., 28 (2011)). Due to the lack of information on the (mBO) flow around the ground state, the energy estimates have to be considerably sharpened in the present paper.

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confidence: 91%

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confidence: 96%

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confidence: 97%

“…the unique even solution of Q ′′ KdV + Q 5 KdV = Q KdV . It follows from [42] and [18] that there exists a solution S KdV on (0, +∞) such that S KdV (t) L 2 = Q KdV L 2 and…”

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confidence: 99%

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confidence: 99%

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Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation

Martel¹,

Pilod²

2016

Math. Ann.

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Full Family of Flattening Solitary Waves for the Critical Generalized KdV Equation

Martel

Pilod

2020

Commun. Math. Phys.

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For the critical generalized KdV equation ∂ t u + ∂ x (∂ 2 x u + u 5) = 0 on R, we construct a full family of flattening solitary wave solutions. Let Q be the unique even positive solution of Q + Q 5 = Q. For any ν ∈ (0, 1 3), there exist global (for t ≥ 0) solutions of the equation with the asymptotic behavior u(t, x) = t − ν 2 Q t −ν (x − x(t)) + w(t, x) where, for some c > 0, x(t) ∼ ct 1−2ν and w(t) H 1 (x> 1 2 x(t)) → 0 as t → +∞. Moreover, the initial data for such solutions can be taken arbitrarily close to a solitary wave in the energy space. The long-time flattening of the solitary wave is forced by a slowly decaying tail in the initial data. This result and its proof are inspired and complement recent blow-up results for the critical generalized KdV equation. This article is also motivated by previous constructions of exotic behaviors close to solitons for other nonlinear dispersive equations such as the energy-critical wave equation.

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Construction of Multibubble Solutions for the Critical GKDV Equation

Combet¹,

Martel²

2018

SIAM J. Math. Anal.

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Sharp asymptotics for the minimal mass blow up solution of the critical gKdV equation

Cited by 5 publications

References 34 publications

Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation

Construction of a minimal mass blow up solution of the modified Benjamin–Ono equation

Full Family of Flattening Solitary Waves for the Critical Generalized KdV Equation

Construction of Multibubble Solutions for the Critical GKDV Equation

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