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

2016/02/10 by Vianney Combet, Combet, Vianney, Yvan Martel +1 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #advanced mathematical theories #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1602.03519

Abstract

Let S be a minimal mass blow up solution of the critical generalized KdV\nequation as constructed by Martel, Merle and Rapha "el in arXiv:1204.4624. We\nprove both time and space sharp asymptotics for S close to the blow up time.\nLet Q be the unique ground state of (gKdV), satisfying Q"+Q⁵=Q.\n First, we show that there exist universal smooth profiles\nQ_k\∈\S(\ℝ) (with Q₀=Q) and a constant\nc₀\∈\ℝ such that, fixing the blow up time at t=0 and appropriate\nscaling and translation parameters, S satisfies, for any m geqslant 0, n
partial_x^m S(t) -
sum_k=0^[m/2]
frac 1t
^{frac 12+m-2k}\nQ_k^(m-k)
left(
frac
cdot+
frac1tt+c₀
right)
to 0
quad
mboxin
L²\n
mboxas
t
downarrow 0. Second, we prove that, for 0<t\≪ 1, x leqslant\n- frac 1t -1, \S(t,x)
sim -
frac 12
|Q
|_L¹ |x|^-3/2, and related\nbounds for the derivatives of S(t) of any order. We also prove\n\∫_\ℝ S(t,x) ,dx=0.\n

Citations

• Nonlinear Schr�dinger equations and sharp interpolation estimates

Cited by

• On the asymptotic dynamics for the L²-supercritical gKDV equation
• Construction of a minimal mass blow up solution of the modified\n Benjamin-Ono equation

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