Construction of multi-bubble solutions for the critical gKdV equation

2017/06/29 by Vianney Combet, Combet, Vianney, Yvan Martel +1 · 1 citation
Mathematics · #35B40 #35B44 #35Q53 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1706.09870

openalex publication_date 2017/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation ∂_t u + ∂_x(∂_xx u + u⁵) = 0 containing an arbitrary number K≥ 2 of blow up bubbles, for any choice of sign and scaling parameters: for any ℓ₁>ℓ₂>⋯>ℓ_K>0 and ε₁,…,ε_K∈\±1\, there exists an H¹ solution u of the equation such that u(t) - ∑_k=1^K \frac ε_kλ_k^\frac12(t) Q( \frac ⋅ - x_k(t)λ_k(t) ) \longrightarrow 0 in H¹ as t\downarrow 0, with λ_k(t)∼ ℓ_k t and x_k(t)∼ -ℓ_k^-2t^-1 as t\downarrow 0. The construction uses and extends techniques developed mainly by Martel, Merle and Raphaël. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved by the authors in arXiv:1602.03519.

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• On the asymptotic dynamics for the L²-supercritical gKDV equation

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