An improved local wellposedness result for the modified KdV equation

2003/12/11 by Axel Gruenrock, Gruenrock, Axel · 4 citations
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.math/0312238

14 pages

arxiv created 2003/12/11 · arxiv updated 2009/12/01

Abstract

The Cauchy problem for the modified KdV equation is shown to be locally well posed for data u₀ in the space (H^r_s) defined by the norm ||u₀||:=||<ξ>^s (u₀)||_L^r', provided 4/3 < r ≤ 2, s ≥ 1/2 - 1/(2r). For r=2 this coincides with the best possible result on the H^s - scale due to Kenig, Ponce and Vega. The proof uses an appropriate variant of the Fourier restriction norm method and linear as well as bilinear estimates for the solutions of the Airy equation.

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