Well-posedness and ill-posedness results for dissipative Benjamin-Ono equations

2008/02/07 by Stéphane Vento, Vento, Stéphane · 3 citations
Mathematics · Physics and Astronomy · #35A05 #35M10 #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons

paper · doi:10.48550/arxiv.0802.1039

openalex publication_date 2008/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Cauchy problem for the dissipative Benjamin-Ono equations u_t+\H u_xx+|D|^αu+uu_x=0 with 0≤α≤ 2. When 0≤α< 1, we show the ill-posedness in H^s(\R), s∈\R, in the sense that the flow map u₀↦ u (if it exists) fails to be \C² at the origin. For 1-α/4. It turns out that this index is optimal.

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