Local and global well-posedness results for the Benjamin-Ono-Zakharov-Kuznetsov equation

2016/01/05 by Francis Ribaud, Ribaud, Francis, Stéphane Vento +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1601.00856

arXiv admin note: text overlap with arXiv:1205.0169 by other authors

arxiv created 2016/01/05 · arxiv updated 2016/01/06

Abstract

We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equationu_t-D_x^αu_x + u_xyy = uu_x, (t,x,y)∈\R³, 1≤ α≤ 2,is locally well-posed in the spaces E^s, s\textgreater\frac 2α-\frac 34, endowed with the norm‖f‖_E^s = ‖⟨ |ξ|^α+μ²⟩^sf‖_L²(\R²).As a consequence, we get the global well-posedness in the energy space E^1/2 as soon as α\textgreater\frac 85. The proof is based on the approach of the short time Bourgain spaces developed by Ionescu, Kenig and Tataru \citeIKT combined with new Strichartz estimates and a modified energy.

Citations

• Well-posedness for the fifth-order KdV equation in the energy space

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