2025/06/21 by Molinet, Luc, Pilod, Didier, Vento, Stéphane · 2 citations
#35A01 #35A02 #35B45 #35E15 #35Q53 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2506.17801
We consider the Cauchy problem associated to a class of dispersive perturbations of Burgers' equations, which contains the low dispersion Benjamin-Ono equation, (also known as low dispersion fractional KdV equation), ∂tu-Dxα∂xu=∂x(u2) , and prove that it is locally well-posed in Hs(\mathbb K), \mathbb K=\mathbb R or \mathbb T, for s>sα, where sα=\begincases 1-\frac3α4 amp; for \frac23 ≤ α≤ 1; \frac 32(1-α) amp; for \frac13 ≤ α≤ \frac23; \frac 32-\fracα1-α amp; for 0 lt; α≤ \frac13 . \endcases The uniqueness is unconditional in Hs(\mathbb K) for s>max\\frac12,sα\. Moreover, we obtain a priori estimates for the solutions at the lower regularity threshold s>\widetildesα where \widetildesα=\begincases \frac 12-\frac α4 amp; for \frac23 ≤ α≤ 1; 1-αamp; for \frac12 ≤ α≤ \frac23; \frac 32-\fracα1-α amp; for 0 lt; α≤ \frac12 . \endcases As a consequence of these results and of the Hamiltonian structure of the equation, we deduce global well-posedness in Hs(\mathbb K) for s>sα when α>\frac23, and in the energy space H\fracα2(\mathbb K) when α>\frac45.