On well-posedness for some dispersive perturbations of Burgers' equation

Molinet, Luc; Pilod, Didier; Vento, Stéphane

doi:10.1016/j.anihpc.2017.12.004

Molinet, Luc ; Pilod, Didier ; Vento, Stéphane

Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 7, pp. 1719-1756

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Résumé

We show that the Cauchy problem for a class of dispersive perturbations of Burgers' equations containing the low dispersion Benjamin–Ono equation

\partial_{t} u - D_{x}^{α} \partial_{x} u = \partial_{x} (u^{2}), 0 < α \leq 1,

is locally well-posed in

H^{s} (R)

when

s > s_{α} : = \frac{3}{2} - \frac{5 α}{4}

. As a consequence, we obtain global well-posedness in the energy space

H^{\frac{α}{2}} (R)

as soon as

\frac{α}{2} > s_{α}

, i.e.

α > \frac{6}{7}

.

MR Zbl

DOI : 10.1016/j.anihpc.2017.12.004

Keywords: Nonlinear dispersive equations, Benjamin–Ono type equations with low dispersion, Global well-posedness, Modified energy

@article{AIHPC_2018__35_7_1719_0,
     author = {Molinet, Luc and Pilod, Didier and Vento, St\'ephane},
     title = {On well-posedness for some dispersive perturbations of {Burgers'} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1719--1756},
     publisher = {Elsevier},
     volume = {35},
     number = {7},
     year = {2018},
     doi = {10.1016/j.anihpc.2017.12.004},
     mrnumber = {3906854},
     zbl = {1459.76024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2017.12.004/}
}

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Molinet, Luc; Pilod, Didier; Vento, Stéphane. On well-posedness for some dispersive perturbations of Burgers' equation. Annales de l'I.H.P. Analyse non linéaire, Tome 35 (2018) no. 7, pp. 1719-1756. doi: 10.1016/j.anihpc.2017.12.004

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