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

  • Stéphane Vento

    Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99, avenue Jean-Baptiste Clément, F-93 430 Villetaneuse, France
  • Luc Molinet

    Institut Denis Poisson, Université de Tours, Université d'Orléans, CNRS (UMR 7013), Parc Grandmont, 37200 Tours, France
  • Didier Pilod

    Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, CEP: 21945-970, Rio de Janeiro, RJ, Brazil
On well-posedness for some dispersive perturbations of Burgers' equation cover
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Abstract

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

tuDxαxu=x(u2),0<α1,\partial _{t}u−D_{x}^{\alpha }\partial _{x}u = \partial _{x}(u^{2})\:,\:0 < \alpha \leq 1,

is locally well-posed in Hs(R)H^{s}(\mathbb{R}) when s>sα:=325α4s > s_{\alpha }: = \frac{3}{2}−\frac{5\alpha }{4}. As a consequence, we obtain global well-posedness in the energy space Hα2(R)H^{\frac{\alpha }{2}}(\mathbb{R}) as soon as α2>sα\frac{\alpha }{2} > s_{\alpha }, i.e. α>67\alpha > \frac{6}{7}.

Cite this article

Stéphane Vento, Luc Molinet, Didier Pilod, On well-posedness for some dispersive perturbations of Burgers' equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 7, pp. 1719–1756

DOI 10.1016/J.ANIHPC.2017.12.004