The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion Article

Bessaih, H, Ferrario, B. (2017). The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion . JOURNAL OF DIFFERENTIAL EQUATIONS, 262(3), 1822-1849. 10.1016/j.jde.2016.10.032

cited authors

  • Bessaih, H; Ferrario, B

authors

abstract

  • In this paper, we study the 3D regularized Boussinesq equations. The velocity equation is regularized à la Leray through a smoothing kernel of order α in the nonlinear term and a β-fractional Laplacian; we consider the critical case α+β=[formula presented] and we assume [formula presented]<β<[formula presented]. The temperature equation is a pure transport equation, where the transport velocity is regularized through the same smoothing kernel of order α. We prove global well posedness when the initial velocity is in Hr and the initial temperature is in Hr−β for r>max⁡(2β,β+1). This regularity is enough to prove uniqueness of solutions. We also prove a continuous dependence of solutions on the initial conditions.

publication date

  • February 5, 2017

published in

Digital Object Identifier (DOI)

start page

  • 1822

end page

  • 1849

volume

  • 262

issue

  • 3