In this paper we study a transport-diffusion model with some logarithmic dissipations. We
look for two kinds of estimates. The first is a maximum principle whose proof is based on
Askey theorem concerning characteristic functions and some tools from the theory of
-semigroups.
The second is a smoothing effect based on some results from harmonic
analysis and submarkovian operators. As an application we prove the global
well-posedness for the two-dimensional Euler–Boussinesq system where the
dissipation occurs only on the temperature equation and has the form
, with
.
This result improves on an earlier critical dissipation condition
needed for global well-posedness.
Keywords
Boussinesq system, logarithmic dissipation, global
existence
