On a maximum principle and its application to logarithmically critical Boussinesq system

TL;DR

This paper establishes a maximum principle and smoothing effects for a logarithmically dissipative transport-diffusion model, leading to improved global well-posedness results for the 2D Euler-Boussinesq system with critical dissipation.

Contribution

It introduces new maximum principle and smoothing estimates for a logarithmic dissipation model, enabling the proof of global well-posedness for a more critical dissipation case in the Boussinesq system.

Findings

Proved a maximum principle using Askey theorem and semigroup theory.

Established smoothing effects via harmonic analysis and sub-Markovian operators.

Achieved global well-posedness for the 2D Euler-Boussinesq system with less dissipation than previously required.

Abstract

In this paper we study a transport-diffusion model with some logarithmic dissipations. We look for two kinds of estimates. The first one is a maximum principle whose proof is based on Askey theorem concerning characteristic functions and some tools from the theory of $C_0$-semigroups. The second one is a smoothing effect based on some results from harmonic analysis and sub-Markovian 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 $\frac{\DD}{\log^\alpha(e^4+\DD)}$, with $\alpha\in[0,\frac12]$. This result improves the critical dissipation $(\alpha=0)$ needed for global well-posedness which was discussed in [15].