Existence of blowup solutions to Boussinesq equations on $\mathbb{R}^3$ with dissipative temperature

TL;DR

This paper constructs finite-time blow-up solutions for the 3D Boussinesq equations with dissipative temperature, extending previous work on Euler systems and addressing challenges posed by temperature diffusion.

Contribution

It introduces a novel time-dependent scaling and weighted Sobolev norms to prove finite-time blow-up solutions for Boussinesq equations with temperature diffusion.

Findings

Constructed $C^{1,eta}$ blow-up solutions for Boussinesq system

Developed new weighted Sobolev norms for analysis

Overcame difficulties due to temperature diffusion smoothing effects

Abstract

The three-dimensional incompressible Boussinesq system is one of the important equations in fluid dynamics. The system describes the motion of temperature-dependent incompressible flows. And the temperature naturally has diffusion. Recently, Elgindi, Ghoul and Masmoudi constructed a $C^{1,\alpha}$ finite time blow-up solutions for Euler systems with finite energy. Inspired by their works, we constructed $C^{1,\alpha}$ finite time blow-up solution for Boussinesq equations where the temperature has diffusion and finite energy. Generally speaking, the diffusion of temperature smooths the solution of the system which is against the formations of singularity. The main difficulty is that the Laplace operator of the temperature equation is not coercive under the Sobolev weighted norm introduced by Elgindi. We introduced a new time depending scaling formulation and new weighted Sobolev norms, under which we obtain the nonlinear estimate. The new norm is well-coupled with the original norm, which enables us to finish the proof.