On the global well-posedness of a generalized 2D Boussinesq equations

• Junxiong Jia
• Jigen Peng
• Kexue Li
Article

Abstract

In this paper, we consider the global solutions to a generalized 2D Boussinesq equation
$$\left \{ \begin{array}{ll}\partial_{t} \omega + u \cdot \nabla \omega + \nu \Lambda^{\alpha} \omega = \theta_{x_{1}} , \quad \\ u = \nabla^{\bot} \psi = (-\partial_{x_{2}} , \partial_{x_{1}}) \psi , \quad \Delta \psi = \Lambda^{\sigma} (\log (I-\Delta))^{\gamma} \omega , \quad \\ \partial_{t} \theta + u\cdot \nabla \theta + \kappa \Lambda^{\beta} \theta = 0, \quad \\ \omega(x,0) = \omega_{0}(x) , \quad \theta(x,0) = \theta_{0}(x),\end{array}\right.$$
with $${\sigma \geq 0}$$, $${\gamma \geq 0}$$, $${\nu > 0}$$, $${\kappa > 0}$$, $${\alpha < 1}$$ and $${\beta < 1}$$. When $${\sigma = 0}$$, $${\gamma \geq 0}$$, $${\alpha \in [0.95,1)}$$ and $${\beta \in (1-\alpha,g(\alpha))}$$, where $${g(\alpha) < 1}$$ is an explicit function as a technical bound, we prove that the above equation has a global and unique solution in suitable functional space.

76D03 76D05

Keywords

Generalized 2D Boussinesq equation Global regularity Supercritical Boussinesq equations Regularization effect

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