Small scale formation for the 2D Boussinesq equation

Abstract

We study the 2D incompressible Boussinesq equation without thermal diffusion, and aim to construct rigorous examples of small scale formations as time goes to infinity. In the viscous case, we construct examples of global smooth solutions satisfying ${\sup }_{\tau \in [0,t]}\Vert \nabla \rho (\tau ){\Vert }_{L^2}\gtrsim t^{\alpha }$ for some $\alpha >0$. For the inviscid equation in the strip, we construct examples satisfying $\Vert \omega (t){\Vert }_{L^{\infty }}\gtrsim t^3$ and ${\sup }_{\tau \in [0,t]}\Vert \nabla \rho (\tau ){\Vert }_{L^{\infty }}\gtrsim t^2$ during the existence of a smooth solution. These growth results hold for a broad class of initial data, where we only require certain symmetry and sign conditions. As an application, we also construct solutions to the 3D axisymmetric Euler equation whose velocity has infinite-in-time growth.

arXiv comment26 pages, 3 figures

References in corpus (4)

• Potentially Singular Solutions of the 3D Incompressible Euler Equations
• Finite time blowup of 2D Boussinesq and 3D Euler equations with $C^{1,\alpha }$ velocity and boundary
• Infinite superlinear growth of the gradient for the two-dimensional Euler equation
• Global Regularity and Long-time Behavior of the Solutions to the 2D Boussinesq Equations without Diffusivity in a Bounded Domain

Cited by in corpus (2)

• Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains
• Finite-time singularity via multi-layer degenerate pendula for the 2D Boussinesq equation with uniform $C^{1,\sqrt{\frac{4}{3}}-1-\epsilon }\cap L^2$ force