On echo chains in the linearized Boussinesq equations around traveling waves

TL;DR

This paper investigates the linearized 2D Boussinesq equations around traveling wave solutions, revealing echo chains and norm inflation phenomena despite viscosity, and constructs initial data leading to divergence in temperature and vorticity.

Contribution

It demonstrates the existence of echo chains and norm inflation in linearized Boussinesq equations around traveling waves, and constructs initial data with divergent temperature and vorticity.

Findings

Echo chains and norm inflation occur despite viscosity.

Initial data can cause temperature and vorticity to diverge.

Velocity converges even when temperature and vorticity diverge.

Abstract

We consider the 2D Boussinesq equations with viscous but without thermal dissipation and observe that in any neighborhood of Couette flow and hydrostatic balance (with respect to local norms) there are time-dependent traveling wave solutions of the form $\omega=-1+f(t)\cos(x-ty)$, $\theta=\alpha y + g(t)\sin(x-ty)$. As our main result we show that the linearized equations around these waves for $\alpha=0$ exhibit echo chains and norm inflation despite viscous dissipation of the velocity. Furthermore, we construct initial data in a critical Gevrey 3 class, for which temperature and vorticity diverge to infinity in Sobolev regularity as $t\rightarrow \infty$ but for which the velocity still converges.