Growth of curvature and perimeter of temperature patches in the 2D Boussinesq equations

TL;DR

This paper constructs an example of temperature patches in 2D Boussinesq equations where both curvature and perimeter grow infinitely over time, demonstrating significant geometric evolution.

Contribution

It provides the first explicit example showing unbounded growth of curvature and perimeter in temperature patches for the 2D Boussinesq system.

Findings

Curvature of patches grows to infinity over time.

Perimeter of patches also grows to infinity.

Growth rates are at least algebraic.

Abstract

In this paper, we construct an example of temperature patch solutions for the two-dimensional, incompressible Boussinesq system with kinematic viscosity such that both the curvature and perimeter grow to infinity over time. The presented example consists of two disjoint, simply connected patches. The rates of growth for both curvature and perimeter in this example are at least algebraic.