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

Jaemin Park

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.

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

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.
Paper Structure (10 sections, 6 theorems, 62 equations, 1 figure)

This paper contains 10 sections, 6 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction and the main results
  2. Overview of long-time behavior in the Boussinesq equations
  3. Global well-poseness
  4. Long-time behavior of classical solutions
  5. Temperature patch problem
  6. Main results
  7. prelimineries
  8. Uniform in time estimates
  9. Lemmas for curvature and perimeter
  10. Proof of the main theorem

Key Result

Theorem 1.1

gancedo2017global Let $N\in\mathbb{N}$. For $i=1,\ldots,N$, let us pick real numbers $a_i\in \mathbb{R}$ and simply connected bounded domains $D_i\subset \mathbb{R}^2$ such that $\overline{D}_i$ are disjoint and $\partial D_i \in C^{2+\gamma}$. Let us consider initial data $(\rho_0,u_0)$ such that $ where $X_t$ is the flow map generated by the velocity $u$. Lastly, we have persistence of the curva

Figures (1)

  • Figure 1: Illustration of the initial patch $\rho_0$

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • Remark 3.2
  • proof
  • Lemma 3.3
  • Remark 3.4
  • Lemma 4.1
  • Remark 4.2
  • proof
  • ...and 5 more