Invertibility of a linearized Boussinesq flow: a symbolic approach

TL;DR

This work proves the invertibility of the linearized self-similar Boussinesq flow by showing that the operator \mathfrak{L} has a one-dimensional kernel and no generalized kernel. The analysis reduces to verifying a positivity property for the angular-average Υ(t) via a duality argument, transforming nonlocal angular effects into a Volterra integral equation with computable, positive kernels. A central contribution is the computer-assisted symbolic method that yields rigorous, exact bounds for high-degree polynomial expressions and for the Volterra kernel, enabling a complete positivity verification both for large times (t \geq \log(4)) and for short times through Picard iterations. The results solidify the invertibility needed for the surrounding self-similar blow-up analysis and demonstrate a novel, verifiable symbolic-computational approach to nonlocal PDE operators in fluid dynamics.

Abstract

We develop a computer-assisted symbolic method to show that a linearized Boussinesq flow in self-similar coordinates gives rise to an invertible operator.

Figures (2)

• Figure 1: Evolution of the lower bound for $\Upsilon(t)$ for $1.2 \leq t \leq 3$. Note that the lower bound (plotted in blue) becomes positive around time $t = 1.35$, and it stays positive afterwards, also when $t \geq 3$.
• Figure 2: Evolution of the lower bound for $P_3(t)$ for $0 \leq t \leq \log(4) + 0.5$. The lower bound for $P_3(t)$ is in blue, and the orange graph is the constant zero function. Note: the lower bound becomes negative after $t = 1.4$.

Theorems & Definitions (49)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• proof : Proof of Theorem \ref{['thm:inv']} assuming Theorem \ref{['thm:main']}
• Lemma 1.4: Reduction to angular averages
• proof : Proof of Lemma \ref{['lem:first']}
• Remark 1.5
• Theorem 1.6
• Lemma 2.1
• proof : Proof of Lemma \ref{['lem:weight']}