# Maximum palinstrophy amplification in the two-dimensional Navier-Stokes equations

## TL;DR

This paper derives sharp bounds for the growth of palinstrophy in 2D Navier-Stokes solutions, constructs optimal initial fields, and investigates their evolution to understand potential singularity formation.

## Contribution

It introduces a family of optimal initial fields for maximizing palinstrophy growth and validates the sharpness of analytic bounds through numerical simulations.

## Key findings

- Optimal initial fields maximize palinstrophy growth.
- Numerical results confirm the sharpness of analytic bounds.
- Palinstrophy peaks scale with initial values, supporting theoretical estimates.

## Abstract

We derive and assess the sharpness of analytic upper bounds for the instantaneous growth rate and finite-time amplification of palinstrophy in solutions of the two-dimensional incompressible Navier-Stokes equations. A family of optimal solenoidal fields parametrized by initial values for the Reynolds number $\textrm{Re}$ and palinstrophy $\mathcal{P}$ which maximize $d\mathcal{P}/dt$ is constructed by numerically solving suitable optimization problems for a wide range of $\textrm{Re}$ and $\mathcal{P}$, providing numerical evidence for the sharpness of the analytic estimate $d\mathcal{P}/dt \leq \left(a + b\sqrt{\ln\textrm{Re}+c} \, \right) \mathcal{P}^{3/2}$ with respect to both $\textrm{Re}$ and $\mathcal{P}$. This family of instantaneously optimal fields is then used as initial data in fully resolved direct numerical simulations and the time evolution of different relevant norms is carefully monitored as the palinstrophy is transiently amplified before decaying. The peak values of the palinstrophy produced by these initial data, i.e., $\sup_{t > 0} \mathcal{P} (t)$, are observed to scale with the magnitude of the initial palinstrophy $\mathcal{P}(0)$ in accord with the corresponding $\textit{a priori}$ estimate. Implications of these findings for the question of finite-time singularity formation in the three-dimensional incompressible Navier-Stokes equation are discussed.