Remarks on the Spatial Asymptotic Behavior of Solutions to a 1D Model of Equatorial Oceanic Flows

TL;DR

This paper studies the long-distance behavior of solutions to a new 1D nonlocal nonlinear model for equatorial ocean flows, revealing the influence of Coriolis effects on decay rates and their optimality.

Contribution

It analyzes the spatial asymptotics of solutions to a recently derived nonlocal model, highlighting the impact of Coriolis effects and establishing decay rate optimality.

Findings

Solutions decay at the rate 1/|x| even for rapidly decaying initial data

The decay rate of 1/|x| is shown to be optimal

Coriolis effects significantly influence the asymptotic behavior

Abstract

We consider a new nonlocal and nonlinear one-dimensional evolution model arising in the study of oceanic flows in equatorial regions, recently derived in [A. Constantin and L. Molinet, Global Existence and Finite-Time Blow-Up for a Nonlinear Nonlocal Evolution Equation, Commun. Math. Phys. 402 (2023), 3233-3252]. We investigate the spatial asymptotic behavior of its solutions. In particular, we observe the influence of the Coriolis effect, which, even for rapidly decaying initial data, yields solutions that decay at the rate $1 / |x|$. Thereafter, we shed light on the optimality of this decay rate.