Remarks on the Spatial Asymptotic Behavior of Solutions to a 1D Model of Equatorial Oceanic Flows
Manuel Fernando Cortez, Oscar Jarrin
TL;DR
This work analyzes a new 1D nonlocal nonlinear model for equatorial oceanic flows, focusing on the spatial asymptotics of its solutions. By deriving a detailed kernel structure for the linear part and coupling it with a carefully constructed nonlinear analysis, the authors establish a local well-posedness theory in $H^s(\mathbb{R})$ for $s>\tfrac{3}{2}$ and obtain a sharp pointwise decay bound $|u(t,x)|+|\mathcal{H}u(t,x)| \le \frac{C}{t^{1/2}(1+|x|)^{\min(1,\gamma)}}$, revealing that the Coriolis term $\beta \mathcal{H}u$ enforces a long-range tail $\sim 1/|x|$. They further prove an optimality result: if $\gamma>1$ and the initial mass $M(u_0)\neq 0$, the decay cannot be faster than $1/|x|$, and, under zero-mean conditions, the linear part induces a precise asymptotic profile $u(t,x) \sim x^{-1}\,\Phi(M(u_0),t,u)$. The paper also develops a blow-up criterion tied to $\int_0^{T_*}\|\partial_x u\|_{L^{\infty}}\,dt$, and discusses global-in-time questions on the real line, highlighting open problems and partial results. Overall, the study advances understanding of how nonlocal nonlinearities and Coriolis forcing shape spatial decay and long-time behavior in geophysical models.
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.