On the local well-posedness of fractionally dissipated primitive equations with transport noise

TL;DR

This paper proves local well-posedness of three-dimensional fractionally dissipated primitive equations with transport noise, handling subcritical and critical dissipation regimes by developing new commutator estimates to overcome derivative loss and partial dissipation challenges.

Contribution

It establishes the first local existence and uniqueness results for these equations with transport noise in Sobolev spaces, including small data results in the critical case.

Findings

Local existence of solutions in Sobolev space $H^\sigma$ for $\sigma>3$

Unique pathwise solutions for subcritical and small initial data in critical case

Development of novel commutator estimates involving hydrostatic Leray projection

Abstract

We investigate the three-dimensional fractionally dissipated primitive equations with transport noise, focusing on subcritical and critical dissipation regimes characterized by $ (-\Delta)^{s/2} $ with $ s \in (1,2)$ and $s = 1$, respectively. For $\sigma>3$, we establish the local existence of unique pathwise solutions in Sobolev space $H^\sigma$. This result applies to arbitrary initial data in the subcritical case ($s \in(1,2)$), and to small initial data in the critical case ($s=1$). The analysis is particularly challenging due to the loss of horizontal derivatives in the nonlinear terms and the lack of full dissipation. To address these challenges, we develop novel commutator estimates involving the hydrostatic Leray projection.