Data Assimilation to the Primitive Equations with $L^p$-$L^q$-based Maximal Regularity Approach

TL;DR

This paper provides a rigorous mathematical framework for data assimilation in primitive equations using $L^p$-$L^q$ maximal regularity, demonstrating exponential convergence of approximate solutions to true solutions in a specific Besov space.

Contribution

It introduces a novel $L^p$-$L^q$ maximal regularity approach for data assimilation in primitive equations and proves exponential convergence in a Besov space setting.

Findings

Approximate solutions converge exponentially to true solutions.

The convergence is established in the Besov space $B^{2/q}_{q,p}()$.

The approach is justified mathematically in a periodic layer domain.

Abstract

In this paper, we show mathematical justification of the data assimilation of nudging type in $L^p$-$L^q$ maximal regularity settings. We prove that the approximate solution of the primitive equations by data assimilation converges to the true solution with exponential order on the Besov space $B^{2/q}_{q,p}(\Omega)$ in the periodic layer domain $\Omega = \mathbb{T}^2 \times (-h, 0)$.