Existence, uniqueness and stability of an inverse problem for two-dimensional convective Brinkman-Forchheimer equations with the integral overdetermination

TL;DR

This paper investigates an inverse problem for 2D convective Brinkman-Forchheimer equations, proving existence, uniqueness, and stability of solutions with integral overdetermination, extending mathematical understanding of these complex fluid models.

Contribution

It introduces a novel approach to prove existence, uniqueness, and stability for an inverse problem involving 2D CBF equations with integral overdetermination.

Findings

Existence of solutions established using fixed point theorem.

Uniqueness and Lipschitz stability proven for the inverse problem.

Results hold for r in [1,3], covering various nonlinear regimes.

Abstract

In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations: \begin{align*} \boldsymbol{u}_t-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{F}:=f \boldsymbol{g}, \ \ \ \nabla\cdot\boldsymbol{u}=0, \end{align*} in a bounded domain $\Omega\subset\mathbb{R}^2$ with smooth boundary $\partial\Omega$, where $\alpha,\beta,\mu>0$ and $r\in[1,3]$. The investigated inverse problem consists of reconstructing the vector-valued velocity function $\boldsymbol{u}$, the pressure field $p$ and the scalar function $f$. For the divergence free initial data $\boldsymbol{u}_0 \in \mathbb{L}^2(\Omega)$, we prove the existence of a solution to the inverse problem for two-dimensional CBF equations with the integral overdetermination condition, by showing the existence of a unique fixed point for an equivalent operator equation (using an extension of the contraction mapping theorem). Moreover, we establish the uniqueness and Lipschitz stability results of the solution to the inverse problem for 2D CBF equations with $r \in[1,3]$.