An inverse source problem for convective Brinkman-Forchheimer equations with the final overdetermination

TL;DR

This paper investigates an inverse problem for convective Brinkman-Forchheimer equations, establishing existence, uniqueness, and stability of solutions using Tikhonov fixed point theorem, with results applicable in 2D and 3D for various nonlinear growth conditions.

Contribution

It provides the first comprehensive analysis of inverse problems for CBF equations with final overdetermination, including existence, uniqueness, and stability results.

Findings

Existence of solutions for 2D and 3D CBF inverse problems.

Uniqueness and H"older stability proven using regularity results.

Better results for supercritical growth ($r>3$) than existing literature.

Abstract

In this paper, we examine an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations or damped Navier-Stokes equations: \begin{align*} \boldsymbol{v}_t-\mu \Delta\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+\alpha\boldsymbol{v}+\beta|\boldsymbol{v}|^{r-1}\boldsymbol{v}+\nabla p=\boldsymbol{F}:=\boldsymbol{f} g, \ \ \ \nabla\cdot\boldsymbol{v}=0, \end{align*} on a torus $\mathbb{T}^d$, $d=2,3$. The inverse problem under consideration consists of determining the vector-valued velocity function $\boldsymbol {v}$, the pressure gradient $\nabla p$ and the vector-valued forcing function $\boldsymbol{f} $. Using the Tikhonov fixed point theorem, we prove the existence of a solution for the inverse problem for 2D and 3D CBF equations with the final overdetermination data for the divergence free initial data in the energy space $ \mathbb{L}^2(\mathbb{T}^d)$. A concrete example is also provided to validate the obtained result. Moreover, we overcome the technical difficulties while proving the uniqueness and H\"older type stability results by using the regularity results available for the direct problem for CBF equations. The well-posedness results hold for $r \geq 1$ in two dimensions and for $r \geq 3$ in three dimensions for appropriate values of $\alpha,\mu$ and $\beta$. The nonlinear damping term $|\boldsymbol{v}|^{r-1}\boldsymbol{v}$ plays a crucial role in obtaining the required results. In the case of supercritical growth ($r>3$), we obtain better results than that are available in the literature for 2D Navier-Stokes equations.