论文标题
二维对流Brinkman-Forchheimer方程的存在，唯一性和稳定性，具有整体性的过度确定性
Existence, uniqueness and stability of an inverse problem for two-dimensional convective Brinkman-Forchheimer equations with the integral overdetermination
论文作者
论文摘要
在本文中，我们研究了以下对流Brinkman-Forchheimer（CBF）方程的反问题： \ begin {align*} \ boldsymbol {u} _t-μδ\ boldsymbol {u}+（\ boldsymbol {\ boldSymbol {u} \ cdot \ nabla）\ boldsymbol {u}+α\α\ boldsymbol {u} p = \ boldsymbol {f}：= f \ boldsymbol {g}，\ \ \ \ \ \ \ \ \ \ cdot \ cdot \ boldsymbol {u} = 0， \ end {align*} 在有界域中$ω\ subset \ mathbb {r}^2 $，带有光滑边界$ \partialΩ$，其中$α，β，μ> 0 $和$ r \ in [1,3] $。调查的反问题包括重建矢量值速度函数$ \ boldsymbol {u} $，压力字段$ p $和标量函数$ f $。对于差异免费初始数据$ \ boldsymbol {u} _0 \ in \ mathbb {l}^2（ω）$，我们证明存在解决二维CBF方程的解决方案，该方程与二维CBF方程与整体构成过度确定的过度确定的固定点的存在，以表明对等值的固定点（使用等值序列的固定点）（使用等值序列），该方程（使用等值序列）。此外，我们确定了$ r \ in [1,3] $的2D CBF方程的逆问题解决方案的唯一性和Lipschitz稳定性结果。
In this article, we study an inverse problem for the following convective Brinkman-Forchheimer (CBF) equations: \begin{align*} \boldsymbol{u}_t-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{F}:=f \boldsymbol{g}, \ \ \ \nabla\cdot\boldsymbol{u}=0, \end{align*} in a bounded domain $Ω\subset\mathbb{R}^2$ with smooth boundary $\partialΩ$, where $α,β,μ>0$ 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 $\boldsymbol{u}_0 \in \mathbb{L}^2(Ω)$, 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 $r \in[1,3]$.
