# Inverse Problems for Navier-Stokes Systems

Alexander Chebotarev^{*}

Inverse problems for for fluid dynamics equations are related to the determination of hydrodynamic fields as well as external conditions specifying a flow, and their statement requires additional information on the solution of the Navier--Stokes equations. Consider the abstract formulations of inverse problems for evolution and stationary Navier--Stokes systems describing the dynamics of a viscous incompressible fluid in which a finite number of moments of the solution (i.e., values of certain functionals on the solution) are used as additional data. To state the inverse problems, we consider real Hilbert spaces $V$ and $H$ such that $V\subset H\subset V'$, the embedding $V\subset H$ is continuous and compact, and V is dense in H. Here $V'$ is the dual space of $V$. By $\|\cdot\|$ and $|\cdot |$ we denote the norms on $V$ and $H$, respectively; by $(f, v)$ we denote the value of a functional $f\in V'$ on an element $v \in V$, which coincides with their inner product in $H$ if $f\in H$. Let $A=A_0+A_1$, $A_0:V\to V'$, $A_1:H\to H$ be linear continuous operators such that $$ (A_0y,y)\geq\rho\| y\|^2,\; (A_0y,z)=(A_0z,y), \;(A_1y,z)\leq \beta |y|\cdot |z|,\; \; \rho>0,\beta>0, $$ and, respectively, let $B:V\times V\to V'$ be a bilinear continuous operator such that $(B(y,z),z)=0$; $B[y]=B(y,y)$. Consider a linearly independent system of functionals $\{ Q_1,Q_2,...Q_m\}$ in $V'$. The following problems for the equations with operator coefficients provides a model for inverse problems for the MHD, Navier--Stokes equations and thermal convection of viscous incompressible fluid. $\textbf{Evolution Problem.}$ Find the functions $\alpha_j\in L^1(0,T),\, j=1,2,...,m$ and function $y\in L^2(0,T;V)$, satisfying the conditions: $y'=dy/dt\in L^1(0,T;V')$, $$ y'+Ay+B[y]=f(t)+\sum_1^m\alpha_k(t)Q_k,\;(Q_j,y(t))=q_j(t),\; 1\leq j\leq m,\; t\in (0,T),\; y(0)=y_0. $$ Here, $y_0\in H$ and the functions $f\in L^2(0,T;V')$, $q_j\in L^4(0,T)$ are given. $\textbf{Stationary Problem.}$ Find $\alpha=(\alpha_1,\alpha_2,...,\alpha_m)\in \mathbb{R}^m$, $y\in V$, satisfying the conditions: $$ Ay+B[y]=f_d+\sum_1^m\alpha_kQ_k,\;\; (Q_j,y)=q_j,\; j=1,2,...,m. $$ Here, vector $q=(q_1,q_2,...,q_m)\in \mathbb{R}^m$ and $f_d\in V'$ are given. The result concerning the nonlocal solvability of Evolution Problem, even though its proof is relatively simple, makes it possible to prove the existence of solutions of various inverse problems including statements with unknown boundary conditions. By way of example, a problem for the Boussinesq equations with unknown values of thermal flows across the boundary is considered. Also possible to prove the nonlocal unique solvability of the inverse problem for the three--dimensional thermal convection equations with unknown density of the heat sources in the case when the dimension $m$ of the space of observations is sufficiently large and the initial data are close to the corresponding stationary fields. The sufficient conditions for the solvability of Stationary Problem are established. It is shown that the solution set is homeomorphic to the finite--dimensional compact. The stationary boundary inverse problem for three--dimensional thermal convection equations and the inverse problem of magnetohydrodynamics are considered as an application of the theoretical analysis.

Mathematics Subject Classification: 35Q30 35R30

Keywords: Navierâ€“Stokes equations; thermal convection equations; inverse problems; MHD equations; nonlocal existence and uniqueness theorems

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