A study of the initialization problem for the Navier-Stokes equations (Q1892233)

Let \(v(x,t)\) be the velocity of liquid which occupies a bounded domain \(\Omega \subset\mathbb{R}^2\) with smooth boundary \(\Gamma\). The initial condition of \(v\) is not known. The author considers the problem of recovering the initial condition for the function \(v\) using a certain information of \(v\). More precisely, it is necessary to find the vector \(u_0\) such that the solution to the initial-boundary value problem \[ u_t - \nu \Delta u + (u \cdot \nabla) u + \nabla p = f,\;\nabla \cdot u = 0 \text{ in } \Omega \times (0,T) \quad u |_\Gamma =0,\;u(x,0) = u_0 (x) \] is closed to \(v\) in a certain sense. This problem is formulated as a control problem and the existence of the solution is proved.