On the solvability of a nonstationary problem describing the dynamics of an incompressible viscoelastic fluid

We study the local solvability of the Cauchy–Dirichlet problem for the system
\begin{gather*} (1-\varkappa\nabla ^2)\mathbf v_t=\nu\nabla^2\mathbf v-(\mathbf v\cdot\nabla)\mathbf v-\nabla p+\mathbf f(t), \\ 0=-\nabla(\nabla\cdot\mathbf v), \end{gather*}
which describes the dynamics of an incompressible viscoelastic Kelvin–Voigt fluid. The configuration space of the problem is described.