Generalized solutions to a free boundary problem of motion of a non-newtonian fluid

We consider a problem of the motion of fluids that occupy open sets $\omega_i(t)\subset\mathbf{R}^2$ separated by a compact manifold $\Gamma(t)=\mathbf{R}^2\setminus(\omega_0(t)\cup\omega_1(t))$. The problem is to determine the manifold $\Gamma(t)$ and the solenoidal velocity field $v(x,t)$ so as to satisfy the equations
\begin{gather*} v_t+v\nabla v-\operatorname{div}(b_i(|D(v)|)D(v))+\nabla p=0, \\ [v]=0, \quad [P]\cdot n+kn=0, \\ v(x,0)=v_0(x), \quad \Gamma(0)=\Gamma_0. \end{gather*}
Here $k$ is the curvature of the interface, and $P$ and $D$ are the tensors of tensions and deformation velocities. The functions $b_i$ meet the conditions $c^{-1}s^{p-2}\le b_i(s)\le cs^{p-2}$ and $(sb_i)'\ge0$, $p>2$. Some definition of a generalized solution is given. We prove that the problem has at least one such solution.