On the Existence and Uniqueness of a Local Classical Solution of an Initial Boundary-Value Problem for Incompressible Nonhomogeneous Viscous Fluids

The existence of a unique local solution $\{ u,\nabla p,\rho ,\chi \} $ in the space of Hölder continuous functions of the initial boundary-value problem \[ \begin{gathered} \rho (x,t)\left\{ {\frac{{\partial u}}{{\partial t}} + u \cdot \nabla u} \right\} - \nabla \{ \chi (x,t)\nabla u\} + {\operatorname{grad}}p = \rho f,\quad \nabla \cdot u = 0\quad {\text{on}}\,G \times (0,T), \hfill \\ u(x,t) = 0\quad {\text{on}}\,\partial G \times (0,T),\quad u(x,0) = 0\quad {\text{on}}\,G, \hfill \\ \end{gathered} \] and of the initial-value problem \[ \frac{{\partial \rho }}{{\partial t}} + u \cdot {\operatorname{grad}}\rho = 0,\quad \rho (x,t) > 0\quad {\text{on}}\,G \times (0,T),\quad \rho (x,0) = \rho _0 (x)\quad {\text{on}}\,G\] with \[ \frac{{\partial \chi }}{{\partial t}} + u \cdot {\operatorname{grad }}\chi = 0,\quad \chi (x,t) > 0\quad {\text{on}}\,G \times (0,T),\quad \chi (x,0) = \chi _0 (x,\rho _0 (x))\] is shown. G is a bounded open subset of $R^3 $. The method of successive approximations and Lagrangian coordinates as developed by Solonnikov are used.