Cauchy problem for the Euler equations of a nonhomogeneous ideal incompressible fluid

Let us consider the Cauchy problem $$ {\rho_t + \upsilon \cdot \nabla\rho = 0 {\rho[\upsilon_t + (\upsilon \cdot \nabla)\upsilon] + \nabla p + \rho f {div \upsilon = 0 (1.1) {\rho$\mid$_t = 0 = \rho_0(x) {\upsilon$\mid$_t = 0 = \upsilon_0(x) $$ in $Q_T = R^3 \times [0,T]$, where $f(x,t), \rho_0(x) and \upsilon_0(x)$ are given, while the density $\rho(x,t)$, the velocity vector $\upsilon(x,t) = (\upsilon^1(x,t),\upsilon^2(x,t),\upsilon^3(x,t))$ and the pressure p(x,t) are unknowns. The equations $(1.1)_1 - (1.1)_3$ describe the motion of a nonhomogeneous ideal incompressible fluid.