  In this paper, we establish some local and global solutions for the two phase
incompressible inhomogeneous flows with moving interfaces in $L_p-L_q$ maximal
regularity class. Compared with previous results obtained by V.A.Solonnikov and
by Y.Shibata \& S.Shimizu, we find the local solutions in $L_p-L_q$ class in
some general uniform $W^{2-1\slash r}_r$ domain in $\BBR^N$ by assuming $(p,
q)\in ]2,\infty[\times ]N,\infty[$ or $(p, q)\in ]1,2[ \times ]N,\infty[$
satisfying $1 \slash p + N \slash q>3\slash 2.$ In particular, less regular
initial data are allowed by assuming $p<2.$ In addition, if the density and the
viscosity coefficient are piecewise constant, we can construct the long time
solution from the small initial states in the case of the bounded droplet. This
is due to some decay property for the corresponding linearized problem.
