On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion

The local-in-time existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion is proved. We show the existence of solutions with lowest possible regularity for this problem such that ${\displaystyle u\in W^{2,1}\_r\left(w{\widetilde{\{mit\Omega \}}}^T\right)}$ with r>3. The existence is proved by the method of successive approximations where the solvability of the Cauchy-Neumann problem for the Stokes system is applied. We have to underline that in the ${\displaystyle L_p}$-approach the Lagrangian coordinates must be used. We are looking for solutions with lowest possible regularity because this simplifies the proof and decreases the number of compatibility conditions.