Well-posedness of the equations of a viscoelastic fluid with a free boundary

In this article, we prove the local well-posedness, for arbitrary initial data with certain regularity assumptions, of the equations of a Viscoelastic Fluid of Johnson-Segalman type with a free surface. More general constitutive laws can be easily managed in the same way. The geometry is defined by a solid fixed bottom and an upper free boundary submitted to surface tension. The proof relies on a Lagrangian formulation. First we solve two intermediate problems through a fixed point using mainly G. Allain Appl. Math. Optim. 16 (1987) 37-50 for the Navier-Stokes part. Then we solve the whole Lagrangian problem on $[0,T_0]$ for $T_0$ small enough through a contraction mapping. Since the Lagrangian solution is smooth, we can come back to an Eulerian one.