Title:

Small-time existence for a viscoelastic model with free bounday and a free surface.

Abstract :

We prove the small-time existence of the solution of a Johnson-Segalmann-type viscoelastic fluid for any initial data and a free boundary. We take into account the surface tension.

To that purpose, we split the Navier-Stokes part of the equations, which is solved in G. Allain Appl. Math. Optim. 16 (1987), no. 1, 37--50, and the purely viscoelastic constitutive equation. In the Lagrangian coordinates, the constitutive equation improves the time regularity. This enables to have continuity constants depending on the final time in a convenient way. Then we estimate all the error terms and then prove the result with a fixed point theorem.