A model is derived for the coupling of transient free surface and pressurized flows. The resulting system of equations is written under a conservative form with discontinuous gradient of pressure. We treat the transition point between the two types of flows as a free boundary associated to a discontinuity of the gradient of pressure. The numerical simulation is performed by making use of a Roe-like finite volume scheme that we adapted to such discontinuities in the flux. The validation is performed by comparison with experimental results.

We obtain new fine properties of entropy solutions to scalar nonlinear conservation laws. For this purpose, we study the "fractional BV spaces" denoted by BV s (R) (for 0 < s ≤ 1), which were introduced by Love and Young in 1937 and closely related to the critical Sobolev space W s,1/s (R). We investigate these spaces in connection with one-dimensional scalar conservation laws. The BV s spaces allow one to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with BV s initial data. Furthermore, for the first time, we get the maximal W s,p smoothing effect conjectured by Lions, Perthame and Tadmor for all nonlinear (possibly degenerate) convex fluxes.

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International audienceThis paper deals with a system of two equations which describes heatless adsoption of a gaseous mixture with two species. When one of the components is inert, we obtain an existence result of a weak solution satisfying some entropy condition under some simplifying assumptions. The proposed method makes use of a Godunov-type scheme. Uniqueness is proved in the class of piecewise C^1 functions