Variational Limit of Compressible to Incompressible Fluid

Abstract

We consider the flow with (or without) wake of a stationnary, irrotational, compressible fluid, with non-constant density, in a channel (or in the whole plane), with given velocity at infinity and at the wake.

Considering a sequence of densities ρ _n and letting ρ _n → 1, when n → ∞, we are going to prove that the variational solution u _n of the compressible problem converges (in a sense that will be defined) to the variational solution of the incompressible problem.

Although the linear variational inequalities corresponding to density ρ _n ≠ 1 have non-constant coefficients, it is still possible to show that the free boundaries are graphs of Lipschitz functions. Calling l _n and l the graphs of the free boundaries of problems corresponding to densities ρ _n and 1 respectively, we prove that l _n → l in a convenient space, when n → ∞.

We also obtain a result of continuous dependence on the data, of the solutions.