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| We are concerned with the Cauchy problem for models describing the evolution of nonhomogeneous fluids in the whole space or with periodic boundary conditions. Both the compressible and the incompressible cases are considered. \cr
In Eulerian coordinates, the corresponding systems are of mixed hyperbolic/parabolic type so that uniqueness conditions are stronger than existence conditions.
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We here show that it is no longer the case if those systems are reformulated in Lagrangian coordinates. As a matter of fact, it turns out to be possible to solve the Cauchy problem with critical regularity data, by means of the standard Banach fixed point theorem. As a by-product, in this framework, the flow is Lipschitz continuous. |
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