| Abstract: |
| We present some mathematical results on a PDE system describing the evolution of two-phase fluids in presence of inertial effects at the microscopic level. The model couples the Navier-Stokes equations for the macroscopic velocity $u$ with a variant of the Allen-Cahn equation for the phase parameter $\phi$ where inertial effects are taken into account. As a consequence, we face the presence of
of a second order {\sl material}\/ derivative of $\phi$ in the equation, which gives rise to severe mathematical difficulties. We discuss a number of analytical results for this model considering both the 2D and the 3D case. Due to the intrinsic difficulties of the problem, we can only show partial results whose validity may require simplifications of the equations or the presence of ad hoc regularizing terms. |
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