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For the two-phase incompressible Navier--Stokes equations with surfacetension, we derive an appropriate weak formulation incorporating a variationalformulation using divergence-free test functions. We prove a consistency resultto justify our definition and, under reasonable regularity assumptions, wereconstruct the pressure function from the weak formulation.

Pressure Reconstruction for Weak Solutions of the Two-Phase Incompressible Navier--Stokes Equations with Surface Tension