| Abstract: |
| In this talk, we discuss a diffuse interface model that describes the dynamics of incompressible two-phase flows with chemotaxis effect. The PDE system couples the Navier-Stokes equations for the fluid velocity, a convective Cahn-Hilliard equation for the phase field variable with an advection-diffusion-reaction equation for the nutrient density. In the analysis, we consider a singular (e.g., logarithmic type) potential in the Cahn-Hilliard equation and prove the existence of global weak solutions in both two and three dimensions. In the two dimensional case, we establish a continuous dependence result that implies the uniqueness of global weak solutions. Furthermore, we prove the existence and uniqueness of global strong solutions that are strictly separated from the pure states over time in 2D. |
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