PDE-Seminar
Wednesday, 24 th April 2024, 16:30 am-17:30 pm
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Venue: Ramanujan hall, Department of Mathematics, IIT Bombay
Host: Debanjana Mitra
Speaker: Dr. Dharmatti Sheetal
Affiliation: Department of Mathematics, IISER Thiruvananthapuram
Title: Cahn-Hilliard-Navier-Stokes equations with Nonhomogeneous Boundary: Existence, Uniqueness, Regularity and Optimal Control
Abstract: The evolution of two isothermal, incompressible, immiscible fluids in a bounded domain is governed by Cahn-Hilliard-Navier-Stokes equations (CHNS System). In this work we study the well-posedness results for CHNS systems with nonhomogeneous boundary conditions for the velocity equation. We obtain the existence of global weak solutions in the two dimensional bounded domain using semi Galerkin approximation. We further prove the continuous dependence of the solution on initial conditions and boundary data that will provide the uniqueness of the weak solution. The existence of strong solutions is also established in this work. Furthermore, we study optimal boundary control using the continuous dependance of strong solution. Using Pontryagin's maximum principle we show that the optimal control is characterised as a unique solution of the appropriate adjoint system.