On the stationary nonlocal Cahn–Hilliard–Navier–Stokes system: Existence, uniqueness and exponential stability

Article type: Research Article

Authors: Biswas, Tania^a | Dharmatti, Sheetal^{a; *} | Mohan, Manil T.^b | Perisetti, Lakshmi Naga Mahendranath^a

Affiliations: [a] School of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram (IISER-TVM), Maruthamala PO, Vithura, Thiruvananthapuram, Kerala, 695 551, India. E-mails: tania0003@gmail.com, sheetal@iisertvm.ac.in, plnmn915@iisertvm.ac.in | [b] Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, India. E-mails: maniltmohan@gmail.com, maniltmohan@ma.iitr.ac.in

Correspondence: [*] Corresponding author. E-mail: sheetal@iisertvm.ac.in.

Abstract: The Cahn–Hilliard–Navier–Stokes system describes the evolution of two isothermal, incompressible, immiscible fluids in a bounded domain. In this work, we consider the stationary nonlocal Cahn–Hilliard–Navier–Stokes system in two and three dimensions with singular potential. We prove the existence of a weak solution for the system using pseudo-monotonicity arguments and Browder’s theorem. Further, we establish the uniqueness and regularity results for the weak solution of the stationary nonlocal Cahn–Hilliard–Navier–Stokes system for constant mobility parameter and viscosity. Finally, in two dimensions, we establish that the stationary solution is exponentially stable (for convex singular potentials) under suitable conditions on mobility parameter and viscosity.

Keywords: Steady state nonlocal Cahn–Hilliard–Navier–Stokes systems, maximal monotone operator, pseudo-monotonicity, exponential stability

DOI: 10.3233/ASY-201650

Journal: Asymptotic Analysis, vol. 125, no. 1-2, pp. 59-99, 2021