Partial regularity results for solutions of the Navier–Stokes system

2011 ◽  
pp. 121-145 ◽  
Author(s):  
I. Kukavica
Keyword(s):  
Partial Regularity ◽  
Stokes System ◽  
Navier Stokes ◽  
Regularity Results

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

2020 ◽  
pp. 1-41
Author(s):  
Tania Biswas ◽  
Sheetal Dharmatti ◽  
Manil T. Mohan ◽  
Lakshmi Naga Mahendranath Perisetti
Keyword(s):  
Weak Solution ◽  
Stokes System ◽  
Two Dimensions ◽  
Immiscible Fluids ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Browder’S Theorem ◽  
Exponentially Stable ◽  
Mobility Parameter ◽  
Regularity Results

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.

scholarly journals Some preliminary observations on a defect Navier–Stokes system

2019 ◽  
Vol 347 (10) ◽  
pp. 677-684 ◽  
Author(s):  
Amit Acharya ◽  
Roger Fosdick
Keyword(s):  
Stokes System ◽  
Navier Stokes

scholarly journals Asymptotic analysis of an optimal control problem for a viscous incompressible fluid with Navier slip boundary conditions

2021 ◽  
pp. 1-21
Author(s):  
Claudia Gariboldi ◽  
Takéo Takahashi
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Boundary Conditions ◽  
Control Problem ◽  
Stokes System ◽  
Navier Stokes ◽  
Slip Boundary Conditions ◽  
Navier Slip ◽  
Slip Boundary ◽  
Navier Slip Boundary Conditions

We consider an optimal control problem for the Navier–Stokes system with Navier slip boundary conditions. We denote by α the friction coefficient and we analyze the asymptotic behavior of such a problem as α → ∞. More precisely, we prove that if we take an optimal control for each α, then there exists a sequence of optimal controls converging to an optimal control of the same optimal control problem for the Navier–Stokes system with the Dirichlet boundary condition. We also show the convergence of the corresponding direct and adjoint states.

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