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

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.

Property $$(UW{\scriptstyle \Pi })$$ under perturbations

Property $$(UW {\scriptstyle \Pi })$$(UWΠ), introduced in Berkani and Kachad (Bull Korean Math Soc 49:1027–1040, 2015) and studied more recently in Aiena and Kachad (Acta Sci Math (Szeged) 84:555–571, 2018) may be thought as a variant of Browder’s theorem, or Weyl’s theorem, for bounded linear operators acting on Banach spaces. In this article we study the stability of this property under some commuting perturbations, as quasi-nilpotent perturbation and, more in general, under Riesz commuting perturbations. We also study the transmission of property $$(UW {\scriptstyle \Pi })$$(UWΠ) from T to f(T), where f is an analytic function defined on a neighborhood of the spectrum of T. Furthermore, it is shown that this property is transferred from a Drazin invertible operator T to its Drazin inverse S.

A new characterization of generalized Browder’s theorem and a cline’s formula for generalized Drazin-eromorphic inverses

In this paper we give a new characterization of generalized Browder?s theorem by considering equality between the generalized Drazin-meromorphic Weyl spectrum and the generalized Drazinmeromorphic spectrum. Also, wegeneralize Cline?s formula to the case of generalized Drazin-meromorphic invertibility under the assumption that AkBkAk = Ak+1 for some positive integer k.

A new characterization of Browder’s theorem

We give a new characterization of Browder?s theorem using spectra originated from Drazin-Fredholm theory.

Property (gab) through localized SVEP

In this article we study the property (gab) for a bounded linear operator T 2 L(X) on a Banach space X which is a stronger variant of Browder’s theorem. We shall give several characterizations of property (gab). These characterizations are obtained by using typical tools from local spectral theory. We also show that property (gab) holds for large classes of operators and prove the stability of property (gab) under some commuting perturbations.

A remark on a theorem of Browder

This work deals with a Browder type theorem, and some of its consequences.We consider hX, Y i a dual pair of real normed spaces, C a weakly closed convex subset of X containing 0X, and L a function from C into Y which is monotone, weakly continuous on the line segments in C, and coercive. In the article ,,Nonlinear monotone operators and convex sets in Banach spaces”, Bull. Amer. Math. Soc., 71 (1965), F. E. Browder proved the existence of solutions for variational inequalities with such an operator L provided that X = E is a reflexive Banach space, and Y = E0 is its dual space. It is the object of this note to remark that a similar result is valid when Y = E is a Banach space (not necessary reflexive) and X = E0 (for example in the case of the Lebesgue spaces E = L1 (T), and E0 = L∞(T)). Moreover we shall show that the Browder’s theorem is a consequence of this result, and we shall also prove a Stampacchia type theorem.