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Existence of weak solutions for steady flows of electrorheological fluid with Navier-slip type boundary conditions

Cholmin Sin;  Sin-Il Ri — 2022

Mathematica Bohemica

We prove the existence of weak solutions for steady flows of electrorheological fluids with homogeneous Navier-slip type boundary conditions provided p ( x ) > 2 n / ( n + 2 ) . To prove this, we show Poincaré- and Korn-type inequalities, and then construct Lipschitz truncation functions preserving the zero normal component in variable exponent Sobolev spaces.

Properties of a quasi-uniformly monotone operator and its application to the electromagnetic $p$-$\text {curl}$ systems

Chang-Ho Song;  Yong-Gon Ri;  Cholmin Sin — 2022

Applications of Mathematics

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation $Au=b$. We prove that if $A$ is a quasi-uniformly monotone and hemi-continuous operator, then $A^{-1}$ is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence...

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