Variational and Quasi--variational Inequalities of Navier--Stokes Type with Velocity Constraints

TL;DR

This paper develops a weak variational formulation for Navier-Stokes type inequalities with velocity constraints, including gradient constraints, and applies it to Stefan/Navier-Stokes problems involving phase change and material freezing effects.

Contribution

It introduces a novel weak variational formulation for Navier-Stokes variational inequalities and applies it to complex Stefan/Navier-Stokes problems with phase change effects.

Findings

Established existence of solutions using compactness theorem

Formulated quasi-variational inequalities for phase change problems

Applied the framework to fluid-structure interaction scenarios

Abstract

In this paper we deal with parabolic variational inequalities of Navier-Stokes type with time-dependent constraints on velocity fields, including gradient constraint case. One of the objectives of this paper is to propose a weak variational formulation for variational inequalities of Navier-Stokes type and to solve them by applying the compactness theorem, which was recently developed by the authors (cf. [22]). Another objective is to approach to a class of quasi-variational inequalities associated with Stefan/Navier-Stokes problems in which we are taking into account the freezing effect of materials in fluids. As is easily understood, the phase change from liquid into solid gives a great influence to the velocity field in the fluid. For instance, in the mushy region, the velocity of the fluid is constrained by some obstacle caused by moving solid. We shall challenge to the mathematical modeling of Stefan/Navier-Stokes problem as a quasi-variational inequality and solve it as an application of parabolic variational inequalities of Navier-Stokes type.