Navier-Stokes equations with variable viscosity in variable exponent spaces of Clifford-valued functions

Rui Niu,Binlin Zhang,Hongtao Zheng

doi:10.1186/s13661-015-0291-y

Abstract

In this paper we study the stationary generalized Navier-Stokes equations when the viscosity is not only a constant but also a function which depends on the position and the shear-velocity. For this we establish an improved decomposition of variable exponent Lebesgue spaces of Clifford-valued functions. Using this decomposition together with Clifford operator calculus, we obtain the existence, uniqueness and representation of solutions for the generalized Stokes equations and the generalized Navier-Stokes equations with variable viscosity in the setting of variable exponent spaces of Clifford-valued functions. Furthermore, the equivalences of solutions and weak solutions for the aforementioned equations are justified.

Highlights

In this paper we are concerned with the stationary generalized Navier-Stokes equations:– div(A∇u) + ρ(u · ∇)u + ∇q = ρf in, ( . ) div u = in,u = on ∂, where the operator A is defined by Au = au with a(x, u) : × Rn → R+ and a ∈ C∞( × Rn) and ⊂ Rn (n ≥ ) is a bounded domain with sufficiently smooth boundary ∂, u is the velocity, q the hydrostatic pressure, ρ the density, f the vector of the external forces
As an active branch of mathematics over the past years, Clifford analysis usually studies the solutions of the Dirac equation for functions defined on domains in Euclidean space and taking value in Clifford algebras; see, for example, [ ]
In Section, using similar methods to [ ], we prove the existence and uniqueness of solutions to the generalized Navier-Stokes equations in W,p(x)(, C n) × Lp(x)(, R) under certain hypotheses

Summary

Introduction

As an active branch of mathematics over the past years, Clifford analysis usually studies the solutions of the Dirac equation for functions defined on domains in Euclidean space and taking value in Clifford algebras; see, for example, [ ]. Based on the above consideration, we should study the generalized Navier-Stokes equations in variable exponent spaces of Clifford-valued functions. Fu and Zhang in [ , ] were interested in the existence of weak solutions for the general nonlinear A-Dirac equations with variable growth.

Results

Conclusion