Local solutions for nonhomogeneous Navier-Stokes equations with large flux

2021/11/19 by Joanna Rencławowicz, Wojciech M. Zajączkowski, Rencławowicz, Joanna +1
Mathematics · Engineering · #Navier-Stokes equation solutions #Computational Fluid Dynamics and Aerodynamics #Gas Dynamics and Kinetic Theory

paper · pdf · doi:10.48550/arxiv.2111.10182

Abstract

The local existence of solutions to nonhomogeneous Navier-Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev-Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the x₃-axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that x₃-coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to W_σ ^2+s,1+s/2(Ω^t), the gradient of pressure to W_σ ^s,s/2(Ω^t) and the density to W_r,∞^1,1(Ω^t), where s∈(0,1), σ >3/s, r>5/s, r>σ .

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