Large, global solutions to the three-dimensional the navier-stokes equations without vertical viscosity

2022/02/23 by Isabelle Gallagher, Gallagher, Isabelle, Alexandre Yotopoulos +1
Mathematics · Engineering · #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows #Computational Fluid Dynamics and Aerodynamics

paper · pdf · doi:10.48550/arxiv.2202.11536

Abstract

The three-dimensional, homogeneous, incompressible Navier-Stokes equations are studied in the absence of viscosity in one direction. It is shown that there are arbitrarily large initial data generating a unique global solution, the main feature of which is that they are slowly varying in the direction where viscosity is missing. The difficulty arises from the complete absence of a regularising effect in this direction. The special structure of the nonlinear term, joint with the divergence-free condition on the velocity field, is crucial in obtaining the result.

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