Communications in Mathematical Sciences

Volume 13 (2015)

Number 1

Global existence for two extended Navier-Stokes systems

Pages: 249 – 267

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n1.a12

Authors

Mihaela Ignatova (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

Gautam Iyer (Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

James P. Kelliher (Department of Mathematics, University of California at Riverside)

Robert L. Pego (Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A.)

Arghir D. Zarnescu (Department of Mathematics, University of Sussex, Brighton, United Kingdom)

Abstract

We prove global existence of weak solutions to two systems of equations which extend the dynamics of the Navier-Stokes equations for incompressible viscous flow with no-slip boundary condition. The systems of equations we consider arise as formal limits of time discrete pressure-Poisson schemes introduced by Johnston & Liu [J. Comput. Phys. 199, 221–259, 2004] and by Shirokoff & Rosales [J. Comput. Phys. 230, 8619–8646, 2011] when the initial data does not satisfy the required compatibility condition. Unlike the results of Iyer et al. [J. Math. Phys. 53, 115605, 2012], our approach proves existence of weak solutions in domains with less than $C^1$ regularity. Our approach also addresses uniqueness in 2D and higher regularity.

Keywords

Navier-Stokes, numerics, global well-posedness

2010 Mathematics Subject Classification

35Q30, 65M06, 76D05, 76M25

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Published 16 July 2014