Publication 13-CNA-017

Global existence for two extended Navier-Stokes systems

Mihaela Ignatova
Department of Mathematics
Stanford University
Stanford, CA 94305
mihaelai@stanford.edu

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

James P. Kelliher
Department of Mathematics
University of California, Riverside
Riverside, California
kelliher@math.ucr.edu

Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rpego@andrew.cmu.edu

Arghir Zarnescu
Department of Mathematics
University of Sussex
Pevensey III, Falmer, BN1 9QH, United Kingdom
A.Zarnescu@sussex.ac.uk

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 and Liu (J. Comp. Phys. 199 (2004) 221-259) and by Shirokoff and Rosales (J Comp. Phys 230 (2011) 8619-8646) when the initial data does not satisfy the required compatibility condition. Unlike the results of Iyer et al (J. Math. Phys. 53 (2012) 115605), 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.

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