pEuler equations and pNavierStokes equations
Abstract
We propose in this work new systems of equations which we call pEuler equations and pNavierStokes equations. pEuler equations are derived as the EulerLagrange equations for the action represented by the BenamouBrenier characterization of Wassersteinp distances, with incompressibility constraint. pEuler equations have similar structures with the usual Euler equations but the 'momentum' is the signed (p  1)th power of the velocity. In the 2D case, the pEuler equations have streamfunctionvorticity formulation, where the vorticity is given by the pLaplacian of the streamfunction. By adding diffusion presented by γLaplacian of the velocity, we obtain what we call pNavierStokes equations. If γ = p, the a priori energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a timeshift estimate, we show the global existence of weak solutions for the pNavierStokes equations in R^{d} for γ = p and p ≥ d ≥ 2 through a compactness criterion.
 Publication:

Journal of Differential Equations
 Pub Date:
 April 2018
 DOI:
 10.1016/j.jde.2017.12.023
 arXiv:
 arXiv:1706.05693
 Bibcode:
 2018JDE...264.4707L
 Keywords:

 Wassersteinp geodesics;
 BenamouBrenier functional;
 pmomentum;
 pLaplacian;
 Global weak solutions;
 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 49K30;
 35Q35;
 76D03