$p$-Euler equations and $p$-Navier-Stokes equations

摘要

We propose in this work new systems of equations which we call $p$-Eulerequations and $p$-Navier-Stokes equations. $p$-Euler equations are derived asthe Euler-Lagrange equations for the action represented by the Benamou-Breniercharacterization of Wasserstein-$p$ distances, with incompressibilityconstraint. $p$-Euler equations have similar structures with the usual Eulerequations but the `momentum' is the signed ($p-1$)-th power of the velocity. Inthe 2D case, the $p$-Euler equations have streamfunction-vorticity formulation,where the vorticity is given by the $p$-Laplacian of the streamfunction. Byadding diffusion presented by $\gamma$-Laplacian of the velocity, we obtainwhat we call $p$-Navier-Stokes equations. If $\gamma=p$, the {\it a priori}energy estimates for the velocity and momentum have dual symmetries. Usingthese energy estimates and a time-shift estimate, we show the global existenceof weak solutions for the $p$-Navier-Stokes equations in $\mathbb{R}^d$ for$\gamma=p$ and $p\ge d\ge 2$ through a compactness criterion.