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  1. This paper investigates the global existence of weak solutions for the incompressible \begin{document}$ p $\end{document}-Navier-Stokes equations in \begin{document}$ \mathbb{R}^d $\end{document} \begin{document}$ (2\leq d\leq p) $\end{document}. The \begin{document}$ p $\end{document}-Navier-Stokes equations are obtained by adding viscosity term to the \begin{document}$ p $\end{document}-Euler equations. The diffusion added is represented by the \begin{document}$ p $\end{document}-Laplacian of velocity and the \begin{document}$ p $\end{document}-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-\begin{document}$ p $\end{document} distances with constraint density to be characteristic functions.

     
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