Existence of global weak solutions of $ p $-Navier-Stokes equations

Jian-Guo Liu; Zhaoyun Zhang

doi:10.3934/dcdsb.2021051

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2022, Volume 27, Issue 1: 469-486. Doi: 10.3934/dcdsb.2021051

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Existence of global weak solutions of $ p $-Navier-Stokes equations

Jian-Guo Liu^1, and
Zhaoyun Zhang^2, ,

1.
Department of Mathematics and Department of Physics, Duke University, Durham, NC 27708, USA
2.
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, China, and, Department of Physics, Duke University, Durham, NC 27708, USA

^* Corresponding author: Zhaoyun Zhang
^* Corresponding author: Zhaoyun Zhang

Received: November 2020

Revised: January 2021

Early access: February 2021

Published: January 2022

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Abstract

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

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Mathematics Subject Classification: Primary: 35K92, 76D03, 47J30.

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