Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure

Bum Ja Jin; Kyungkeun Kang

doi:10.3934/dcds.2017207

Article Contents

2017, Volume 37, Issue 9: 4815-4834. Doi: 10.3934/dcds.2017207

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Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure

Bum Ja Jin^1, and
Kyungkeun Kang^2, ,

1.
Department of Mathematics, Mokpo National University, Mokpo, Republic of Korea
2.
Department of Mathematics, Yonsei University, Seoul, Republic of Korea

* Corresponding author: Kyungkeun Kang
* Corresponding author: Kyungkeun Kang

Received: May 31, 2016

Revised: March 31, 2017

Published: October 2017

Bum Ja Jin's work is supported by NRF-2014R1A1A3A04049515 and Kyungkeun Kang's work was partially supported by NRF-2014R1A2A1A11051161 and NRF20151009350

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Abstract

We prove a Caccioppoli type inequality for the solution of a parabolic system related to the nonlinear Stokes problem. Using the method of Caccioppoli type inequality, we also establish the existence of weak solutions satisfying a local energy inequality without pressure for the non-Newtonian Navier-Stokes equations.

Keywords:

Caccioppoli type inequality,
non-Newtonian Stokes and Navier-Stokes equations,
local energy inequality.

Mathematics Subject Classification: Primary: 76A05, 76D03; Secondary: 76D05, 76D07.

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