# American Institute of Mathematical Sciences

September  2017, 37(9): 4815-4834. doi: 10.3934/dcds.2017207

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

 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

Received  June 2016 Revised  April 2017 Published  June 2017

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

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.

Citation: Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
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