ON THE ENERGY EQUALITY FOR WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS

N. V. Giang; D. Q. Khai; N. M. Tri

doi:10.17654/0974324322035

ON THE ENERGY EQUALITY FOR WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS

N. V. Giang Faculty of Basic Sciences, Thai Nguyen University of Technology, 666, 3/2 Street, Tich Luong, Thai Nguyen, Vietnam
D. Q. Khai Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Cau Giay, Hanoi, Vietnam
N. M. Tri Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Cau Giay, Hanoi, Vietnam

Article ID: 2564

DOI: https://doi.org/10.17654/0974324322035

Keywords: Navier-Stokes equations; energy equality; energy inequality; weak solution

Abstract

In this paper, we first introduce the concept of absolutely continuous functions of order $s(0<s \leq 1)$. Next, we prove the energy equality for weak solutions of the Navier-Stokes equations (NSE) in bounded three dimensional domains if and only if u is an absolutely continuous solution of order 1/2. Finally, we present a sufficient condition for the energy equality of weak solutions to NSE. Here, we prove that if $u \in L^2\left(0, T ; H^s\right) \cap L^4\left(0, T ; L^{\frac{12}{2 s+1}}\right)\left(1 \leq s<\frac{5}{2}\right)$, then the energy equality holds.

Published

2022-12-12

How to Cite

Giang, N. V., Khai, D. Q., & Tri, N. M. (2022). ON THE ENERGY EQUALITY FOR WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS. Advances in Differential Equations and Control Processes, 29, 101-115. https://doi.org/10.17654/0974324322035

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Vol. 29 (2022)

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Article

This work is licensed under a Creative Commons Attribution 4.0 International License.

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