ON THE ENERGY EQUALITY FOR WEAK SOLUTIONS OF THE NAVIER-STOKES EQUATIONS
Keywords:
Navier-Stokes equations, energy equality, energy inequality, weak solution.DOI:
https://doi.org/10.17654/0974324322035Abstract
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.
Received: October 11, 2022
Accepted: November 15, 2022
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