On the Conserved Quantities for the Weak Solutions of the Euler Equations and the Quasi-geostrophic Equations

Abstract

In this paper we obtain sufficient conditions on the regularity of the weak solutions to guarantee conservation of the energy and the helicity for the incompressible Euler equations. The regularity of the weak solutions are measured in terms of the Triebel-Lizorkin type of norms, \(\dot {\mathcal F}^s_{p,q}\) and the Besov norms, \(\dot {\mathcal B}^s_{p,q}\). In particular, in the Besov space case, our results refine the previous ones due to Constantin-E-Titi (energy) and the author of this paper (helicity), where the regularity is measured by a special class of the Besov space norm \(\dot {\mathcal B}^s_{p,\infty}= \dot {\mathcal N}^s_p\), which is the Nikolskii space. We also obtain a sufficient regularity condition for the conservation of the Lp-norm of the temperature function in the weak solutions of the quasi-geostrophic equation.