The energy equality of the Navier–Stokes equations in the framework of Besov-Lorentz type spaces, and its application to the MHD system

We find a new refined sufficient condition to establish the energy equality of the incompressible Navier–Stokes equations in the framework of the Besov–Lorentz type space. By virtue of the larger Besov–Lorentz type space than the usual Besov space, our result may include the previous result of Cheskidov–Luo (2020). As an application of our results on the Navier–Stokes equations, we obtain a new sufficient condition to establish the energy equality of the MHD system. Our result may also cover author’s previous result (2024) on the validity of the energy equality of the MHD system. Moreover, it should be noted that our sufficient condition of the magnetic field is strictly weaker than that of the Navier–Stokes equations.