Energy equality and inviscid limit of the fractional Navier–Stokes equations

Abstract We find a new criterion for the validity of the energy equality of the 3D fractional Navier–Stokes equations in the framework of the Lorentz–Besov spaces. Note that our sufficient condition is strictly weaker than that of Cheskidov et al. (Nonlinearity 21:1233–1252, 2008) related to the largest class $$L^3(0,T;B^{1/3}_{3,\infty })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>;</mml:mo> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for the validity of the energy conservation law of the Euler equations. Moreover, taking the inviscid limit of the fractional Navier–Stokes equations, we obtain the energy conservation law of the Euler equations in the framework of the same Lorentz–Besov spaces. Our result covers the recent work of Cheskidov and Luo (Nonlinearity 33:1388–1403, 2020) for the Navier–Stokes equations. Furthermore, we mention the relation between our new criterion and the Onsager conjecture.