Energy estimates for the Euler equations

Develop an alternative argument that yields an energy estimate similar to the new Navier–Stokes energy estimate for the Euler equations, despite the absence of viscosity for pressure control.

Background

The paper’s principal energy estimate for the three-dimensional homogeneous Navier–Stokes system relies crucially on viscosity to control the pressure term. The authors note that analogous computations may apply in broader settings, but specifically leave unresolved whether a comparable estimate can be obtained for the inviscid Euler equations. Such an estimate could potentially provide regularity or uniqueness information without relying on viscous smoothing.