Abstract
We study vanishing viscosity solutions to the axisymmetric Euler equations without swirl with (relative) vorticity in L^p with p>1. We show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions. Moreover, we show that the kinetic energy is preserved provided that p>3/2 and the vorticity is nonnegative and has finite second moments.
Highlights
Our aim is to study renormalization and energy conservation of solutions to the Euler equations that are obtained as vanishing viscosity solutions from the axisymmetric Navier–Stokes equations
The analogous studies for the two-dimensional planar equations have been conducted quite recently: As long as the vorticity is L p integrable with exponent p ≥ 2, DiPerna and Lions’s theory for transport equations ensures that the vorticity is a renormalized solution of the corresponding vorticity equation [39]
Whenever the fluid has locally finite kinetic energy, which will be the case in the regularity framework considered in this paper, the Euler equations can be interpreted in the sense of distributions
Summary
Theorem 1 (Compactness and convergence to Euler) Let uν be the unique solution to the Navier–Stokes equations (10), with initial datum u0. Such that the associated relative vorticity ξ0 belongs to L1 ∩ L p(R3) for some p > 1. This is in principle not required by our method of proving Theorem 3, and any estimate of the form r 2ω(t) L1(H) r 2ω0 L1(H) would be sufficient It is, not clear to us whether such an estimate holds true under our integrability assumptions apart from the special case considered in Lemma 8, that is, for nonnegative (or nonpositive) vorticity fields. This work, contains an appendix in which a helpful interpolation estimate is provided
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