Inviscid limit for axisymmetric flows without swirl in a critical Besov space

Abstract

In this paper, we study the inviscid limit for the 3-D axisymmetric incompressible fluid flows without swirl and prove the convergence rate. We will also prove the uniform persistence of the initial regularity for 3-D axisymmetric Navier-Stokes equations in a critical Besov space. (2000). 35Q30 · 76D05 · 76C05. Mathematics Subject Classification

Local well-posedness of the incompressible Euler equations in B∞,11 and the inviscid limit of the Navier–Stokes equations

“…The two dimensional results were further generalized to other Besov spaces B 2/p+1 p,1 with convergence in L p , see section 3.4 in [11]. In three dimension a similar result was proved in [17] for axis-symmetric flows without swirl. By interpolation with the uniform estimates, one can get the convergence in all intermediate spaces.…”

Renormalization and energy conservation for axisymmetric fluid flows

“…Indeed, the statement in (49) immediately follows from (50) and Hölder's inquality. Let a and b be Hölder dual exponents given by a = 4(1−λ) q = 7 p−4 7 p−6 and b = 4(1−λ)…”

“…The novelty in the above result is the kinetic energy may be unbounded. For earlier and related convergence results for non-classical solutions, we refer to [1,31,35,47,49] and references therein. Our next statement concerns the renormalization property of the relative vorticity.…”

A remark on the Lipschitz estimates of solutions to Navier-Stokes equations

“…In [13,14], the estimate in the Lipschitz norm of the velocity is of double exponential-type. In this paper, we will prove that the estimate is of polynomial-type although it depends on the viscosity.…”

“…2,1 (R 3 ) have been studied independently in [13,14] using different methods. In [13,14], the estimate in the Lipschitz norm of the velocity is of double exponential-type.…”