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Uniqueness of axisymmetric viscous flows originating from circular vortex filaments. (English. French summary) Zbl 1461.35182

Summary: The incompressible Navier-Stokes equations in \(\mathbb R^3\) are shown to admit a unique axisymmetric solution without swirl if the initial vorticity is a circular vortex filament with arbitrarily large circulation Reynolds number. The emphasis is on uniqueness, as existence has already been established in [H. Feng and the second author, Arch. Ration. Mech. Anal. 215, No. 1, 89–123 (2015; Zbl 1314.35098)]. The main difficulty which has to be overcome is that the nonlinear regime for such flows is outside of applicability of standard perturbation theory, even for short times. The solutions we consider are archetypal examples of viscous vortex rings, and can be thought of as axisymmetric analogs of the self-similar Lamb-Oseen vortices in two-dimensional flows. Our method provides the leading term in a fixed-viscosity short-time asymptotic expansion of the solution, and may in principle be extended so as to give a rigorous justification, in the axisymmetric situation, of higher-order formal asymptotic expansions that can be found in [A. J. Callegari and L. Ting, SIAM J. Appl. Math. 35, 148–175 (1978; Zbl 0395.76024)].

MSC:

35Q30 Navier-Stokes equations
35B07 Axially symmetric solutions to PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
76D05 Navier-Stokes equations for incompressible viscous fluids
76D17 Viscous vortex flows
35B40 Asymptotic behavior of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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