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The limit of vanishing viscosity for the incompressible 3D Navier-Stokes equations with helical symmetry. (English) Zbl 1398.35148

Summary: In this paper, we are concerned with the vanishing viscosity problem for the three-dimensional Navier-Stokes equations with helical symmetry, in the whole space. We choose viscosity-dependent initial \(\mathbf{u}_0^\nu\) with helical swirl, an analogue of the swirl component of axisymmetric flow, of magnitude \(\mathcal{O}(\nu)\) in the \(L^2\) norm; we assume \(\mathbf{u}_0^\nu \rightarrow \mathbf{u}_0\) in \(H^1\). The new ingredient in our analysis is a decomposition of helical vector fields, through which we obtain the required estimates.

MSC:

35Q30 Navier-Stokes equations
35Q31 Euler equations
35D30 Weak solutions to PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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References:

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