Inviscid Limit of Vorticity Distributions in the Yudovich Class

We prove that given initial data ω0∈L∞T2, forcing g∈L∞0,T;L∞T2, and any T > 0, the solutions uν of Navier‐Stokes converge strongly in L∞0,T;W1,pT2 for any p ∈ [1, ∞) to the unique Yudovich weak solution u of the Euler equations. A consequence is that vorticity distribution functions converge to t...

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Published inCommunications on pure and applied mathematics Vol. 75; no. 1; pp. 60 - 82
Main Authors Constantin, Peter, Drivas, Theodore D., Elgindi, Tarek M.
Format Journal Article
LanguageEnglish
Published Melbourne John Wiley & Sons Australia, Ltd 01.01.2022
John Wiley and Sons, Limited
Subjects
Convergence
Distribution functions
Euler-Lagrange equation
Topology
Viscous fluids
Vorticity
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Summary:We prove that given initial data ω0∈L∞T2, forcing g∈L∞0,T;L∞T2, and any T > 0, the solutions uν of Navier‐Stokes converge strongly in L∞0,T;W1,pT2 for any p ∈ [1, ∞) to the unique Yudovich weak solution u of the Euler equations. A consequence is that vorticity distribution functions converge to their inviscid counterparts. As a by‐product of the proof, we establish continuity of the Euler solution map for Yudovich solutions in the Lp vorticity topology. The main tool in these proofs is a uniformly controlled loss of regularity property of the linear transport by Yudovich solutions. Our results provide a partial foundation for the Miller‐Robert statistical equilibrium theory of vortices as it applies to slightly viscous fluids. © 2020 Wiley Periodicals LLC.
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ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.21940