Ill-Posedness of Leray Solutions for the Hypodissipative Navier–Stokes Equations

We prove the ill-posedness of Leray solutions to the Cauchy problem for the hypodissipative Navier–Stokes equations, when the dissipative term is a fractional Laplacian ( - Δ ) α with exponent α < 1 5 . The proof follows the “convex integration methods” introduced by the second author and László...

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Bibliographic Details
Published inCommunications in mathematical physics Vol. 362; no. 2; pp. 659 - 688
Main Authors Colombo, Maria, De Lellis, Camillo, De Rosa, Luigi
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2018
Springer Nature B.V
Subjects
Cauchy problems
Classical and Quantum Gravitation
Complex Systems
Euler-Lagrange equation
Fluid dynamics
Fluid flow
Mathematical analysis
Mathematical and Computational Physics
Mathematical Physics
Navier-Stokes equations
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
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Summary:We prove the ill-posedness of Leray solutions to the Cauchy problem for the hypodissipative Navier–Stokes equations, when the dissipative term is a fractional Laplacian ( - Δ ) α with exponent α < 1 5 . The proof follows the “convex integration methods” introduced by the second author and László Székelyhidi Jr. for the incompressible Euler equations. The methods yield indeed some conclusions even for exponents in the range [ 1 5 , 1 2 [ .
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-018-3177-x