Infinitely many Leray-Hopf solutions for the fractional Navier-Stokes equations

We prove the ill-posedness for the Leray-Hopf weak solutions of the incompressible and ipodissipative Navier-Stokes equations, when the power of the diffusive term is . We construct infinitely many solutions, starting from the same initial datum, which belong to and strictly dissipate their energy i...

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Bibliographic Details
Published inCommunications in partial differential equations Vol. 44; no. 4; pp. 335 - 365
Main Author De Rosa, Luigi
Format Journal Article
LanguageEnglish
Published Philadelphia Taylor & Francis 03.04.2019
Taylor & Francis Ltd
Subjects
Advection-diffusion equation
Computational fluid dynamics
Convex integration
Euler-Lagrange equation
Fluid flow
Formulas (mathematics)
fractional dissipation
Leray-Hopf
Mathematical analysis
Navier-Stokes equations
non-uniqueness
Viscosity
Well posed problems
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Summary:We prove the ill-posedness for the Leray-Hopf weak solutions of the incompressible and ipodissipative Navier-Stokes equations, when the power of the diffusive term is . We construct infinitely many solutions, starting from the same initial datum, which belong to and strictly dissipate their energy in small time intervals. The proof exploits the "convex integration scheme" introduced by C. De Lellis and L. Székelyhidi for the incompressible Euler equations, joining these ideas with new stability estimates for a class of non-local advection-diffusion equations and a local (in time) well-posedness result for the fractional Navier-Stokes system. Moreover, we show the existence of dissipative Hölder continuous solutions of Euler equations that can be obtained as a vanishing viscosity limit of Leray-Hopf weak solutions of suitable fractional Navier-Stokes equations.
ISSN:0360-5302
1532-4133
DOI:10.1080/03605302.2018.1547745