An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes. (10th October 2019)

• Record Type: Journal Article
• Authors: Drivas, Theodore D; Eyink, Gregory L
• Is Part Of: Nonlinearity. Volume 32:Number 11(2019)
• Page Start: 4465
• Page End: 4482
• Publication Date: 2019-10-10
• Subjects: Onsager's conjecture -- fluid turbulence -- anomalous dissipation; 35Q30 -- 35Q31 -- 76F02 -- 35Q35; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab2f42
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier–Stokes on the torus, assuming that the solutions have norms for Besov space that are bounded in the L 3 -sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form vanishing as if A consequence is that Onsager-type 'quasi-singularities' are required in the Leray solutions, even if the total energy dissipation vanishes in the limit, as long as it does so sufficiently slowly. We also give two sufficient conditions which guarantee the existence of limiting weak Euler solutions u which satisfy a local energy balance with possible anomalous dissipation due to inertial-range energy cascade in the Leray solutions. For the anomalous dissipation vanishes and the weak Euler solutions may be spatially 'rough' but conserve energy.