Dissipation anomaly and anomalous dissipation in incompressible fluid flows

2023-11-14 14:05:16



ZOOM ID: 946 0347 2439


主讲人:Alexey Cheskidov, Westlake University

报告题目:Dissipation anomaly and anomalous dissipation in incompressible fluid flows

报告摘要:Dissipation anomaly, a phenomenon predicted by Kolmogorov's theory of turbulence, is the persistence of a non-vanishing energy dissipation for solutions of the Navier-Stokes equations as the viscosity goes to zero. Anomalous dissipation, predicted by Onsager, is a failure of solutions of the limiting Euler equations to preserve the energy balance. Motivated by a recent dissipation anomaly construction for the 3D Navier-Stokes equations by Bru\`e and De Lellis (2023), we prove the existence of various scenarios in the limit of vanishing viscosity: the total and partial loss of the energy due to dissipation anomaly, dissipation anomaly and anomalous dissipation with a continuous energy limit, anomalous dissipation without dissipation anomaly, and the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. We also discover a relation between dissipation anomaly and the discontinuity of the energy of the limit solution.

Finally, expanding on the obtained total dissipation anomaly construction, we show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias (2002) upper bound is sharp.