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Mini-course. Mathematical aspects of turbulence: Part IV
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Abstract:
In Lecture 1 & 2, we will discuss some foundational aspects of three-dimensional incompressible fluid turbulence, including guiding experimental observations, Kolmogorov's 1941 theory on the structure of a turbulent flow, Onsager's 1949 conjecture on anomalous dissipation and weak Euler solutions, and Landau’s Kazan remark concerning intermittency. Mathematical examples and constructions that exhibit features of turbulent behavior will be discussed. In Lecture 3 & 4, we will discuss the formation of small and large scales in two-dimensional fluids (both viscous and inviscid). In the inviscid setting, we will discuss the mixing process which creates infinitely fine scales of motion at long times and serves as the dynamical mechanism for the direct enstrophy cascade. Rigorous statements can be made in this setting near steady states (Nadirashvili, Koch). For viscous fluids, we will discuss the stability and instability of Kolmogorov flow on two-dimensional flat tori (Meshalkin-Sinai) and a related example of non-uniqueness of smooth steady states of the Navier-Stokes equations (Yudovich). Destabilization of this laminar regime relates to the transition to turbulence.
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