Invariant measures of the 2D Euler and Vlasov equations

We discuss invariant measures of partial differential equations such as the 2D Euler or Vlasov equations. For the 2D Euler equations, starting from the Liouville theorem, valid for N-dimensional approximations of the dynamics, we define the microcanonical measure as a limit measure where N goes to infinity. When only the energy and enstrophy invariants are taken into account, we give an explicit computation to prove the following result: the microcanonical measure is actually a Young measure corresponding to the maximization of a mean-field entropy. We explain why this result remains true for more general microcanonical measures, when all the dynamical invariants are taken into account. We give an explicit proof that these microcanonical measures are invariant measures for the dynamics of the 2D Euler equations. We describe a more general set of invariant measures, and discuss briefly their stability and their consequence for the ergodicity of the 2D Euler equations. The extension of these results to the Vlasov equations is also discussed, together with a proof of the uniqueness of statistical equilibria, for Vlasov equations with repulsive convex potentials. Even if we consider, in this paper, invariant measures only for Hamiltonian equations, with no fluxes of conserved quantities, we think this work is an important step towards the description of non-equilibrium invariant measures with fluxes.Comment: 40 page

Organizing risk: organization and management theory for the risk society

Risk has become a crucial part of organizing, affecting a wide range of organizations in all sectors. We identify, review and integrate diverse literatures relevant to organizing risk, building on an existing framework that describes how risk is organized in three ‘modes’ – prospectively, in real-time, and retrospectively. We then identify three critical issues in the existing literature: its fragmented nature; its neglect of the tensions associated with each of the modes; and its tendency to assume that the meaning of an object in relation to risk is singular and stable. We provide a series of new insights with regard to each of these issues. First, we develop the concept of a risk cycle that shows how organizations engage with all three modes and transition between them over time. Second, we explain why the tensions have been largely ignored and show how studies using a risk work perspective can provide further insights into them. Third, we develop the concept of risk translation to highlight the ways in the meanings of risks can be transformed and to identify the political consequences of such translations. We conclude the paper with a research agenda to elaborate these insights and ideas further

On topological Morse theory

Starting from the concept of Morse critical point, introduced in [A. Ioffe and E. Schwartzman, J. Math. Pures Appl. (9), 75 (1996), 125-153], we propose a purely topological approach to Morse theory