Inviscid symmetry breaking with non-increasing energy

In a recent article, C. Bardos et. al. constructed weak solutions of the three-dimensional incompressible Euler equations which emerge from two-dimensional initial data yet become fully three-dimensional at positive times. They asked whether such symmetry-breaking solutions could also be constructed under the additional condition that they should have non-increasing energy. In this note, we give a positive answer to this question and show that such a construction is possible for a large class of initial data. We use convex integration techniques as developed by De Lellis-Sz\'ekelyhidi.Comment: To appear in C. R. Math. Acad. Sci. Pari

Generalized Entropy Method for the Renewal Equation with Measure Data

We study the long-time asymptotics for the so-called McKendrick-Von Foerster or renewal equation, a simple model frequently considered in structured population dynamics. In contrast to previous works, we can admit a bounded measure as initial data. To this end, we apply techniques from the calculus of variations that have not been employed previously in this context. We demonstrate how the generalized relative entropy method can be refined in the Radon measure framework

Statistical solutions and Onsager's conjecture

We prove a version of Onsager's conjecture on the conservation of energy for the incompressible Euler equations in the context of statistical solutions, as introduced recently by Fjordholm et al. As a byproduct, we also obtain a new proof for the conservative direction of Onsager's conjecture for weak solutions. Dedicated to Edriss S. Titi on the occasion of his 60th birthday

Matching Code and Law: Achieving Algorithmic Fairness with Optimal Transport

Increasingly, discrimination by algorithms is perceived as a societal and legal problem. As a response, a number of criteria for implementing algorithmic fairness in machine learning have been developed in the literature. This paper proposes the Continuous Fairness Algorithm (CFA$\theta$) which enables a continuous interpolation between different fairness definitions. More specifically, we make three main contributions to the existing literature. First, our approach allows the decision maker to continuously vary between specific concepts of individual and group fairness. As a consequence, the algorithm enables the decision maker to adopt intermediate ``worldviews'' on the degree of discrimination encoded in algorithmic processes, adding nuance to the extreme cases of ``we're all equal'' (WAE) and ``what you see is what you get'' (WYSIWYG) proposed so far in the literature. Second, we use optimal transport theory, and specifically the concept of the barycenter, to maximize decision maker utility under the chosen fairness constraints. Third, the algorithm is able to handle cases of intersectionality, i.e., of multi-dimensional discrimination of certain groups on grounds of several criteria. We discuss three main examples (credit applications; college admissions; insurance contracts) and map out the legal and policy implications of our approach. The explicit formalization of the trade-off between individual and group fairness allows this post-processing approach to be tailored to different situational contexts in which one or the other fairness criterion may take precedence. Finally, we evaluate our model experimentally.Comment: Vastly extended new version, now including computational experiment

Existence of Weak Solutions for the Incompressible Euler Equations

Using a recent result of C. De Lellis and L. Sz\'{e}kelyhidi Jr. we show that, in the case of periodic boundary conditions and for dimension greater or equal 2, there exist infinitely many global weak solutions to the incompressible Euler equations with initial data $v_0$, where $v_0$ may be any solenoidal $L^2$-vectorfield. In addition, the energy of these solutions is bounded in time.Comment: 5 page

Weak and Measure-Valued Solutions of the Incompressible Euler Equations

This thesis is concerned with the existence problem for weak solutions of the incompressible Euler equations in arbitrary dimension, and with the relationship between weak solutions and other "very weak'' concepts of solution. In particular, measure-valued solutions as introduced by R. DiPerna and A. Majda (Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys., 108(4):667-689, 1987) are studied. There are three main results of this thesis: Theorem 1.1 asserts the global existence of weak solutions for the incompressible Euler equations. However, these solutions are physically not admissible since their kinetic energy increases at least at the initial time. Moreover, the solutions constructed are highly non-unique in the sense that there exist infinitely many weak solutions with the same initial data. Concerning admissible weak solutions (i.e. such whose energy never exceeds the initial energy), the second result, Theorem 1.2, shows that they exist globally in time at least for a dense subset of initial data. The last result, Theorem 1.3, elucidates the relationship between weak and measure-valued solutions: It is shown that every measure-valued solution is generated by a sequence of weak solutions and that therefore, surprisingly, weak solutions are as flexible as measure-valued solutions. A common feature of these results is their relying on methods recently developed by C. De Lellis and L. Székelyhidi Jr. (On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal., 195(1):225-260, 2010). This thesis includes a fairly detailed presentation of these methods