Our starting point is a selection-mutation equation describing the adaptive dynamics of a quantitative trait under the influence of an ecological feedback loop. Based on the assumption of small (but frequent) mutations we employ asymptotic analysis to derive a Hamilton-Jacobi equation. Well-established and powerful numerical tools for solving the Hamilton-Jacobi equations then allow us to easily compute the evolution of the trait in a monomorphic population when this evolution is continuous but also when the trait exhibits a jump. By adapting the numerical method we can, at the expense of a significantly increased computing time, also capture the branching event in which a monomorphic population turns dimorphic and subsequently follow the evolution of the two traits in the dimorphic population. From the beginning we concentrate on a caricatural yet interesting model for competition for two resources. This provides the perhaps simplest example of branching and has the great advantage that it can be analyzed and understood in detail.

We prove global existence of appropriate weak solutions for the compressible Navier-Stokes equations for more general stress tensor than those covered by P.-L. Lions and E. Feireisl's theory. More precisely we focus on more general pressure laws which are not thermodynamically stable; we are also able to handle some anisotropy in the viscous stress tensor. To give answers to these two longstanding problems, we revisit the classical compactness theory on the density by obtaining precise quantitative regularity estimates: This requires a more precise analysis of the structure of the equations combined to a novel approach to the compactness of the continuity equation. These two cases open the theory to important physical applications, for instance to describe solar events (virial pressure law), geophysical flows (eddy viscosity) or biological situations (anisotropy).

Abstract. In this paper, we provide results about the long time behavior of integrodifferential equations appearing in the study of populations structured with respect to a quantitative (continuous) trait, which are submitted to selection (or competition).

We prove the mean field limit and the propagation of chaos for a system of particles interacting with a singular interaction force of the type 1/|x| α , with α < 1 in dimension d ≥ 3. We also provide results for forces with singularity up to α < d − 1 but with a large enough cut-off. This last result thus almost includes the case of Coulombian or gravitational interaction, but it also allows for a very small cut-off when the strength of the singularity α is larger but close to one.

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