Triangular Gross-Pitaevskii breathers and Damski-Chandrasekhar shock waves

The recently proposed map [5] between the hydrodynamic equations governing the two-dimensional triangular cold-bosonic breathers [1] and the high-density zero-temperature triangular free-fermionic clouds, both trapped harmonically, perfectly explains the former phenomenon but leaves uninterpreted the nature of the initial (t=0) singularity. This singularity is a density discontinuity that leads, in the bosonic case, to an infinite force at the cloud edge. The map itself becomes invalid at times t<0t<0. A similar singularity appears at t = T/4t=T/4, where T is the period of the harmonic trap, with the Fermi-Bose map becoming invalid at t > T/4t>T/4. Here, we first map—using the scale invariance of the problem—the trapped motion to an untrapped one. Then we show that in the new representation, the solution [5] becomes, along a ray in the direction normal to one of the three edges of the initial cloud, a freely propagating one-dimensional shock wave of a class proposed by Damski in [7]. There, for a broad class of initial conditions, the one-dimensional hydrodynamic equations can be mapped to the inviscid Burgers’ equation, which is equivalent to a nonlinear transport equation. More specifically, under the Damski map, the t=0 singularity of the original problem becomes, verbatim, the initial condition for the wave catastrophe solution found by Chandrasekhar in 1943 [9]. At t=T/8t=T/8, our interpretation ceases to exist: at this instance, all three effectively one-dimensional shock waves emanating from each of the three sides of the initial triangle collide at the origin, and the 2D-1D correspondence between the solution of [5] and the Damski-Chandrasekhar shock wave becomes invalid.

A special form of solution to half-wave equations

<p style='text-indent:20px;'>We investigate a special form of solution to the one-dimensional half-wave equations with particular forms of nonlinearities. Using the special form of solution involving Hilbert transform, the half-wave equations reduce to nonlocal nonlinear transport equation which can be solved explicitly.</p>

Modeling of crowds in regions with moving obstacles

<p style='text-indent:20px;'>We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to <inline-formula><tex-math id="M1">\begin{document}$ r $\end{document}</tex-math></inline-formula>-prox-regular sets. The crowd motion obeys a nonlinear transport equation outside the obstacle and a normal cone condition (similar to that of the classical sweeping processes theory) on the boundary. We prove the well-posedness of the model, give an application to environment optimization problems, and provide some results of numerical computations.</p>

A mixed-primal finite element method for the coupling of Brinkman–Darcy flow and nonlinear transport

This paper is devoted to the mathematical and numerical analysis of a model describing the interfacial flow-transport interaction in a porous-fluidic domain. The medium consists of a highly permeable material, where the flow of an incompressible viscous fluid is governed by Brinkman equations (written in terms of vorticity, velocity and pressure), and a porous medium where Darcy’s law describes fluid motion using filtration velocity and pressure. Gravity and the local fluctuations of a scalar field (representing for instance, the solids volume fraction or the concentration of a contaminant) are the main drivers of the fluid patterns on the whole domain, and the Brinkman-Darcy equations are coupled to a nonlinear transport equation accounting for mass balance of the scalar concentration. We introduce a mixed-primal variational formulation of the problem and establish existence and uniqueness of solution using fixed-point arguments and small-data assumptions. A family of Galerkin discretizations that produce divergence-free discrete velocities is also presented and analysed using similar tools to those employed in the continuous problem. Convergence of the resulting mixed-primal finite element method is proven, and some numerical examples confirming the theoretical error bounds and illustrating the performance of the proposed discrete scheme are reported.

Dynamics of fitness distributions in the presence of a phenotypic optimum: an integro-differential approach

We propose an integro-differential description of the dynamics of the fitness distribution in an asexual population under mutation and selection, in the presence of a phenotype optimum. Due to the presence of this optimum, the distribution of mutation effects on fitness depends on the parent’s fitness, leading to a non-standard equation with “context-dependent" mutation kernels.Under general assumptions on the mutation kernels, which encompass the standardndimensional Gaussian Fisher’s geometrical model (FGM), we prove that the equation admits a unique time-global solution. Furthermore, we derive a nonlocal nonlinear transport equation satisfied by the cumulant generating function of the fitness distribution. As this equation is the same as the equation derived by Martin and Roques (2016) while studying stochastic Wright-Fisher-type models, this shows that the solution of the main integro-differential equation can be interpreted as the expected distribution of fitness corresponding to this type of microscopic models, in a deterministic limit. Additionally, we give simple sufficient conditions for the existence/non-existence of a concentration phenomenon at the optimal fitness value, i.e, of a Dirac mass at the optimum in the stationary fitness distribution. We show how it determines a phase transition, as mutation rates increase, in the value of the equilibrium mean fitness at mutation-selection balance. In the particular case of the FGM, consistently with previous studies based on other formalisms (Waxman and Peck, 1998, 2006), the condition for the existence of the concentration phenomenon simply requires that the dimensionnof the phenotype space be larger than or equal to 3 and the mutation rateUbe smaller than some explicit threshold.The accuracy of these deterministic approximations are further checked by stochastic individual-based simulations.