Mathematical Models and Methods in Applied Sciences

Models with a vanishing anisotropic viscosity in the vertical direction are of relevance for the study of turbulent flows in geophysics. This motivates us to study the two-dimensional Boussinesq system with horizontal viscosity in only one equation. In this paper, we focus on the global existence issue for possibly large initial data. We first examine the case where the Navier–Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transport–diffusion equation with diffusion in the horizontal direction only. For both systems, we construct global weak solutions à la Leray and strong unique solutions for more regular data. Our results rest on the fact that the diffusion acts perpendicularly to the buoyancy force.

Starting from the Euler equations for incompressible flows, we propose an extension of the classical Saint-Venant system where the non-hydrostatic pressure terms are taken into account in the asymptotic expansion. We also derive a multilayer version of this vertically averaged Euler system. The multilayer approach allows mass exchanges between the neighboring layers. Finally we give a kinetic type interpretation for the proposed models.

In this paper, we describe an incompressible Unified Continuum (UC) model in Euler (laboratory) coordinates with a moving mesh for tracking the fluid-structure interface as part of the discretization, allowing simple and general formulation and efficient computation. The model consists of conservation equations for mass and momentum, a phase convection equation and a Cauchy stress and phase variable θ as data for defining material properties and constitutive laws. We target realistic 3D turbulent fluid-structure interaction (FSI) applications, where we show simulation results of a flexible flag mounted in the turbulent wake behind a cube as a qualitative test of the method, and quantitative results for 2D benchmarks, leaving adaptive error control for future work. We compute piecewise linear continuous discrete solutions in space and time by a general Galerkin (G2) finite element method (FEM). We introduce and compensate for mesh motion by defining a local arbitrary Euler–Lagrange (ALE) map on each space-time slab as part of the discretization, allowing a sharp phase interface given by θ on cell facets. The Unicorn implementation is published as part of the FEniCS Free Software system for automation of computational mathematical modeling. Simulation results are given for a 2D stationary convergence test, indicating quadratic convergence of the displacement, a simple 2D Poiseuille test for verifying correct treatment of the fluid-structure interface, showing quadratic convergence to the exact drag value, an established 2D dynamic flag benchmark test, showing a good match to published reference solutions and a 3D turbulent flag test as indicated above.

We present existence, uniqueness and continuous dependence results for some kinetic equations motivated by models for the collective behavior of large groups of individuals. Models of this kind have been recently proposed to study the behavior of large groups of animals, such as flocks of birds, swarms, or schools of fish. Our aim is to give a well-posedness theory for general models which possibly include a variety of effects: an interaction through a potential, such as a short-range repulsion and long-range attraction; a velocity-averaging effect where individuals try to adapt their own velocity to that of other individuals in their surroundings; and self-propulsion effects, which take into account effects on one individual that are independent of the others. We develop our theory in a space of measures, using mass transportation distances. As consequences of our theory, we show also the convergence of particle systems to their corresponding kinetic equations, and the local-in-time convergence to the hydrodynamic limit for one of the models.

Artificial interface conditions parametrized by a complex number θ₀ are introduced for 1D-Schrödinger operators. When this complex parameter equals the parameter θ ∈ iℝ of the complex deformation which unveils the shape resonances, the Hamiltonian becomes dissipative. This makes possible an adiabatic theory for the time evolution of resonant states for arbitrarily large time scales. The effect of the artificial interface conditions on the important stationary quantities involved in quantum transport models is also checked to be as small as wanted, in the polynomial scale (h^N)_N∈ℕ as h → 0, according to θ₀.