Razvan C. Fetecau

Nonsmooth lagrangian mechanics and variational collision integrators

Variational techniques are used to analyze the problem of rigid-body dynamics with impacts. The theory of smooth Lagrangian mechanics is extended to a nonsmooth context appropriate for colli- sions, and it is shown in what sense the system is symplectic and satisfies a Noether-style momentum conservation theorem. Discretizations of this nonsmooth mechanics are developed by using the methodology of vari- ational discrete mechanics. This leads to variational integrators which are symplectic-momentum preserving and are consistent with the jump conditions given in the continuous theory. Specific examples of these methods are tested numerically, and the long-time stable energy behavior typical of variational methods is demonstrated.

Swarm dynamics and equilibria for a nonlocal aggregation model

We consider the aggregation equation ρt −∇ ·(ρ∇K ∗ ρ) = 0i nR n , where the interaction potential K models short-range repulsion and long-range attraction. We study a family of interaction potentials for which the equilibria are of finite density and compact support. We show global well-posedness of solutions and investigate analytically and numerically the equilibria and their global stability. In particular, we consider a potential for which the corresponding equilibrium solutions are of uniform density inside a ball of R n and zero outside. For such a potential, various explicit calculations can be carried out in detail. In one dimension we fully solve the temporal dynamics, and in two or higher dimensions we show the global stability of this steady state within the class of radially symmetric solutions. Finally, we solve the following restricted inverse problem: given a radially symmetric density ¯ ρ that is zero outside some ball of radius R and is polynomial inside the ball, construct an interaction potential K for which ¯ ρ is the steady-state solution of the corresponding aggregation equation. Throughout the paper, numerical simulations are used to motivate and validate the analytical results.

A Hamiltonian Regularization of the Burgers Equation

We consider a quasilinear equation that consists of the inviscid Burgers equation plus O(α2) nonlinear terms. As we show, these extra terms regularize the Burgers equation in the following sense: for smooth initial data, the α > 0 equation has classical solutions globally in time. Furthermore, in the zero-α limit, solutions of the regularized equation converge strongly to weak solutions of the Burgers equation. We present numerical evidence that the zero-α limit satisfies the Oleinik entropy inequality. For all α ≥ 0, the regularized equation possesses a nonlocal Poisson structure. We prove the Jacobi identity for this generalized Hamiltonian structure.

LAGRANGIAN AVERAGING FOR COMPRESSIBLE FLUIDS

This paper extends the derivation of the Lagrangian averaged Euler (LAE-

Equilibria of biological aggregations with nonlocal repulsive-attractive interactions

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COLLECTIVE BEHAVIOR OF BIOLOGICAL AGGREGATIONS IN TWO DIMENSIONS: A NONLOCAL KINETIC MODEL

) equations to the case of barotropic compressible flows. The aim of Lagrangian averaging is to regularize the compressible Euler equations by adding dispersion instead of artificial viscosity. Along the way, the derivation of the isotropic and anisotropic LAE-

An investigation of a nonlocal hyperbolic model for self-organization of biological groups

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Variational Multisymplectic Formulations of Nonsmooth Continuum Mechanics

equations is simplified and clarified. The derivation in this paper involves averaging over a tube of trajectories

Stationary States and Asymptotic Behavior of Aggregation Models with Nonlinear Local Repulsion

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Anisotropic interactions in a first-order aggregation model

centered around a given Lagrangian flow