29,931 research outputs found
Weak solutions for Euler systems with non-local interactions
We consider several modi cations of the Euler system of uid dynamics including its pressureless variant driven by non-local interaction repulsive-attractive and alignment forces in the space dimension N = 2 ; 3. These models arise in the study of self-organisation in collective behavior modeling of animals and crowds. We adapt the method of convex integration to show the existence of in nitely many global-in-time weak solutions for any bounded initial data. Then we consider the class of dissipative solutions satisfying, in addition, the associated global energy balance (inequality). We identify a large set of initial data for which the problem admits in nitely many dissipative weak solutions. Finally, we establish a weak-strong uniqueness principle for the pressure driven Euler system with non-local interaction terms as well as for the pressureless system with Newtonian interaction
Singular Cucker-Smale Dynamics
The existing state of the art for singular models of flocking is overviewed,
starting from microscopic model of Cucker and Smale with singular communication
weight, through its mesoscopic mean-filed limit, up to the corresponding
macroscopic regime. For the microscopic Cucker-Smale (CS) model, the
collision-avoidance phenomenon is discussed, also in the presence of bonding
forces and the decentralized control. For the kinetic mean-field model, the
existence of global-in-time measure-valued solutions, with a special emphasis
on a weak atomic uniqueness of solutions is sketched. Ultimately, for the
macroscopic singular model, the summary of the existence results for the
Euler-type alignment system is provided, including existence of strong
solutions on one-dimensional torus, and the extension of this result to higher
dimensions upon restriction on the smallness of initial data. Additionally, the
pressureless Navier-Stokes-type system corresponding to particular choice of
alignment kernel is presented, and compared - analytically and numerically - to
the porous medium equation
Dynamic Density Functional theory for steady currents: Application to colloidal particles in narrow channels
We present the theoretical analysis of the steady state currents and density
distributions of particles moving with Langevin dynamics, under the effects of
an external potential displaced at constant rate. The Dynamic Density
Functional (DDF) formalism is used to introduce the effects of the molecular
interactions, from the equilibrium Helmholtz free energy density functional. We
analyzed the generic form of the DDF for one-dimensional external potentials
and the limits of strong and weak potential barriers. The ideal gas case is
solved in a closed form for generic potentials and compared with the numerical
results for hard-rods, with the exact equilibrium free energy. The results may
be of relevance for microfluidic devices, with colloidal particles moving along
narrow channels, if external driving forces have to compete with the brownian
fluctuations and the interaction forces of the particles
Mean Field Limit for Coulomb-Type Flows
We establish the mean-field convergence for systems of points evolving along
the gradient flow of their interaction energy when the interaction is the
Coulomb potential or a super-coulombic Riesz potential, for the first time in
arbitrary dimension. The proof is based on a modulated energy method using a
Coulomb or Riesz distance, assumes that the solutions of the limiting equation
are regular enough and exploits a weak-strong stability property for them. The
method can handle the addition of a regular interaction kernel, and applies
also to conservative and mixed flows. In the appendix, it is also adapted to
prove the mean-field convergence of the solutions to Newton's law with Coulomb
or Riesz interaction in the monokinetic case to solutions of an Euler-Poisson
type system.Comment: Final version with expanded introduction, to appear in Duke Math
Journal. 35 page
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