Flocking Hydrodynamics with External Potentials
- PDF / 512,309 Bytes
- 35 Pages / 439.37 x 666.142 pts Page_size
- 83 Downloads / 178 Views
Flocking Hydrodynamics with External Potentials Ruiwen Shu & Eitan Tadmor Communicated by P. Rabinowitz
Abstract We study the large-time behavior of a hydrodynamic model which describes the collective behavior of continuum of agents, driven by pairwise alignment interactions with additional external potential forcing. The external force tends to compete with the alignment which makes the large time behavior very different from the original Cucker–Smale (CS) alignment model, and far more interesting. Here we focus on uniformly convex potentials. In the particular case of quadratic potentials, we are able to treat a large class of admissible interaction kernels, φ(r ) (1+r 2 )−β with ‘thin’ tails β 1—thinner than the usual ‘fat-tail’ kernels encountered in CS flocking β 1/2; we discover unconditional flocking with exponential convergence of velocities and positions towards a Dirac mass traveling as harmonic oscillator. For general convex potentials, we impose a stability condition, requiring a large enough alignment kernel to avoid crowd scattering. We then prove, by hypocoercivity arguments, that both the velocities and positions of a smooth solution must flock. We also prove the existence of global smooth solutions for one and two space dimensions, subject to critical thresholds in initial configuration space. It is interesting to observe that global smoothness can be guaranteed for sub-critical initial data, independently of the apriori knowledge of large time flocking behavior.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Statement of Main Results: Flocking with Quadratic Potentials . . . 3. Statement of Main Results: Flocking with General Convex Potentials 4. Existence of Global Smooth Solutions . . . . . . . . . . . . . . . . 5. Proof of Main Results: Hypocoercivity Bounds . . . . . . . . . . . . 6. Proof of Main Results: Existence of Global Smooth Solutions . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
R. Shu, E. Tadmor
1. Introduction We are concerned with the hydrodynamic alignment model with external potential forcing: ⎧ ⎨ ∂t ρ + ∇x · (ρu) = 0, (1.1) ⎩ ∂t u + u · ∇x u = φ(|x − y|)(u(y, t) − u(x, t))ρ(y, t) dy − ∇U (x). Here (ρ(x, t), u(x, t)) are the local density and velocity field of a continuum of agents, depending on the spatial variables x ∈ = Rd or Td and time t ∈ R0 . The integral term on the right represents the alignment between agents, quantified in terms of the pairwise interaction kernel φ = φ(r ) 0. In many realistic scenarios, agents driven by alignment are also subject to other forces—external forces from environment, pairwise attractive-repulsive forces, etc. Such forces may compete with alignment, which makes the large time behavior very different from the original potential-free model and far more interesting. One of the simplest types of external forces is potential force, given by the fixed external p
Data Loading...
