scholarly journals A Hamiltonian Interacting Particle System for Compressible Flow

Water ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 2109
Author(s):  
Simon Hochgerner
Keyword(s):  
Particle System ◽  
Mean Field ◽  
Interacting Particle System ◽  
Navier Stokes ◽  
Mean Field Limit ◽  
Navier Stokes Equation ◽  
Interacting Particle ◽  
The Mean ◽  
Dependent Viscosity ◽  
Circulation Theorem

The decomposition of the energy of a compressible fluid parcel into slow (deterministic) and fast (stochastic) components is interpreted as a stochastic Hamiltonian interacting particle system (HIPS). It is shown that the McKean–Vlasov equation associated to the mean field limit yields the barotropic Navier–Stokes equation with density-dependent viscosity. Capillary forces can also be treated by this approach. Due to the Hamiltonian structure, the mean field system satisfies a Kelvin circulation theorem along stochastic Lagrangian paths.

scholarly journals A Hamiltonian mean field system for the Navier–Stokes equation

2018 ◽  
Vol 474 (2218) ◽  
pp. 20180178 ◽  
Author(s):  
Simon Hochgerner
Keyword(s):  
Energy Dissipation ◽  
Stokes Equation ◽  
Particle System ◽  
Mean Field ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Field System ◽  
Interacting Particle ◽  
Incompressible Navier Stokes Equation ◽  
Circulation Theorem

We use a Hamiltonian interacting particle system to derive a stochastic mean field system whose McKean–Vlasov equation yields the incompressible Navier–Stokes equation. Since the system is Hamiltonian, the particle relabeling symmetry implies a Kelvin Circulation Theorem along stochastic Lagrangian paths. Moreover, issues of energy dissipation are discussed and the model is connected to other approaches in the literature.

scholarly journals The Microscopic Derivation and Well-Posedness of the Stochastic Keller–Segel Equation

2020 ◽  
Vol 31 (1) ◽  
Author(s):  
Hui Huang ◽  
Jinniao Qiu
Keyword(s):  
Existence Of Solutions ◽  
Particle System ◽  
Mean Field ◽  
Field Limit ◽  
Mean Field Limit ◽  
Unique Existence ◽  
Interacting Particle ◽  
Microscopic Derivation ◽  
The Mean ◽  
Limit Result

AbstractIn this paper, we propose and study a stochastic aggregation–diffusion equation of the Keller–Segel (KS) type for modeling the chemotaxis in dimensions $$d=2,3$$ d = 2 , 3 . Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncratic and common noises. Both the unique existence of solutions to the stochastic KS equation and the mean-field limit result are addressed.

scholarly journals On the Mean-Field Limit for the Vlasov–Poisson–Fokker–Planck System

2020 ◽  
Vol 181 (5) ◽  
pp. 1915-1965
Author(s):  
Hui Huang ◽  
Jian-Guo Liu ◽  
Peter Pickl
Keyword(s):  
Initial Data ◽  
Particle System ◽  
Mean Field ◽  
Newtonian Potential ◽  
Empirical Measure ◽  
Field Limit ◽  
Mean Field Limit ◽  
Brownian Motions ◽  
The Mean ◽  
Fokker Planck

AbstractWe rigorously justify the mean-field limit of an N-particle system subject to Brownian motions and interacting through the Newtonian potential in $${\mathbb {R}}^3$$ R 3 . Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by $$N^{-\frac{1}{3}+\varepsilon }$$ N - 1 3 + ε ($$\frac{1}{63}\le \varepsilon <\frac{1}{36}$$ 1 63 ≤ ε < 1 36 ) with a blob size of $$N^{-\delta }$$ N - δ ($$\frac{1}{3}\le \delta <\frac{19}{54}-\frac{2\varepsilon }{3}$$ 1 3 ≤ δ < 19 54 - 2 ε 3 ) up to a probability of $$1-N^{-\alpha }$$ 1 - N - α for any $$\alpha >0$$ α > 0 . Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions.

scholarly journals Instantaneous control of interacting particle systems in the mean-field limit

2020 ◽  
Vol 405 ◽  
pp. 109181 ◽  
Author(s):  
Martin Burger ◽  
René Pinnau ◽  
Claudia Totzeck ◽  
Oliver Tse ◽  
Andreas Roth
Keyword(s):  
Interacting Particle Systems ◽  
Mean Field ◽  
Particle Systems ◽  
Field Limit ◽  
Mean Field Limit ◽  
Interacting Particle ◽  
The Mean ◽  
Instantaneous Control

SELF-PROPELLED INTERACTING PARTICLE SYSTEMS WITH ROOSTING FORCE

2010 ◽  
Vol 20 (supp01) ◽  
pp. 1533-1552 ◽  
Author(s):  
JOSÉ A. CARRILLO ◽  
AXEL KLAR ◽  
STEPHAN MARTIN ◽  
SUDARSHAN TIWARI
Keyword(s):  
Stochastic Model ◽  
Collective Behavior ◽  
Food Source ◽  
Interacting Particle Systems ◽  
Particle System ◽  
Mean Field ◽  
Particle Systems ◽  
Interacting Particle System ◽  
Hydrodynamic Equations ◽  
Interacting Particle

We consider a self-propelled interacting particle system for the collective behavior of swarms of animals, and extend it with an attraction term called roosting force, as it has been suggested in Ref. 30. This new force models the tendency of birds to overfly a fixed preferred location, e.g. a nest or a food source. We include roosting to the existing individual-based model and consider the associated mean-field and hydrodynamic equations. The resulting equations are investigated analytically looking at different asymptotic limits of the corresponding stochastic model. In addition to existing patterns like single mills, the inclusion of roosting yields new scenarios of collective behavior, which we study numerically on the microscopic as well as on the hydrodynamic level.

scholarly journals The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow

2008 ◽  
Vol 131 (5) ◽  
pp. 941-967 ◽  
Author(s):  
Laurent Desvillettes ◽  
François Golse ◽  
Valeria Ricci
Keyword(s):  
Stokes Flow ◽  
Mean Field ◽  
Solid Particles ◽  
Navier Stokes ◽  
Field Limit ◽  
Mean Field Limit ◽  
The Mean

scholarly journals A Local Barycentric Version of the Bak–Sneppen Model

2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Philip Kennerberg ◽  
Stanislav Volkov
Keyword(s):  
Real Number ◽  
Uniform Distribution ◽  
Particle System ◽  
Interacting Particle System ◽  
Interacting Particle

AbstractWe study the behaviour of an interacting particle system, related to the Bak–Sneppen model and Jante’s law process defined in Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018). Let $$N\ge 3$$ N ≥ 3 vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called fitness (we use this term, as it is quite standard for Bak–Sneppen models). Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution $$\zeta $$ ζ . We show that in case where $$\zeta $$ ζ is a finitely supported or continuous uniform distribution, all the fitnesses except one converge to the same value.

scholarly journals CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY

2010 ◽  
Vol 20 (07) ◽  
pp. 1049-1087 ◽  
Author(s):  
BORIS HASPOT
Keyword(s):  
Shallow Water ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Water System ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Viscosity Coefficients ◽  
Global Existence Of Solutions ◽  
Dependent Viscosity

In this paper, we consider the compressible Navier–Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier–Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

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