The hydrodynamic limit of a kinetic Cucker-Smale flocking model is investigated. The starting point is the model considered in [Existence of weak solutions to kinetic flocking models, SIAM Math. Anal. 45 (2013) 215-243], which in addition to free transport of individuals and a standard Cucker-Smale alignment operator, includes Brownian noise and strong local alignment. The latter was derived in [On strong local alignment in the kinetic Cucker-Smale equation, in Hyperbolic Conservation Laws and Related Analysis with Applications (Springer, 2013), pp. 227-242] as the singular limit of an alignment operator first introduced by Motsch and Tadmor in [A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys. 141 (2011) 923-947].The objective of this work is the rigorous investigation of the singular limit corresponding to strong noise and strong local alignment. The proof relies on a relative entropy method. The asymptotic dynamics is described by an Euler-type flocking system.

Abstract. We establish the global existence of weak solutions to a class of kinetic flocking equations. The models under conideration include the kinetic Cucker-Smale equation [6,7] with possibly non-symmetric flocking potential, the Cucker-Smale equation with additional strong local alignment, and a newly proposed model by Motsch and Tadmor [14]. The main tools employed in the analysis are the velocity averaging lemma and the Schauder fixed point theorem along with various integral bounds.

This paper presents a new numerical method for the compressible Navier-Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix-Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax-Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.

We establish existence of global-in-time weak solutions to the one dimensional, compressible Navier-Stokes system for a viscous and heat conducting ideal polytropic gas (pressure p = Kθ/τ , internal energy e = cvθ), when the viscosity µ is constant and the heat conductivity κ depends on the temperature θ according to κ(θ) =κθ β , with 0 ≤ β < 3 2 . This choice of degenerate transport coefficients is motivated by the kinetic theory of gasses.Approximate solutions are generated by a semi-discrete finite element scheme. We first formulate sufficient conditions that guarantee convergence to a weak solution. The convergence proof relies on weak compactness and convexity, and it applies to the more general constitutive relations µ(θ) =μθ α , κ(θ) =κθ β , with α ≥ 0, 0 ≤ β < 2 (μ,κ constants). We then verify the sufficient conditions in the case α = 0 and 0 ≤ β < 3 2 . The data are assumed to be without vacuum, mass concentrations, or vanishing temperatures, and the same holds for the weak solutions.

This paper studies global existence, hydrodynamic limit, and large-time behavior of weak solutions to a kinetic flocking model coupled to the incompressible Navier-Stokes equations. The model describes the motion of particles immersed in a Navier-Stokes fluid interacting through local alignment. We first prove the existence of weak solutions using energy and L p estimates together with the velocity averaging lemma. We also rigorously establish a hydrodynamic limit corresponding to strong noise and local alignment. In this limit, the dynamics can be totally described by a coupled compressible Euler -incompressible Navier-Stokes system. The proof is via relative entropy techniques. Finally, we show a conditional result on the large-time behavior of classical solutions. Specifically, if the mass-density satisfies a uniform in time integrability estimate, then particles align with the fluid velocity exponentially fast without any further assumption on the viscosity of the fluid.