Convergence to Equilibrium in Wasserstein Distance for Damped Euler Equations with Interaction Forces

Abstract

We develop tools to construct Lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped Euler system under the influence of external and interaction potential forces with respect to the 2-Wasserstein distance. We also discuss the overdamped limit to a nonlocal equation used in the modelling of granular media with respect to the 2-Wasserstein distance, and provide rigorous proofs for particular examples in one spatial dimension.2010 Mathematics Subject Classification. 49K20, 76N99, 35L40 .

Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces

“…The final step, corresponding to the bottom part of the diagram in Figure 1, is inspired on a recent work of part of the authors [7]. Actually, we can estimate the error between the solutions ρ γ and ρ to (1.3) and (1.2), respectively, in the 2-Wasserstein distance again.…”

Mean-Field Limits: From Particle Descriptions to Macroscopic Equations

“…In fact, we can easily show that 14) This shows that the convergence of the modulated kinetic energy (1.12) implies the convergence of the modulated macro energy (1.13). We notice that if f is a monokinetic distribution, f (x, v, t) = ρ f (x, t)δ u f (x,t) (v), then the second term on the right hand side of (1.14) becomes zero, and the two modulated energies (1.12) and (1.13) coincide.…”

“…The hydrodynamic system (1.9) has a rich variety of phenomena compared to the plain pressureless Euler system. This fact is due to the competition between attraction/repulsion and alignment leading to sharp thresholds for the global existence of strong solutions versus finite time blow-up and decay to equilibrium, see [13][14][15]26,63,68]. We emphasize that the additional alignment, linear damping and attraction/repulsion terms can promote the existence of global solutions depending on the intial data.…”

Relative entropy method for the relaxation limit of hydrodynamic models

“…The main objective of this work is to deduce the following equilibrium equation ∂ tρ = div x ρ∇ x δE(ρ) δρ (1.4) by taking the overdamped limit ε → 0 in system (1.1) under the framework of relative entropy method. This method is an efficient mathematical tool for establishing the limiting processes and stabilities among thermomechanical theories, see [6,7,16,17,19,24,31,32] for instance. With the various choices of the functional E(ρ), the corresponding models spanned from the system of isentropic gas dynamics and variants of the Euler-Poisson system [29,33,35] leading to the porous medium equation and nonlinear aggregation-diffusion equations in the overdamped limit, see [15,26,27,28,30,34] and references therein.…”

“…7) where ϕ andφ are Lipschitz test functions compactly supported in [0, ∞) in time andφ•ν = 0 on ∂Ω when Ω = R d . Using the definition of θ(τ ) in (2.2), we introduce the test functions in the above relations…”