[Paper Review] Hydrodynamic limit of granular gases to pressureless Euler in dimension 1

Pierre‐Emmanuel Jabin, Thomas Rey|arXiv (Cornell University)|Feb 29, 2016

Gas Dynamics and Kinetic Theory37 references18 citations

TL;DR

This paper rigorously proves the hydrodynamic limit of one-dimensional granular gases with strong inelasticity toward the pressureless Euler system as the Knudsen number tends to zero. By introducing a novel dissipative functional and leveraging dispersive relations at the kinetic level, the authors establish the Oleinik entropy condition, ensuring convergence to the monokinetic limit and validating the pressureless Euler equations as the macroscopic description of inelastic particle dynamics.

ABSTRACT

We investigate the behavior of granular gases in the limit of small Knudsen number, that is very frequent collisions. We deal with the strongly inelastic case, in one dimension of space and velocity. We are able to prove the convergence toward the pressureless Euler system. The proof relies on dispersive relations at the kinetic level, which leads to the so-called Oleinik property at the limit.

Motivation & Objective

• To justify the hydrodynamic limit of granular gases with strong inelasticity toward the pressureless Euler system in one space dimension.
• To overcome the failure of classical Hilbert and Chapman-Enskog expansions due to energy dissipation and system singularity.
• To establish the convergence of the kinetic solution to a monokinetic distribution in the small Knudsen number limit.
• To prove that the limit system satisfies the Oleinik entropy condition via a new dissipative functional at the kinetic level.
• To provide a rigorous mathematical foundation for the pressureless Euler system as a hydrodynamic model of inelastic particle systems.

Proposed method

• The authors introduce a new dissipative functional that captures the energy dissipation and dispersion effects in the kinetic equation.
• They use dispersive relations derived from the collision operator to control the evolution of velocity moments and enforce the Oleinik property.
• The proof relies on uniform bounds in $L^p$ and moment estimates to control the concentration of mass in velocity space.
• A key step involves showing that the $L^2$-type functional $\Lambda_{f_\varepsilon,k}$ vanishes in the limit $\varepsilon \to 0$, implying monokinetic behavior.
• The convergence is established via weak compactness and the use of a modified Lyapunov functional $\mathcal{L}_{\eta,\mu,0}$ to control logarithmic singularities in velocity differences.
• The argument combines energy estimates, moment propagation, and uniform integrability to pass to the limit in the kinetic equation.

Experimental results

Research questions

• RQ1 Can the hydrodynamic limit of granular gases with strong inelasticity be rigorously derived in one dimension?
• RQ2 Does the pressureless Euler system emerge as the macroscopic limit of the inelastic Boltzmann equation in the small Knudsen number regime?
• RQ3 How can dispersive and dissipative effects at the kinetic level enforce the Oleinik entropy condition in the limit?
• RQ4 What functional framework allows control of the singular behavior of the limit system despite the absence of pressure?
• RQ5 Is the convergence to a monokinetic distribution stable and uniform under weak convergence of initial data?

Key findings

• The solution $f^\varepsilon$ of the inelastic Boltzmann equation converges weakly to a monokinetic distribution $\rho(x)\delta_{u(x)}(v)$ as $\varepsilon \to 0$.
• The macroscopic fields $\rho$ and $u$ satisfy the pressureless Euler system: $\partial_t \rho + \partial_x(\rho u) = 0$, $\partial_t(\rho u) + \partial_x(\rho u^2) = 0$.
• The Oleinik entropy condition is recovered at the limit through the dispersive properties of the collision operator and the new dissipative functional.
• The $L^p$ and moment bounds on $f^\varepsilon$ are uniform in $\varepsilon$, ensuring precompactness in the weak topology.
• The functional $\Lambda_{f_\varepsilon,k}$, measuring velocity concentration, vanishes in the limit, confirming monokineticity.
• The initial data are assumed in $L^p$ with $p>2$ and bounded moments, and the convergence holds uniformly in time on compact intervals.