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The hydrodynamic limit of viscoelastic granular gases

R. Alonso, B. Lods, I. Tristani

TL;DR

The paper derives a rigorous hydrodynamic limit for viscoelastic granular gases by modeling inelastic collisions with a velocity-dependent restitution coefficient and applying a self-similar scaling that balances energy input and dissipation. It develops a non-autonomous, rescaled Boltzmann equation, constructs a nonlinear analysis around the elastic operator via a precise splitting, and proves convergence to an incompressible Navier–Stokes–Fourier system with self-consistent forcing. The authors also establish a quantitative Haff-type law for both global and local granular temperatures and provide explicit expressions for transport and forcing coefficients. This work advances the rigorous connection between kinetic models of viscoelastic granular media and macroscopic hydrodynamics, with implications for understanding dissipative flows in granular materials under realistic restitution laws.

Abstract

We obtain the first rigorous derivation of an incompressible Navier-Stokes-Fourier system with self-consistent and time-dependent forcing terms from the inelastic hard-spheres Boltzmann equation associated to the relevant case of viscoelastic granular gases. The model's inelasticity is measured by the so-called restitution coefficient which, for viscoelastic particles, depends on the relative velocities of particles. Through a suitable self-similar change of variables, a balanced dynamic between energy inflow and outflow naturally emerges in the model which permits its analysis. In contrast, such balanced dynamic does not emerge naturally in the constant restitution case and has to be imposed in our previous contribution (Alonso, Lods, Tristani, Mémoires SMF). The exact self-similar rescaling, which allows to capture nontrivial inelastic-hydrodynamic effects, presents itself explicitly in terms of the Knudsen number and the restitution coefficient. The consequence of such scaling is a non-autonomous rescaled Boltzmann equation whose solutions converge, in a specific weak sense, towards the aforementioned hydrodynamic limit. The incompressible Navier-Stokes-Fourier system obtained by this process appears to be new in this context. As a byproduct of the analysis, we determine the exact dissipation rate of the granular temperature known as \emph{Haff's law}.

The hydrodynamic limit of viscoelastic granular gases

TL;DR

The paper derives a rigorous hydrodynamic limit for viscoelastic granular gases by modeling inelastic collisions with a velocity-dependent restitution coefficient and applying a self-similar scaling that balances energy input and dissipation. It develops a non-autonomous, rescaled Boltzmann equation, constructs a nonlinear analysis around the elastic operator via a precise splitting, and proves convergence to an incompressible Navier–Stokes–Fourier system with self-consistent forcing. The authors also establish a quantitative Haff-type law for both global and local granular temperatures and provide explicit expressions for transport and forcing coefficients. This work advances the rigorous connection between kinetic models of viscoelastic granular media and macroscopic hydrodynamics, with implications for understanding dissipative flows in granular materials under realistic restitution laws.

Abstract

We obtain the first rigorous derivation of an incompressible Navier-Stokes-Fourier system with self-consistent and time-dependent forcing terms from the inelastic hard-spheres Boltzmann equation associated to the relevant case of viscoelastic granular gases. The model's inelasticity is measured by the so-called restitution coefficient which, for viscoelastic particles, depends on the relative velocities of particles. Through a suitable self-similar change of variables, a balanced dynamic between energy inflow and outflow naturally emerges in the model which permits its analysis. In contrast, such balanced dynamic does not emerge naturally in the constant restitution case and has to be imposed in our previous contribution (Alonso, Lods, Tristani, Mémoires SMF). The exact self-similar rescaling, which allows to capture nontrivial inelastic-hydrodynamic effects, presents itself explicitly in terms of the Knudsen number and the restitution coefficient. The consequence of such scaling is a non-autonomous rescaled Boltzmann equation whose solutions converge, in a specific weak sense, towards the aforementioned hydrodynamic limit. The incompressible Navier-Stokes-Fourier system obtained by this process appears to be new in this context. As a byproduct of the analysis, we determine the exact dissipation rate of the granular temperature known as \emph{Haff's law}.

Paper Structure

This paper contains 29 sections, 44 theorems, 485 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Boltzmann equation for granular gases
  3. Related literature and strategy
  4. Self-similar change of variables
  5. Notations and definitions
  6. Main results
  7. Outline of the paper
  8. Summary of useful results about the collision operator
  9. Collision mechanism in granular gases
  10. Strong form and alternative representation of the Boltzmann operator
  11. Estimates on the difference between inelastic and elastic collisions
  12. Derivation of the sharp scaling
  13. The complete inelastic linearized operator
  14. Decomposition of $\mathscr{L}_{1}$
  15. Splitting of $\mathcal{G}_{\varepsilon,t}$
  16. ...and 14 more sections

Key Result

Theorem 1.1

There exists a pair $(\varepsilon_0,{\alpha_{0}})$ of positive constants that depend on the mass and energy of $F_{\mathrm{in}}^\varepsilon$, $m$, and $q$ introduced in eq:defE, such that for $\varepsilon\in(0,\varepsilon_0)$ and if then, the equation eq:heps has a unique solution $h_{\varepsilon} \in\mathcal{C}([0,\infty); \mathcal{E}) \cap L^1_{\rm{loc}}((0,\infty);\mathcal{E}_1)$ satisfying f

Figures (1)

  • Figure 1: Different scales and the hydrodynamic limit.

Theorems & Definitions (91)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Haff's law -- local and global
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 81 more