[Paper Review] Concentration versus absorption for the Vlasov-Navier-Stokes system on bounded domains

Lucas Ertzbischoff, Daniel Han-Kwan+1 more|arXiv (Cornell University)|Jan 13, 2021

Navier-Stokes equation solutions38 references18 citations

TL;DR

This paper establishes exponential convergence to monokinetic behavior for small-data solutions of the Vlasov-Navier-Stokes system on a bounded 3D domain, where fluid velocity homogenizes to zero and the kinetic distribution concentrates as a Dirac mass at zero velocity. The analysis accounts for competing boundary effects—absorption at the phase-space boundary and no-slip fluid conditions—using energy-dissipation estimates and refined trace and interpolation inequalities to prove concentration with exponential rate despite domain constraints.

ABSTRACT

We study the large time behavior of small data solutions to the Vlasov-Navier-Stokes system set on $\Omega imes \mathbb{R}^3$, for a smooth bounded domain $\Omega$ of $\mathbb{R}^3$, with homogeneous Dirichlet boundary condition for the fluid and absorption boundary condition for the kinetic phase. We prove that the fluid velocity homogenizes to $0$ while the distribution function concentrates towards a Dirac mass in velocity centered at $0$, with an exponential rate. The proof, which follows the methods introduced in [Han-Kwan - Moussa - Moyano, arXiv:1902.03864v2], requires a careful analysis of the boundary effects. We also exhibit examples of classes of initial data leading to a variety of asymptotic behaviors for the kinetic density, from total absorption to no absorption at all.

Motivation & Objective

• To analyze the large-time behavior of small-data global weak solutions to the Vlasov-Navier-Stokes system on a bounded 3D domain with absorption and no-slip boundary conditions.
• To resolve the competition between particle absorption at the boundary and velocity concentration toward zero in the kinetic phase space.
• To establish exponential decay of fluid velocity and exponential concentration of the distribution function toward a Dirac mass at zero velocity.
• To characterize the asymptotic spatial profile as highly dependent on initial data, ranging from total absorption to no absorption.
• To extend the monokinetic asymptotic scenario—previously shown on the torus and in whole space—to bounded domains with nontrivial boundary effects.

Proposed method

• Formalizing the system on a bounded domain Ω⊂R³ with homogeneous Dirichlet boundary condition for fluid velocity and absorption condition for the kinetic function on the incoming phase-space boundary Σ⁻.
• Defining the total energy E(t) and dissipation D(t) to derive a priori energy-dissipation inequalities for weak solutions satisfying a natural energy-dissipation inequality.
• Applying a variant of Gronwall's lemma under integral form (Lemma A.8) to derive exponential decay of key quantities, including fluid velocity and velocity-space dispersion.
• Using maximal LpLq regularity theory for the Stokes system (Theorem A.11) and parabolic regularization (Theorem A.12) to control the fluid velocity in H¹ and L∞ norms.
• Employing Agmon and Gagliardo-Nirenberg-Sobolev inequalities (Proposition A.9, Theorem A.10) to control pointwise behavior and trace terms on the boundary.
• Constructing explicit examples of initial data to demonstrate the full spectrum of asymptotic behaviors, from complete absorption to no absorption, under the same system.

Experimental results

Research questions

• RQ1 Can monokinetic behavior—convergence of the kinetic distribution to a Dirac mass at zero velocity—be rigorously established for the Vlasov-Navier-Stokes system on a bounded domain with absorption and no-slip boundaries?
• RQ2 How do competing boundary effects (absorption at Σ⁻ and no-slip at ∂Ω) influence the long-time dynamics of the system?
• RQ3 What is the rate of convergence to monokinetic behavior, and can it be shown to be exponential under small-data assumptions?
• RQ4 How does the initial data influence the final asymptotic spatial profile, and can one construct examples showing total absorption, partial absorption, or no absorption?
• RQ5 To what extent can the methods used on the torus and in R³ be adapted to bounded domains with nontrivial geometry and boundary conditions?

Key findings

• The fluid velocity u(t) converges to zero in L²(Ω) with exponential rate, uniformly in time.
• The kinetic distribution function f(t,x,v) concentrates toward a Dirac mass in velocity at v=0, with exponential rate in the L¹ norm of the velocity dispersion.
• The total energy E(t) decays exponentially, and the dissipation D(t) remains bounded, confirming the system's stability under small perturbations from equilibrium.
• The concentration of f toward a Dirac mass in velocity is robust and occurs even in the presence of boundary absorption, provided the initial data are sufficiently small.
• Explicit examples of initial data are constructed that lead to a full range of asymptotic behaviors: from complete particle absorption (f→0 in L¹) to no absorption (f→0 only in velocity, not in space).
• The proof relies on a delicate balance between energy-dissipation estimates, parabolic regularization of the Navier-Stokes equation, and boundary trace control via Agmon and Gagliardo-Nirenberg inequalities.