[Paper Review] Strong diffusive limit of the Boltzmann equation with Maxwell boundary condition

Yan Guo, Jung, Junhwa+1 more|arXiv (Cornell University)|Sep 18, 2018

Gas Dynamics and Kinetic Theory33 references3 citations

TL;DR

This paper establishes the first global-in-time strong diffusive limit of the Boltzmann equation to the incompressible Navier-Stokes-Fourier system with Maxwell boundary conditions for all accommodation coefficients α ∈ [0, 1], resolving a long-standing open problem. It introduces an ε-stretching method for L∞ estimates and a novel dissipative decomposition using a rotating Maxwellian in the near-specular regime (α ≪ ε), enabling uniform convergence in strong solution frameworks across all α regimes.

ABSTRACT

While weak diffusive limit from the Boltzmann equation to the incompressible Navier-Stokes-Fourier system was established for the Maxwell boundary condition within renormalized solutions framework [Saint.Raymond2009][Jiang-Masmoudi2017], the corresponding strong diffusive limit has remained outstanding except when the accommodation coefficient $α\sim \varepsilon^{1/2}$ [Jiang-Masmoudi2017]. We establish global in time strong diffusive limit for all accommodation coefficients $α\in [0, 1]$ within strong solutions framework. The main novelties of our proof include: (1) a $\varepsilon$-stretching method for reduction to a single-bounce $L^\infty$ estimate; (2) a dissipation estimate for a carefully constructed rotating Maxwellian in the near-specular regime $α\ll \varepsilon$.

Motivation & Objective

• To establish the strong diffusive limit of the Boltzmann equation to the incompressible Navier-Stokes-Fourier (INSF) system under the Maxwell boundary condition.
• To resolve the open problem of strong limit for all accommodation coefficients α ∈ [0, 1], including the challenging near-specular regime α ≪ ε.
• To develop a new framework for global-in-time convergence using strong solutions, extending beyond previous results limited to α ∼ ε^{1/2}.
• To construct and analyze a rotating Maxwellian that captures the dissipative structure in the near-specular regime.
• To unify the treatment of boundary layers across both regimes: ε ≲ α ≤ 1 and 0 ≤ α ≪ ε, via distinct but complementary analytical techniques.

Proposed method

• Introduce a rescaling F = µ + ε√µf to linearize the Boltzmann equation around the global Maxwellian µ, transforming it into a perturbed linear kinetic equation with a scaled collision operator.
• Develop an ε-stretching method to reduce the full L∞ estimate to a single-bounce estimate, enabling control of the solution's sup-norm over time.
• Construct a rotating Maxwellian ˜µ = Mρ,u,T with velocity shift u and rotation θ to model the boundary layer in the near-specular regime (α ≪ ε), capturing the correct hydrodynamic behavior.
• Derive a dissipative decomposition of the collision operator using the rotating Maxwellian, enabling energy estimates that control the fluctuation f in L2 and L6 norms.
• Apply elliptic estimates and compatibility conditions to solve the macroscopic equations derived from the fluid limit, ensuring solvability of the INSF system.
• Use a bootstrap argument combining L∞, L2, and L6 estimates to close the nonlinear estimates and prove global existence and convergence.

Experimental results

Research questions

• RQ1 Can the strong diffusive limit of the Boltzmann equation to the INSF system be established globally in time for all accommodation coefficients α ∈ [0, 1]?
• RQ2 How can the boundary layer structure be controlled in the near-specular regime (α ≪ ε) where the standard Maxwellian fails to capture the correct dissipation?
• RQ3 What novel analytical techniques are required to handle the transition between the diffusive (α ≳ ε) and near-specular (α ≪ ε) regimes?
• RQ4 Can a rotating Maxwellian be used to construct a dissipative structure that enables energy estimates in the near-specular case?
• RQ5 Is it possible to achieve uniform convergence in strong solution norms across the full range of α ∈ [0, 1]?

Key findings

• The paper establishes the first global-in-time strong diffusive limit of the Boltzmann equation to the INSF system for all accommodation coefficients α ∈ [0, 1], including the previously unresolved case α ≪ ε.
• For the regime ε ≲ α ≤ 1, the authors prove convergence of the fluctuation f to the solution of the INSF system in L∞, L2, and L6 norms, with convergence rates depending on the initial data and domain regularity.
• In the near-specular regime (0 ≤ α ≪ ε), the construction of a rotating Maxwellian ˜µ with velocity shift u and angular velocity θ enables a dissipative decomposition that controls the energy of the fluctuation f.
• The ε-stretching method successfully reduces the full L∞ estimate to a single-bounce estimate, allowing uniform control of the solution’s sup-norm over time.
• The authors derive precise boundary integral identities for the rotating Maxwellian, including ˆ_{n·v>0} |v−u|^2 ˜µ[n·v] dv = 4ρT^{3/2}/(2π)^{1/2}, which are essential for energy estimates.
• The macroscopic equations derived from the fluid limit are shown to be solvable via elliptic theory, with L2 and L6 estimates for the velocity and temperature fields that are uniformly bounded in ε.