[Paper Review] From Vlasov-Maxwell-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law: convergence for classical solutions

Ning Jiang, Yi-Long Luo|arXiv (Cornell University)|May 12, 2019

Gas Dynamics and Kinetic TheoryMathematics63 references20 citations

TL;DR

This paper establishes the rigorous convergence of classical solutions of the two-species Vlasov-Maxwell-Boltzmann (VMB) system to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm’s law in the small Knudsen number limit. Using a nonlinear energy method, the authors prove uniform energy bounds for fluctuations in the VMB system and justify the hydrodynamic limit from kinetic to fluid description for classical solutions, extending prior results on renormalized solutions to the classical regime.

ABSTRACT

For the two-species Vlasov-Maxwell-Boltzmann (VMB) system with the scaling under which the moments of the fluctuations to the global Maxwellians formally converge to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell (NSFM) system with Ohm's law, we prove the uniform estimates with respect to Knudsen number $\eps$ for the fluctuations. As consequences, the existence of the global in time classical solutions of VMB with all $\eps \in (0,1]$ is established. Furthermore, the convergence of the fluctuations of the solutions of VMB to the classical solutions of NSFM with Ohm's law is rigorously justified. This limit was justified in the recent breakthrough of Ars\'enio and Saint-Raymond \cite{Arsenio-SRM-2016} from renormalized solutions of VMB to dissipative solutions of incompressible viscous electro-magneto-hydrodynamics under the corresponding scaling. In this sense, our result gives a classical solution analogue of the corresponding limit in \cite{Arsenio-SRM-2016}.

Motivation & Objective

• To establish the existence of global-in-time classical solutions for the two-species Vlasov-Maxwell-Boltzmann (VMB) system for all Knudsen numbers ε ∈ (0,1].
• To prove the uniform energy estimates with respect to ε for the fluctuations of VMB solutions around global Maxwellians.
• To justify the hydrodynamic limit from the VMB system to the two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm’s law for classical solutions.
• To provide a classical solution analogue of the renormalized solution limit previously established by Arsénio and Saint-Raymond.

Proposed method

• Employing a nonlinear energy method to derive uniform energy estimates for the fluctuation components of the VMB system.
• Introducing a decomposition of the distribution functions into fluid and kinetic (non-fluid) parts using projection operators P and P⊥.
• Establishing a hierarchy of energy estimates through induction on the order of derivatives, controlling both fluid and kinetic components.
• Using the hard sphere collision kernel and assuming equal mass and charge magnitude for both species to simplify the analysis without loss of generality.
• Deriving a modified energy functional that includes time-integrated dissipation terms to control the ε-dependent singularities.
• Applying Young’s inequality and Gronwall-type arguments to close the energy estimates and ensure uniform boundedness in ε.

Experimental results

Research questions

• RQ1 Can global classical solutions be constructed for the two-species Vlasov-Maxwell-Boltzmann system uniformly in the Knudsen number ε ∈ (0,1]?
• RQ2 Do the fluctuations of VMB solutions converge to classical solutions of the two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm’s law as ε → 0?
• RQ3 What uniform energy bounds can be established for the VMB system that are independent of ε and ensure the convergence of classical solutions?
• RQ4 How does the nonlinear energy method handle the coupling between kinetic and fluid dynamics in the small Knudsen number regime?
• RQ5 Can the convergence result be established in the classical solution framework, extending the prior renormalized solution results of Arsénio and Saint-Raymond?

Key findings

• The authors establish the existence of global-in-time classical solutions for the two-species VMB system for all ε ∈ (0,1], under suitable smallness assumptions on initial fluctuations.
• Uniform energy bounds are proven for the fluctuations of the VMB solutions with respect to the Knudsen number ε, independent of ε ∈ (0,1].
• The fluctuations of the VMB solutions converge strongly to classical solutions of the two-fluid incompressible NSFM system with Ohm’s law as ε → 0.
• The convergence is justified in the classical solution framework, providing a rigorous analogue to the renormalized solution limit established by Arsénio and Saint-Raymond.
• The energy estimates are closed via an induction argument on the derivative order, with ε-dependent dissipation terms carefully controlled using Gronwall-type inequalities.
• The modified energy functional includes time-integrated dissipation terms that ensure uniform control even as ε → 0, enabling the hydrodynamic limit.