Relativistic Vlasov-Maxwell-Landau System

• The relativistic Vlasov-Maxwell-Landau system is a kinetic model that governs collisional relativistic plasmas through coupled electromagnetic fields and relativistic Landau collision dynamics.
• It uses Hilbert expansion to rigorously derive hydrodynamic limits, demonstrating convergence to the relativistic Euler-Maxwell system in the strong-collision regime.
• Analytical tools such as weighted energy estimates, macro-micro decompositions, and tailored boundary conditions ensure global well-posedness and decay properties.

The relativistic Vlasov-Maxwell-Landau (r-VML) system is the fundamental kinetic model for collisional relativistic plasmas coupled self-consistently with electromagnetic fields. It governs the time evolution of the phase space particle density for each species, incorporating the Lorentz force and relativistic Landau (Coulomb) collision dynamics. The model rigorously describes the hydrodynamic-to-kinetic transition in weakly-coupled, relativistic electron or ionized gases and provides the foundational analytic structure for understanding the emergence of relativistic fluid systems, limits such as the non-relativistic or electrostatic regimes, and the role of boundaries and singularities.

1. Mathematical Formulation of the Relativistic Vlasov-Maxwell-Landau System

Let $f^\varepsilon(t,x,p)$ denote the one-species phase space electron density with $(x,p)\in\mathbb R^3_x\times\mathbb R^3_p$, where $p^0=\sqrt{m^2c^2+|p|^2}$ and $\hat p=p/p^0$ are the relativistic energy and velocity, respectively. The self-consistent electromagnetic field is $$(E^\varepsilon,B^\varepsilon)(t,x)\in\mathbb R^3\times\mathbb R^3$$, and the Knudsen number $\varepsilon>0$ measures the mean free path relative to system size. For normalized $c=m=e=k_B=1$, the r-VML system reads (Ouyang et al., 2022):

$$\begin{aligned} &\partial_t f^\varepsilon + \hat p\cdot\nabla_x f^\varepsilon - (E^\varepsilon+\hat p\times B^\varepsilon)\cdot\nabla_p f^\varepsilon = \frac{1}{\varepsilon} Q_L(f^\varepsilon,f^\varepsilon), \ &\partial_t E^\varepsilon - \nabla_x\times B^\varepsilon = 4\pi e \int_{\mathbb R^3} \hat p\,f^\varepsilon\,dp, \quad \partial_t B^\varepsilon +\nabla_x\times E^\varepsilon=0, \ &\nabla_x\cdot E^\varepsilon =-4\pi e \left(n_{\rm ion}-\int_{\mathbb R^3}f^\varepsilon\,dp\right),\quad \nabla_x\cdot B^\varepsilon=0. \end{aligned}$$

The relativistic Landau collision operator is

$$Q_L(f,g)(p) = \nabla_p\cdot\int_{\mathbb R^3} \Phi(p,q) \left[\nabla_p f(p)\,g(q) - f(p)\,\nabla_q g(q)\right]\,dq,$$

with the symmetric kernel $\Phi(p,q)$ enforcing conservation of mass, momentum, and energy.

For multi-species, the two-component RVML system is formulated analogously, with suitably coupled collision and field terms (Cao et al., 23 Nov 2025). The system incorporates both fully relativistic kinetic transport and Maxwell's equations, with the collision operator reflecting the instantaneous grazing Coulomb limit.

2. Hydrodynamic Limits and Hilbert Expansion

The central analytic program is the Hilbert expansion, which demonstrates that the r-VML system, in the strong-collision (small Knudsen) limit $\varepsilon\to0$, converges to relativistic fluid dynamics, specifically the Euler-Maxwell system. The expansion ansatz is (Ouyang et al., 2022):

$$f^\varepsilon = \sum_{k=0}^N \varepsilon^k f_k(t,x,p) + \varepsilon^{N+1} r^\varepsilon(t,x,p), \quad E^\varepsilon = \sum_{k=0}^N \varepsilon^k E_k + \varepsilon^{N+1} e_R^\varepsilon,$$

with analogous expansion for $B^\varepsilon$. The leading term $f_0$ is a Jüttner-Maxwellian

$$M[n,u,T]=\frac{n}{4\pi T K_2(\gamma)}\exp\left(\frac{u^\mu p_\mu}{T}\right),$$

where the fluid parameters $(n,u,T)$ satisfy the compressible relativistic Euler-Maxwell PDEs. The higher-order terms $f_k,E_k,B_k$ are solvable iteratively via moment methods and elliptic inversion, with uniformly controlled remainder (Ouyang et al., 2022).

The main results rigorously establish global-in-time existence, regularity of the expansion, and convergence rates for appropriate weighted Sobolev energies and show that $\|f^\varepsilon-f_0\|\rightarrow 0$ as $\varepsilon\to0$ on timescales $O(\varepsilon^{-1/3})$.

3. Functional Analysis: Weighted Energy and Coercivity

Analysis of r-VML systems fundamentally relies on weighted, time-dependent energy functionals and microscopic/macroscopic decompositions. The critical innovation is the introduction of exponential-in-momentum time weights,

$$w^\ell(p,t) = (p^0)^{2(N_c-\ell)}\exp\left(\frac{p^0}{5T_c \ln(e+t)}\right),\quad \ell=0,1,2,$$

chosen such that the time derivative of $w^\ell$ introduces a $p^0(w^\ell)^2$ damping term, offsetting the unbounded velocity growth from $\hat p\cdot\nabla_x f$ (Ouyang et al., 2022, Cao et al., 23 Nov 2025). Energy estimates are closed in norms

$$\|f\|_{H^s_{w,\sigma}}^2 = \sum_{|\alpha|\le s} \int_x \langle w^\ell \partial_x^\alpha f,\,w^\ell \partial_x^\alpha f \rangle_\sigma dx,$$

where the Landau $\sigma$-norm quantifies collision-mediated dissipation in momentum. Coercivity of the linearized Landau operator $L$, essential for stability and decay, holds uniformly in key parameters (notably the speed of light $c\ge1$) (Cao et al., 23 Nov 2025).

The proof framework relies on spectral gap/coercivity, commutator estimates (since $L$ and hydrodynamic projections $P$ do not commute with derivatives or weights), and macro-micro decompositions to treat kinetic-field coupling and exploit Maxwell dissipative structure.

4. Boundary Conditions and Bounded Domain Theory

The theory for r-VML in bounded domains addresses specular reflection for the kinetic component and perfect conductor conditions for the field (Dong et al., 2023, Dong et al., 2023). For a spatial domain $\Omega\subset\mathbb R^3$ with smooth boundary:

• Specular reflection for $f$ on inflow: $f(t,x,p)|_{p\cdot n_x<0}=f(t,x,p-2(p\cdot n_x)n_x)$.
• Field boundary: $E\times n_x=0$, $B\cdot n_x=0$.

Local and global well-posedness in such settings are established under mild geometric/topological assumptions, and at high regularity ($m\ge 20$ derivatives), for small initial perturbations in instant energy norms. Proofs address singularities generated by the Lorentz force at the wall, via velocity averaging, mirror extensions, and a functional framework based on velocity-weighted Sobolev norms (Dong et al., 2023, Dong et al., 2023).

The global theory ensures uniqueness, polynomial-in-time decay of energies (for $k$ derivatives, decay rate $(1+t)^{-k}$), and control over boundary contributions, providing a rigorous basis for modeling plasma-wall interactions in, e.g., fusion designs.

5. Classical and Nonrelativistic Limits

The precise transition from r-VML to the nonrelativistic Vlasov-Poisson-Landau (VPL) system occurs as $c\to\infty$, where electromagnetic fields become effectively instantaneous and particle transport reduces to Newtonian kinematics (Cao et al., 23 Nov 2025). The main technical achievements are:

• Uniform coercivity of the relativistic Landau collision operator in $c\ge1$.
• Weighted energy dissipation estimates that accommodate the weakening Maxwellian damping as $c$ increases.
• Quantitative convergence: for solutions $[F^c,E^c,B^c]$ of (RVML) and $[F^\infty,E^\infty]$ of (VPL) with matched initial data,

$$\sup_{t\ge0} \left\{ \|F^c(t)-F^\infty(t)\|_{L^2_{x,p}} + \|E^c(t)-E^\infty(t)\|_{L^2_x} + \|B^c(t)-\bar B^0\|_{L^2_x} \right\} \le \frac{C}{c}.$$

• Decay rate analysis and convergence of the full electromagnetic field $B^c$ to a constant background.
• Stability and applicability to more general collisional models where uniform bounds and weighted energy arguments are available.

This regime is relevant for describing plasmas where the characteristic velocities are small compared to the speed of light, justifying the use of classical theories.

6. Well-posedness, Regularity, and Decay

Comprehensive theory has been developed for local and global well-posedness, uniqueness, and asymptotic decay, both in periodic and bounded domains (Ouyang et al., 2022, Dong et al., 2023, Dong et al., 2023, Cao et al., 23 Nov 2025). The analysis is based on:

• High-order instant energy $\mathcal E_m(t)$ and dissipation $\mathcal D_m(t)$ functionals, encompassing time/space derivatives, velocity weights, and macroscopic moments.
• Macro-micro decomposition, leveraging dissipation in the complement of the collision invariants.
• Velocity averaging, div–curl estimates, and Sobolev embeddings to treat boundary and kinetic–field coupling.
• Mirror extension and odd-Sobolev arguments to treat specular boundaries and control singular behavior.

Well-posedness statements require smallness in weighted Sobolev norms of the initial perturbations, high smoothness, and domain geometric constraints for bounded cases. Solutions enjoy uniqueness, propagation of regularity, and temporal decay of energies, with polynomial rates reflecting the dissipative structure and absence of uniform spectral gaps for the full nonlinear system.

7. Open Problems and Research Directions

Current research continues to address several open directions:

• Extension to the whole-space case, especially for weak decay or large initial data (Cao et al., 23 Nov 2025).
• Justification of hydrodynamic and nonrelativistic limits for more general collision kernels (e.g., non-cutoff Boltzmann, soft potentials).
• Bounded domains with nontrivial topology, boundary layers, and full two-species ion-electron coupling (Ouyang et al., 2022, Dong et al., 2023).
• Decay and regularity under reduced regularity or compatibility assumptions.
• Numerical and computational challenges associated with unbounded velocity growth, boundary singularities, and kinetic–field coupling.

The rigorous completion of the relativistic Hilbert program, the quantification of rates at physical limits, and the control of plasma dynamics in experimental geometries remain active fields of investigation.