Math Conjectures

Global classical solutions of the relativistic Vlasov–Maxwell system

OPENMajorOpen problemProposed c. 1980 · Canonical special case

Canonical statement

Let f0±∈Cc∞(Rx3×Rp3)f^\pm_0\in C_c^\infty(\mathbb R^3_x\times\mathbb R^3_p) be nonnegative and let E0,B0∈Cc∞(R3;R3)E_0,B_0\in C_c^\infty(\mathbb R^3;\mathbb R^3) satisfy ∇⋅E0=∫(f0+−f0−) dp\nabla\cdot E_0=\int(f^+_0-f^-_0)\,dp and ∇⋅B0=0\nabla\cdot B_0=0. With v(p)=p/1+∣p∣2v(p)=p/\sqrt{1+|p|^2}, the system
∂tf±+v⋅∇xf±±(E+v×B)⋅∇pf±=0,∂tE−∇×B=−j,∂tB+∇×E=0,∇⋅E=ρ,∇⋅B=0,ρ=∫R3(f+−f−) dp,j=∫R3v(p)(f+−f−) dp \begin{aligned} &\partial_tf^\pm+v\cdot\nabla_xf^\pm \pm(E+v\times B)\cdot\nabla_pf^\pm=0,\\ &\partial_tE-\nabla\times B=-j,\quad \partial_tB+\nabla\times E=0,\quad \nabla\cdot E=\rho,\quad\nabla\cdot B=0,\\ &\rho=\int_{\mathbb R^3}(f^+-f^-)\,dp,\qquad j=\int_{\mathbb R^3}v(p)(f^+-f^-)\,dp \end{aligned}
has a unique classical solution for all t≥0t\ge0.
View source LaTeX
Let \(f^\pm_0\in C_c^\infty(\mathbb R^3_x\times\mathbb R^3_p)\) be nonnegative and let \(E_0,B_0\in C_c^\infty(\mathbb R^3;\mathbb R^3)\) satisfy \(\nabla\cdot E_0=\int(f^+_0-f^-_0)\,dp\) and \(\nabla\cdot B_0=0\). With \(v(p)=p/\sqrt{1+|p|^2}\), the system \[ \begin{aligned} &\partial_tf^\pm+v\cdot\nabla_xf^\pm \pm(E+v\times B)\cdot\nabla_pf^\pm=0,\\ &\partial_tE-\nabla\times B=-j,\quad \partial_tB+\nabla\times E=0,\quad \nabla\cdot E=\rho,\quad\nabla\cdot B=0,\\ &\rho=\int_{\mathbb R^3}(f^+-f^-)\,dp,\qquad j=\int_{\mathbb R^3}v(p)(f^+-f^-)\,dp \end{aligned} \] has a unique classical solution for all \(t\ge0\).
PDE-006:[GlasseyStrauss1986VlasovMaxwell][DiPernaLions1989VlasovMaxwell][LukStrain2014VlasovCriterion][Wang2026CylindricalVlasov]
Partial differential equationsMathematical physicsProbability theory

Notes

The relativistic Vlasov–Maxwell system describes a two-species collisionless plasma: particle densities f±f^\pm on phase space are transported along relativistic trajectories bent by the Lorentz force, while the charge and current they carry source the Maxwell equations. The problem, which took shape around 1980, asks whether smooth, compactly supported initial data always yield a unique global classical solution.

The structural result of Glassey and Strauss shows that a singularity could form only if the momentum support of the particle densities becomes unbounded, so global regularity reduces to controlling how fast particles are accelerated [GlasseyStrauss1986VlasovMaxwell]; Luk and Strain later gave a new continuation criterion [LukStrain2014VlasovCriterion]. Global weak solutions were constructed by DiPerna and Lions [DiPernaLions1989VlasovMaxwell], classical solutions are global for small data, and large-data global results are available under lower-dimensional or symmetry restrictions, most recently for cylindrically symmetric data [Wang2026CylindricalVlasov].

What remains open is the unrestricted three-dimensional large-data case: no argument yet controls the momentum support in general, and the global existence of classical solutions is unresolved.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.