[Paper Review] On the relativistic Vlasov-Poisson system

Michael K.‐H. Kiessling, A. Shadi Tahvildar‐Zadeh|ArXiv.org|Aug 13, 2007

Gas Dynamics and Kinetic Theory16 references3 citations

TL;DR

This paper establishes the existence of unique global classical solutions to the relativistic Vlasov-Poisson system in the attractive case ($\sigma = -1$) under specific initial conditions: spherically symmetric, compactly supported initial data in momentum space, vanishing on zero-angular-momentum characteristics, and $L^\beta$-norm below a critical threshold $C_\beta$. It further proves that if the $L^\beta$-norm exceeds $C_\beta$, finite-time blow-up occurs, with the sharp threshold $C_{3/2} = \frac{3}{8}\left(\frac{15}{16}\right)^{1/3}$ when $\|f_0\|_{L^1} = 1$, and shows that $L^{3/2}$ is the optimal integrability condition for global existence.

ABSTRACT

The Cauchy problem is revisited for the so-called relativistic Vlasov-Poisson system in the attractive case. Global existence and uniqueness of spherical classical solutions is proved under weaker assumptions than previously used. A new class of blowing up solutions is found when these conditions are violated. A new, non-gravitational physical vindication of the model which (unlike the gravitational one) is not restricted to weak fields, is also given.

Motivation & Objective

• To re-examine the Cauchy problem for the relativistic Vlasov-Poisson system in the attractive case ($\sigma = -1$), originally studied by Glassey and Schaeffer (1985).
• To determine the sharp threshold for global existence of classical solutions in terms of the $L^\beta$-norm of the initial data $f_0$.
• To show that the $L^{3/2}$-norm condition is optimal, as any weaker $L^\beta$-bound with $\beta < 3/2$ fails to prevent blow-up.
• To provide a non-gravitational physical justification for the model, valid beyond the weak-field limit.

Proposed method

• The analysis relies on the relativistic Vlasov-Poisson system with $\sigma = -1$, where $f_t(p,q)$ is the phase-space density, $v = p / \sqrt{1 + |p|^2}$, and $\phi_t$ satisfies Poisson's equation $\Delta_q \phi_t = 4\pi \int f_t \, dp$.
• The initial data $f_0$ are assumed spherically symmetric, compactly supported in momentum space, and vanish on characteristics with zero angular momentum.
• The existence of global classical solutions is proven when $\|f_0\|_{L^\beta} < C_\beta$ for $\beta \geq 3/2$, with $C_\beta$ explicitly computed for $\beta = 3/2$.
• For $\beta > 3/2$, upper and lower bounds on $C_\beta$ are derived, which converge as $\beta \downarrow 3/2$, confirming the optimality of the $L^{3/2}$ threshold.
• A non-gravitational physical interpretation is provided via a mean-field limit of a regularized relativistic Vlasov-Maxwell system with equal numbers of positive and negative charges.
• The derivation involves a scaling limit where $N \to \infty$, $\epsilon \to 0$, and the self-force vanishes, leading to a decoupled Vlasov equation with an effective potential $\phi_t^\epsilon \to \phi_t$.

Experimental results

Research questions

• RQ1 What is the sharp $L^\beta$-norm threshold $C_\beta$ below which the relativistic Vlasov-Poisson system admits global classical solutions for $\beta \geq 3/2$?
• RQ2 Can the $L^{3/2}$-norm condition be weakened to $L^\beta$ with $\beta < 3/2$ while still ensuring global existence?
• RQ3 What happens to solutions when the $L^\beta$-norm of the initial data exceeds the critical threshold $C_\beta$?
• RQ4 Is there a physical justification for the attractive relativistic Vlasov-Poisson system beyond the weak-field gravitational limit?
• RQ5 How does the spherical symmetry and compact support in momentum space influence the global existence and blow-up behavior?

Key findings

• A unique global classical solution exists for the relativistic Vlasov-Poisson system in the attractive case if the initial data $f_0$ are spherically symmetric, compactly supported in momentum space, vanish on zero-angular-momentum characteristics, and satisfy $\|f_0\|_{L^\beta} < C_\beta$ for $\beta \geq 3/2$.
• The sharp value of the critical threshold is $C_{3/2} = \frac{3}{8}\left(\frac{15}{16}\right)^{1/3}$ when $\|f_0\|_{L^1} = 1$, and this value is optimal in the sense that no $L^\beta$-bound with $\beta < 3/2$ can replace it.
• For $\beta \in (1, 3/2)$, the critical threshold $C_\beta = 0$, meaning that no non-zero initial data in $L^\beta$ can yield global solutions.
• When the $L^\beta$-norm of $f_0$ exceeds $C_\beta$ for any $\beta \in (1, \infty)$, classical initial data exist that lead to finite-time blow-up.
• The $L^{3/2}$-norm condition is optimal: the system cannot be globally well-posed under any weaker $L^\beta$-integrability assumption with $\beta < 3/2$, regardless of the bound.
• A non-gravitational physical interpretation is established via a mean-field limit of a regularized relativistic Vlasov-Maxwell system with equal numbers of positive and negative charges, where the effective potential arises from a net attractive force due to imbalance in charge counts.