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Stability estimates for magnetized Vlasov equations

Alexandre Rege

TL;DR

This work studies stability of magnetized Vlasov–Poisson and Vlasov equations using optimal transport tools. It extends Loeper's $2$-Wasserstein stability and Iacobelli's nonlinear control to the magnetized VP system with non-uniform magnetic field, revealing anisotropy that requires stronger velocity decay in one datum, and it derives two explicit stability statements under suitable smallness assumptions. It also provides an improved Dobrushin-type $W_1$-estimate for the magnetized Vlasov equation with a uniform magnetic field, with explicit dimension-dependent bounds that recover the unmagnetized limits as the magnetic field vanishes. Together, these results illuminate how external magnetic fields influence stability in kinetic models and offer quantitative, transport-based stability criteria applicable to quasi-neutral and related regimes.

Abstract

We present two results related to magnetized Vlasov equations. Our first contribution concerns the stability of solutions to the magnetized Vlasov-Poisson system with a non-uniform magnetic field using the optimal transport approach introduced by Loeper [24]. We show that the extra magnetized terms can be suitably controlled by imposing stronger decay in velocity on one of the distribution functions, illustrating how the external magnetic field creates anisotropy in the evolution. This allows us to generalize the classical 2-Wasserstein stability estimate by Loeper [24, Theorem 1.2] and the recent stability estimate using a kinetic Wasserstein distance by Iacobelli [20, Theorem 3.1] to the magnetized Vlasov-Poisson system. In our second result, we extend the improved Dobrushin estimate by Iacobelli [20, Theorem 2.1] to the magnetized Vlasov equation with a uniform magnetic field.

Stability estimates for magnetized Vlasov equations

TL;DR

This work studies stability of magnetized Vlasov–Poisson and Vlasov equations using optimal transport tools. It extends Loeper's -Wasserstein stability and Iacobelli's nonlinear control to the magnetized VP system with non-uniform magnetic field, revealing anisotropy that requires stronger velocity decay in one datum, and it derives two explicit stability statements under suitable smallness assumptions. It also provides an improved Dobrushin-type -estimate for the magnetized Vlasov equation with a uniform magnetic field, with explicit dimension-dependent bounds that recover the unmagnetized limits as the magnetic field vanishes. Together, these results illuminate how external magnetic fields influence stability in kinetic models and offer quantitative, transport-based stability criteria applicable to quasi-neutral and related regimes.

Abstract

We present two results related to magnetized Vlasov equations. Our first contribution concerns the stability of solutions to the magnetized Vlasov-Poisson system with a non-uniform magnetic field using the optimal transport approach introduced by Loeper [24]. We show that the extra magnetized terms can be suitably controlled by imposing stronger decay in velocity on one of the distribution functions, illustrating how the external magnetic field creates anisotropy in the evolution. This allows us to generalize the classical 2-Wasserstein stability estimate by Loeper [24, Theorem 1.2] and the recent stability estimate using a kinetic Wasserstein distance by Iacobelli [20, Theorem 3.1] to the magnetized Vlasov-Poisson system. In our second result, we extend the improved Dobrushin estimate by Iacobelli [20, Theorem 2.1] to the magnetized Vlasov equation with a uniform magnetic field.
Paper Structure (9 sections, 6 theorems, 95 equations)

This paper contains 9 sections, 6 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Stability for the magnetized Vlasov--Poisson system
  3. Main result
  4. Preliminaries and notations
  5. Proof of \ref{['theo:main']}
  6. Stability estimates when $B$ is log-lipschitz in position
  7. An improved Dobrushin estimate for the magnetized Vlasov equation
  8. Main result and preliminaries
  9. Proof of \ref{['theo:2']}

Key Result

THEOREM 2.1

Let $f_1,f_2$ be two weak solutions of sys:VPwB with respective densities $\rho_1$ and $\rho_2$. We define the function We also define For $T>0$, assume that $B$ verifies reguB and that $A$ satisfies, Assume also that there exists a universal constant $C_0$ such that for all $k\geq 1$ Then we can write the two following statements:

Theorems & Definitions (22)

  • DEFINITION 1.1
  • THEOREM 2.1
  • REMARK 2.2
  • REMARK 2.3
  • REMARK 2.4
  • REMARK 2.5
  • REMARK 2.6
  • LEMMA 2.7
  • proof
  • LEMMA 2.8
  • ...and 12 more