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On the semi-classical limit for the Landau-Fermi-Dirac equation

Paulo Sampaio

TL;DR

We address the semiclassical limit of the inhomogeneous Landau–Fermi–Dirac equation with Coulomb potential by constructing suitable approximations and proving a diagonal compactness result as $\\varepsilon\\to0$. The main achievement is that, up to subsequences, the LFD solutions converge to a renormalized Landau solution with a defect measure in the Villani sense, while conservation laws and the entropy inequality are preserved in the limit. A companion result handles the diffusion-quadratic term via a square-root truncation scheme, yielding convergence with an accompanying defect measure. The work rigorously connects the quantum LFD dynamics to the classical Landau dynamics in the inhomogeneous setting and justifies using LFD as an accurate semiclassical approximation.

Abstract

We study sequences of solutions to the inhomogeneous Landau-Fermi-Dirac equation with Coulomb potential in which the quantum parameter converges to zero. Our main result establishes the compactness of these sequences, which allows us to show that, up to a subsequence, these solutions converge to a renormalized solution of the classical Landau equation with a defect measure, as defined by Villani. To do this, we work in the class of solutions that are obtained through approximation procedures. For these solutions, we were able to show compactness in the vanishing quantum parameter limit through a diagonal argument, which combines techniques from the study of Cauchy problems for both the classical Landau and the Landau-Fermi-Dirac equations.

On the semi-classical limit for the Landau-Fermi-Dirac equation

TL;DR

We address the semiclassical limit of the inhomogeneous Landau–Fermi–Dirac equation with Coulomb potential by constructing suitable approximations and proving a diagonal compactness result as . The main achievement is that, up to subsequences, the LFD solutions converge to a renormalized Landau solution with a defect measure in the Villani sense, while conservation laws and the entropy inequality are preserved in the limit. A companion result handles the diffusion-quadratic term via a square-root truncation scheme, yielding convergence with an accompanying defect measure. The work rigorously connects the quantum LFD dynamics to the classical Landau dynamics in the inhomogeneous setting and justifies using LFD as an accurate semiclassical approximation.

Abstract

We study sequences of solutions to the inhomogeneous Landau-Fermi-Dirac equation with Coulomb potential in which the quantum parameter converges to zero. Our main result establishes the compactness of these sequences, which allows us to show that, up to a subsequence, these solutions converge to a renormalized solution of the classical Landau equation with a defect measure, as defined by Villani. To do this, we work in the class of solutions that are obtained through approximation procedures. For these solutions, we were able to show compactness in the vanishing quantum parameter limit through a diagonal argument, which combines techniques from the study of Cauchy problems for both the classical Landau and the Landau-Fermi-Dirac equations.
Paper Structure (12 sections, 11 theorems, 204 equations)

This paper contains 12 sections, 11 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Weak compactness of diagonal sequences
  4. From weak to renormalized solutions
  5. Diagonal compactness of approximating schemes
  6. Velocity Averaging
  7. Almost everywhere compactness
  8. Proof of Theorem 1
  9. Convergence to renormalized solutions
  10. Conservation laws and entropy inequality
  11. Entropy inequality
  12. Proof of Theorem 2

Key Result

Theorem 1

Let $\varepsilon_n \to 0$. For each $n$, let $f^n$ be a suitable solution of the LFD equation with quantum parameter $\varepsilon_n$ and initial data $0 \leq f^n_0 \leq \varepsilon^{-1}_n$. For each $n$, let $f^n_m \to f^n$ be an approximating scheme with diffusion $(\overline{a}^{n,m})_m$. Suppose with $\Gamma_m$ such that $\Gamma_m(|z|) \to \Gamma(|z|)$ in $L^1_{loc}(\mathbb{R}^N)$ and for ever

Theorems & Definitions (23)

  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2
  • Remark 4
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • ...and 13 more