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Stability of the spatially homogeneous Landau equation in relative entropy and applications to score-based numerical methods

Vasily Ilin

TL;DR

This work analyzes the spatially homogeneous Landau equation with Coulomb collisions in 3D, establishing stability of strong solutions in relative entropy and providing quantitative, explicit KL bounds. The authors derive a concise entropy-based differential inequality for $KL(f_t|g_t)$ whose coefficients are governed by moments, $L^p$ norms, and weighted Fisher information, yielding a KL decay control up to a growth factor. The same computation delivers a KL-based a posteriori error bound for score-based deterministic solvers, linking the training loss to the KL error and enabling practical numerical guarantees for methods such as SBTM and blob-based approximations. The main result shows KL-stability $\frac{d}{dt}KL(f_t|g_t) \le C(1+t)KL(f_t|g_t)$, giving $KL(f_t|g_t) \le C e^{t^2} KL(f_0|g_0)$, and extends to a score-matching framework where numerical approximations satisfy a similar differential inequality driven by a weighted score-matching loss. This provides a velocity-space, tensor-free analytic framework that connects Landau stability to score-based transport methods with actionable error bounds.

Abstract

We give a short and elementary proof of stability for strong solutions of the spatially homogeneous Landau equation with Coulomb collisions, measured in relative entropy. The argument yields an explicit differential inequality for relative entropy under natural moment and regularity assumptions. The same computation provides an a posteriori error bound for score-based transport modeling and related deterministic numerical schemes, linking the training loss to the relative-entropy error.

Stability of the spatially homogeneous Landau equation in relative entropy and applications to score-based numerical methods

TL;DR

This work analyzes the spatially homogeneous Landau equation with Coulomb collisions in 3D, establishing stability of strong solutions in relative entropy and providing quantitative, explicit KL bounds. The authors derive a concise entropy-based differential inequality for whose coefficients are governed by moments, norms, and weighted Fisher information, yielding a KL decay control up to a growth factor. The same computation delivers a KL-based a posteriori error bound for score-based deterministic solvers, linking the training loss to the KL error and enabling practical numerical guarantees for methods such as SBTM and blob-based approximations. The main result shows KL-stability , giving , and extends to a score-matching framework where numerical approximations satisfy a similar differential inequality driven by a weighted score-matching loss. This provides a velocity-space, tensor-free analytic framework that connects Landau stability to score-based transport methods with actionable error bounds.

Abstract

We give a short and elementary proof of stability for strong solutions of the spatially homogeneous Landau equation with Coulomb collisions, measured in relative entropy. The argument yields an explicit differential inequality for relative entropy under natural moment and regularity assumptions. The same computation provides an a posteriori error bound for score-based transport modeling and related deterministic numerical schemes, linking the training loss to the relative-entropy error.
Paper Structure (7 sections, 8 theorems, 72 equations)

This paper contains 7 sections, 8 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Related Work and Contributions.
  3. Organization.
  4. Known results
  5. Key estimates
  6. Main results
  7. Proof idea.

Key Result

Lemma 2.1

Let $f_t$ satisfy the Landau equation with $-4 < \gamma < -2$ and $\gamma \ge -d$. Assume the initial data $f_0$ has finite energy, entropy, Fisher information and enough moments: Then the solution to the Landau equation $f_t$ is unique, smooth, and satisfies and for any $k,\kappa>0$, and any $\varepsilon>0$, there exist $\alpha>0$ and $s>0$ such that for all $t>0$, If we assume further that $\

Theorems & Definitions (14)

  • Lemma 2.1: Regularity of solutions
  • Lemma 2.2: Coercivity of $A\ast f$
  • Lemma 2.3: $L^2$ version of Pinsker's inequality
  • proof
  • Theorem 3.1: Growth estimates on weighted Fisher information
  • proof
  • Lemma 3.2: Sufficient assumption on initial data
  • proof
  • Lemma 3.3
  • proof
  • ...and 4 more