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Weak-strong uniqueness for the Landau equation by a relative entropy method

Côme Tabary

TL;DR

This work establishes a weak-strong uniqueness principle for the space-homogeneous Landau equation in soft potentials, including Coulomb interactions, by measuring the distance between a weak H-solution and a strong solution via the relative entropy H(f|g). A novel relative-entropy estimate is derived, relying on ln g as a test function and a careful decomposition into a positive entropy-production term and a controllable error, with a Gronwall-type bound that depends on conditional logarithmic bounds on the strong solution g. The authors provide sufficient initial-data conditions ensuring the strong solution satisfies these logarithmic bounds in very soft potentials, using a combination of parabolic maximum principles, ellipticity estimates, and Schauder theory to propagate Maxwellian tails and obtain polynomial bounds on ln g and its derivatives. By coupling these tools, the paper advances a quantitative stability framework in L^1 for the Landau equation that complements existing integrability-based uniqueness results and clarifies the regularity requirements needed for weak-strong stability in the grazing-collision regime. The results have potential implications for rigorously passing grazing collision limits and for understanding stability of plasmas under very soft interactions.

Abstract

We derive a weak-strong uniqueness and stability principle for the Landau equation in the soft potentials case (including Coulomb interactions). The distance between two solutions is measured by their relative entropy, which to our knowledge was never used before in stability estimates. The logarithm of the strong solution is required to have polynomial growth while the weak solution can be any H-solution with sufficiently many moments at initial time. Since we require a substantial amount of regularity on the strong solution, we also provide an example of sufficient conditions on the initial data that ensure this regularity in the Coulomb (and very soft potentials) case.

Weak-strong uniqueness for the Landau equation by a relative entropy method

TL;DR

This work establishes a weak-strong uniqueness principle for the space-homogeneous Landau equation in soft potentials, including Coulomb interactions, by measuring the distance between a weak H-solution and a strong solution via the relative entropy H(f|g). A novel relative-entropy estimate is derived, relying on ln g as a test function and a careful decomposition into a positive entropy-production term and a controllable error, with a Gronwall-type bound that depends on conditional logarithmic bounds on the strong solution g. The authors provide sufficient initial-data conditions ensuring the strong solution satisfies these logarithmic bounds in very soft potentials, using a combination of parabolic maximum principles, ellipticity estimates, and Schauder theory to propagate Maxwellian tails and obtain polynomial bounds on ln g and its derivatives. By coupling these tools, the paper advances a quantitative stability framework in L^1 for the Landau equation that complements existing integrability-based uniqueness results and clarifies the regularity requirements needed for weak-strong stability in the grazing-collision regime. The results have potential implications for rigorously passing grazing collision limits and for understanding stability of plasmas under very soft interactions.

Abstract

We derive a weak-strong uniqueness and stability principle for the Landau equation in the soft potentials case (including Coulomb interactions). The distance between two solutions is measured by their relative entropy, which to our knowledge was never used before in stability estimates. The logarithm of the strong solution is required to have polynomial growth while the weak solution can be any H-solution with sufficiently many moments at initial time. Since we require a substantial amount of regularity on the strong solution, we also provide an example of sufficient conditions on the initial data that ensure this regularity in the Coulomb (and very soft potentials) case.

Paper Structure

This paper contains 17 sections, 25 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Review of the literature and discussion
  4. Definitions and notation
  5. Acknowledgments
  6. The weak-strong relative entropy estimate
  7. Setup
  8. Using $\ln g$ as a test function
  9. Evolution of the relative entropy
  10. Control of the bad term by the good one
  11. Logarithmic bounds
  12. Building the solution
  13. Ellipticity and propagation of Maxwellian bounds
  14. Using Schauder estimates to derive logarithmic bounds
  15. Maximum principle for the Landau equation
  16. ...and 2 more sections

Key Result

Theorem 1.1

Fix $\gamma\in [-3,0)$ and a final time $T>0$. Let $g=(f_t)_{t\in[0,T]}$ be a H-solution of the Landau equation on $[0,T]$ with initial data $f_0$. Let $g=(g_t)_{t\in[0,T]}$ be a classical solution to the Landau equation for $C^2$ initial data $g_0>0$ (with unit mass, finite energy and finite entrop Further assume that the $(\rho-\gamma)$-th moment of $f_0$ is also bounded: Then the relative entr

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Lemma 2.1: Propagation of moments
  • Lemma 2.2: Entropy production estimate
  • Lemma 2.3
  • proof
  • ...and 44 more