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On the monotonicity of the entropy production in the Landau-Maxwell equation

Côme Tabary

Abstract

We study the homogeneous Landau equation with Maxwell molecules and prove that the entropy production is non-increasing provided the directional temperatures are well-distributed and the solution admits a moment of order $\ell$, for some $\ell$ arbitrarily close to $2$. It implies that for an initial condition with finite moment of order $\ell$, the entropy production is guaranteed to be non-increasing after a certain time, that we explicitly compute. This is the first partial answer to a conjecture made by Henry P. McKean in 1966 on the sign of the time-derivatives of the entropy. We also obtain algebraic decay estimates for the entropy production for large time; as well as a short-time estimate without moment assumptions.

On the monotonicity of the entropy production in the Landau-Maxwell equation

Abstract

We study the homogeneous Landau equation with Maxwell molecules and prove that the entropy production is non-increasing provided the directional temperatures are well-distributed and the solution admits a moment of order , for some arbitrarily close to . It implies that for an initial condition with finite moment of order , the entropy production is guaranteed to be non-increasing after a certain time, that we explicitly compute. This is the first partial answer to a conjecture made by Henry P. McKean in 1966 on the sign of the time-derivatives of the entropy. We also obtain algebraic decay estimates for the entropy production for large time; as well as a short-time estimate without moment assumptions.
Paper Structure (13 sections, 18 theorems, 162 equations)

This paper contains 13 sections, 18 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Background and main result
  3. Literature review and discussion
  4. Notation and layout
  5. Properties of the Landau-Maxwell equation
  6. Moments
  7. The entropy production as a Fisher information
  8. The lifted Landau operator through vector fields
  9. A formula for the derivative of the entropy production
  10. Analysis of the formula and conclusion
  11. Introducing relative Fisher informations
  12. Exploiting the second-order term
  13. Conclusion

Key Result

Theorem 1.1

Let $f_0\geq 0$ with finite mass and energy satisfying the normalization and let $T_{0,\max}=\max (T_{0,i})_{1\leq i \leq d}\geq 1$. Also suppose that for some $\ell>2$, $f_0$ has a finite moment of order $\ell$: Let $f=(f_t)_{t\geq 0}$ be the solution to the Landau equation eq:landau with Maxwell molecules $\alpha=1$. Then,

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 42 more