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Exponential reference-relative entropy bound for homogeneous BGK

Statement

Assume Countable Choice ${A C}_{ω}$ : every $N$ -indexed family of nonempty sets has a choice function. Let $v_{*} = 1 m s^{- 1}$ , $ω_{3} (v) = \sum_{j = 0}^{3} (∣ v ∣ / v_{*})^{j}$ , let $τ > 0$ be a relaxation time in seconds, and let $ϵ > 0$ be dimensionless. Let $f_{0} : R^{3} \to [0, \infty)$ be a finite-valued measurable velocity density with $\int ω_{3} f_{0} d v < \infty$ , number density $n = \int f_{0} d v > 0$ , and strictly positive centered second moment $\int ∣ v - u_{0} ∣^{2} f_{0} d v > 0$ , where $u_{0} = n^{- 1} \int v f_{0} d v$ . Fix a constant reference density $f_{r e f} > 0$ with the same units as $f_{0}$ , and assume $\int f_{0} ∣ \log (f_{0} / f_{r e f}) ∣ d v < \infty$ , using $0 \log 0 : = 0$ and assigning $f_{0} ∣ \log (f_{0} / f_{r e f}) ∣$ the value zero where $f_{0} = 0$ . These are the hypotheses of the homogeneous BGK relaxation theorem; the fixed $m > 0$ and $k_{B} > 0$ and the Gaussian $M = M [f_{0}]$ are as in the single-species BGK definition. Set $f^{ϵ} (t) = M + e^{- t / (ϵ τ)} (f_{0} - M), t \geq 0.$ For each nonnegative velocity density $g$ with the same units as $f_{0}$ and for which the integral is absolutely finite, define $H (g) : = \int_{R^{3}} g (v) \log (g (v) / f_{r e f}) d v, 0 \log 0 : = 0.$ Then for every time $t \geq 0$ measured in seconds, $0 \leq H (f^{ϵ} (t)) - H (M) \leq e^{- t / (ϵ τ)} (H (f_{0}) - H (M)) .$ The lower bound is relative entropy with respect to the moment-matched Maxwellian. This is a deduction for the adopted BGK model and its explicit solution, not an empirical entropy measurement or a claim about the binary collision operator.

Statement references

Countable Choice inherited from the kinetic measure and Gaussian-moment conventions

Tong, §2.5.1, full lecture notes

Golse, §§3.1–3.2, full chapter

Facts & Assumptions

Given: Countable Choice and the exact solution $f^{ϵ} = M + a (t) (f_{0} - M)$ with $a (t) = e^{- t / (ϵ τ)}$ , and equality of its mass, momentum, and energy moments with those of $M$ .

[F1]

Exact homogeneous BGK fast relaxation: The explicit solution is a convex combination of $f_{0}$ and $M$ and preserves their defining moments.

[F2]

Nonnegative kinetic density and moment domain and Gaussian Maxwellian normalization, stress, and heat flux: $\log (M / f_{r e f})$ is a constant plus a linear term and a quadratic term in velocity, and all its products with $f_{0}$ or $M$ are integrable.

[F3]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t and Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : The logarithm derivative and product rule give $Φ' (z) = \log z$ ; its sign shows the minimum at $1$ , hence $Φ (z) \geq 0$ .

[F4]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: On $(0, \infty)$ , $Φ^{''} (z) = 1 / z > 0$ , so $Φ$ is convex.

[F5]

The exponential dominates every fixed nonnegative integer power at $+ \infty$ : Setting $z = e^{- y}$ gives $z \log z = - y e^{- y} \to 0$ as $y \to \infty$ ; hence $Φ (0) = 1$ gives the continuous convex extension on $[0, \infty)$ .

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): the assumed Countable Choice is the kinetic-density and Gaussian-moment integration convention required by [F2].

[A1]

The nonnegative initial data has finite absolute third moment and finite absolute reference-relative entropy as in the homogeneous BGK theorem.

Proof

1.1A1A2F1F2F5algebra

Put $z_{0} = f_{0} / M$ and $Φ (z) = z \log z - z + 1$ , with $Φ (0) = 1$ . Since $M$ is a positive Gaussian, $\log (M / f_{r e f})$ is a quadratic polynomial plus a constant. The assumed finite absolute entropy and second moments make $\int f_{0} \log (f_{0} / M)$ finite; the negative part is bounded by $M / e$ . Equal mass, momentum, and energy moments then give $H (f_{0}) - H (M) = \int M Φ (z_{0}) d v < \infty .$ At zeros of $f_{0}$ , the formula uses $0 \log 0 = 0$ ; continuity of $z \log z$ at zero follows from [F5].

2.1F1F3F4step 1.1algebra

Let $a = e^{- t / (ϵ τ)}$ . By [F1], $f^{ϵ} / M = a z_{0} + (1 - a)$ . Convexity [F4] and $Φ (1) = 0$ imply pointwise $0 \leq Φ (f^{ϵ} / M) \leq a Φ (z_{0}) .$ Integrating against $M$ and using step 1.1 shows the relative-entropy integral is finite and $\int M Φ (f^{ϵ} / M) d v \leq a (H (f_{0}) - H (M)) .$

3.1A1F1F2F3step 1.1step 2.1algebra∎

The same moment cancellation as in step 1.1 gives $H (f^{ϵ}) - H (M) = \int [f^{ϵ} \log (f^{ϵ} / M) - f^{ϵ} + M] d v = \int M Φ (f^{ϵ} / M) d v .$ The identity also proves $H (f^{ϵ})$ is finite: the relative-entropy integrand is integrable by step 2.1, $\log (M / f_{r e f})$ grows at most quadratically, and $f^{ϵ}$ has finite second moment. Nonnegativity follows from [F3]. Combining this equality with step 2.1 yields both asserted bounds.

Exponential reference-relative entropy bound for homogeneous BGK

Statement

Statement references

Facts & Assumptions

Proof

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