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Non-equilibrium BGK data create an initial layer

Example

Adopt Countable Choice, denoted ${A C}_{ω}$ , as in the kinetic-density definition. Let $v_{*} = 1 m s^{- 1}$ , $ω_{3} (v) = \sum_{j = 0}^{3} (∣ v ∣ / v_{*})^{j}$ , $θ > 0$ be a squared speed, choose a speed $a > 0$ with $a^{2} < θ$ , and let $n > 0$ . For a positive-definite covariance matrix $Σ$ , define the normalized Gaussian $G_{μ, Σ} (v) : = \frac{n}{(2 π)^{3 / 2} \sqrt{\det Σ}} \exp ⁣ (- \frac{1}{2} (v - μ)^{T} Σ^{- 1} (v - μ)) .$ Set $Σ = diag (θ - a^{2}, θ, θ)$ and $f_{0} (v) : = \frac{1}{2} G_{a e_{1}, Σ} (v) + \frac{1}{2} G_{- a e_{1}, Σ} (v) .$ This positive density has mass $n$ , mean velocity $0$ , and normalized covariance $θ I_{3}$ (equivalently, second centered moment tensor $n θ I_{3}$ ), so its moment-matched Maxwellian is $M = M_{n, 0, T}$ with $k_{B} T = m θ$ . It is not itself Maxwellian and satisfies $\int f_{0} ∣ \log (f_{0} / f_{r e f}) ∣ d v < \infty$ for every fixed positive constant reference density $f_{r e f}$ with the same units as $f_{0}$ , interpreting the integrand as zero where $f_{0} = 0$ . For $ϵ, τ > 0$ , its homogeneous BGK evolution is $f^{ϵ} (t, v) = M (v) + e^{- t / (ϵ τ)} (f_{0} (v) - M (v)) .$ For every $t > 0$ this approaches $M$ as $ϵ \to 0$ , and on every finite interval $[0, T_{*}]$ its $ω_{3}$ weighted spacetime distance to $M$ is at most $ϵ τ \int ω_{3} ∣ f_{0} - M ∣ d v$ . It does not converge uniformly in time on $[0, T_{*}]$ , since the distance at $t = 0$ is strictly positive.

Example references

Tong, §2.5.1, full lecture notes

Zamponi, §3.2, full lecture notes

Facts & Assumptions

Given: Countable Choice, $0 < a^{2} < θ$ , $n, m, k_{B}, ϵ, τ > 0$ , $k_{B} T = m θ$ , and a constant reference density $f_{r e f} > 0$ with the same units as $f_{0}$ .

[F1]

The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ , Change of variable in an improper integral, and Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ : Each one-dimensional Gaussian factor with variance $σ_{i}^{2} > 0$ normalizes to one after $z = (v_{i} - μ_{i}) / σ_{i}$ .

[F2]

If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ and A nonnegative improper Riemann integral on a half-line agrees with the Lebesgue integral: The centered one-dimensional Gaussian has mean zero by oddness and variance $σ_{i}^{2}$ by integration by parts on compact truncations and passage to the limit. Under the adopted Countable Choice, the nonnegative improper integrals equal their Lebesgue values; their factors are continuous on every compact truncation, with reflection used on the negative half-line.

[F3]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product applied iteratively to the nonnegative absolute products establishes their integrability from the finite one-dimensional Gaussian moments; Fubini's theorem for L^1 functions on a sigma-finite product then factors the signed moments.

[F4]

Gaussian Maxwellian normalization, stress, and heat flux: The isotropic Gaussian with density $n$ , mean $u$ , and variance $θ I_{3}$ is $M_{n, u, T}$ when $k_{B} T = m θ$ .

[F5]

Nonnegative kinetic density and moment domain: Countable Choice is adopted, and $ω_{3} (v) = \sum_{j = 0}^{3} (∣ v ∣ / v_{*})^{j}$ for $v_{*} = 1 m s^{- 1}$ .

[F6]

The exponential dominates every fixed nonnegative integer power at $+ \infty$ : Polynomial factors times a Gaussian are integrable at infinity.

[F7]

Exact homogeneous BGK fast relaxation: The exact BGK solution satisfies $∥ f^{ϵ} (t) - M ∥_{L^{1} (ω_{3} d v)} = e^{- t / (ϵ τ)} ∥ f_{0} - M ∥_{L^{1} (ω_{3} d v)}$ .

[A1]

$f_{0}$ is the positive Gaussian mixture in the Example.

Verification

1.1givenA1F1F2F3F4F5F6algebra

Write $σ_{1}^{2} = θ - a^{2}$ and $σ_{2}^{2} = σ_{3}^{2} = θ$ . Each factor $g_{μ_{i}, σ_{i}} (v_{i}) = (2 π σ_{i}^{2})^{- 1 / 2} e^{- (v_{i} - μ_{i})^{2} / (2 σ_{i}^{2})}$ has mass one, mean $μ_{i}$ , and centered second moment $σ_{i}^{2}$ by [F1]–[F2]. Product Fubini [F3] therefore gives mass $n$ , mean $\pm a e_{1}$ , and covariance $Σ$ for each $G_{\pm a e_{1}, Σ}$ . Their equal mixture has mean zero and covariance $Σ + a^{2} e_{1} \otimes e_{1} = diag (θ, θ, θ) = θ I_{3} .$ Its moments identify the target as $M_{n, 0, T}$ with $k_{B} T = m θ$ by [F4]. Since $Σ$ is positive definite, eigenvalue bounds give constants $C, c > 0$ such that $f_{0} (v) \leq C e^{- c ∣ v ∣^{2}}$ and $f_{0} (v) \geq C^{- 1} e^{- C (1 + ∣ v ∣^{2})}$ . For the fixed constant $f_{r e f} > 0$ , these bounds imply $∣ \log (f_{0} / f_{r e f}) ∣ \leq C^{'} (1 + ∣ v ∣^{2})$ , with $C^{'}$ allowed to depend on $f_{r e f}$ , so the absolute entropy and weighted third moment are finite by [F6] and [F3].

2.1A1F3F4F5F7step 1.1algebra∎

Along $v = (s, 0, 0)$ , the mixture is a positive constant times $e^{- s^{2} / (2 (θ - a^{2}))} \cosh ⁣ (\frac{a s}{θ - a^{2}}) .$ The second derivative of its logarithm contains ${(\frac{a}{θ - a^{2}})}^{2} {sech}^{2} ⁣ (\frac{a s}{θ - a^{2}}),$ which is nonconstant because $a > 0$ . The logarithm of any Maxwellian is quadratic with constant Hessian, so $f_{0} \neq M$ . Both densities are continuous, so they differ on a nonempty open set of positive Lebesgue measure. Thus $C : = \int ω_{3} ∣ f_{0} - M ∣ d v > 0$ ; it is finite by step 1.1 and the Gaussian moment bounds. Step 1.1 verifies the weighted-moment, positivity, and constant-reference entropy hypotheses of [F7], which gives the weighted distance at time $t$ as $e^{- t / (ϵ τ)} C$ . It tends to zero for each fixed $t > 0$ , equals $C$ at $t = 0$ , and its time integral is at most $ϵ τ C$ . Therefore convergence is not uniform on $[0, T_{*}]$ , while the stated spacetime convergence holds.

Non-equilibrium BGK data create an initial layer

Example

Example references

Facts & Assumptions

Verification

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