1. How do you rigorously define H ( t ) H(t) (entropy) for Navier–Stokes?

 

 1. How do you rigorously define $H(t)$ (entropy) for Navier–Stokes?

Option A (Velocity PDF Entropy):
If $u(x,t)$ is a velocity field, define a probability density function $\rho_u(v,t)$ for the velocity magnitude or direction. Then:

$$H(t) = -\int_{\mathbb{R}^3} \rho_u(v,t) \log \rho_u(v,t)\, dv$$

This requires a smoothed or projected distribution, often constructed via kernel density estimates on simulation data.

Option B (Ensemble / Statistical Mechanics):
In turbulence or ensemble frameworks, define entropy on the space of flow realizations. Then:

$$H(t) = -\mathbb{E}[\log \mathbb{P}(u(x,t))]$$

This aligns with existing statistical treatments of turbulence and connects to entropy solutions in hyperbolic PDEs.

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🔍 2. What ensures $S(t)$ remains bounded away from zero?

Approach: Model $S(t)$ as a stochastic process:

$$dS = -a(S - \bar{S})dt + \sigma dW_t$$

This is an Ornstein–Uhlenbeck process, mean-reverting around $\bar{S} > 0$. Under standard assumptions (e.g. $\sigma^2 < 2a\bar{S}$), the process remains bounded away from 0 with high probability.

You can add a reflecting barrier at $S = \epsilon > 0$ if strict positivity is needed.

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🔍 3. Can you prove $\int_0^\infty \kappa(t)\,dt = \infty$ for generic initial data?

This is the central difficulty—to rigorously prove this requires:

• Bounds on how fast $E(t)$ and $Z(t)$ decay (available via standard energy inequalities).

• Probabilistic estimates on $S(t)$, ensuring it doesn't suppress $\kappa(t)$ too quickly.

• Entropy decay $H(t) \to \infty$ slowly enough so that $N(t) = e^{-\lambda H(t)}$ doesn’t vanish too fast.

Sketch of argument:

From energy dissipation:

$$\frac{dE}{dt} = -2\nu Z(t), \quad D(t) = 2\nu Z(t)$$

So $\kappa(t) = \frac{E(t)+Z(t)}{\nu S(t) D(t) N(t)} \approx \frac{E(t)+Z(t)}{2\nu^2 Z(t) S(t) N(t)}$

As long as:

• $S(t) \ge s_1 > 0$

• $N(t) \ge n_1 > 0$

• $E(t)$ does not decay faster than $Z(t)$

then $\kappa(t) \sim \frac{1}{\text{bounded}}$, and $\int_0^\infty \kappa(t)dt = \infty$.

This still needs formal quantification of decay rates.

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🔍 4. How does this compare to existing Lyapunov function approaches?

• Classical Lyapunov: Often energy-based or involve norms of $u$, $\nabla u$, or vorticity. They yield a priori bounds but don’t guarantee global regularity.

• This Framework: Introduces recursive, stochastically modulated, and entropy-weighted Lyapunov-type control, adapting over time via:

  $$\kappa(t) = \frac{E(t)+Z(t)}{\nu S(t) D(t) N(t)}$$

This dynamic feedback mechanism isn't standard in PDE literature and potentially offers tighter control near critical events.