Inviscid Limit for Yudovich solution to heat conductive Boussinesq equation on two-dimensional periodic domain

This paper establishes the inviscid limit of Yudovich solutions for the heat-conductive Boussinesq equations on a two-dimensional periodic domain, proving that diffusive solutions converge to the Euler-Boussinesq solutions in $L^\infty(0,T; W^{1,p})$ as viscosity vanishes, provided the initial data converge strongly in $L^2$.

The Gist

The Big Picture: The "Perfect Fluid" vs. The "Real Fluid"

Imagine you are watching a pot of soup simmer on a stove.

• The Real Fluid (Viscous): In reality, soup is thick. It has "stickiness" (viscosity) and heat moves through it slowly (conductivity). If you stir it, the spoon drags against the liquid, creating friction. This is the Boussinesq equation with viscosity (Equation 1 in the paper).
• The Perfect Fluid (Inviscid): Now, imagine a magical version of that soup where there is absolutely no stickiness. It flows like a ghost. It doesn't rub against itself or the spoon. This is the Euler-Boussinesq equation (Equation 2).

The Big Question:
If you take a real pot of soup and slowly make it less and less sticky (until the stickiness is almost zero), does the behavior of the real soup eventually look exactly like the behavior of the magical, perfect soup?

Mathematicians call this the "Inviscid Limit." It's a fundamental question in physics and math: Does the messy, real world smoothly turn into the clean, ideal world when we remove friction?

The Problem: The "Yudovich" Challenge

For a long time, mathematicians could prove this works for smooth, gentle flows. But real fluids can get chaotic. They can have "rough" spots where the swirls (vorticity) are very intense, even if the total energy is finite.

The paper focuses on Yudovich solutions. Think of these as fluids that are "rough" but not too rough.

• The Analogy: Imagine a crowd of people.
  • Smooth: Everyone is walking in perfect, straight lines. Easy to predict.
  • Rough (Yudovich): People are jostling, pushing, and swirling. It's chaotic, but no one is screaming or running in a panic. It's messy, but manageable.
  • Too Rough: A riot. The math breaks down.

The author, Siran Li, wants to prove that even for these "messy but manageable" fluids, if you slowly remove the stickiness, the real fluid will eventually match the perfect fluid perfectly.

The Setting: A Tiled Room

The paper takes place on a 2D Periodic Domain ($T^2$).

• The Analogy: Imagine a video game screen (like Pac-Man). If you walk off the right edge, you instantly appear on the left edge. There are no walls, no corners, and no boundaries.
• Why this matters: In real life, fluids hit walls (like the side of a pipe), creating "boundary layers" (friction zones) that make the math very hard. By using a "Pac-Man world," the author removes the walls to focus purely on the fluid's internal chaos.

The Main Result: "Yes, it works!"

The paper proves Theorem 2:
If you start with a real fluid (with a tiny bit of stickiness) and a perfect fluid (no stickiness), and they start out looking almost the same, then as the stickiness vanishes, the real fluid will converge to the perfect fluid.

They don't just get close; they get very close in a specific mathematical sense (measuring the "swirls" or vorticity).

How They Proved It: The "Two-Step" Dance

The proof is complex, but it relies on two main tricks (Propositions 8 and 9) that act like a safety net.

1. The "Roughness" Safety Net (Proposition 8)

• The Problem: When you have a rough fluid, the heat (buoyancy) can create wild spikes in the flow. Usually, this makes the math explode.
• The Trick: The author shows that even if the heat creates spikes, the "roughness" of the fluid (the swirls) is strong enough to contain them.
• Analogy: Imagine a wild dog (the heat) trying to pull a leash. The dog is strong, but the person holding the leash (the fluid's structure) is even stronger. The person can control the dog for a while, even if the dog pulls hard. The paper proves this control holds up even when the fluid is "rough."

2. The "Small Difference" Amplifier (Proposition 9)

• The Problem: You have two fluids: one with a tiny bit of stickiness ($\nu$) and one with none. They start slightly different. Will that small difference grow into a huge disaster?
• The Trick: The author proves that for a specific amount of time, that small difference shrinks or stays small.
• Analogy: Imagine two runners starting a race. One is wearing heavy boots (viscosity), and one is barefoot. If they start side-by-side, the heavy boots might slow the runner down slightly. The paper proves that for a certain distance, the runner in boots won't fall so far behind that they can never catch up. The "gap" between them stays tiny.

The "Secret Weapon": Adapting Old Math

The author didn't invent a whole new universe of math. Instead, they took a famous proof by Constantin, Drivas, and Elgindi (from 2022) which solved this problem for simple fluids.

• The Innovation: The old proof assumed the "heat force" was perfectly smooth. The new proof had to handle "rough" heat forces (mathematically, moving from $L^\infty$ to $L^1$).
• The Metaphor: It's like taking a recipe for a perfect cake and modifying it to work with slightly burnt flour. The author showed that with a few clever adjustments (using "ODE-type arguments"), the recipe still works perfectly.

Why Does This Matter?

1. Physics: It confirms that our idealized models of the atmosphere and oceans (which assume no friction) are actually good approximations of reality, even when things get messy.
2. Math: It pushes the boundary of what we know about "rough" fluids. It tells us that chaos has limits; even in a messy fluid, removing friction leads to a predictable, smooth outcome.
3. Limitations: The author admits this only works in the "Pac-Man world" (periodic domain). If you put the fluid in a real box with walls, the fluid sticks to the walls, creating a "boundary layer" that breaks the math. But for the open ocean or atmosphere (where walls are far away), this result is a big win.

Summary in One Sentence

Siran Li proved that even if a fluid is messy and rough, as long as you slowly remove its stickiness, it will eventually behave exactly like the perfect, frictionless fluid we imagine in our ideal models.

Technical Summary

1. Problem Statement

The paper addresses the inviscid limit problem for the two-dimensional (2D) heat-conductive Boussinesq equations on a periodic domain $\mathbb{T}^2$. Specifically, it investigates the convergence of solutions to the viscous, diffusive Boussinesq system (Equation 1) to the inviscid, non-diffusive Euler–Boussinesq system (Equation 2) as the viscosity coefficient $\nu$ tends to zero.

The systems are defined as:

• Viscous System (1): Includes viscosity $\nu > 0$ and thermal diffusivity $\kappa > 0$.
• Inviscid System (2): Sets $\nu = 0$ (Euler–Boussinesq), retaining $\kappa > 0$.

Key Challenge: The study focuses on Yudovich-class solutions, where the initial vorticity $\omega_0$ is in $L^\infty$ (bounded), and the initial velocity/temperature are in $L^2$. This is a critical regime because:

1. Standard energy estimates are insufficient to prove strong convergence in high regularity norms ($W^{1,p}$) without additional assumptions.
2. The forcing term in the vorticity equation, $\partial_1 \theta$ (the vertical derivative of temperature), lacks the $L^\infty_t L^\infty_x$ regularity typically required by previous inviscid limit theorems (e.g., for Navier-Stokes). Instead, for Yudovich solutions, the forcing term only belongs to $L^1_t L^\infty_x$.

2. Methodology

The author adapts and extends the analytical framework established by Constantin, Drivas, and Elgindi (CDE) in their 2022 work on the Navier-Stokes inviscid limit. The proof strategy involves several sophisticated steps:

A. Vorticity Formulation

The problem is reformulated in terms of vorticity $\omega = \text{rot}(u)$. The difference between the viscous and inviscid vorticities is analyzed via the transport-diffusion equations:
$$\partial_t \omega^\nu + u^\nu \cdot \nabla \omega^\nu - \nu \Delta \omega^\nu = \partial_1 \theta^\nu$$
$$\partial_t \omega + u \cdot \nabla \omega = \partial_1 \theta$$
The forcing terms $\partial_1 \theta^\nu$ and $\partial_1 \theta$ are treated as external forces.

B. Extension of Key Estimates (The Core Innovation)

The paper's primary technical contribution is extending the CDE arguments to handle $L^1_t L^\infty_x$ forcing terms rather than $L^\infty_t L^\infty_x$.

• Proposition 8 (Quantified Loss of Regularity): The author proves a variant of the CDE Lemma 2. It establishes that for a scalar $\sigma$ transported by a Yudovich velocity field with $L^1_t L^\infty_x$ forcing, the $L^2$ norm remains controlled over a time interval that degenerates as the initial regularity approaches the limit. This relies on Trudinger–Moser inequalities and a carefully constructed time-dependent exponent $p(t)$ in energy estimates.
• Proposition 9 (Propagation of Smallness): This extends CDE Lemma 4. It demonstrates that if the initial velocity difference $u^\nu_0 - u_0$ and viscosity $\nu$ are sufficiently small, the $L^2$ difference $\|u^\nu(t) - u(t)\|_{L^2}$ remains small for a short time interval. The proof involves splitting the domain into "large" and "small" sets based on the velocity difference and utilizing the exponential integrability of the gradient of the velocity field (via the Moser–Trudinger inequality).

C. Mollification and Iteration

To handle the lack of smoothness in the forcing terms:

1. Mollification: The vorticity equations are regularized using a standard mollifier $\phi_\ell$. This allows the use of higher regularity estimates (e.g., $H^1$ bounds) for the mollified solutions.
2. Time-Stepping: The convergence is proven on a small time interval $[0, T^*]$ where $T^* \sim \gamma / \Omega_\infty$. The result is then extended to any finite time $T$ by iterating the argument, using the terminal state of one interval as the initial state for the next.

3. Key Contributions

1. Generalization of Inviscid Limit Theory: The paper successfully extends the inviscid limit results from the Navier-Stokes equations (with $L^\infty$ forcing) to the Boussinesq equations where the forcing term (temperature gradient) is only in $L^1_t L^\infty_x$. This is a natural and necessary extension for Yudovich solutions to the Euler–Boussinesq system.
2. Rigorous Proof for Periodic Domains: It establishes strong convergence in the space $L^\infty(0, T; W^{1,p}(\mathbb{T}^2))$ for any $p \in [1, \infty)$, provided the initial data converges strongly in $L^2$.
3. Refined Estimates: The derivation of Propositions 8 and 9 provides new quantitative estimates for transport equations with low-regularity forcing, which may be applicable to other fluid dynamics problems involving Yudovich-type data.

4. Main Results

Theorem 2 (Main Theorem):
Let $(u, \theta)$ be the unique Yudovich weak solution to the inviscid Euler–Boussinesq equation and $(u^\nu, \theta^\nu)$ be the solution to the viscous Boussinesq equation. If the initial data $(u^\nu_0, \theta^\nu_0)$ converges strongly to $(u_0, \theta_0)$ in $L^2$ (and consequently the initial vorticities $\omega^\nu_0 \to \omega_0$ in $L^2$), then:
$$\lim_{\nu \searrow 0} \sup_{t \in [0, T]} \| \omega^\nu(t) - \omega(t) \|_{L^p(\mathbb{T}^2)} = 0$$
for any finite time $T > 0$ and $p \in [1, \infty)$. Consequently, the velocity fields converge strongly in $L^\infty(0, T; W^{1,p}(\mathbb{T}^2))$.

Remarks on Limitations:

• The result holds for finite time $T$. Global existence ($T \to \infty$) is not guaranteed due to the potential for finite-time blow-up in the Euler–Boussinesq system (though none is known to occur for this specific data class, the convergence rate estimates degrade).
• The result is specific to the periodic domain $\mathbb{T}^2$. On bounded domains with no-slip boundary conditions, boundary layers (Prandtl layers) are expected to form, preventing strong convergence in $L^p$ near the boundary.
• Convergence does not hold in $L^\infty_t L^\infty_x$ (the vorticity does not converge uniformly in space).

5. Significance

• Physical Relevance: The Boussinesq equations model atmospheric and oceanic turbulence where buoyancy drives flow. Understanding the inviscid limit is crucial for validating numerical simulations that often neglect viscosity or use very low viscosity.
• Mathematical Rigor: The paper resolves a gap in the literature regarding the inviscid limit for systems with $L^1_t L^\infty_x$ forcing. Previous results were limited to cases with higher regularity forcing or different boundary conditions.
• Turbulence Onset: The result suggests that for 2D periodic Boussinesq flows with bounded vorticity, the inviscid limit does not induce "turbulence onset" (i.e., the solution remains well-behaved and converges to the unique inviscid solution) as long as the initial data is sufficiently regular in the Yudovich sense.

In summary, Siran Li provides a rigorous proof that the viscous Boussinesq equations converge to the Euler–Boussinesq equations in the inviscid limit for Yudovich data, overcoming the technical hurdle of lower regularity forcing terms through refined energy estimates and the adaptation of ODE-type arguments to PDEs.