Corrigendum to "Optimal time-decay estimates for an Oldroyd-B model with zero viscosity [J. Differential Equations, 306(2022), 456--491]"

This paper provides a corrigendum to the authors' previous work in the *Journal of Differential Equations* by making minor corrections to the assumptions and proof of Theorem 1.2 regarding optimal time-decay estimates for an Oldroyd-B model with zero viscosity.

The Gist

Imagine you and a team of scientists built a very sophisticated weather forecast model. This model predicts how a special, stretchy fluid (like a mix of water and polymer chains) moves and slows down over time. You published your findings in a major journal, claiming you had found the "perfect" way to predict how fast this fluid would calm down.

However, after the paper was published, you realized you made a small but important mistake in the rules you used to build your model. This document is your Corrigendum—a polite and professional "oops, we fixed it" note to the world.

Here is the story of what happened, explained simply:

1. The Original Mistake: The "Empty Bucket" Paradox

In your original paper, you wanted to prove that the fluid slows down at a specific, optimal speed. To do this, you set a rule for the starting conditions of the fluid. You said: "The fluid must start with a certain amount of 'oomph' (energy) in its initial state."

You tried to define this "oomph" by saying the fluid's starting speed must be integrable (mathematically, it belongs to a space called $L^1$). In simple terms, this means if you added up all the speed of the fluid everywhere in the universe, it would equal a specific, finite number.

The Problem:
You also required the fluid to be incompressible (like water, it can't be squished). In the world of fluid dynamics, there is a famous rule (like a law of physics) that says: If a fluid is incompressible and you add up its speed everywhere, the total must be zero. It's like a perfectly balanced scale; the positive speeds cancel out the negative speeds exactly.

So, you had a contradiction:

• Rule A: The starting speed must be a specific, non-zero number (to prove your decay rates).
• Rule B: Because the fluid is incompressible, the total starting speed must be zero.

It was like trying to build a bridge that requires the river to be both "dry" and "flowing with water" at the same time. The math didn't work because the starting condition you asked for was impossible to achieve.

2. The Fix: Changing the Lens

You realized you didn't need to look at the fluid's speed in the "real world" (where the total must be zero). Instead, you needed to look at it through a different lens: the Frequency Lens (mathematically, the Fourier transform).

Think of it like this:

• The Original View: Looking at a song and trying to measure the total volume of the sound waves. If the song has no bass (zero frequency), the total volume might be zero, making your measurement useless.
• The New View: Looking at the song's sheet music (the frequencies). Even if the total volume is zero, you can still see that the high notes and low notes are there and strong.

The Solution:
Instead of demanding the fluid's starting speed be "integrable" in the real world, you changed the rule to demand that its frequency profile is "bounded" (mathematically, $L^\infty$).

This is a much weaker, more flexible rule. It allows the fluid to be incompressible (total speed = 0) while still having enough "structure" in its frequency makeup to prove your time-decay estimates.

3. The "Recipe" Correction

The paper includes a table (Table 1) which acts like a "Recipe Correction."

• Old Recipe: "Add 1 cup of 'Total Real-World Speed' ($L^1$)."
• New Recipe: "Add 1 cup of 'Maximum Frequency Strength' ($L^\infty$)."

Every time the original proof used the old ingredient, you simply swapped it for the new one. The rest of the cooking (the logic and the final result) remains delicious and valid.

4. Proof That It Works

To show the world that this new rule isn't just a theoretical trick, you provided a specific example (Remark 4). You constructed a "perfect" starting fluid that:

1. Is incompressible (obeys the laws of physics).
2. Has a smooth, finite energy.
3. Has a strong "frequency signal" near zero (satisfying your new rule).

This proves that your corrected theorem is not only mathematically sound but also physically possible.

The Bottom Line

The scientists found a tiny logical loop-hole in their own work. They realized they asked for something impossible (a non-zero total speed for an incompressible fluid). They fixed it by changing the question from "How much total speed is there?" to "How strong is the pattern of the speed?"

The result? The original conclusion—that the fluid slows down at a specific, optimal rate—still stands true. The math just needed a slight adjustment to make sense of the real world.

Technical Summary

Based on the provided corrigendum, here is a detailed technical summary of the paper.

1. Problem Statement

The paper addresses a logical inconsistency found in the original publication (Huang et al., J. Differential Equations, 2022) regarding the optimal time-decay estimates for the Oldroyd-B model with zero viscosity.

The specific system under study is the Cauchy problem for the Oldroyd-B fluid in $\mathbb{R}^3$:
$$\begin{cases} \partial_t u + u \cdot \nabla u + \nabla p - \epsilon \Delta u = \kappa \text{div}\tau, \\ \partial_t \tau + u \cdot \nabla \tau - \mu \Delta \tau + \beta \tau = Q(\nabla u, \tau) + \alpha D u, \\ \text{div} u = 0, \\ (u, \tau)(x, 0) = (u_0, \tau_0), \end{cases}$$
where $u$ is the velocity, $\tau$ is the elastic stress tensor, and $p$ is pressure. The study covers two cases:

• Case I: $\mu > 0$ (stress diffusion) and $\epsilon \ge 0$ (zero fluid viscosity).
• Case II: $\epsilon > 0$ and $\mu \ge 0$.

The Core Issue:
In the original Theorem 1.2, the authors claimed that optimal time-decay estimates hold under the assumption that the initial data $(u_0, \tau_0) \in L^1(\mathbb{R}^3) \cap H^3(\mathbb{R}^3)$ and that the Fourier transform of the initial velocity satisfies a lower bound condition near the origin:
$$\inf_{0 \le |\xi| \le R} |\hat{u}_0(\xi)| \ge c_0 > 0.$$
The corrigendum identifies that these assumptions are mutually contradictory.

• If $u_0 \in L^1(\mathbb{R}^3)$ and $\text{div} u_0 = 0$, then by the properties of the Fourier transform and the divergence-free condition, $\hat{u}_0(0) = \int_{\mathbb{R}^3} u_0 dx = 0$.
• However, the condition $\inf_{0 \le |\xi| \le R} |\hat{u}_0| \ge c_0 > 0$ implies that $|\hat{u}_0(0)| \ge c_0 > 0$.
• Therefore, no such function $u_0$ exists that satisfies both the divergence-free $L^1$ condition and the non-vanishing low-frequency condition.

2. Methodology

The authors employ a correction and reconstruction methodology:

1. Identification of Contradiction: They utilize a known lemma (Lemma B from Schonbek et al.) stating that for any divergence-free vector field in $L^1 \cap L^2$, the integral over $\mathbb{R}^3$ is zero, implying $\hat{u}_0(0) = 0$.
2. Relaxation of Assumptions: Instead of requiring the initial data to be in $L^1(\mathbb{R}^3)$, the authors propose replacing this with a weaker condition on the Fourier transform of the initial data: $(\hat{u}_0, \hat{\tau}_0) \in L^\infty(\mathbb{R}^3)$.
3. Existence Proof: They construct a specific example (Remark 4) to demonstrate that functions satisfying the new, corrected assumptions actually exist. They define $\hat{u}_0$ using a smooth, compactly supported function $g(\xi)$ such that $\hat{u}_0$ is divergence-free in Fourier space ($i\xi \cdot \hat{u}_0 = 0$) and satisfies the low-frequency lower bound, while ensuring $u_0 \in H^3$.
4. Proof Adjustment: The authors verify that the original proof techniques remain valid by simply substituting the $L^1$ norms of the initial data with $L^\infty$ norms of their Fourier transforms in the energy estimates.

3. Key Contributions

• Correction of Theorem 1.2: The primary contribution is the formal correction of the assumptions in the main theorem regarding time-decay estimates.
• Resolution of Logical Flaw: The paper resolves the contradiction between the $L^1$ integrability of divergence-free fields and the non-vanishing low-frequency assumption.
• Refined Assumptions: The paper establishes that the optimal decay rates can be derived under the condition $(\hat{u}_0, \hat{\tau}_0) \in L^\infty(\mathbb{R}^3)$ rather than $(u_0, \tau_0) \in L^1(\mathbb{R}^3)$.
• Constructive Example: The authors provide an explicit construction (Remark 4) proving that the new assumptions are not vacuous and that valid initial data exists.

4. Results

The corrected theorem (Theorem 3 in the corrigendum) states that for solutions $(u_{\epsilon, \mu}, \tau_{\epsilon, \mu})$ satisfying the corrected assumptions:

Upper Decay Estimates:
If $(\hat{u}_0, \hat{\tau}_0) \in L^\infty(\mathbb{R}^3)$, then for $t \ge 0$:
$$\|\nabla^k u_{\epsilon, \mu}(t)\|_{L^2} \le C_2 (1+t)^{-\frac{3}{4} - \frac{k}{2}}, \quad k=0,1,2$$
$$\|\nabla^{k_1} \tau_{\epsilon, \mu}(t)\|_{L^2} \le C_2 (1+t)^{-\frac{5}{4} - \frac{k_1}{2}}, \quad k_1=0,1$$
Note: The constant $C_2$ now depends on $\|(\hat{u}_0, \hat{\tau}_0)\|_{L^\infty}$ instead of the $L^1$ norm.

Lower Decay Estimates:
If, in addition, $\inf_{0 \le |\xi| \le R} |\hat{u}_0| \ge c_0 > 0$, then for sufficiently large $t$:
$$\|\nabla^k u_{\epsilon, \mu}(t)\|_{L^2} \ge c (1+t)^{-\frac{3}{4} - \frac{k}{2}}$$
$$\|\nabla^{k_1} \tau_{\epsilon, \mu}(t)\|_{L^2} \ge c (1+t)^{-\frac{5}{4} - \frac{k_1}{2}}$$

Technical Corrections:
The corrigendum provides a table (Table 1) detailing specific replacements required in the original paper's proofs (pages 475–487). Specifically, terms like $\|U_0\|_{L^1}$ and $\|(u_0, \tau_0)\|_{L^1}$ must be replaced with $\|\hat{U}_0\|_{L^\infty_\xi}$ and $\|(\hat{u}_0, \hat{\tau}_0)\|_{L^\infty_\xi}$, respectively.

5. Significance

• Mathematical Rigor: The corrigendum restores the mathematical validity of the original results. Without this correction, the main theorem of the original paper would be vacuously true (as no such initial data exists) or logically flawed.
• Physical Relevance: The Oldroyd-B model is crucial for understanding viscoelastic fluids. Ensuring that decay estimates are derived under physically and mathematically consistent assumptions is vital for the reliability of theoretical predictions in fluid dynamics.
• Methodological Impact: The shift from $L^1$ spatial assumptions to $L^\infty$ frequency assumptions is a subtle but important distinction in the analysis of decay rates for partial differential equations, particularly for divergence-free systems where the zero-frequency mode is constrained. This correction clarifies the precise conditions required for optimal decay in zero-viscosity regimes.