On the continuous properties for the 3D incompressible rotating Euler equations

1. Problem Statement

The paper investigates the Cauchy problem for the 3D incompressible Euler equations with the Coriolis force in the whole space $\mathbb{R}^3$. The system is given by:
$$\begin{cases} \partial_t u + \Omega e_3 \times u + u \cdot \nabla u + \nabla P = 0, \\ \text{div } u = 0, \\ u(0, x) = u_0(x), \end{cases}$$
where $u$ is the velocity field, $P$ is the pressure, $\Omega \in \mathbb{R}$ is the rotation speed (Coriolis parameter), and $e_3 = (0,0,1)$.

The primary objective is to analyze the well-posedness of this system in Besov spaces $B^s_{p,r}(\mathbb{R}^3)$. Specifically, the authors aim to determine:

1. Local existence and uniqueness of solutions.
2. The continuity properties of the data-to-solution map $u_0 \mapsto S_t(u_0)$.
3. Whether the solution map is uniformly continuous with respect to initial data.
4. Whether the solution is Hölder continuous with respect to the time variable $t$.

2. Methodology and Technical Framework

Functional Setting

The analysis is conducted in non-homogeneous Besov spaces $B^s_{p,r}(\mathbb{R}^3)$. The authors utilize:

• Littlewood-Paley Decomposition: To define dyadic blocks $\Delta_j$ and characterize the norms of the function spaces.
• Leray Projection ($P$): To handle the incompressibility constraint and eliminate the pressure term.
• Commutator Estimates: To handle the nonlinear terms $u \cdot \nabla u$ and the interaction between the projection operator and the nonlinearity.
• Product Laws in Besov Spaces: To estimate nonlinear interactions (e.g., Lemma 2.2 and 2.3).

Construction of Counter-Examples

To prove ill-posedness (lack of uniform continuity and Hölder continuity), the authors employ a specific high-frequency perturbation strategy:

• They construct specific initial data sequences involving high-frequency oscillations (using functions like $\cos(\lambda x_1)$ modulated by smooth cutoff functions $\theta(x)$).
• For Non-uniform Continuity (Theorem 1.2): They construct two sequences of initial data, $u_{0,n} = f_n + g_n$ and $v_{0,n} = f_n$, where $g_n$ is a small perturbation in the $B^s_{p,r}$ norm, but the difference in the resulting solutions remains large in a specific component.
• For Hölder Continuity Failure (Theorem 1.3 & 1.4): They construct a single initial datum $u_0$ composed of a sum of high-frequency modes. They analyze the Taylor expansion of the solution $S_t(u_0)$ around $t=0$ to show that the remainder term dominates, preventing Hölder continuity.

Key Analytical Tools

• Perturbation Analysis: Decomposing the solution difference $S_t(u_0) - u_0$ into a linear part, a quadratic part (convective term), and a remainder.
• Dispersion vs. Nonlinearity: The authors note that while the Coriolis force introduces a dispersive mechanism ($\Omega e_3 \times u$), the nonlinear convective term $u \cdot \nabla u$ dominates the short-time behavior regarding continuity properties, similar to the classical Euler equations.
• Bernstein's Inequalities: Used to estimate the norms of high-frequency functions and their derivatives.

3. Key Contributions and Results

The paper establishes four main theorems:

Theorem 1.1: Local Well-Posedness

• Result: For initial data $u_0 \in B^s_{p,r}(\mathbb{R}^3)$ satisfying $s > 3/p + 1$ (or the critical case $s=3/p+1, q=1$), there exists a unique local solution $u \in C([0, T]; B^s_{p,r})$.
• Significance: This extends known well-posedness results for the classical Euler equations to the rotating case, confirming that the Coriolis force does not destroy local existence or uniqueness in these spaces.

Theorem 1.2: Lack of Uniform Continuity

• Result: The data-to-solution map $u_0 \mapsto S_t(u_0)$ is not uniformly continuous from any bounded subset of $B^s_{p,r}$ into $C([0, T]; B^s_{p,r})$.
• Detail: There exist sequences of initial data $f_n + g_n$ and $f_n$ such that $\|g_n\|_{B^s_{p,r}} \to 0$, but the distance between the second components of their solutions satisfies:
  $$\liminf_{n \to \infty} \| (S_t(f_n + g_n) - S_t(f_n))^{(2)} \|_{B^s_{p,r}} \gtrsim t$$
• Significance: This extends the work of Himonas and Misiołek (2010) from Sobolev spaces to the more general Besov framework. It implies that the solution map is not Lipschitz continuous, and standard contraction mapping arguments cannot be used to prove well-posedness in these spaces.

Theorem 1.3: Failure of Hölder Continuity in Time

• Result: For any $\alpha \in (0, 1)$, there exists an initial datum $u_0 \in B^s_{p,r}$ such that the solution is not Hölder continuous in time. Specifically:
  $$\limsup_{t \to 0^+} \frac{\| S_t(u_0) - u_0 \|_{B^s_{p,r}}}{t^\alpha} = +\infty$$
• Significance: This is a novel contribution. To the authors' knowledge, this is the first work addressing the failure of Hölder continuity in time for the classical Euler equations (with or without Coriolis force). It shows that even though the solution is continuous in time, it lacks higher regularity in the time variable within the same Besov space.

Theorem 1.4: Discontinuity at $t=0$ in Weaker Spaces

• Result: In the weaker Besov spaces $B^s_{p,\infty}(\mathbb{R}^3)$, the solution map is discontinuous at $t=0$. There exists $u_0$ such that:
  $$\limsup_{t \to 0^+} \| S_t(u_0) - u_0 \|_{B^s_{p,\infty}} \geq \epsilon_0 > 0$$
• Significance: This indicates a stronger form of ill-posedness in the $r=\infty$ case, where the solution does not even return to the initial data continuously in the metric of the space.

4. Significance and Impact

1. Extension to Rotating Fluids: The paper confirms that the Coriolis force does not regularize the Euler equations in terms of the continuity of the solution map. Despite the dispersive nature of the linear part, the nonlinear instability (ill-posedness) persists in the same manner as the non-rotating Euler equations.
2. Generalization of Framework: By working in $B^s_{p,r}$ spaces, the results generalize previous findings restricted to Sobolev spaces ($H^s$) or specific $L^p$ settings.
3. New Regularity Insight: The proof of the failure of Hölder continuity (Theorem 1.3) provides a deeper understanding of the "roughness" of the solution map for quasilinear hyperbolic systems. It highlights that the solution map is continuous but "very" non-smooth.
4. Methodological Advancement: The construction of specific initial data involving high-frequency oscillations and the precise estimation of the convective term's contribution to the time derivative offer a robust template for analyzing ill-posedness in other fluid dynamics equations.

5. Conclusion and Future Directions

The authors conclude that the 3D rotating Euler equations are ill-posed in the sense of Hadamard regarding the uniform continuity of the solution map and the Hölder regularity in time.

Future Work: The authors note that the case $p=1$ or $p=\infty$ remains open. Due to the unboundedness of the Riesz transform on $L^\infty$ and $L^1$, and the specific dispersive effects of rotation, they conjecture that the system is ill-posed in $L^\infty$-based Besov spaces, a problem they intend to investigate further.