A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation

1. Problem Statement

The paper addresses the non-uniqueness of solutions to the forced 2D Euler equations in the vorticity class $L^1 \cap L^p$ for $2 < p < \infty$.

• Context: Yudovich's theorem guarantees uniqueness for 2D Euler equations if the vorticity is in $L^\infty$ (or $L^1 \cap L^\infty$). However, it remains an open question whether uniqueness holds for $L^p$ with finite $p$ in the absence of forcing.
• The Challenge: While Vishik [54, 55] previously proved non-uniqueness for the forced 2D Euler equation, his proof relied on a highly intricate construction of an unstable vortex based on power-law profiles ($\omega \sim |x|^{-\alpha}$).
• Goal: The authors aim to provide a simpler, more accessible proof of Vishik's theorem by constructing a different type of unstable vortex (compactly supported and piecewise constant initially) and demonstrating that it leads to non-uniqueness for the forced system.

2. Methodology

The proof follows Vishik's three-step strategy but introduces significant simplifications in the construction of the unstable vortex. The proof is divided into three main phases:

Step 1: Construction of an Unstable Vortex (Eulerian Instability)

Instead of modifying power-law vortices, the authors construct a piecewise constant vortex profile and then regularize it.

1. Piecewise Constant Ansatz: They define a radially symmetric vorticity $\bar{\omega}$ that is constant in two annular regions and zero elsewhere:
   $$\bar{w}(r) = \begin{cases} c & 0 < r \le r_1 \\ -1 & r_1 < r \le r_2 \\ 0 & r > r_2 \end{cases}$$
   Parameters $c, r_1, r_2$ are chosen such that the mean vorticity is zero (ensuring $L^2$ velocity) and the support is compact.
2. Linear Stability Analysis: They analyze the linearized Euler equation around this background state. For radially symmetric vortices, the linear stability reduces to the Rayleigh stability equation.
   • For a piecewise constant profile, the derivative $\partial_r \bar{w}$ is a sum of Dirac deltas at $r_1$ and $r_2$.
   • This reduces the infinite-dimensional eigenvalue problem to a finite-dimensional linear system ($2 \times 2$ matrix $A$).
   • They explicitly compute the characteristic polynomial of $A$ and show that for specific parameters (and frequency $n \ge 2$), the eigenvalues have a positive real part ($\Re \lambda > 0$), proving linear instability.
3. Regularization (Fixed Point Argument): Since the piecewise constant vortex is not smooth, they construct a smooth approximation $\bar{\omega}_\epsilon$ by mollifying the velocity profile near the discontinuities ($r_1, r_2$).
   • They use a fixed-point argument in rescaled variables to show that for sufficiently small $\epsilon$, the regularized vortex retains the unstable eigenvalue ($\Im z_\epsilon > 0$).
   • This yields a smooth, compactly supported, unstable vortex $\bar{\Omega} \in C^\infty_c$.

Step 2: Self-Similar Instability

The authors embed the unstable vortex $\bar{\Omega}$ into a self-similar solution of the forced Euler equation.

• Scaling: They introduce self-similar coordinates $(\tau, X)$ where $\tau = \frac{1}{ab} \log t$ and $X = x / (abt)^{1/a}$.
• Background Solution: They define a background solution $\omega_0(t,x) = (abt)^{-1} \bar{\Omega}(X)$ and a corresponding force $f$ such that $\omega_0$ is an exact solution.
• Spectral Continuity: They analyze the linearized operator $L_b$ in self-similar coordinates. Using operator theory (Fredholm properties and compact perturbations), they prove that as the parameter $b \to 0$, the spectrum of the self-similar operator $L_b$ converges to the spectrum of the Eulerian operator $L$.
• Result: Since $L$ has an unstable eigenvalue, $L_b$ also possesses an unstable eigenvalue $\lambda$ with $\Re \lambda > 0$ for small $b > 0$. This implies the existence of a solution growing exponentially in self-similar time $\tau$ (which corresponds to $t \to 0$).

Step 3: Nonlinear Instability

The final step controls the nonlinear terms to ensure the linear instability persists in the full nonlinear equation.

• Decomposition: They seek a solution of the form $\omega = \omega_0 + \epsilon \omega_{lin} + \epsilon^2 \omega_{cor}$.
• Duhamel Formula: Using the Duhamel formula, they treat the nonlinear terms as a forcing term for the correction $\omega_{cor}$.
• Energy Estimates: They establish rigorous bounds in a weighted Sobolev space $Y$ (involving $L^2$, $L^4$ gradients, and weighted norms).
  • Crucially, the compact support of their constructed vortex simplifies these estimates compared to Vishik's original work.
  • They prove that the correction term $\omega_{cor}$ decays faster than the linear instability term as $t \to 0$ (specifically, $\|\omega_{cor}\| = o(e^{\Re \lambda \tau})$).
• Conclusion: This allows the construction of two distinct solutions ($\omega_0$ and $\omega_0 + \epsilon \omega_{lin} + \dots$) starting from the same zero initial data, proving non-uniqueness.

3. Key Contributions

1. Simplified Construction of Unstable Vortices: The primary contribution is replacing Vishik's complex power-law vortex modification with a piecewise constant ansatz. This reduces the spectral analysis of the linearized operator to elementary linear algebra (solving a $2 \times 2$ matrix eigenvalue problem).
2. Fixed Point Regularization: They provide a streamlined fixed-point argument to regularize the piecewise constant vortex into a smooth one while preserving instability, avoiding the intricate asymptotic matching required in previous works.
3. Self-Contained Proof: The paper offers a more transparent and self-contained proof of Vishik's theorem, making the mechanism of non-uniqueness (via forcing and self-similar instability) more accessible to the mathematical community.
4. Robustness for Generalizations: The authors note that their construction is robust and flexible, suggesting it could be adapted to more singular contexts like the Generalized Surface Quasi-Geostrophic (SQG) equations, where the Biot-Savart law is non-local and singular.

4. Main Results

• Theorem 1.1: For any $2 < p < \infty$, there exists a smooth, compactly supported force $f \in L^1_t(L^1 \cap L^p)$ such that the forced 2D Euler equation with zero initial data ($\omega|_{t=0} = 0$) admits at least two distinct solutions in the class $L^\infty_t(L^1 \cap L^p)$.
• Sharpness: This result demonstrates the sharpness of Yudovich's uniqueness theorem, showing that uniqueness fails immediately if the vorticity is allowed to be in $L^p$ ($p < \infty$) rather than $L^\infty$, provided a suitable force is present.
• Regularity: The constructed solutions are smooth ($C^\infty$) for $t > 0$, though the regularity deteriorates as $t \to 0$.

5. Significance

• Theoretical Clarity: By simplifying the construction of the "seed" unstable vortex, the paper demystifies the mechanism behind non-uniqueness in forced fluid equations. It clarifies that the core difficulty lies in the spectral properties of the linearized operator, which can be captured by simple geometric profiles.
• Impact on Navier-Stokes: The authors highlight that their unstable vortex construction is compatible with the requirements for proving non-uniqueness of Leray solutions for the forced 3D Navier-Stokes equations (as done by Albritton, Brué, and Colombo), offering an alternative and potentially simpler route to that result.
• Future Directions: The methodology opens the door to investigating non-uniqueness in other active scalar equations, specifically the SQG equation and its generalizations, where the lack of smoothing in the velocity-vorticity relation presents new challenges.

In summary, this paper provides a rigorous, simplified, and elegant proof of non-uniqueness for the forced 2D Euler equation, leveraging a piecewise constant vortex construction to bypass the technical complexities of previous proofs while maintaining the core physical insight of self-similar instability.