Spectral Selection in Sphere-Constrained Flows Generated by Polynomials of the Dirichlet Laplacian

Technical Summary: Spectral Selection in Sphere-Constrained Flows Generated by Polynomials of the Dirichlet Laplacian

Problem Statement
The paper investigates the asymptotic behavior of a constrained linear evolution equation on the unit sphere of $L^2(\Omega)$, where $\Omega \subset \mathbb{R}^d$ is a bounded smooth domain. The flow is generated by a polynomial $P$ of the positive Dirichlet Laplacian $B = -\Delta_D$. The equation is given by:
$$u_t = -\Pi_u P(B)u, \quad \|u(0)\|_{L^2} = 1,$$
where $\Pi_u$ denotes the orthogonal projection onto the tangent space of the sphere at $u$. Unlike standard gradient flows where the spectral ordering is fixed by the eigenvalues of the operator, this system's dynamics are governed by the values of the polynomial $P$ evaluated at the Dirichlet eigenvalues $\{\lambda_j\}$. The central question is how the choice of $P$ alters the "spectral selection"—the specific eigenspace to which the trajectory converges—and the rate of this convergence.

Methodology
The analysis relies on the spectral realization of the operator $P(B)$. Since $P$ is a real polynomial with a positive leading coefficient, $P(B)$ is self-adjoint but not necessarily positive. To facilitate the use of semigroup theory, the author introduces a constant shift $\kappa$ such that $A = P(B) + \kappa I$ is strictly positive. Crucially, Lemma 2.5 demonstrates that this shift is invariant under the tangent projection $\Pi_u$, meaning the constrained dynamics remain unchanged.

The solution is explicitly represented as a normalized semigroup orbit:
$$u(t) = \frac{e^{-tP(B)}u_0}{\|e^{-tP(B)}u_0\|_{L^2}} = \frac{e^{-tA}u_0}{\|e^{-tA}u_0\|_{L^2}}.$$
By expanding the initial datum $u_0$ in the orthonormal basis of eigenfunctions $\{e_j\}$, the author analyzes the time evolution of the spectral coefficients. The methodology involves:

1. Spectral Decomposition: Factoring out the dominant exponential term corresponding to the minimal active polynomial value.
2. Hilbert Scale Analysis: Establishing the equivalence of the domains $D(A^\theta)$ and $D(B^{m\theta})$ to derive convergence rates in fractional Sobolev spaces.
3. Perturbation Theory: Analyzing the stability of the selected spectral set under small perturbations of the polynomial $P$.

Key Contributions and Results

1. Spectral Selection Theorem:
   The trajectory $u(t)$ converges strongly in $L^2(\Omega)$ to the normalized projection of the initial datum $u_0$ onto the eigenspace corresponding to the minimal value of $P(\lambda_j)$ among the "active" spectrum (those $\lambda_j$ for which the initial coefficient $c_j \neq 0$).

   • Let $\mu^*(u_0) = \min \{P(\lambda_j) : c_j \neq 0\}$.
   • The limit is $u_\infty = \frac{E^* u_0}{\|E^* u_0\|}$, where $E^*$ projects onto $\text{Ker}(P(B) - \mu^*(u_0)I)$.
   • The active spectral support is preserved; no new spectral components are created, and existing ones are not removed in finite time.

2. Convergence Rates and Regularity:

   • The convergence is exponential with a rate determined by the "active spectral gap" $\delta^*(u_0) = \min \{P(\lambda_j) - \mu^*(u_0) : P(\lambda_j) > \mu^*(u_0)\}$.
   • A significant result is that for any $t \ge \tau > 0$, the solution converges in the fractional domain $D(A^\theta)$ (and thus $D(B^{m\theta})$) with the same exponential rate, even if the initial datum $u_0$ lacks fractional regularity. This "instantaneous smoothing" is a consequence of the analytic semigroup properties.

3. Spectral Design and Stability:

   • Design: The paper shows that any finite set of distinct Dirichlet levels can be made the global minimizing set by choosing a polynomial $P(s) = \prod (s - \nu_k)^2$.
   • Stability: If a set of active levels is isolated by a positive gap, the selection remains stable under sufficiently small polynomial perturbations. If the selected set is degenerate (multiple levels share the minimal value), a perturbation can split this degeneracy, with the surviving level determined by the perturbation restricted to that finite set.

4. Specific Cases:

   • Polyharmonic Flow ($P(s) = s^m$): The polynomial is strictly increasing on the positive spectrum. Consequently, the flow selects the lowest active Dirichlet eigenvalue, recovering the standard behavior of the heat flow but with higher-order boundary conditions (Navier type).
   • Swift–Hohenberg Type Flow ($P(s) = (s-\rho)^2$): The selection is based on the distance of the eigenvalues from a parameter $\rho$. The flow selects active levels closest to $\rho$. Cross-level degeneracy (selecting two distinct levels simultaneously) occurs only if $\rho$ is exactly the midpoint of two active eigenvalues. The set of such "midpoints" is countable, implying that for generic $\rho$, the selected level is unique.

Significance and Claims
The paper claims to provide a complete characterization of constrained linear flows driven by polynomial operators. Its primary significance lies in demonstrating that the asymptotic state is not determined solely by the degree of the polynomial or the lowest eigenvalue of the Laplacian, but by the specific values $P(\lambda_j)$ on the active spectrum.

The author emphasizes that:

• Non-monotone polynomials allow for the selection of "interior" spectral levels, deviating from the standard preference for the ground state.
• The framework allows for the precise prescription of minimizing spectral sets and quantifies the robustness of these selections against perturbations.
• The results extend to fractional domains without requiring regularity assumptions on the initial data, a feature attributed to the smoothing properties of the analytic semigroup.

The paper concludes by noting that while the linear case allows for a mode-by-mode analysis due to spectral invariance, the introduction of nonlinear reaction terms would couple these modes, requiring different techniques (referenced as being treated in a separate work [4]). The current work establishes the foundational spectral selection mechanism for the linear, polynomial-driven constrained flow.