C¹ Inertial Manifolds

• C¹ inertial manifolds are finite-dimensional, C¹-smooth invariant manifolds constructed as graphs over low modes that exponentially attract trajectories in infinite-dimensional dissipative systems.
• They rely on a sharp spectral gap and use the Lyapunov–Perron method to ensure global C¹ regularity and uniform bounds for the manifold’s first derivative.
• These structures enable a rigorous finite-dimensional reduction of complex PDEs, facilitating numerical simulations and deeper analytical insights into long-term dynamics.

A $C^1$ inertial manifold is a finite-dimensional, $C^1$-smooth, exponentially attracting, invariant manifold for an infinite-dimensional evolutionary PDE or dynamical system, usually associated with a parabolic equation. The $C^1$ regularity signifies that the manifold, constructed as a graph over a finite set of “low modes,” possesses globally bounded and continuous first derivatives. $C^1$ inertial manifolds are central to understanding long-time dynamics of dissipative systems, providing a rigorous finite-dimensional reduction and enabling the use of dynamical systems theory in infinite-dimensional settings.

1. Foundational Definition and Key Properties

Given an abstract semilinear parabolic equation in a real Hilbert space $H$: $\partial_t u + A u = F(u), \qquad u(0)=u_0,$ where $A:D(A)\to H$ is self-adjoint, positive, sectorial, with compact inverse and spectrum $0<\dots\leq \lambda_n \leq \lambda_{n+1}\to\infty$; $F:H\to H$ is globally Lipschitz and $C^1$, often $C^\infty$ with globally bounded derivatives. An $N$-dimensional inertial manifold $\mathcal{M}_N\subset H$ is defined as:

• A graph $\mathcal{M}_N = \{\,p + M(p)\,:\,p\in H_N\,\}$ with $H_N = \text{span}\{e_1,\dots, e_N\}$ and $M: H_N \to Q_N H$ ($Q_N=I-P_N$) at least $C^{1,\varepsilon}$, i.e., differentiable with Hölder-continuous derivative.
• Invariant under the solution semigroup: $S(t)\mathcal{M}_N=\mathcal{M}_N$ for all $t\geq 0$.
• Exponentially attracting: for every trajectory $u(t)$, there exists a “shadow” $\bar u(t)\in \mathcal{M}_N$ with $\|u(t)-\bar u(t)\| \leq C e^{-\alpha t}$ for some $\alpha>0$. This structure enables rigorous reduction of the (potentially infinite-dimensional) system to a finite number of ODEs for the “slow modes” (Kostianko et al., 2021, Kostianko et al., 2020).

2. Existence and the Spectral Gap Condition

A classical sufficient condition for the existence of $C^1$ inertial manifolds is the spectral gap criterion: $\lambda_{N+1} - \lambda_N > 2L,$ where $L$ is the global Lipschitz constant of $F$ (Kostianko et al., 2021, Kostianko et al., 2020). This ensures a splitting into finitely many “slow modes” and infinitely many “fast modes” with strong enough separation to “squeeze out” the fast dynamics. The Lyapunov–Perron method is then employed on backward time: $$u(t) = \int_{-\infty}^t e^{-A(t-s)}F(u(s))ds + e^{-At}p, \quad t\leq 0,$$ for $p\in H_N$, yielding a fixed point in a weighted Banach space and yielding the graph function $M$ (Kostianko et al., 2021). The spectral gap is sharp: under the construction in both autonomous and nonautonomous settings, a graph with global $C^1$-regularity exists for $\gamma - \rho > 2L$, where $\gamma$ and $\rho$ are “decay” and “growth” rates of the linear splitting (Czaja et al., 31 Jul 2025).

Alternative mechanisms for constructing $C^1$ inertial manifolds, such as spatial averaging (Sell–Mallet-Paret), allow for the existence of inertial manifolds when the spectral gap is not uniform but the nonlinearity is “almost scalar” on intermediate modes (Kostianko et al., 2020). In this framework, the C^1 manifold persists provided the perturbation from scalar is small enough in a suitable norm.

3. Construction and Regularity: Lyapunov–Perron and Cone Methods

The Lyapunov–Perron method remains the principal approach: for each $p\in H_N$, a unique solution $u(\cdot;p)$ is found as a fixed point in an exponentially weighted function space. The manifold is recovered as $M(p) = Q_N u(0;p)$. The $C^1$ regularity is guaranteed by analyzing the variation equation along backward trajectories: $$\partial_t v + A v - F'(u(t)) v = 0, \quad t\leq 0, \quad P_N v(0) = \xi\in H_N.$$ If, in addition to a sufficient spectral gap, the first derivative $M'$ exists and is Hölder-continuous with exponent $\varepsilon$, then the inertial manifold is $C^{1,\varepsilon}$ (Kostianko et al., 2021).

• A further extension to $C^{1+\varepsilon}$ is possible under higher-order cone estimates, such as

$$\|F(u_1) - F(u_2) - F'(u_1)(u_1 - u_2)\|_H \leq C\,\|u_1 - u_2\|_H^{1+\varepsilon}$$

on the manifold (Kostianko et al., 2020).

A crucial obstruction appears for $C^2$ regularity: the second variation equation demands a further large gap (e.g., $\lambda_{N+1} - 2\lambda_N > 3L$) which is rarely met in generic spectra. The consequence is that $C^{1,\varepsilon}$-manifolds are generally the optimal regularity for fixed dimension and unmodified nonlinearities (Kostianko et al., 2021).

4. Extensions: Higher Regularity and the Whitney Principle

The limitations imposed by the spectral gap for higher smoothness ($C^n$) can be bypassed under certain conditions. If infinitely many arbitrarily large gaps appear in the spectrum,

$$\limsup_{N\rightarrow\infty} (\lambda_{N+1} - \lambda_N) = \infty,$$

then one can construct inertial manifolds of increasing dimension with arbitrarily high $C^{n,\varepsilon}$ smoothness by passing to larger $N$ and suitably modifying the nonlinearity $F$ (e.g., cutoff outside the global attractor), or by employing the Whitney extension theorem to match Taylor jets up to desired order. This guarantees the existence of $C^{n,\varepsilon}$ inertial manifolds arbitrarily close to the original $C^{1,\varepsilon}$ manifold on the attractor (Kostianko et al., 2021).

Table: Regularity vs. Spectral Gap

Regularity	Spectral Gap Condition	Additional Assumption
Lipschitz	$\lambda_{N+1}-\lambda_N > L$	$F$ globally Lipschitz
$C^{1,\varepsilon}$	$$\lambda_{N+1}-(1+\varepsilon)\lambda_N > (2+\varepsilon)L$$	$F$ $C^1$, uniform bounds
$C^{n,\varepsilon}$	Infinitely many gaps $>2L$ and possibly modified $F$	Whitney extension employed

A plausible implication is that, for typical equations like the 1D heat equation with Dirichlet boundary ($A=-\partial_{xx}$, $\lambda_k = k^2$), the growing gaps $2k+1\to\infty$ as $k\to\infty$ allow construction of $C^n$ inertial manifolds of arbitrary dimension on modified systems (Kostianko et al., 2021).

5. Non-Autonomous and Stochastic Generalizations

For time-dependent (non-autonomous) systems,

$u_t = A(t) u + f(t,u),\qquad u\in X,$

a $C^1$ inertial manifold exists under a time-dependent exponential dichotomy (splitting) and a sharp gap condition, with all constructions and estimates made in moving, adapted norms. The Lyapunov–Perron fixed-point theory extends, guaranteeing a unique, exponentially attracting $C^1$-manifold that depends smoothly on the fiber variable, provided $f$ is $C^1$ in $u$ (Czaja et al., 31 Jul 2025).

In stochastic frameworks, for systems with (e.g.) non-Gaussian Lévy noise, one conjugates away the noise component and applies similar backward Lyapunov–Perron methods on random Banach spaces. Under an appropriate spectral gap (in expectation or almost surely), one constructs random invariant manifolds (RIMs) with $C^1$ regularity and proves convergence in appropriate senses (e.g., in probability as parameters of the noise law converge) (Wu et al., 23 Nov 2025).

6. Perturbation Theory and Convergence of C^1 Inertial Manifolds

The $C^1$ regularity is robust under perturbations. If two parabolic problems are posed on nearby phase spaces, with elliptic operators $A_0$, $A_\varepsilon$ and nonlinearities $F_0$, $F_\varepsilon$, and the data converge suitably (in operator norm, in $C^1$ for nonlinearities, and with uniform spectral gap), their inertial manifolds $\mathcal{M}_\varepsilon$, $\mathcal{M}_0$ satisfy

$$\|\Phi_\varepsilon - \Phi_0\|_{C^1} + [D\Phi_\varepsilon - D\Phi_0]_{C^{0,\theta}} \leq C \{ T(\varepsilon)|\ln T(\varepsilon)| + p(\varepsilon) + B(\varepsilon)\}$$

for $0<\theta<1$ and small $\varepsilon$, where $T(\varepsilon)$, $p(\varepsilon)$, $B(\varepsilon)$ quantify the operator and nonlinearity differences. Thus, the distance between the inertial manifolds converges to zero in $C^{1,\theta}$ under convergence of the underlying problems (Arrieta et al., 2017).

7. Applications and Broader Relevance

$C^1$ inertial manifolds provide a rigorous finite-dimensional reduction for a wide range of PDEs:

• Parabolic equations (reaction–diffusion, Cahn–Hilliard, Kuramoto–Sivashinsky)
• Damped wave equations
• Generalized (modified) Navier–Stokes equations under spatial averaging They are routinely constructed in periodic or bounded domains, and their existence is central for both the numerical simulation of infinite-dimensional dynamics and the analysis of bifurcation, pattern formation, and long-time behavior in dissipative systems.

The extension of $C^1$ inertial manifold theory to non-autonomous, stochastic, or perturbed settings supports the analysis of multiscale and random systems in mathematical physics, fluid dynamics, and beyond (Kostianko et al., 2020, Czaja et al., 31 Jul 2025, Wu et al., 23 Nov 2025). The sharpness of the spectral gap condition ($\gamma-\rho>2L$) is established and known to be optimal (Czaja et al., 31 Jul 2025), and the existence and convergence estimates underpin a unified approach to invariant manifold theory in infinite dimensions.