Rigorous $C_1$ integration of dissipative PDEs

Abstract: We introduce a new $C^1$ algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with respect to initial data. As applications, we establish the existence of locally attracting periodic orbits for initial and boundary value problems for two non-autonomous dissipative PDEs: the Chafee-Infante equation and the Burgers equation with a fractional Laplacian.

• The paper introduces a rigorous C¹ integration algorithm that extends computer-assisted proofs for dissipative PDEs.
• It employs a novel phase space decomposition and interval arithmetic to obtain tight estimates for both solution trajectories and Fréchet derivatives.
• Validated on the Chafee-Infante and fractional Burgers equations, the method guarantees the existence and local stability of periodic orbits even under sign-changing nonlinearities.

Rigorous $C_1$ Integration Algorithm for Dissipative PDEs

Introduction

The paper "Rigorous $C_1$ integration of dissipative PDEs" (2604.01046) develops a novel computer-assisted algorithm for rigorous $C_1$ integration of dissipative PDEs. The technique advances validated numerics for infinite-dimensional systems by enabling reliable control over both the flow and its derivative with respect to initial data. The approach is tailored for use in the validation of qualitative features such as local attractivity and periodic orbits—a cornerstone in the computability of global dynamics in certain dissipative systems.

The validation pipeline is exemplified through comprehensive applications to two non-autonomous dissipative PDEs: the Chafee-Infante equation and the forced Burgers equation with a fractional Laplacian. Notably, these results establish, for the first time, the existence of locally attracting periodic orbits in parameter regimes where nonlinearity may change sign, overcoming persistent open difficulties related to blowup and instability.

Mathematical Formulation and Algorithmic Architecture

Abstract Evolution Problems

The analysis is conducted in the general setting of semilinear evolutionary equations in Banach and Hilbert spaces. The PDEs are recast as abstract ODEs:

$\frac{dx}{dt} = Lx + f(t, x), \quad x(t_0) = x_0,$

with $L$ a linear (diagonalizable) generator on $Y$ densely embedded in $W$, and $f$ a time-dependent nonlinearity.

Variational Equation

A defining element is the explicit handling of the variational equation,

$$\frac{dh}{dt} = Lh + D_x f(t, x(t)) h, \quad h(t_0) = h_0,$$

which describes the linearization about a reference solution and underpins $C_1$-type results (control of the derivative of the flow with respect to initial data).

The paper proves that, under standard dissipativity, regularity, and embedding conditions, both the base flow and its variational equation possess unique local solutions, with propagation estimates uniform over high- (tail) modes.

Representation of Infinite-Dimensional Sets

Rigorous integration is implemented via a hybrid finite-tail/infinite-mode framework. Spatial discretization is effected via Galerkin projections onto a $C_1$0-space (finitely many modes) and $C_1$1-space (tail). Sets in $C_1$2 or $C_1$3 are efficiently encoded by infinite interval vectors (sequence of intervals specifying bounds on Fourier coefficients), with the polynomial decay of high modes governing enclosure radii.

For the variational system, initial sets can be unbounded in the tail, enabling coverage of genuinely infinite-dimensional perturbation directions—critical for $C_1$4 validation.

The $C_1$5 Integration Algorithm

The algorithm symmetrically integrates both the base and variational system. For explicit modes, the variational equation is solved using the standard $C_1$6 algorithm along each basis direction, then combined linearly. Unbounded sets are controlled for the tail using new rigorous estimates (extensions of self-consistent bounds), and convolutions of power-law decaying Fourier components are handled using sharp interval techniques (see Lemmas 5.3–5.6).

Nonlinear terms (e.g., cubic and quadratic convolutions) and their derivatives are bounded using explicit algebraic estimates on products of Fourier coefficients, accommodating even slow or nondecaying tails.

Enclosures are validated using iterative interval arithmetic, advancing set inclusions forward in time with explicit control of local truncation and wrapping errors.

Applications to Canonical Dissipative PDEs

Non-autonomous Chafee-Infante Equation

This equation presents a time-dependent cubic nonlinearity that can change sign, allowing for periods of potential blowup. Previous results have only addressed the globally attractive equilibria in parameter regimes where non-autonomous modulations are strictly positive.

By constructing explicit enclosures for the initial value map and its $C_1$7 variational derivative using the described algorithm, the paper establishes:

• Existence of a periodic orbit in $C_1$8 with period 1 for $C_1$9 sign-changing (e.g., $C_1$0).
• Local exponential attractivity with explicit rates, even when coefficients become negative, so the term $C_1$1 can promote blowup transiently.
• Fine structure of the attractor, extending analysis to parameter regimes with multiple unstable directions.

These results address previously unresolved cases in the PDE literature. The estimates are numerically tight: for $C_1$2 with negative intervals, this is the first global existence result for nonzero non-autonomous solutions in such regimes with rigorous attractivity constants.

Forced Fractional Burgers Equation

The non-autonomous forced Burgers equation with fractional Laplacian, studied for $C_1$3 and odd-periodic $C_1$4, exhibits a delicate balance between nonlocal dissipation and quadratic convection. Previous results only handled finite-time existence in certain cases, with global attractors demonstrated under strong smoothing.

Using the $C_1$5 algorithm and high-mode interval arithmetic, the paper provides:

• Validation of locally attracting periodic orbits for a broad range of $C_1$6 values, approaching the critical threshold where regularity can break down.
• For $C_1$7, tight operator norm estimates are shown to be below unity, directly yielding local exponential convergence in the $C_1$8 topology.

At lower $C_1$9, enclosure blowup prevents validation within standard norms, but the approach provides computable thresholds for attractivity.

Algorithmic Novelty and Numerical Results

• Control over unbounded initial data in the variational direction is a critical innovation, enabling fully infinite-dimensional $\frac{dx}{dt} = Lx + f(t, x), \quad x(t_0) = x_0,$0 propagation with rigorous error control.
• The method handles non-diagonal (nonlocal) linear terms and time-dependent nonlinearities without recourse to the logarithmic norm, allowing for generality beyond classic parabolic PDEs.
• Results match theoretical predictions for linearized stability, and the framework accommodates validation of instability/hyperbolicity using rigorous eigenvalue enclosure (for Krawczyk/Newton-based proofs).
• Operator norm bounds for derivatives in the entire phase space are computed, with exponential rates aligned with numerically obtained eigenvalue spectra.
• Extensions to compute rigorous invariants (e.g., for Poincaré maps) and validate the stability of orbits in systems such as the Brusselator are described.

Broader Implications and Future Directions

The capacity for rigorous $\frac{dx}{dt} = Lx + f(t, x), \quad x(t_0) = x_0,$1 integration in infinite dimensions fundamentally expands the reach of computer-assisted proofs in dissipative PDEs. The algorithm is directly applicable to:

• Validation of periodic orbits and their hyperbolicity in broad classes of semilinear PDEs, including systems with non-diagonal linear parts (e.g., complex Ginzburg-Landau).
• Rigorous bifurcation analysis, including period-doubling cascades and heteroclinic network validation in non-autonomous environments.
• Automated, reliable validation of attractors and their local structure (stable, unstable, or center manifolds) in settings where traditional functional analytic approaches encounter inherent limitations (e.g., due to sign-changing nonlinearities, as in the Chafee-Infante case).

Further, the explicit design of the algorithm with interval and rigorous numerics packages (CAPD) ensures reproducibility and transparency in all computational proofs.

Conclusion

This work establishes a mathematically rigorous, computer-assisted framework for validated $\frac{dx}{dt} = Lx + f(t, x), \quad x(t_0) = x_0,$2 integration in dissipative PDEs. By overcoming limitations in the control of variational derivatives and enabling the inclusion of unbounded perturbations, the algorithm enables computer-assisted proofs of the existence, stability, and, crucially, attractivity of periodic orbits in complex infinite-dimensional dynamical systems. The approach closes several open questions regarding global non-autonomous solutions in classical nonlinear PDEs and will serve as a foundation for further developments in rigorous PDE numerics, including generalized attractor characterization and validated bifurcation analysis.