Computer-Assisted Proofs of Blow-Up

• Computer-assisted proofs of blow-up are rigorous methods combining symbolic computation with analytical techniques to verify finite-time singularities in nonlinear PDEs.
• They leverage tools such as convex integration, MATLAB-based minimization, and physics-informed neural networks to construct and validate singular blow-up profiles.
• These strategies not only yield explicit examples in fluid dynamics but also guide future rigorous verification approaches for complex nonlinear systems.

Computer-assisted proofs of blow-up employ rigorous numerical and symbolic computation to establish the formation of singularities (finite-time blow-up) in nonlinear partial differential equations (PDEs). Such proofs combine analytical constructions (e.g., convex integration, self-similar reduction) with computer-aided searches for algebraic or functional structures that are either out of reach or infeasibly complex for hand calculation. Recent advances have yielded the first explicit constructions of bounded weak solutions with instantaneous blow-up of relevant norms, as well as numerical profiles for self-similar blow-up in fundamental fluid models, thus laying the foundation for the next generation of rigorous, computer-validated singularity results.

1. Mathematical Frameworks for Blow-Up

Blow-up in the context of PDEs refers to solutions that develop singularities in finite time, such as unbounded gradients, $L^\infty$-norm, or total variation (BV-norm). Canonical systems include hyperbolic conservation laws (e.g., the $p$-system) and hydrodynamic equations (e.g., Euler and Boussinesq). Rigorous constructions may target different notions of solutions (weak, bounded, entropy-admissible) and different types of blow-up (e.g., divergence of $\|u\|_\infty$, $\|\omega\|_\infty$, or $\mathrm{TV}(u)$), with or without the formation of shocks.

A prototypical result concerns the $p$-system in Lagrangian coordinates,

$$\partial_t v + \partial_x u = 0, \quad \partial_t u + \partial_x p(v) = 0,$$

where $p:\mathbb{R}\to\mathbb{R}$ is strictly increasing and strictly concave. For open sets of Riemann data, there exist uncountably many bounded weak solutions with instantaneous, infinite BV-norm for all $t>0$, yet no shocks—vacuously satisfying Lax and Liu admissibility conditions (Krupa, 2024). In fluid dynamics, computer-assisted analysis is central to investigating finite-time singularity in the three-dimensional Euler equations and proxies such as the two-dimensional Boussinesq model (Wang et al., 2022).

2. Analytic and Algebraic Ingredients

The proof of BV blow-up for the $p$-system leverages classical tools (Riemann problems, shock admissibility, differential inclusions) and modern convex integration techniques. The central analytic object is a Lipschitz “stream function” whose derivative takes exactly five prescribed values $p$0 in $p$1, with no two differing by a rank-one matrix—precluding shocks entirely. The construction is completed by mapping the function's derivatives to the physical variables $p$2 and showing that in each wedge region of spacetime, the solution oscillates infinitely fast among all five states, forcing infinite total variation.

This analytic structure requires the explicit identification, via computation, of a “large $p$3 configuration,” i.e., a quintuple of matrices within the constitutive set satisfying key algebraic relations—none rank-one connected, multiple $p$4 cycle orderings, linear independence of the connecting directions, and two additional inequalities that guarantee the existence of a suitable pressure law $p$5 via a convex-extension lemma (Krupa, 2024).

3. Computer-Assisted Strategies

The core of the computer-assisted approach is a rigorous search for the required algebraic configuration. The procedure involves:

• Parameterizing rank-one “jumps” as outer products, encoding all nonlinear constraints of the $p$6 cycles, and posing the realization as a constrained minimization problem.
• Employing MATLAB's \texttt{fmincon} (interior-point algorithm) to obtain a floating-point solution.
• Converting the numerical solution to exact rational or symbolic form and verifying all algebraic constraints—rank-one disconnect, inequalities, symmetries—symbolically.
• Automated verification that at least three permutations of the quintuple form valid $p$7 cycles, using algebraic criteria as in Székelyhidi–Weber.
• Construction of a pressure law $p$8 via explicit implementation of a convex-extension lemma, checked symbolically for strict convexity.

The process is implemented in a modular MATLAB workflow, with reproducibility and symbolic validation central to the method (see https://github.com/sammykrupa/BV-blowup-for-p-system). Once the algebraic cornerstone is obtained, the remainder of the proof—analytic convex integration and extraction of the singular solution—is carried out rigorously without further computational input (Krupa, 2024).

4. Neural-Network Based Verification of Self-Similar Blow-Up

A distinct paradigm harnesses machine learning, in particular physics-informed neural networks (PINNs), to numerically approximate self-similar blow-up profiles in hydrodynamic PDEs. In the work of Wang–Lai–Gómez-Serrano, this approach is used to identify a smooth self-similar profile for the 2D Boussinesq and axisymmetric 3D Euler equations, as well as unstable self-similar solutions for the Córdoba–Cordoba–Fontelos equation (Wang et al., 2022).

Key aspects include:

• Representing solution components and the critical exponent as neural networks with tailored symmetry constraints.
• Constructing compound loss functions to enforce the stationary self-similar PDE, boundary/decay conditions, and regularity.
• Training via Adam and L-BFGS optimizers, employing domain decomposition for efficient collocation.
• Validation by robust convergence, comparison to known analytical and prior numerical results, and exploration of both stable and unstable branches.

The resulting profiles exhibit vanishingly small PDE residuals and precise determination of blow-up exponents (e.g., $p$9), and the approach has yielded the first unstable CCF self-similar solutions with $\|u\|_\infty$0 (Wang et al., 2022).

5. Toward Fully Rigorous, Computer-Assisted Proofs

The neural-network-based outputs yield highly accurate, explicit multidimensional blow-up profiles. The roadmap to a fully computer-assisted proof consists of:

1. Substituting the trained network with validated rational or Chebyshev expansions and using interval arithmetic to rigorously bound PDE residuals.
2. Decomposing the full PDE solution as the sum of the self-similar profile plus a remainder, linearizing and establishing evolution equations for the remainder.
3. Computing the spectral gap for the linearized operator and bounding it rigorously to control perturbation growth.
4. Establishing nonlinear stability by means of energy inequalities and bootstrap arguments.
5. Truncating the infinite-energy self-similar profile to obtain finite-energy initial data, verifying that solutions remain close to the singularity-forming core (Wang et al., 2022).

This suggests that data-driven numerical discovery, when combined with interval analysis and spectral verification, forms a viable pathway to computer-assisted proofs of finite-time singularity in multidimensional fluids.

6. Significance and Limitations

The computer-assisted construction of blow-up in the $\|u\|_\infty$1-system delivers the first example of bounded, Liu-admissible (entropy) solutions with instantaneous BV blow-up in a genuinely nonlinear hyperbolic system (Krupa, 2024). The neural network approach provides explicit, numerically validated self-similar profiles and exponents for hydrodynamic systems, with small residuals and verified robustness, setting the stage for interval-validated proofs (Wang et al., 2022).

However, such proofs are currently limited to settings where the algebraic and computational constraints are tractable, and where symbolic or interval arithmetic can guarantee rigor. Analytical gaps remain open in establishing full nonlinear stability, especially in the presence of multiple unstable modes, and in extending methods to equations of higher complexity or less symmetry.

7. Open Problems and Future Directions

Ongoing research targets generalizing convex integration and computer-assisted search to broader classes of PDEs and initial data, with interest in:

• Extending to higher-dimensional and non-symmetric configurations.
• Incorporating verified a posteriori error bounds into neural network-based solution profiles.
• Bridging from numerical profiles to rigorous energy and stability estimates, particularly for the 3D Euler singularity question.
• Developing computational tools for spectral gap analysis and interval arithmetic at scale.

Advances in these directions will be essential for full computer-assisted proofs of blow-up in critical physical systems, further integrating analytic and computational perspectives.