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Area of Science:

  • Fluid Dynamics
  • Computational Mathematics
  • Applied Physics

Background:

  • The existence of finite-time blow-up solutions for the 2D Boussinesq and 3D Euler equations remains a critical unsolved problem in fluid mechanics.
  • Understanding blow-up phenomena is essential for predicting turbulent behavior and instabilities in fluid systems.

Purpose of the Study:

  • To develop a novel numerical framework capable of discovering self-similar blow-up solutions for complex fluid dynamics equations.
  • To investigate the potential of physics-informed neural networks (PINNs) in finding both stable and unstable solutions to nonlinear partial differential equations.

Main Methods:

  • Implementation of a new numerical framework utilizing physics-informed neural networks (PINNs).
  • The PINNs were trained to discover smooth, self-similar solution profiles for the 2D Boussinesq and 3D Euler equations.
  • Application of the framework to find an unstable self-similar solution for the Córdoba-Córdoba-Fontelos equation.

Main Results:

  • Successfully discovered the first smooth self-similar blow-up profile for both the 2D Boussinesq and 3D Euler equations.
  • Demonstrated the capability of PINNs to identify unstable self-similar solutions, exemplified by the Córdoba-Córdoba-Fontelos equation.
  • Validated the robustness and adaptability of the developed numerical framework across different fluid equations.

Conclusions:

  • The discovered self-similar solutions provide a foundation for potential computer-assisted proofs of blow-up in fluid dynamics.
  • Physics-informed neural networks represent a powerful and versatile tool for exploring complex solutions in fluid mechanics and beyond.
  • The numerical framework offers a robust approach for finding challenging solutions to nonlinear partial differential equations.