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Remarks on Regularization by Noise, Convex Integration and Spontaneous Stochasticity.

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This study explores connections and distinctions between regularization by noise, convex integration, and spontaneous stochasticity. These methods examine how small perturbations in fluid dynamics affect large-scale behaviors, aiming to clarify their relationships and differences.

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Area of Science:

  • Fluid dynamics
  • Stochastic processes
  • Mathematical physics

Background:

  • Small-scale perturbations in fluid dynamic equations can significantly impact large-scale behaviors.
  • Existing research has explored regularization by noise, convex integration, and spontaneous stochasticity independently.
  • The precise relationships and differences between these phenomena are not fully understood.

Purpose of the Study:

  • To discuss the potential links and differences between regularization by noise, convex integration, and spontaneous stochasticity.
  • To provide a comparative examination of these three topics.
  • To stimulate new research into the effects of small-scale perturbations on fluid dynamics.

Main Methods:

  • Comparative analysis of theoretical frameworks.
  • Examination of the impact of small-scale perturbations on fluid dynamic equations.
  • Literature review and conceptual synthesis.

Main Results:

  • Identified commonalities between convex integration and spontaneous stochasticity.
  • Highlighted contrasting effects, such as in regularization by noise.
  • Acknowledged the lack of rigorous links and precise explanations for the observed phenomena.

Conclusions:

  • The study provides a foundational comparison of three distinct but related concepts in fluid dynamics.
  • Further research is needed to rigorously define the links and differences between these perturbation effects.
  • This comparative examination serves as a catalyst for future investigations in the field.