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Global solutions for stochastically controlled fluid dynamics models

Dan Crisan, Oana Lang

Abstract

For a class of evolution equations that possibly have only local solutions, we introduce a stochastic component that ensures that the solutions of the corresponding stochastically perturbed equations are global. The class of partial differential equations amenable for this type of treatment includes the 3D Navier-Stokes equation, the rotating shallow water equation (viscous and inviscid), 3D Euler equation (in vorticity form), 2D Burgers' equation and many other fluid dynamics models.

Global solutions for stochastically controlled fluid dynamics models

Abstract

For a class of evolution equations that possibly have only local solutions, we introduce a stochastic component that ensures that the solutions of the corresponding stochastically perturbed equations are global. The class of partial differential equations amenable for this type of treatment includes the 3D Navier-Stokes equation, the rotating shallow water equation (viscous and inviscid), 3D Euler equation (in vorticity form), 2D Burgers' equation and many other fluid dynamics models.
Paper Structure (10 sections, 8 theorems, 166 equations)

This paper contains 10 sections, 8 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Setting and assumptions
  3. Main results
  4. Uniform control
  5. Tightness
  6. Applications to models from stochastic fluid dynamics
  7. Viscous compressible and incompressible models
  8. Inviscid compressible models
  9. Inviscid incompressible models
  10. Appendix

Key Result

Theorem 7

Under assumptions $(A1)-(A3)$, the stochastic partial differential equation eqnew:maingeneralnew admits a global strong solution $X$ with paths that belong to the space $C\left( [0,\infty ),\mathscr{G}\right) ,$$\mathbb{P}$-almost surely, such that for any $T>0$,

Theorems & Definitions (22)

  • Definition 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 7
  • Remark 8
  • Remark 9
  • Lemma 10
  • ...and 12 more