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Well-posedness of rough 2D Euler equation with bounded vorticity

Leonardo Roveri, Francesco Triggiano

Abstract

We consider the 2D Euler equation with bounded initial vorticity and perturbed by rough transport noise. We show that there exists a unique solution, which coincides with the starting condition advected by the Lagrangian flow. Moreover, the stability of the solution map with respect to the initial vorticity and the rough perturbation yields a Wong-Zakai result for fractional Brownian driving paths.

Well-posedness of rough 2D Euler equation with bounded vorticity

Abstract

We consider the 2D Euler equation with bounded initial vorticity and perturbed by rough transport noise. We show that there exists a unique solution, which coincides with the starting condition advected by the Lagrangian flow. Moreover, the stability of the solution map with respect to the initial vorticity and the rough perturbation yields a Wong-Zakai result for fractional Brownian driving paths.

Paper Structure

This paper contains 18 sections, 25 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Localized variation spaces
  5. Localized controlled rough paths
  6. Unbounded rough drivers and smoothing operators
  7. Existence of solutions
  8. A priori estimates
  9. Existence
  10. Euler Lagrangian flows
  11. Log-Lipschitz RDE
  12. RDEs with non-local drift
  13. Uniqueness of the solution
  14. Doubling variable
  15. Uniqueness result
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $w_0\in L^{\infty}(\mathbb{T}^2)$, $\mathbf{Z}$ be a $p$-geometric rough path with $p \in [2,3)$ and $\{\sigma_j\}_{j=1}^M\subset C^3(\mathbb{T}^2,\mathbb{R}^2)$ be divergence-free vector fields. Then, the following statements hold:

Theorems & Definitions (54)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 44 more