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Random dynamical system generated by the 3D Navier-Stokes equation with rough transport noise

Jorge Cardona, Martina Hofmanova, Torstein Nilssen, Nimit Rana

Abstract

We consider the Navier-Stokes system in three dimensions perturbed by a transport noise which is sufficiently smooth in space and rough in time. The existence of a weak solution was proved recently, however, as in the deterministic setting the question of uniqueness remains a major open problem. An important feature of systems with uniqueness is the semigroup property satisfied by their solutions. Without uniqueness, this property cannot hold generally. We select a system of solutions satisfying the semigroup property with appropriately shifted rough path. In addition, the selected solutions respect the well accepted admissibility criterium for physical solutions, namely, maximization of the energy dissipation. Finally, under suitable assumptions on the driving rough path, we show that the Navier-Stokes system generates a measurable random dynamical system. To the best of our knowledge, this is the first construction of a measurable single-valued random dynamical system in the state space for an SPDE without uniqueness.

Random dynamical system generated by the 3D Navier-Stokes equation with rough transport noise

Abstract

We consider the Navier-Stokes system in three dimensions perturbed by a transport noise which is sufficiently smooth in space and rough in time. The existence of a weak solution was proved recently, however, as in the deterministic setting the question of uniqueness remains a major open problem. An important feature of systems with uniqueness is the semigroup property satisfied by their solutions. Without uniqueness, this property cannot hold generally. We select a system of solutions satisfying the semigroup property with appropriately shifted rough path. In addition, the selected solutions respect the well accepted admissibility criterium for physical solutions, namely, maximization of the energy dissipation. Finally, under suitable assumptions on the driving rough path, we show that the Navier-Stokes system generates a measurable random dynamical system. To the best of our knowledge, this is the first construction of a measurable single-valued random dynamical system in the state space for an SPDE without uniqueness.

Paper Structure

This paper contains 13 sections, 15 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Helmholtz--Leray projection
  4. Smoothing operators
  5. Rough path theory
  6. Definition of weak solution
  7. Semiflow Selection
  8. Sequential stability
  9. Shift invariance and continuation property
  10. General ansatz
  11. Selection sequence
  12. Random dynamical system
  13. A priori estimate and compactness

Key Result

Theorem 1

The Navier-Stokes equation eq:classicalForm-eq:aa3 admits a semiflow selection in the class of weak solutions, that is, there is a measurable mapping which assigns to every initial condition $[u_{0},E_{0}]$ and a rough path $\textbf{Z}$ one solution trajectory $[u,E]$ so that the following semigroup property holds true where $\tilde{\textbf{Z}}_{t_1}(\cdot)$ denotes the shifted rough path $\tild

Theorems & Definitions (32)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5: Weak solution
  • Theorem 2.6
  • Remark 2.7
  • Definition 3.1: Admissible weak solution
  • ...and 22 more