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Anomalous and total dissipation due to advection by solutions of randomly forced Navier-Stokes equations

Martina Hofmanová, Umberto Pappalettera, Rongchan Zhu, Xiangchan Zhu

Abstract

We propose a novel approach to induce anomalous dissipation through advection driven by turbulent fluid flows. Specifically, we establish the existence of a velocity field $v$ satisfying randomly forced Navier-Stokes equations, leading to total dissipation of kinetic energy in finite time when advecting a passive scalar. This dissipation phenomenon is uniform across viscosity parameters and initial conditions, representing a case of anomalous dissipation. We further explore dissipation induced by individual realizations of $v$. Our results extend to scenarios where the passive scalar is replaced by solutions to two or three-dimensional deterministic Navier-Stokes equations advected by $v$.

Anomalous and total dissipation due to advection by solutions of randomly forced Navier-Stokes equations

Abstract

We propose a novel approach to induce anomalous dissipation through advection driven by turbulent fluid flows. Specifically, we establish the existence of a velocity field satisfying randomly forced Navier-Stokes equations, leading to total dissipation of kinetic energy in finite time when advecting a passive scalar. This dissipation phenomenon is uniform across viscosity parameters and initial conditions, representing a case of anomalous dissipation. We further explore dissipation induced by individual realizations of . Our results extend to scenarios where the passive scalar is replaced by solutions to two or three-dimensional deterministic Navier-Stokes equations advected by .
Paper Structure (18 sections, 11 theorems, 160 equations)

This paper contains 18 sections, 11 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Known deterministic results
  4. Known stochastic results
  5. Our contribution
  6. Organization of the paper
  7. Main ideas of the construction
  8. Structure of the noise
  9. Reintroducing roughness at small time scales
  10. Total dissipation by transport noise
  11. Stratonovich corrector
  12. Dissipation
  13. Non dissipating solutions
  14. Total dissipation by solution of randomly forced Navier-Stokes equations
  15. Estimates on $v$
  16. ...and 3 more sections

Key Result

Theorem 1.1

There exist a countable family of Brownian motions $\{W^{k,\alpha}\}$, time-dependent velocity fields $\{\sigma_{k,\alpha}\}$, and a weak solution $v$ of the Navier-Stokes equations with large friction and additive noise: where $\varepsilon=\varepsilon(t)$ depends on $t$ and is piecewise constant, such that and every progressively measurableBy weak continuity of weak solutions to eq:passive_scal

Theorems & Definitions (25)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 2.1
  • Proposition 3.1
  • Definition 3.2
  • Remark 3.3
  • ...and 15 more