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Unbounded rough drivers, rough PDEs and applications

Antoine Hocquet, Martina Hofmanova, Torstein Nilssen

TL;DR

This paper surveys the unbounded rough driver (URD) framework for rough PDEs driven by linear rough input and presents a variational approach that yields well-posedness and energy-type estimates for parabolic RPDEs with transport noise. It develops concrete operator-valued URDs, including differential and bounded-operator cases, and analyzes geometric versus weak-geometric enhancements and their implications (e.g., Lévy areas). The survey then applies these methods to nonlinear parabolic RPDEs, nonlocal transport models, and key fluid models—stochastic Landau–Lifshitz–Gilbert, Navier–Stokes, and Euler equations—obtaining pathwise solutions, energy inequalities, uniqueness in certain regimes, large deviation principles, and random dynamical-system properties. The results demonstrate the versatility of URD-based variational methods in handling rough transports, preserving structural features (like incompressibility and vorticity conservation), and enabling robust well-posedness theories with implications for stochastic fluid dynamics and micromagnetism.

Abstract

A summary of recent contributions in the field of rough partial differential equations is given. For that purpose we rely on the formalism of ``unbounded rough driver''. We present applications to concrete models including Landau-Lifshitz-Gilbert, Navier-Stokes and Euler equations.

Unbounded rough drivers, rough PDEs and applications

TL;DR

This paper surveys the unbounded rough driver (URD) framework for rough PDEs driven by linear rough input and presents a variational approach that yields well-posedness and energy-type estimates for parabolic RPDEs with transport noise. It develops concrete operator-valued URDs, including differential and bounded-operator cases, and analyzes geometric versus weak-geometric enhancements and their implications (e.g., Lévy areas). The survey then applies these methods to nonlinear parabolic RPDEs, nonlocal transport models, and key fluid models—stochastic Landau–Lifshitz–Gilbert, Navier–Stokes, and Euler equations—obtaining pathwise solutions, energy inequalities, uniqueness in certain regimes, large deviation principles, and random dynamical-system properties. The results demonstrate the versatility of URD-based variational methods in handling rough transports, preserving structural features (like incompressibility and vorticity conservation), and enabling robust well-posedness theories with implications for stochastic fluid dynamics and micromagnetism.

Abstract

A summary of recent contributions in the field of rough partial differential equations is given. For that purpose we rely on the formalism of ``unbounded rough driver''. We present applications to concrete models including Landau-Lifshitz-Gilbert, Navier-Stokes and Euler equations.
Paper Structure (27 sections, 20 theorems, 127 equations)

This paper contains 27 sections, 20 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Unbounded rough drivers
  3. Formal definition
  4. Specific examples
  5. Operator-valued paths of order one
  6. The case of differential operators
  7. Geometric enhancement property
  8. Weak versus strong URD geometricity
  9. Incompressible Navier-Stokes
  10. Operator-valued paths of order zero
  11. A priori estimates
  12. Controlled rough paths estimates
  13. Behavior with respect to products
  14. The main Sobolev (energy) estimates
  15. Parabolic rough PDEs
  16. ...and 12 more sections

Key Result

Lemma 2.4

Let ${\mathbf A}$ be a geometric enhancement of B1_diff. There is a $2$-index family of coefficents $(X\mathbin{\vcenter{\hbox{$\m@th\bullet$}}}\nabla X^i)_{i=0,1,\dots,d}$ which is subject to the Chen's-type relation for each $x\in {\mathbb{R}}^d$, $i=0,1,\dots ,d$, such that the second level of ${\mathbf A}$ is given by

Theorems & Definitions (30)

  • Example 2.2
  • Definition 2.3: Unbounded rough driver on $E$
  • Lemma 2.4
  • Example 2.5
  • Proposition 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Definition 3.1: Solution
  • Proposition 3.2: Remainder estimates
  • ...and 20 more