On the Navier–Stokes equation perturbed by rough transport noise

Abstract

We consider the Navier–Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space–time-dependent noise that is smooth in space and rough in time. We study the system within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of a weak solution of the Navier–Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we prove that the solution is unique and stable with respect to the driving noise.

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Correspondence to Torstein Nilssen.

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M. Hofmanová and T. Nilssen: Financial support by the DFG via Research Unit FOR 2402 is gratefully acknowledged. T. Nilssen: Funded by the Norwegian Research Council (Project 230448/F20).

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Hofmanová, M., Leahy, JM. & Nilssen, T. On the Navier–Stokes equation perturbed by rough transport noise. J. Evol. Equ. 19, 203–247 (2019). https://doi.org/10.1007/s00028-018-0473-z

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Mathematics Subject Classification

  • 60H15
  • 76D05
  • 47J30
  • 60H05
  • 35A15

Keywords

  • Rough paths
  • Stochastic PDEs
  • Navier–Stokes equation
  • Variational method