Navier-Stokes equations with stochastic lie transport: well-posedness and inviscid limit
File(s)
Author(s)
Goodair, Daniel
Type
Thesis
Abstract
The Navier-Stokes Equations are the fundamental model for viscous fluid dynamics, but no prediction is perfect. Introducing randomness into differential equations has long been an answer to uncertainty quantification; recent developments in the modelling literature point to the significance of transport noise, where the stochastic integral depends on the gradient of the solution. Analytically, this unbounded noise breaks classical frameworks built for studying nonlinear stochastic partial differential equations (SPDEs).
The primary goal of this thesis is to develop a rigorous solution theory for the Navier-Stokes Equations under Stochastic Advection by Lie Transport, in a general manner which extends to a wide class of nonlinear SPDEs with unbounded noise. Particular attention is given to the case of a domain with physical boundary, where existence results for strong solutions with transport noise were previously unknown. Our theory pertains to both the most classical no-slip boundary condition, as well as the Navier boundary conditions. In particular, strong solutions are obtained for the Navier boundary conditions.
Beyond well-posedness theory, we consider the inviscid limit of solutions under the no-slip and Navier boundary conditions. The deterministic theory is markedly different in these two scenarios, the core results of which are recovered in our stochastic setting. For the no-slip condition, the vanishing viscosity and noise limit is shown to provide a solution of the deterministic Euler equation if and only if the energy dissipation in a boundary layer of width approaching zero with viscosity likewise tends to zero. Whilst this property is unknown in general, for a special case of the Navier boundary conditions, the inviscid limit is demonstrated to exist and give a weak solution of the corresponding Stochastic Euler Equation.
The primary goal of this thesis is to develop a rigorous solution theory for the Navier-Stokes Equations under Stochastic Advection by Lie Transport, in a general manner which extends to a wide class of nonlinear SPDEs with unbounded noise. Particular attention is given to the case of a domain with physical boundary, where existence results for strong solutions with transport noise were previously unknown. Our theory pertains to both the most classical no-slip boundary condition, as well as the Navier boundary conditions. In particular, strong solutions are obtained for the Navier boundary conditions.
Beyond well-posedness theory, we consider the inviscid limit of solutions under the no-slip and Navier boundary conditions. The deterministic theory is markedly different in these two scenarios, the core results of which are recovered in our stochastic setting. For the no-slip condition, the vanishing viscosity and noise limit is shown to provide a solution of the deterministic Euler equation if and only if the energy dissipation in a boundary layer of width approaching zero with viscosity likewise tends to zero. Whilst this property is unknown in general, for a special case of the Navier boundary conditions, the inviscid limit is demonstrated to exist and give a weak solution of the corresponding Stochastic Euler Equation.
Version
Open Access
Date Issued
2024-07
Date Awarded
2024-09
Copyright Statement
Creative Commons Attribution NonCommercial NoDerivatives Licence
Advisor
Crisan, Dan
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
2478902
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)