arXiv subjects

Daniel Goodair

Publications and source records attributed to Daniel Goodair.

14 recordsLinked to original sources

arXiv2026

Global well-posedness for the 2D stochastic hypoviscous Navier-Stokes equations

We study stochastic hypoviscous Navier--Stokes equations on the torus with dissipation $(-\Delta)^\gamma$ for $\gamma \in (\frac{1}{2},1]$ and multiplicative noise. Relying on stochastic maximal regularity results for the linear equation, we establish local well-posedness for this problem in arbitrary dimensions with initial data in a range of scaling-critical Besov spaces. With the aid of $L^q$-energy estimates for the vorticity equation, we also prove global well-posedness of the stochastic hypoviscous Navier--Stokes equation in 2D with linear multiplicative noise.

math.PR↗

arXiv2026

Anisotropic Inviscid Limit for the Navier-Stokes Equations with Transport Noise Between Two Plates

We investigate an anisotropic vanishing viscosity limit of the 3D stochastic Navier-Stokes equations posed between two horizontal plates, with Dirichlet no-slip boundary condition. The turbulent viscosity is split into horizontal and vertical directions, each of which approaches zero at a different rate. The underlying Cylindrical Brownian Motion driving our transport-stretching noise is decomposed into horizontal and vertical components, which are scaled by the square root of the respective directional viscosities. We prove that if the ratio of the vertical to horizontal viscosities approaches zero, then there exists a sequence of weak martingale solutions convergent to the strong solution of the deterministic Euler equation on its lifetime of existence. A particular challenge is that the anisotropic scaling ruins the divergence-free property for the spatial correlation functions of the noise.

math.PR↗

arXiv2025

It\^{o}-Stratonovich Conversion in Infinite Dimensions for Unbounded, Time-Dependent, Nonlinear Operators

We prove that a solution, in a variational framework, to the Stratonovich stochastic partial differential equation with noise $G\left(t, \Psi_t\right) \circ dW_t$ is given by a solution to the It\^{o} equation with It\^{o}-Stratonovich corrector $\frac{1}{2}\sum_{i=1}^\infty D_uG_i\left(t, \Psi_t\right)\left[G_i(t,\Psi_t)\right]dt$. Here $G_i$ denotes the action of $G$ on the $i^{th}$ component of the cylindrical noise, and $D_uG_i$ its Fr\'{e}chet partial derivative in the Hilbert space for which the It\^{o} form is satisfied. The noise operator $G$ may be time-dependent, nonlinear, and unbounded in the sense of differential operators; in the latter case, one must pass to a larger space in order to solve the Stratonovich equation. Our proof relies on martingale techniques, and the results apply to fluid equations with time-dependent and nonlinear transport noise.

math.PR↗

arXiv2025

Closed Estimates of Leray Projected Transport Noise and Strong Solutions of the Stochastic Euler Equations

We consider the incompressible Euler and Navier-Stokes equations on the three dimensional torus, in velocity form, perturbed by a transport or transport-stretching Stratonovich noise. Closed control of the noise contributions in energy estimates are demonstrated, for any positive integer ordered Sobolev Space and the equivalent Stokes Space; difficulty arises due to the presence of the Leray Projector disrupting cancellation of the top order derivative. This is particularly pertinent in the case of a transport noise without stretching, where the vorticity form cannot be used. As a consequence we obtain, for the first time, the existence of a local strong solution to the corresponding stochastic Euler equation. Furthermore, smooth solutions are shown to exist until blow-up in $L^1\left([0,T];W^{1,\infty}\right)$.

math.AP↗

arXiv2025

Infinite Boundary Friction Limit for Weak Solutions of the Stochastic Navier-Stokes Equations

We address convergence of the unique weak solutions of the 2D stochastic Navier-Stokes equations with Navier boundary conditions, as the boundary friction is taken uniformly to infinity, to the unique weak solution under the no-slip condition. Our result is that for initial velocity in $L^2_x$, the convergence holds in probability in $C_tW^{-\varepsilon,2}_x \cap L^2_tL^2_x$ for any $0 < \varepsilon$. The noise is of transport-stretching type, although the theorem holds with other transport, multiplicative and additive noise structures. This seems to be the first work concerning the large boundary friction limit with noise, and convergence for weak solutions, due to only $L^2_{x}$ initial data, appears new even deterministically.

math.PR↗

arXiv2024

High Order Smoothness for Stochastic Navier-Stokes Equations with Transport and Stretching Noise on Bounded Domains

We obtain energy estimates for a transport and stretching noise under Leray Projection on a 2D bounded convex domain, in Sobolev Spaces of arbitrarily high order. The estimates are taken in equivalent inner products, defined through powers of the Stokes Operator with a specific choice of Navier boundary conditions. We exploit fine properties of the noise in relation to the Stokes Operator to achieve cancellation of derivatives in the presence of the Leray Projector. As a result, we achieve an additional degree of regularity in the corresponding Stochastic Navier-Stokes Equation to attain a true strong solution of the original Stratonovich equation. Furthermore for any order of smoothness, we can construct a strong solution of a hyperdissipative version of the Stochastic Navier-Stokes Equation with the given regularity; hyperdissipation is only required to control the nonlinear term in the presence of a boundary. We supplement the result by obtaining smoothness without hyperdissipation on the torus, in 2D and 3D on the lifetime of solutions.

math.AP↗

arXiv2024

Improved Blow-Up Criterion in a Variational Framework for Nonlinear SPDEs

We extend recent existence and uniqueness results for maximal solutions of SPDEs through an improved blow-up criterion. Whilst the maximal time of existence is typically characterised by blow-up in the energy norm of solutions, we show instead that solutions exist until blow-up in the larger spaces of the variational framework. The result is applied to show that solutions of 2D and 3D Stochastic Navier-Stokes Equations retain the higher order regularity of the initial condition on their time of existence.

math.AP↗

arXiv2024

Weak and Strong Solutions to Nonlinear SPDEs with Unbounded Noise

We introduce an extended variational framework for nonlinear SPDEs with unbounded noise, defining three different solution types of increasing strength along with criteria to establish their existence. The three notions can be understood as probabilistically and analytically weak, probabilistically strong and analytically weak, as well as probabilistically and analytically strong. Our framework facilitates several well-posedness results for the Navier-Stokes Equation with transport noise, equipped with the no-slip and Navier boundary conditions.

math.AP↗

arXiv2023

Navier-Stokes Equations with Navier Boundary Conditions and Stochastic Lie Transport: Well-Posedness and Inviscid Limit

We prove the existence and uniqueness of global, probabilistically strong, analytically strong solutions of the 2D Stochastic Navier-Stokes Equation under Navier boundary conditions. The choice of noise includes a large class of additive, multiplicative and transport models. We emphasise that with a transport type noise, the Navier boundary conditions enable direct energy estimates which appear to be prohibited for the usual no-slip condition. The importance of the Stochastic Advection by Lie Transport (SALT) structure, in comparison to a purely transport Stratonovich noise, is also highlighted in these estimates. In the particular cases of SALT noise, the free boundary condition and a domain of non-negative curvature, the inviscid limit exists and is a global, probabilistically weak, analytically weak solution of the corresponding Stochastic Euler Equation.

math.PR↗

arXiv2023

The Zero Viscosity Limit of Stochastic Navier-Stokes Flows

We introduce an analogue to Kato's Criterion regarding the inviscid convergence of stochastic Navier-Stokes flows to the strong solution of the deterministic Euler equation. Our assumptions cover additive, multiplicative and transport type noise models. This is achieved firstly for the typical noise scaling of $\nu^\frac{1}{2}$, before considering a new parameter which approaches zero with viscosity but at a potentially different rate. We determine the implications of this for our criterion and clarify a sense in which the scaling by $\nu^\frac{1}{2}$ is optimal. To enable the analysis we prove the existence of probabilistically weak, analytically weak solutions to a general stochastic Navier-Stokes Equation on a bounded domain with no-slip boundary condition in three spatial dimensions, as well as the existence and uniqueness of probabilistically strong, analytically weak solutions in two dimensions. The criterion applies for these solutions in both two and three dimensions, with some technical simplifications in the 2D case.

math.PR↗

arXiv2022

On the Navier-Stokes Equations with Stochastic Lie Transport

We prove the existence and uniqueness of maximal solutions to the 3D SALT (Stochastic Advection by Lie Transport, [Holm arXiv:1410.8311]) Navier-Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively. The current work partners the paper [Goodair et al, arXiv:2209.09137] as an application of the abstract framework presented there, justifying the results first announced in [Goodair, arXiv:2202.09242v2]. In particular this represents the first well-posedness result for a fluid equation perturbed by a general transport type noise on a bounded domain.

math.AP↗

arXiv2022

Existence and Uniqueness of Maximal Solutions to SPDEs with Applications to Viscous Fluid Equations

We present two criteria to conclude that a stochastic partial differential equation (SPDE) posseses a unique maximal strong solution. This paper provides the full details of the abstract well-posedness results first given in arXiv:2202.09242v2, and partners a paper which rigorously addresses applications to the 3D SALT (Stochastic Advection by Lie Transport) Navier-Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively. Each criterion has its corresponding set of assumptions and can be applied to viscous fluid equations with additive, multiplicative or a general transport type noise.

math.AP↗

arXiv2022

Stochastic Calculus in Infinite Dimensions and SPDEs

These notes rigorously construct the stochastic integral of a Hilbert Space valued process driven by a Cylindrical Brownian Motion. We expand upon this stochastic calculus to present an introduction to stochastic differential equations in infinite dimensions, with a particular focus on Stratonovich equations due to their physical importance as well as unbounded noise operators (with applications to transport noise). Furthermore we explore techniques in the existence theory for nonlinear stochastic partial differential equations.

math.PR↗

arXiv2022

Existence and Uniqueness of Maximal Solutions to a 3D Navier-Stokes Equation with Stochastic Lie Transport

We present here a criterion to conclude that an abstract SPDE posseses a unique maximal strong solution, which we apply to a three dimensional Stochastic Navier-Stokes Equation. Inspired by the work of [Kato and Lai,1984] in the deterministic setting, we provide a comparable result here in the stochastic case whilst facilitating a variety of noise structures such as additive, multiplicative and transport. In particular our criterion is designed to fit viscous fluid dynamics models with Stochastic Advection by Lie Transport (SALT) as introduced in [Holm,2015]. Our application to the Incompressible Navier-Stokes equation matches the existence and uniqueness result of the deterministic theory. This short work summarises the results and announces two papers [Goodair et al, 2022] which give the full details for the abstract well-posedness arguments and application to the Navier-Stokes Equation.

math.PR↗