Stochastic partial differential equations (SPDEs) play an important role in modeling fluid dynamics when uncertainty or random effects must be considered. These equations extend traditional partial differential equations (PDEs) by incorporating stochastic processes to account for random fluctuations, noise, or uncertain parameters in fluid flows. This approach is particularly useful in describing turbulence, environmental fluid flows, and other complex systems where deterministic models are insufficient.

1. Overview of SPDEs

SPDEs are equations that combine aspects of both PDEs and stochastic processes, involving partial derivatives with respect to spatial and temporal variables and a stochastic component that represents random influences. An SPDE in fluid dynamics often takes the form:

$$ \frac{\partial u(x, t)}{\partial t} = \mathcal{L}(u) + \mathcal{N}(u) + \sigma(x, t) \dot{W}(x, t), $$

where:

• $u(x, t)$ is the state variable (e.g., velocity field),
• $\mathcal{L}(u)$ represents a linear operator (e.g., diffusion),
• $\mathcal{N}(u)$ represents non-linear terms (e.g., advection),
• $\sigma(x, t)$ is a function describing the intensity of the noise,
• $\dot{W}(x, t)$ is a space-time white noise or another stochastic process (e.g., colored noise).

2. Applications in Fluid Dynamics

SPDEs are used in various fluid dynamics contexts, such as:

• Turbulence Modeling: Traditional deterministic approaches to turbulence, like the Navier–Stokes equations, do not fully capture the unpredictable nature of turbulent flows. By adding stochastic terms to these equations, researchers can model the random nature of turbulence more effectively.
• Atmospheric and Oceanic Flows: Environmental fluid dynamics often involve uncertainties due to incomplete data, modeling assumptions, or inherent natural variability. SPDEs can be used to simulate realistic flows that account for these uncertainties.
• Subgrid-Scale Modeling: In large-eddy simulations (LES) of turbulent flows, SPDEs can represent the effect of smaller, unresolved scales of motion on the larger scales.

3. Examples of SPDEs in Fluid Dynamics

• Stochastic Navier–Stokes Equations: The Navier–Stokes equations describe the motion of viscous fluids. The stochastic version incorporates a noise term to model random external forces or uncertainties in initial and boundary conditions: $\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\nabla P + \nu \nabla^2 \mathbf{u} + \mathbf{f} + \sigma \dot{W},$ where $\mathbf{u}$ is the velocity field, $P$ is pressure, $\nu$ is the kinematic viscosity, and $\mathbf{f}$ is a deterministic forcing term.
• Stochastic Heat Equation: A simpler form relevant for diffusion processes with noise: $\frac{\partial u}{\partial t} = \kappa \nabla^2 u + \sigma \dot{W}(x, t),$ where $\kappa$ is the diffusion coefficient.

4. Noise Types in SPDEs

• White Noise: Represents a purely random process with no temporal or spatial correlation. Mathematically, it is the derivative of a Brownian motion and can lead to technical challenges due to its non-smooth nature.