Set

context $ n\in\mathbb N $
context $ \rho: \mathbb R^n\times\mathbb R\to\mathbb R^n $
context $ p \in C(\mathbb R^n\times\mathbb R,\mathbb R) $
context $ \boldsymbol{\mathsf{T}} \in C(\mathbb R^n\times\mathbb R\times\mathbb R^n,\mathbb R^{n^2}) $
context $ \mathbf{f} \in C(\mathbb R^n\times\mathbb R,\mathbb R^n) $
range $ ::\rho(\mathbf{x},t) $
range $ ::p(\mathbf{x},t) $
range $ ::\boldsymbol{\mathsf{T}}(\mathbf{x},t,\mathbf{v}) $
range $ ::\mathbf{K}(\mathbf{x},t) $
definiendum $ \mathbf{v} \in \mathrm{it} $
postulate $ \mathbf{v} \in C^2(\mathbb R^n\times\mathbb R,\mathbb R^n) $
range $ ::\mathbf{v}(\mathbf{x},t) $
postulate $ \rho\ \left(\frac{\partial}{\partial t} + \mathbf{v} \cdot \nabla \right)\ \mathbf{v} = -\mathrm{grad}(p) + \nabla \cdot \boldsymbol{\mathsf{T}} + \mathbf{K} $

Discussion

Parents

Reference

Wikipedia: Navier–Stokes equations

Subset of

PDE system

Macroscopic observables from kinetic theory