Periodic incompressible Newtonian idealization

Let $I\subseteq \mathbb{R}$ be an open time interval measured in seconds. Fix $L>0$ as the numerical period in the declared SI metre unit, so the physical side length is $L\mathrm{m}$, and use the flat periodic cell ${\mathbb{T}}_L^3=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^3$ with normalized spatial measure $L^{-3}dx$. Spatial coordinates in physical volume integrals are measured in metres. Assume Countable Choice for the stated Fourier and Sobolev space conventions. The velocity is a vector field $u:I\times {\mathbb{T}}_L^3\to {\mathbb{R}}^3$ with SI units $\mathrm{m}{\mathrm{s}}^{-1}$; the mass density is a constant primitive parameter ${\rho }_0>0$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$; the dynamic viscosity is a constant primitive parameter $\mu >0$ in $\mathrm{P}\mathrm{a}\mathrm{s}$; and the mechanical pressure $p$ has units $\mathrm{P}\mathrm{a}$. The body-force density is zero (its SI unit is $\mathrm{N}{\mathrm{m}}^{-3}$), and there is no temperature variable in this model.

For classical fields take $u\in C^1(I;C^2({\mathbb{T}}_L^3;{\mathbb{R}}^3))$ and $p\in C^1(I;C^1({\mathbb{T}}_L^3))$. Adopt incompressibility $\mathrm{div}u=0$ and the local momentum balance ${\rho }_0({\mathrm{\partial }}_{t}u+\mathrm{div}(u\otimes u))=\mathrm{div}\sigma ,\sigma =-p{I}_3+2\mu D(u),D(u)=\frac{1}{2}(\mathrm{\nabla }u+(\mathrm{\nabla }u{)}^{\mathsf{T}})\mathrm{.}$ Here $\sigma$ is the Cauchy stress, so its traction on an oriented surface with unit normal $n$ is $\sigma n$; $D(u)$ is the symmetric rate-of-deformation tensor; and $\mathrm{\nabla }$ has units ${\mathrm{m}}^{-1}$. Every term in the momentum balance has units $\mathrm{N}{\mathrm{m}}^{-3}$. Since $\mathrm{div}u=0$, this equation is equivalently ${\mathrm{\partial }}_{t}u+\mathrm{div}(u\otimes u)+\mathrm{\nabla }\pi =\nu \mathrm{\Delta }u,\nu =\mu \mathrm{/}{\rho }_0>0,\pi =p\mathrm{/}{\rho }_0\mathrm{.}$ where kinematic viscosity $\nu$ and kinematic pressure $\pi$ have units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$ and ${\mathrm{m}}^2{\mathrm{s}}^{-2}$, respectively. All fields are periodic in each spatial coordinate; this is an ideal cell, not a wall-bounded vessel.

For the weak regime, use the global unforced periodic Leray–Hopf class. The spaces are $H$, the real $L^2$ vector fields whose divergence vanishes distributionally on ${\mathbb{T}}_L^3$, with unrestricted constant Fourier mode, and $V=H\cap H^1$, where $H^1$ consists of fields and first weak derivatives in $L^2$; use normalized spatial norms. Here $C_w([0,\mathrm{\infty });H)$ means that $t\mapsto (u(t),h{)}_H$ is continuous for every fixed $h\in H$; the time-dependent $L^p$ spaces are the corresponding Bochner spaces. A weak velocity has $u\in C_w([0,\mathrm{\infty });H)\cap L^{\mathrm{\infty }}(0,\mathrm{\infty };H)\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(0,\mathrm{\infty };V)$, strongly attains $u_0\in H$ as $t\downarrow 0$, and satisfies, for every smooth real periodic divergence-free test field $\varphi$ compactly supported in time,

$${\int }_0^{\mathrm{\infty }}\text{\hspace{0.17em}⁣}[(u,{\mathrm{\partial }}_t\varphi )+(u\otimes u,\mathrm{\nabla }\varphi )-\nu (\mathrm{\nabla }u,\mathrm{\nabla }\varphi )]dt+(u_0,\varphi (0))=0.$$

The tensor pairing is $L^{-3}{\int }_{[0,L{]}^3}{u}_i{u}_j{\mathrm{\partial }}_j{\varphi }_{i}dx$, and the gradient pairing is $L^{-3}{\int }_{[0,L{]}^3}{\mathrm{\partial }}_j{u}_i{\mathrm{\partial }}_j{\varphi }_{i}dx$. The class also requires

$$\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2+2\nu {\int }_0^t\mathrm{\parallel }\mathrm{\nabla }u(s){\mathrm{\parallel }}_2^{2}ds\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2^2(t\ge 0),$$

and a full-measure set $S\subset (0,\mathrm{\infty })$ such that

$$\mathrm{\parallel }u(t){\mathrm{\parallel }}_2^2+2\nu {\int }_s^t\mathrm{\parallel }\mathrm{\nabla }u(\tau ){\mathrm{\parallel }}_2^{2}d\tau \le \mathrm{\parallel }u(s){\mathrm{\parallel }}_2^2(s\in S\cup \{0\},t\ge s)\mathrm{.}$$

This weak convention asserts neither uniqueness nor pressure regularity. These equations and the solution class are adopted model assumptions; they are not deductions that every physical fluid obeys them.

Periodic solenoidal Fourier spaces: normalized torus, mean mode, and spaces H,V

Unforced periodic Leray–Hopf solution: weak identity and initial/restart energy inequalities

Cartesian inertial-frame, SI, and Cauchy-stress conventions supplies only the formulation conventions used here: Cartesian inertial coordinates, velocity in metres per second, density in kilograms per cubic metre, stress and pressure in pascals, dynamic viscosity in pascal seconds, the row-gradient $L_{ij}={\mathrm{\partial }}_j{u}_i$, its symmetric part $D=(L+L^{\mathsf{T}})\mathrm{/}2$, and tension-positive traction $\sigma n$. On the periodic cell these Cartesian conventions are read on local periodic lifts, not as a global affine chart on the compact torus. The smooth local momentum law and the weak solution class in this Postulate are separately adopted model assumptions; no weak balance, energy inequality, boundary trace, or pressure regularity is inferred from a smooth material-balance theorem.

Lenya Ryzhik, Lecture Notes for Math 256B, Version 2024, §1.3, printed pp. 9–11

Berselli and Spirito, On the Existence of Leray–Hopf Weak Solutions to the Navier–Stokes Equations, §1, printed pp. 1–3