Local periodic incompressible model evolution

Assume the Axiom of Countable Choice. Let $d\in \{1,2,3\}$, let $L>0$, $U_{\ast }>0$, and ${\rho }_{\ast }>0$, and fix an integer $s\ge 4$. Use SI units: $L$ is measured in metres, $U_{\ast }$ in metres per second, physical position $X$ in metres, time $t$ in seconds, mass density ${\rho }_{\ast }$ in kilograms per cubic metre, velocity $u$ in metres per second, pressure $p$ and stress $\sigma$ in pascals, and dynamic shear viscosity $\mu$ in pascal seconds. To specialize the Newtonian/Fourier closure, fix an auxiliary constant temperature $T_{\ast }>0$ in kelvins and an open state set $\mathcal{S}\subset (0,\mathrm{\infty }{)}^2$ containing $({\rho }_{\ast },T_{\ast })$; take the auxiliary constitutive state throughout the branch to be the constant pair $(\rho ,T)=({\rho }_{\ast },T_{\ast })$ and the shear-viscosity state law to have value $\mu$ there. The bulk coefficient may be any admissible value because ${\mathrm{\nabla }}_d\cdot u=0$, and any admissible conductivity law gives zero Fourier flux for this constant temperature. No total-energy or entropy-balance solution is asserted. Let $I\subset \mathbb{R}$ be an open time interval containing $0$ and ${\mathbb{T}}_L^d=(\mathbb{R}\mathrm{/}(2\pi L\mathbb{Z}){)}^d$; this is the physical periodic cell for $d=3$ and the reduced coordinate torus for $d=1,2$, whose physical cell is ${\mathbb{T}}_L^3$. The smooth model fields are $u\in C^{\mathrm{\infty }}(I\times {\mathbb{T}}_L^d;{\mathbb{R}}^d)$ and $p\in C^{\mathrm{\infty }}(I\times {\mathbb{T}}_L^d;\mathbb{R})$. Torus fields are understood as smooth periodic lifts to Euclidean coordinate space when the local continuum equations are applied. Adopt a force-free incompressible Newtonian model with constant density ${\rho }_{\ast }$, constant shear viscosity $\mu \ge 0$, velocity field $u$, pressure multiplier $p$ enforcing ${\mathrm{\nabla }}_d\cdot u=0$, and reduced stress ${\sigma }^{(d)}=-p{I}_d+2\mu D_{d}u,D_{d}u=\frac{1}{2}({\mathrm{\nabla }}_{d}u+({\mathrm{\nabla }}_{d}u{)}^{\mathsf{T}})\mathrm{.}$ On the retained $d$ coordinates, the local momentum equation and constraint are ${\rho }_{\ast }({\mathrm{\partial }}_{t}u+(u\cdot {\mathrm{\nabla }}_d)u)={\mathrm{\nabla }}_d\cdot {\sigma }^{(d)},{\mathrm{\nabla }}_d\cdot u=0.$ For $d=3$, this is the direct physical equation on ${\mathbb{T}}_L^3$ with stress ${\sigma }^{(3)}=-p{I}_3+2\mu D_{3}u$. For $d=1,2$, ${\mathbb{T}}_L^d$ is the reduced coordinate torus and the corresponding physical cell is ${\mathbb{T}}_L^3$. Let $J_d:{\mathbb{T}}_L^3\to {\mathbb{T}}_L^d$ be $J_d(X_1,X_2,X_3)=(X_1,\dots ,X_d)$, with coordinates understood modulo $2\pi L$. A reduced velocity $u:{\mathbb{T}}_L^d\to {\mathbb{R}}^d$ and pressure $p:{\mathbb{T}}_L^d\to \mathbb{R}$ lift to $\tilde{u}(X)=(u_1(J_{d}X),\dots ,u_d(J_{d}X),0,\dots ,0),\tilde{p}(X)=p(J_{d}X)\mathrm{.}$ The physical three-dimensional stress is $\tilde{\sigma }=-\tilde{p}{I}_3+2\mu D_3\tilde{u}$, where $D_3\tilde{u}=\frac{1}{2}({\mathrm{\nabla }}_3\tilde{u}+({\mathrm{\nabla }}_3\tilde{u}{)}^{\mathsf{T}})$. For ${\mathrm{\nabla }}_d\cdot u=0$, direct differentiation gives ${\mathrm{\nabla }}_3\cdot \tilde{u}=({\mathrm{\nabla }}_d\cdot u)\circ J_d,(\tilde{u}\cdot {\mathrm{\nabla }}_3)\tilde{u}=((u\cdot {\mathrm{\nabla }}_d)u,0,\dots ,0)\circ J_d,{\mathrm{\nabla }}_3\cdot \tilde{\sigma }=(({\mathrm{\nabla }}_d\cdot {\sigma }^{(d)}),0,\dots ,0)\circ J_d\mathrm{.}$ Thus the reduced equation is exactly the restriction of the three-dimensional equation to fields independent of the omitted coordinates and with the omitted velocity components zero. Conversely, every smooth solution in this invariant class reduces to the displayed $d$-dimensional equation. When $d=3$, no lift or reduction is used. Assume $u_0:{\mathbb{T}}_L^d\to {\mathbb{R}}^d$ is smooth, real, periodic, and divergence-free. Set $x=X\mathrm{/}L\in {\mathbb{T}}^d=(\mathbb{R}\mathrm{/}(2\pi \mathbb{Z}){)}^d$, $\tau =t{U}_{\ast }\mathrm{/}L$, $U_0(x)=u_0(Lx)\mathrm{/}{U}_{\ast }$, $\bar{\nu }=\mu \mathrm{/}({\rho }_{\ast }{U}_{\ast }L)$, and $\mathrm{\Pi }=p\mathrm{/}({\rho }_{\ast }{U}_{\ast }^2)$. For a smooth periodic field $f$, use normalized measure $d{\mu }_x=(2\pi {)}^{-d}dx$, coefficients $\widehat{f}(k)={\int }_{{\mathbb{T}}^d}f(x)e^{-ik\cdot x}d{\mu }_x$, and $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^s}^2={\sum }_{k\in {\mathbb{Z}}^d}(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^s\mathrm{\mid }\widehat{f}(k){\mathrm{\mid }}^2\mathrm{.}$ For this integer $s$, there is $T>0$, depending only on $d$, $s$, and $\mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{H^s}$, and independent of $\bar{\nu }$ and all higher Sobolev norms of $U_0$, such that the normalized system has a unique smooth solution $(U,\mathrm{\Pi })$ on $[0,T]\times {\mathbb{T}}^d$: ${\mathrm{\partial }}_{\tau }U+(U\cdot {\mathrm{\nabla }}_x)U=-{\mathrm{\nabla }}_x\mathrm{\Pi }+\bar{\nu }{\mathrm{\Delta }}_{x}U,{\mathrm{\nabla }}_x\cdot U=0,U(0,\cdot )=U_0,$ with $\mathrm{\Pi }$ fixed by zero spatial mean at each time. Every integer Sobolev order $m\ge s$ remains bounded on this same interval. Returning to physical variables gives a unique smooth model evolution on $0\le t\le (L\mathrm{/}{U}_{\ast })T$. When $\mu =0$, the adopted reduced stress is $-p{I}_d$ (its 3D lift is $-\tilde{p}{I}_3$) and the mechanical model is inviscid Euler; when $\mu >0$, it is viscous Navier–Stokes. This is a conditional model deduction and makes no claim of empirical validity, wall evolution, variable-viscosity evolution, or thermal evolution.

Brian F. Morton, Local-in-Time Well-Posedness of the Incompressible Euler and Navier–Stokes Equations.

Emil Wiedemann, Lecture Notes: Navier–Stokes Equations.

David Tong, Fluid Mechanics, Chapter 3: The Navier–Stokes Equation.

Sergei Kuksin and Armen Shirikyan, Mathematics of Two-Dimensional Turbulence.