Global two-dimensional periodic viscous model evolution

Assume the Axiom of Countable Choice. Let $L>0$ be measured in metres, $U_{\ast }>0$ in metres per second, ${\rho }_{\ast }>0$ be a constant mass density in kilograms per cubic metre, and $\mu >0$ be a constant dynamic shear viscosity in pascal seconds. Fix an auxiliary constant temperature $T_{\ast }>0$ in kelvins and an open constitutive 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 $(\rho ,T)=({\rho }_{\ast },T_{\ast })$, where the adopted shear-viscosity law has value $\mu$. The bulk coefficient may be any admissible value because the velocity is divergence-free, and any admissible conductivity law gives zero Fourier flux for the constant temperature; no total-energy or entropy-balance solution is asserted. Use SI position $X$ in metres, time $t$ in seconds, velocity $u$ in metres per second, and pressure $p$ in pascals. On the reduced two-dimensional periodic torus ${\mathbb{T}}_L^2=(\mathbb{R}\mathrm{/}(2\pi L\mathbb{Z}){)}^2$, let the smooth model fields be $u\in C^{\mathrm{\infty }}(I\times {\mathbb{T}}_L^2;{\mathbb{R}}^2)$ and $p\in C^{\mathrm{\infty }}(I\times {\mathbb{T}}_L^2;\mathbb{R})$ for an open time interval $I$ containing $0$, and adopt the force-free incompressible Newtonian equations. Identify torus fields with smooth periodic lifts to Euclidean coordinates when applying the local continuum equations. ${\rho }_{\ast }({\mathrm{\partial }}_{t}u+(u\cdot {\mathrm{\nabla }}_2)u)={\mathrm{\nabla }}_2\cdot {\sigma }^{(2)},{\mathrm{\nabla }}_2\cdot u=0,{\sigma }^{(2)}=-p{I}_2+2\mu D_{2}u,D_{2}u=\frac{1}{2}({\mathrm{\nabla }}_{2}u+({\mathrm{\nabla }}_{2}u{)}^{\mathsf{T}})\mathrm{.}$ These are the invariant reduction of the adopted three-dimensional continuum and Newtonian stress on physical cell ${\mathbb{T}}_L^3$. Let $J_2:{\mathbb{T}}_L^3\to {\mathbb{T}}_L^2$ be $J_2(X_1,X_2,X_3)=(X_1,X_2)$ modulo $2\pi L$, and lift $u=(u_1,u_2)$ and $p$ to $\tilde{u}=(u_1\circ J_2,u_2\circ J_2,0)$ and $\tilde{p}=p\circ J_2$. The three-dimensional stress is $\tilde{\sigma }=-\tilde{p}{I}_3+2\mu D_3\tilde{u}$. For ${\mathrm{\nabla }}_2\cdot u=0$, direct differentiation yields ${\mathrm{\nabla }}_3\cdot \tilde{u}=0$, ${\mathrm{\nabla }}_3\cdot \tilde{\sigma }=({\mathrm{\nabla }}_2\cdot {\sigma }^{(2)},0)$, and ${\mathrm{\nabla }}_3\cdot (2\mu D_3\tilde{u})=(\mu {\mathrm{\Delta }}_{2}u,0)$. The third momentum component vanishes, and the first two are exactly the displayed equations. Conversely, every smooth 3D solution with this transverse invariance and zero third velocity component reduces to the displayed planar equations. The normalized 3D mean of the lifted pressure equals its normalized 2D mean, so the pressure gauge is preserved. Set $x=X\mathrm{/}L\in {\mathbb{T}}^2=(\mathbb{R}\mathrm{/}(2\pi \mathbb{Z}){)}^2$, $\tau =t{U}_{\ast }\mathrm{/}L$, $U=u\mathrm{/}{U}_{\ast }$, $\mathrm{\Pi }=p\mathrm{/}({\rho }_{\ast }{U}_{\ast }^2)$, and $\bar{\nu }=\mu \mathrm{/}({\rho }_{\ast }{U}_{\ast }L)>0$. Use normalized measure $d\mu =(2\pi {)}^{-2}dx$ and fix the pressure gauge by ${\int }_{{\mathbb{T}}^2}\mathrm{\Pi }(\tau ,x)d\mu (x)=0$ at each time. For any smooth real periodic divergence-free initial velocity $U_0:{\mathbb{T}}^2\to {\mathbb{R}}^2$, the adopted model has a unique smooth velocity $U$ and the unique pressure $\mathrm{\Pi }$ with this zero-mean gauge on all normalized times $\tau \ge 0$, satisfying ${\mathrm{\partial }}_{\tau }U+(U\cdot {\mathrm{\nabla }}_x)U=-{\mathrm{\nabla }}_x\mathrm{\Pi }+\bar{\nu }{\mathrm{\Delta }}_{x}U,{\mathrm{\nabla }}_x\cdot U=0,U(0,\cdot )=U_0\mathrm{.}$ Here global means smooth on $[0,\mathrm{\infty })$ with finite Sobolev norms on every compact time interval; no uniform-in-time norm bound is asserted. Scaling back gives a unique global smooth physical model evolution for all $t\ge 0$. This conclusion applies to the adopted two-dimensional positive-viscosity periodic model and, by the stated lift, to the corresponding transversely invariant three-dimensional data. It asserts no global regularity for arbitrary three-dimensional data or inviscid flow, and no empirical validity claim.

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