Local existence, uniqueness and pressure reconstruction for the incompressible fluid model on the periodic cell

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Adopt the constant-density incompressible Newtonian or inviscid branch and periodic-cell convention of Adopted fluid constitutive branches: incompressible Newtonian, barotropic inviscid, and the periodic-cell idealization clauses (a), (a0), (c), with $\rho >0$, $\eta \ge 0$, and no body force. Fix the model scales $L_{\ast },U_{\ast }>0$, $T_{\ast }=L_{\ast }\mathrm{/}{U}_{\ast }$ and put ${\nu }_{\ast }:=\eta \mathrm{/}(\rho U_{\ast }{L}_{\ast })$. Let $s\ge 4$ be an integer and $U_0$ a smooth divergence-free field on the dimensionless unit torus, with $Y_0:=\mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{H^s}<\mathrm{\infty }$. Set $T:=\{\begin{array}{ll}(2C_s{Y}_0{)}^{-1},& Y_0>0,\\ 1,& Y_0=0,\end{array}$ where $C_s$ is the constant used by Local smooth solutions of the periodic incompressible Euler and constant-viscosity equations on ${\mathbb{T}}^3$. Then:

(i) Reduction and physical-time conversion. Under $\tau =t\mathrm{/}{T}_{\ast }$, $y=x\mathrm{/}{L}_{\ast }$ and the scaling of the postulate, the physical continuity and momentum equations with $\sigma =-P_{\mathrm{m}}{I}_3+\eta (D_x{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+D_x{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\mathsf{T}})$ are equivalent to ${\mathrm{\partial }}_{\tau }U+(U\cdot {\mathrm{\nabla }}_y)U=-{\mathrm{\nabla }}_y\mathrm{\Pi }+{\nu }_{\ast }{\mathrm{\Delta }}_{y}U,{\mathrm{div}}_{y}U=0.$ For smooth fields with zero-mean pressure, these are equivalent to $$\begin{array}{cccc}& {\mathrm{\partial }}_{\tau }U+P((U\cdot {\mathrm{\nabla }}_y)U)={\nu }_{\ast }{\mathrm{\Delta }}_{y}U& & \text{(PE)}\end{array}$$ and the pressure reconstruction below. A solution on $[0,T]$ is a physical solution on $[0,T_{\ast }T]$.

(ii) Existence and pressure. There is $U\in C^{\mathrm{\infty }}([0,T]\times {\mathbb{T}}^3;{\mathbb{R}}^3)$ with $U(0)=U_0$, ${\mathrm{div}}_{y}U=0$ and (PE), and ${\sup }_{[0,T]}\mathrm{\parallel }U{\mathrm{\parallel }}_{H^m}\le \mathrm{\parallel }{U}_0{\mathrm{\parallel }}_{H^m}{e}^{4C_m{Y}_{0}T}$ for every $m\ge s$. There is a unique smooth dimensionless zero-mean pressure $\mathrm{\Pi }$ giving the primitive equations in (i). The physical fields are $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(t,x)=U_{\ast }U(t\mathrm{/}{T}_{\ast },x\mathrm{/}{L}_{\ast }),P_{\mathrm{m}}(t,x)=\rho U_{\ast }^2\mathrm{\Pi }(t\mathrm{/}{T}_{\ast },x\mathrm{/}{L}_{\ast })\mathrm{.}$ They satisfy the adopted field equations for both $\eta >0$ and the separately adopted $\eta =0$ Euler branch.

(iii) Uniqueness and dependence on data. Fix $B>0$. For all smooth divergence-free data $U_0,V_0$ with both $H^s$ norms at most $B$, the corresponding solutions exist on the common dimensionless interval $[0,T_B]$, $T_B=(2C_{s}B{)}^{-1}$, with $U(0)=U_0$ and $V(0)=V_0$. If $U_0=V_0$ then $U=V$. On this bounded data set the solution map is continuous from the $L^2$ topology into $C([0,T_B],H^{s-1})$, as in Uniqueness of smooth periodic solutions and continuous dependence in $H^{s-1}$ on the common local interval. The common physical interval is $[0,T_{\ast }{T}_B]$ for the same fixed scales and material parameters.

(iv) Scope. The Sobolev norm and $T,T_B$ are dimensionless; physical time also depends on the chosen scales. The theorem concerns smooth Eulerian fields on the adopted periodic cell. It constructs no material label map, no nonperiodic or rough-data solution and no global-in-time bound; no measured parameter is used.

The positive scales and material coefficients are model data. A comparison with measurements would additionally require an apparatus model, measured inputs and uncertainties. The global status of other fluid problems is not settled by this local theorem; the dated scope discussion in Scope of the local periodic result: what is proved and what is not distinguishes announcements from proved implications for this exact unforced periodic model.