Existenceand Uniqueness of Time-Periodic Physically Reasonable Navier-StokesFlow Past an Obstacle

Let $\Omega$ be a three-dimensional exterior domain of class $C^2$. We consider the following Navier-Stokes problem: \[\begin{equation}\begin{gathered}\partial_t v+v\cdot \operatorname{grad}v = \Delta v+\operatorname{grad}p+\operatorname{div}f, \quad \operatorname{div} v = 0, \text{ in } \Omega\times (0,\infty) \\ v(x,t)|_{\partial\Omega}=0, \quad \lim_{|x|\to\infty} v(x,t)=0, \quad t\in [0,\infty),\end{gathered} \label{eq1:522}\end{equation}\] where $f=\{f_{ij}(x,t)\}$ is a second-order tensor field such that $f(x,t)=f(x,t+T)$, for all $t\geq 0$, and some $T\gt 0$.

The objective of this talk is to show that, if $f$ satisfies suitable regularity conditions and its norm, in appropriate function spaces, is sufficiently small, problem ($\ref{eq1:522}$) admits at least one time-periodic strong solution $v, p$. Moreover, the velocity field $v$ decays to zero for large $x$ as $|x|^{-1}$ and its spatial gradient decays as $|x|^{-2}$, both uniformly in time. In addition, the pressure $p$ decays like $|x|^{-2}$ and its gradient like $|x|^{-3}$, for almost all $t\in [0,T]$. If, in particular, $f$ is time-independent, the corresponding solution is also time-independent and coincides with Finn's “physically reasonable” solution.

Finally, we show that the above solutions are unique in a class of weak solutions satisfying the “energy inequality” and with corresponding pressure field satisfying certain summability conditions in $\Omega\times [0,T]$.