Periodic solutions of the Navier-Stokes equations in unbounded domains (Q1915240)

The authors consider the Navier-Stokes equations in an unbounded domain \(\Omega \subset \mathbb{R}^n\), i.e. \[ {\partial u \over \partial t} - \Delta u+(u \cdot \nabla)u + \nabla p=f, \quad \text{div} u=0\quad\text{ in } \Omega \times\mathbb{R}, \quad u=0 \text{ on } \partial \Omega \times \mathbb{R}. \] Here, \(\Omega\) is allowed to be the whole space \(\mathbb{R}^n\) or the half-space \(\mathbb{R}^n_+\) if \(n\geq 3\); in case \(n\geq 4\), \(\Omega\) may also be an exterior domain. The external force \(f\) is assumed to be periodic in time. Under suitable smallness conditions on \(f\) they prove the existence and uniqueness of a periodic strong solution. The idea is to rewrite the Navier-Stokes equations as an integral equation \[ u(t)= \int^t_{-\infty} e^{-(t-s)A} Pf(s)ds - \int^t_{-\infty} e^{-(t-s)A} P \bigl((u\cdot \nabla) u\bigr) (s)ds \] where \(A\) denotes the Stokes operator and \(e^{-tA}\) is the semigroup generated by \(A\). The main ingredients of the existence proof then are the \(L^p-L^r\) estimates for \(e^{-tA}\) and an iteration procedure.