Stokes and Navier–Stokes problems in the half-space: existence and uniqueness of solutions non converging to a limit at infinity

We study the Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations. We furnish existence and uniqueness of classical solutions $(u,\pi)$ (meaning at least $C^2\times C^1$ smooth with respect to the space variable and $C^1\times C^0$ smooth with respect to the time variable) without requiring of convergence at infinity. A priori the fields $u$ and $\pi$ are nondecreasing at infinity. In the case of the Stokes problem we prove the existence, for any $t>0$, and uniqueness of solutions with kinetic field $u=O([1+t^\frac\beta2][1+|x|^\beta])$ and pressure field $\pi=O([1+t^\frac\beta2][1+|x|^\beta]|x|^\gamma)$, for some $\beta\in(0,1)$ and $\gamma\in(0,1-\beta)$. In the case of the Navier–Stokes equations we prove the existence (local in time) and the uniqueness of classical solutions to the Navier–Stokes equations assuming an initial data only continuous and bounded, proving that, for any $t\in(0,T)$, the kinetic field $u(x,t)$ is bounded and, for any $\gamma\in(0,1)$, the pressure field $\pi(x,t)=O(1+|x|^\gamma)$. Bibl. – 20 titles.