On the strong solutions of the inhomogeneous incompressible Navier-Stokes equations in a thin domain. (Q265185)

The author studies the inhomogeneous incompressible Navier-Stokes equations in the layer \(\Omega = (0,\varepsilon) \times \mathbb R^2 \subset \mathbb R^3\) with homogeneous Dirichlet boundary conditions on the top and the bottom of the layer. Under the assumption that the initial density is bounded away from zero and infinity, the initial velocity \(u_0 \in H^2(\Omega)\cap H^1_0(\Omega)\), and NEWLINE\[NEWLINE\varepsilon^{\frac 12} \|\nabla u_0\|_{L^2(\Omega)} \leq c_0NEWLINE\]NEWLINE for some fixed \(c_0\) and \(\varepsilon \ll 1\) (i.e., the layer is sufficiently thin), the author proves existence and uniqueness of the global-in-time regular solution to the above mentioned problem. Moreover, for any \(\tau>0\), NEWLINE\[NEWLINE\lim_{\varepsilon \to 0^+} u=0NEWLINE\]NEWLINE in \(C([\tau,+\infty]; H^2(\Omega))\).NEWLINENEWLINE NEWLINEThe proof is based on suitable \(\epsilon\)-dependent estimates of the Gagliardo-Nirenberg and Poincaré-type, local-in-time existence of smooth solutions and a priori estimates in \(L^2(\Omega)\), \(H^1(\Omega)\) and \(H^2(\Omega)\), where the smallness of \(\epsilon\) plays an essential role. The paper extends similar results known for the incompressible homogeneous Navier-Stokes equations to the case of incompressible density-dependent flows.