On the uniqueness of Leray-Hopf solutions for the flow through an aperture
The boundary value problem for the steady Navier-Stokes system is considered in \(\Omega=\mathbb{R}^ 3\backslash \{x\in\mathbb{R}^ 3: x_ 3=0,\;x'=(x_ 1,x_ 2)\in V\}\), where \(V\) is a bounded Lipschitz domain in the plane \(x_ 3= 0\), \[ -\nu\Delta u+ u\cdot\nabla u+\nabla p=f,\quad\text{div }u=0\quad\text{in }\Omega,\quad u=0\quad\text{on }\partial\Omega.\tag{1} \] A solution must satisfy either the additional flux condition (2) \(\int_ V u_ 3(x',x_ 3)dx'= F\) or the pressure drop condition (3) \(p^ +- p^ -=p_*\), where \(F\) or \(p_*\) are given numbers, \(p^ += \lim_{| x|\to\infty,x_ 3>0} p(x)\), \(p^ - =\lim_{| x|\to\infty,x_ 3<0} p(x)\). Let \({\mathcal D}^ 1_ 0(\Omega)\) be the completion of \(C^ \infty_ 0(\Omega)\) in the Dirichlet integral norm, \({\mathcal D}^ 1_{0,\sigma}(\Omega)\) the subspace of solenoidal functions from \({\mathcal D}^ 1_ 0(\Omega)\) and \({\mathcal D}^{-1}(\Omega)\) the dual space to \({\mathcal D}^ 1_ 0(\Omega)\). The authors establish the following result: Let \(f\in{\mathcal D}^{- 1}(\Omega)\), the functional \(\eta\to\langle f,\eta\rangle\) admit the estimate \[ |\langle f,\eta\rangle|\leq C_ f\left(\int_ \Omega|\nabla \eta|^ 2\;(1+| x|^{-1- \varepsilon}\right)^{1/2} \] for some \(\varepsilon>0\) and all \(\eta\in {\mathcal D}^ 1_ 0(\Omega)\). Then the solution \(u\in{\mathcal D}^ 1_{0,\sigma}(\Omega)\) either of the problem (1)--(2) or of the problem (1), (3) satisfying the energy inequality \[ \int_ \Omega (\nabla u: \nabla u)dx\leq \langle f,u\rangle- p_* F \] is unique provided \(C_ f\) and either \(| F|\) or \(| p_*|\) are sufficiently small.
scientific article; zbMATH DE number 399276
scientific article