Hadamard variational formula for the Green function of the Stokes equations under the general second order perturbation. (English) Zbl 1370.76037

The Green function of the three-dimensional Stokes equations is studied. Let the vector-function \(v(x)=(v_1,v_2,v_3)\) and the scalar function \(p(x)\) be the solution to the Stokes problem \[ \begin{aligned} -\Delta v+\nabla p=f,\quad \text{div}\,v=0,\quad x\in\Omega, \\ v=0\quad \text{on}\;\partial\Omega,\end{aligned} \] where \(\Omega\subset\mathbb{R}^3\) is a bounded domain with a smooth boundary \(\partial\Omega\). \(\Omega_\varepsilon \) is the perturbed domain with the boundary \[ \partial\Omega_{\varepsilon}=\left\{x+\rho_1(x)\nu(x)\varepsilon+\frac{1}{2}\rho_2(x)\nu(x)\varepsilon^2,\;x\in\partial\Omega \right\}, \] where \(\nu(x)\) is the unit outer normal to \(\partial\Omega\) at \(x\in\partial\Omega\), \(\varepsilon\geq0\) is a parameter, \(\rho_1\) and \(\rho_2\) are the given smooth functions.
The Green function \(\{G_{\varepsilon,m},P_{\varepsilon,m}\}_{m=1,2,3}\) of the Stokes equations on \(\Omega_\varepsilon \) satisfies \[ \begin{aligned} -\Delta G_{\varepsilon,m}(x,z) +\nabla P_{\varepsilon,m}(x,z) =\delta(x-z)e_m,\quad \text{div}\,G_{\varepsilon,m}(x,z)=0, \quad (x,z)\in\Omega_\varepsilon\times\Omega_\varepsilon, \\ G_{\varepsilon,m}(x,z)=0,\quad x\in\partial\Omega_\varepsilon,\;z\in\Omega_\varepsilon,\end{aligned} \] where \(\{e_m\}_{m=1,2,3}\) is the canonical basis in \(\mathbb{R}^3\), \(m=1,2,3\). The author constructs a representation formula for the first and second variation of the Green function with respect to \(\varepsilon\). These variations are presented in the form of integrals over the surface \(\partial\Omega\).


76D07 Stokes and related (Oseen, etc.) flows
35Q35 PDEs in connection with fluid mechanics
35B20 Perturbations in context of PDEs
Full Text: DOI