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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 \(\partial\Omega\).

MSC:

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