## 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

### Keywords:

domain perurbation; integral representation
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