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Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods. (English) Zbl 0766.35034
Let \(\Omega\) be a bounded connected open set in \(\mathbb{R}^ n\) with smooth boundary \(\partial\Omega=\Gamma\) and denote by \(\vec u=(u_ 1,\dots,u_ n)\) the unit interior normal vector field defined near \(\Gamma\). Let \(I=(0,b)\), where \(b\in(0,\infty)\), let \(Q=\Omega\times I\), and let \(S=\Gamma\times I\). Denote \(f\mid_ \Gamma=\gamma_ 0 f\) and \(\gamma_ 0 \partial_ \nu^ k f=\gamma_ k f\), where \(\partial_ \nu f=\sum_{j=1}^ n n_ j\partial_ j f\).
The authors consider the Navier-Stokes problem \[ \partial_ t u-\Delta u+\sum_{i=1}^ n u_ i\partial_ i u+\text{grad }p=f, \qquad (x,t)\in Q,\tag{1} \] \[ (2)\quad \text{div }u=0,\quad (x,t)\in Q; \qquad (3)\quad T_ k{u \choose p}=\varphi_ k,\quad (x,t)\in Q; \qquad (4)\quad u\mid_{t=0}=u_ 0,\quad x\in\Omega, \] where \(u\) is the velocity vector \(u=(u_ 1,\dots,u_ n)\) and \(p\) is the pressure. Here \(T_ k\) is one of the boundary operators; \[ T_ 1{u \choose p}=\chi_ 1 u-\gamma_ 0 p\vec n, \qquad T_ 0{u \choose p}=\gamma_ 0 u, \qquad T_ 2{u \choose p} =(\chi_ 1 u)+\gamma_ 0 u_ \nu \vec n, \] \(\vartheta_ \nu\vec n\), respectively, \(\vartheta_ \tau\) stand for the normal, resp., tangential components of a vector field \(\vartheta\) defined near \(\Gamma\), and \(\chi_ 1\) is the first order boundary operator defined via the strain tensor \(S(u)=(\partial_ i u_ j+\partial_ j u_ i)_{i,j=1,\dots,n}\) as \(\chi_ 1 u=\gamma_ 0 S(u)\vec n=\gamma_ 0(\sum_ j(\partial_ i u_ j+\partial_ j u_ i))_{i=1,\dots,n}\).
Together with (1)–(4) the authors consider the Stokes problem \(\partial_ t u-\Delta u+\text{grad } p=f\), \((x,t)\in Q\) with (2), (3), (4), which is used in the treatment of (1)–(4).
For investigation of these problems the authors use the treatment connected with algebras of pseudo-differential boundary problems (Boutet- de-Monvel’s algebra).

MSC:
35Q30 Navier-Stokes equations
35S15 Boundary value problems for PDEs with pseudodifferential operators
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