# zbMATH — the first resource for mathematics

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
Full Text: