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A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $$W^{-1/q,q}$$. (English) Zbl 1064.35133
The following boundary value problem to the stationary Stokes and Navier-Stokes equations are considered \begin{aligned} &-\Delta u+\nabla p =\operatorname{div}F,\quad \operatorname{div}u=h\;\text{in }\Omega, \quad u| _{\partial\Omega}=g \tag{1}\\ &-\Delta u+u\cdot\nabla u+\nabla p =\operatorname{div} F,\quad \operatorname{div}u=h\;\text{in }\Omega, \quad u| _{\partial\Omega}=g \tag{2} \end{aligned} Here $$\Omega\subset \mathbb R^n$$ is a bounded domain with boundary $$\partial\Omega$$ of class $$C^{2,1}$$.
The authors introduce a very weak solution to the problems (1), (2). A very weak solution $$u$$ is a function from $$L^q(\Omega)$$ with $$\operatorname{div}u\in L^q(\Omega)$$, and $$u$$ satisfies a certain $$L^q - L^{q'}$$ duality which does not include derivatives of $$u$$. It is proved that for any data $F\in L^r(\Omega),\quad h\in L^q(\Omega),\quad g\in W^{-1/q,q}(\partial\Omega),\quad 1<r\leq q<\infty,\quad \frac 13+\frac 1q\geq \frac 1r$ problem (1) has a unique very weak solution. If $$3\leq q<\infty$$, $$r\leq\frac q2$$ and suitable norms of the data are sufficiently small then problem (2) has a unique very weak solution. The way in which the solution $$u$$ attains the boundary data $$g$$ is analyzed in detail. It is proved that very weak solutions become regular if the data are regular.

##### MSC:
 35Q30 Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 76D05 Navier-Stokes equations for incompressible viscous fluids
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