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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.

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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