On some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations. (Russian. English summary) Zbl 0346.35084

Ladyzhenskaya, O. A.; Solonnikov, V. A.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 59, 81-116 (1976).

Authors’ summary: We consider the problem of finding the restrictions on the domain \(\Omega\subset\mathbb R^n\), \(n=2,3\), under which the space \(\widehat{\overset\circ J}{}_2^1(\Omega)\) of the solenoidal vector fields from \(\overset\circ{W}{}_2^1(\Omega)\) coincides with the space \(\overset\circ{J}{}_2^1(\Omega)\), the closure in \(W_2^1(\Omega)\) of the set of all solenoidal vectors from \(\widehat{\overset\circ{J}}{}^\infty(\Omega)\). We give domains \(\Omega\subset\mathbb R^n\), for which the factor space \(\widehat{\overset\circ{J}}{}_2^1(\Omega)/\overset\circ{J}{}_2^1(\Omega)\) has a finite nonzero dimension. A similar problem is considered for the spaces of solenoidal vectors with a finite Dirichlet integral. Based on this, one compares two generalized formulations of boundary-value problems for the Stokes and Navier-Stokes systems. The following auxiliary problems are studied:
\[ (1)\quad \text{div}\,\vec u=\varphi, \vec u|_{\partial\Omega}=0; \quad (2)\quad \text{div}\,\vec u=0, \vec u|_{\partial\Omega}=\vec\alpha; \quad (3) \quad \text{grad}\,p=\sum_{k=1}^n\frac{\partial\vec R_k}{\partial x_k}+\vec f. \]

Reviewer: Vasile Oproiu (Iaşi)

MSC:

35Q30	Navier-Stokes equations
76D05	Navier-Stokes equations for incompressible viscous fluids
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D07	Stokes and related (Oseen, etc.) flows