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Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations

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Abstract

We consider the problem of finding the restrictions on the domain Ω⊂R n,n=2,3, under which the space

of the solenoidal vector fields from

coincides with the space

, the closure in W 12 (Ω) of the set of all solenoidal vectors from

. We give domains Ω⊂Rn, for which the factor space

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:

,

.

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Authors
  1. O. A. Ladyzhenskaya
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  2. V. A. Solonnikov
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Additional information

Dedicated to J. Leray on his 70th anniversary.

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 81–116, 1976.

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Ladyzhenskaya, O.A., Solonnikov, V.A. Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations. J Math Sci 10, 257–286 (1978). https://doi.org/10.1007/BF01566606

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  • DOI: https://doi.org/10.1007/BF01566606

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