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Transactions of the American Mathematical Society

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Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications


Authors: A. Fursikov, M. Gunzburger and L. Hou
Journal: Trans. Amer. Math. Soc. 354 (2002), 1079-1116
MSC (2000): Primary 46E35, 35K50, 76D05, 76D07
DOI: https://doi.org/10.1090/S0002-9947-01-02865-3
Published electronically: November 2, 2001
MathSciNet review: 1867373
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Abstract: We study trace theorems for three-dimensional, time-dependent solenoidal vector fields. The interior function spaces we consider are natural for solving unsteady boundary value problems for the Navier-Stokes system and other systems of partial differential equations. We describe the space of restrictions of such vector fields to the boundary of the space-time cylinder and construct extension operators from this space of restrictions defined on the boundary into the interior. Only for two exceptional, but useful, values of the spatial smoothness index, the spaces for which we construct extension operators is narrower than the spaces in which we seek restrictions. The trace spaces are characterized by vector fields having different smoothnesses in directions tangential and normal to the boundary; this is a consequence of the solenoidal nature of the fields. These results are fundamental in the study of inhomogeneous boundary value problems for systems involving solenoidal vector fields. In particular, we use the trace theorems in a study of inhomogeneous boundary value problems for the Navier-Stokes system of viscous incompressible flows.


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Additional Information

A. Fursikov
Affiliation: Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia
Email: fursikov@dial01.msu.ru

M. Gunzburger
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011-2064
Email: gunzburg@iastate.edu

L. Hou
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011-2064 and Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada
Email: hou@math.iastate.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02865-3
Received by editor(s): September 29, 1999
Received by editor(s) in revised form: May 23, 2000, and March 19, 2001
Published electronically: November 2, 2001
Article copyright: © Copyright 2001 American Mathematical Society