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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications
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by A. Fursikov, M. Gunzburger and L. Hou PDF
Trans. Amer. Math. Soc. 354 (2002), 1079-1116 Request permission

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
  • MR Author ID: 78360
  • 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
  • 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
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1867373