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Stationary problem of second‐grade fluids in three dimensions: existence, uniqueness and regularity
Author(s) -
Bernard J. M.
Publication year - 1999
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19990525)22:8<655::aid-mma56>3.0.co;2-z
Subject(s) - mathematics , uniqueness , mathematical analysis , mathematical economics
This paper is devoted to the stationary problem of second‐grade fluids, in the case where α 1 + α 2 = 0, in three dimensions. In relation to the problem in two dimensions, studied by E. H. Ouazar, the H 3 norm of the velocity, in three dimensions, is not bounded for all data. However, by a special method, using together a H 1 bound of the velocity, a ‘pseudo‐continuous dependence’ with respect to the data (effective for a small H 3 norm of the velocity) and a polynomial inequality (verified by the H 3 norm of the velocity), we show existence of solutions, uniqueness, continuous dependence with respect to the data, with small data. We also prove further regularity results establishing that this is a classical solution when the datum is small enough and smooth enough. Copyright © 1999 John Wiley & Sons, Ltd.