Stationary problem of second-grade fluids in three dimensions: existence, uniqueness and regularity
J.M. Bernard · Mathematical Methods in the Applied Sciences · 1999
This paper is devoted to the stationary problem of second-grade fluids, in the case where α1 + α2 = 0, in three dimensions. In relation to the problem in two dimensions, studied by E. H. Ouazar, the H3 norm of the velocity, in three dimensions, is not bounded for all data. However, by a special method, using together a H1 bound of the velocity, a ‘pseudo-continuous dependence’ with respect to the data (effective for a small H3 norm of the velocity) and a polynomial inequality (verified by the H3 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.