Well-posedness of The Prandtl Equation in Sobolev Spaces

We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow

Diaqua­bis­[5-(2-pyridyl­meth­yl)tetra­zol­ato-κ2 N 1,N 5]zinc(II)

In the title mononuclear complex, [Zn(C7H6N5)2(H2O)2], the ZnII atom, located on an inversion centre, is in a distorted octa­hedral coordination geometry formed by four N atoms from two chelating 5-(2-pyridyl­meth­yl)tetra­zolate ligands and two O donors from two water mol­ecules. Inter­molecular O—H⋯N hydrogen bonds between the coordinated water mol­ecule and the tetra­zolyl group of the 5-(2-pyridyl­meth­yl)tetra­zolate ligand lead to the formation of a three-dimensional network

N-[2-(2-Chloro­phen­yl)-2-hy­droxy­eth­yl]propan-2-aminium benzoate

In the title compound, C11H17ClNO+·C7H5O2 −, obtained by the reaction of chlorprenaline {or 1-(2-chlorophenyl)-2-[(1-methylethyl)amino]ethanol} and benzoic acid, the chlorprenaline is twisted moderately [C—C—C—C torsion angle = −76.00 (17)°] compared with related compounds. The mol­ecules as usual form dimers. In the crystal structure, the two components are connected by classical O—H⋯O and N—H⋯O hydrogen bonds