We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical NavierStokes equations for barotropic gases; thus the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, w...

The celebrated 100-year old Phragmén-Lindelöf theorem, [15, 16], is a far reaching extension of the maximum modulus theorem for holomorphic functions that in its simplest form can be stated as follows: Theorem 1.1. Let Ω ⊂ C be a simply connected domain whose boundary contains the point at infinity. If f is a bounded holomorphic function on Ω and lim supz→z0 |f(z)| ≤ M at each finite boundary p...

We present two companion results: Phragmén–Lindelöf type tight bounds on the minimal possible growth of subharmonic functions with a recurrent zero set, and maximal decay harmonic measure outer boundary coland

We show that the issue of the drag exerted by an incompressible fluid on a body in uniform motion has played a major role in the early development of fluid dynamics. In 1745 Euler came close, technically, to proving the vanishing of the drag for a body of arbitrary shape; for this he exploited and significantly extended existing ideas on decomposing the flow into thin fillets; he did not howeve...

A comparative analysis of harmonic and biharmonic boundary-value problems for 2D problems on a rectangle is given. Some common features of two types of problems are emphasized and special attention is given to the basic distinction between them. This distinction was thoroughly studied for the first time by L. N. G. Filon with respect to some plane problems in the theory of elasticity. The analy...

We consider deformations in R3 of an infinite general nanotube of atoms where each atom interacts with all the other through a two-body potential. We compute the effect of an external force applied to the nanotube. At the equilibrium, the positions of the atoms satisfy an Euler-Lagrange equation. For large classes of potentials (including Lennard-Jones potential) and under suitable stability as...