Solvability of the two-dimensional kelvin-neumann problem for a submerged circular cylinder

Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations

“…The result obtained in [46], which is of course rather special, seems, however, to be the only known uniqueness theorem for the boundary value problem just stated not relying on any restrictions on ν.…”

“…In [46], the two-dimensional problem on the steady flow of infinite depth about a submerged cylinder is considered and the existence of the unique solution of any velocity v of the undisturbed flow upstream in the case of an arbitrary circular cylinder is proved.…”

On the plane problem of the flow around a submerged beam

“…Even in dimension two, one of the still open questions in the linear theory is to determine whether the problem for a given obstacle in a current is uniquely solvable for all values of the flux velocity. A positive answer is known for special geometries [7,8], but there are examples of nontrivial, finite energy solutions of the homogeneous problem (trapped modes) in the presence of multiple obstacles and in the case of a submerged hollow [9]. In general, the connection between unique solvability and the geometry of the obstacles is not completely understood.…”

The steady two-dimensional flow over a rectangular obstacle lying on the bottom

“…It is a known fact that the answer depends on the geometry of the obstacle; for example, there exists a sufficient condition on the body profile [4] for the uniqueness of a solution with a finite Dirichlet integral. However, such condition seems to be applicable only in special cases [5] since the solutions of the plane problem cannot be assumed to have finite energy for every value of the velocity. In fact, the (a priori) asymptotic properties of these solutions depend critically on the value of the Froude number F r , defined by…”

“…Correspondingly, with a supercritical flow there is unique solvability of the Neumann-Kelvin problem for an arbitrary number of obstacles of generic shape, totally or partially immersed [8]. In the subcritical regime, instead, existence and uniqueness (for every subcritical value of the velocity) have been proved for a submerged cylinder [5] and for a surface-piercing obstacle with symmetric, non bulbous profile [9]. A particularly interesting class of obstacles is represented by localized variations of the bottom topography.…”