Negative or dead-end results in my desk drawer
Many ideas lead, after exploring them as far as possible, into results which, though perhaps somewhat interesting, seem unlikely to ever yield productive results, and so are probably not interesting enough to be worth submitting for publication. This is what I mean by a "dead end" paper (a pejorative term, but I am applying it to my own work, and I am not insulted). These include many negative results that show that an approach does not work, without proving that the desired goal cannot be reached another way (via, typically, a counterexample). These are they types of results that perhaps most belong in one's desk drawer.
In a landmark 2014 paper, Yasunori Maekawa showed that solutions to the Navier-Stokes equations in a half-plane converge to a solution to the Euler equations as the viscosity vanishes if the initial vorticity is supported away from the boundary and the initial velocity vanishes on the boundary. (Moreover, a Prandtl expansion was shown to hold.) Since that publication, there have been many refinements of Maekawa's result, but his paper and all others exploit some degree of analyticity of the Navier-Stokes solutions in the boundary layer and involve a deep and quite involved analysis of those solutions. It made me wonder whether there is a simpler method that exploits the very special properties enjoyed by a solution to the Euler equations that would avoid the need for such involved arguments. My conclusion is a firm, though not quite definitive, no, as I argue in this short paper. I would categorize this paper as a negative result.
In his 1969 text, J. L. Lions used solutions to the 2D Navier-Stokes equations satisfying a vanishing vorticity boundary condition to obtain, in the vanishing viscosity limit (exploiting compactness), a weak (Yudovich) solution to the 2D Euler equations in a bounded domain. With the usual no-slip boundary conditions, it is easy to see that in the very special case in which the Navier-Stokes vorticity happens to vanish on the boundary, the vanishing viscosity limit holds. In this paper, I show that if the Navier-Stokes vorticty does not depart too far from vanishing on the boundary, then the vanishing viscosity limit holds. My measure of the vorticity departing from vanishing on the boundary is indirect, being in terms of projecting the velocity field into the space of divergence-free vector fields having vanishing boundary vorticity. The paper yields a clear-cut result, but applying the result is most likely infeasible. The virtue of the paper is that the properties of the projection operators are interesting, being non-orthogonal in the most-desired norms. I would categorize this paper as a dead end, though not a negative result.
Two things could make this paper publishable, I think. The first would be to show that the condition I develop is not only sufficient but also necessary. The second would be to show that the condition developed in this paper is satisfied by at least a few of the known instances in which the vanishing viscosity limit holds because of special symmetries of the data and/or geometry or because of partial analyticity. That would demonstrate its applicability.
I describe a connection between the Pompeiu problem (known to be equivalent to Schiffer's conjecture), “pressureless” eigenfunctions of the Stokes operator, eigenfunctions of the Neumann and Dirichlet Laplacians, steady state solutions to the Euler equations, and the vanishing viscosity limit. You need to read this short paper to find out more. I would categorize this paper as a dead end.
This is a negative result that shows how a simple argument to extend Yudovich's proof of uniqueness of solutions to the Euler equations for unbounded initial vorticity does not, in fact, work. I know this sounds like an off-putting topic for a paper (hey, it's in my desk drawer), but it sheds a little light on just how delicate Yudovich's uniqueness argument is, and how strange the Osgood condition really is.
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