2 research outputs found
Partial regularity for a surface growth model
We prove two partial regularity results for the scalar equation
, a model of surface growth arising from the
physical process of molecular epitaxy. We show that the set of space-time
singularities has (upper) box-counting dimension no larger than and
-dimensional (parabolic) Hausdorff measure zero. These parallel the results
available for the three-dimensional Navier--Stokes equations. In fact the
mathematical theory of the surface growth model is known to share a number of
striking similarities with the Navier--Stokes equations, and the partial
regularity results are the next step towards understanding this remarkable
similarity. As far as we know the surface growth model is the only
lower-dimensional "mini-model" of the Navier--Stokes equations for which such
an analogue of the partial regularity theory has been proved. In the course of
our proof, which is inspired by the rescaling analysis of Lin (1998) and
Ladyzhenskaya & Seregin (1999), we develop certain nonlinear parabolic
Poincar\'e inequality, which is a concept of independent interest. We believe
that similar inequalities could be applicable in other parabolic equations.Comment: 29 page
Well-posedness of logarithmic spiral vortex sheets
We consider a family of 2D logarithmic spiral vortex sheets which include the
celebrated spirals introduced by Prandtl (Vortr\"age aus dem Gebiete der Hydro-
und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We prove that for
each such spiral the normal component of the velocity field remains continuous
across the spiral. Moreover, we give a complete characterization of such
spirals in terms of weak solutions of the 2D incompressible Euler equations.
Namely, we show that a spiral gives rise to such a solution if and only if two
conditions hold across every spiral: a velocity matching condition and a
pressure matching condition. Furthermore we show that these two conditions are
equivalent to the imaginary part and the real part, respectively, of a single
complex constraint on the coefficients of the spirals. This in particular
provides a rigorous mathematical framework for logarithmic spirals, an issue
that has remained open since their introduction by Prandtl in 1922. Another
consequence of the main result is well-posedness of the symmetric Alexander
spiral with two branches, despite recent evidence for the contrary. Our main
tools are new explicit formulas for the velocity field and for the pressure
function, as well as a notion of a winding number of a spiral, which gives a
robust way of localizing the spirals' arms with respect to a given point in the
plane.Comment: 25 pages, 3 figure