14 research outputs found
On the flow map for 2D Euler equations with unbounded vorticity
In Part I, we construct a class of examples of initial velocities for which
the unique solution to the Euler equations in the plane has an associated flow
map that lies in no Holder space of positive exponent for any positive time. In
Part II, we explore inverse problems that arise in attempting to construct an
example of an initial velocity producing an arbitrarily poor modulus of
continuity of the flow map.Comment: http://iopscience.iop.org/0951-7715/24/9/013/ for published versio
Remarks on the Cauchy functional equation and variations of it
This paper examines various aspects related to the Cauchy functional equation
, a fundamental equation in the theory of functional
equations. In particular, it considers its solvability and its stability
relative to subsets of multi-dimensional Euclidean spaces and tori. Several new
types of regularity conditions are introduced, such as a one in which a complex
exponent of the unknown function is locally measurable. An initial value
approach to analyzing this equation is considered too and it yields a few
by-products, such as the existence of a non-constant real function having an
uncountable set of periods which are linearly independent over the rationals.
The analysis is extended to related equations such as the Jensen equation, the
multiplicative Cauchy equation, and the Pexider equation. The paper also
includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the
acknowledgments section in the official paper exists, but it appears before
the appendix and not before the references as in the arXiv version);
correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of
Theorem 2.1; a few small improvements in various sections; added thank
Investigation of the existence and uniqueness of differentiable solutions of a functional equation
Euler's Beta function diagonalized and a related functional equation
Euler's Gamma function is the unique logarithmically convex solution of the functional equation (1), cf. the Proposition. In this paper we deal with the function , where B(x, y) is the Euler Beta function. We prove that, whenever a function h is asymptotically comparable at the origin with the function a log +b, a > 0, if satisfies equation (5) and the function is continuous and ultimately convex, then