Analytical and Geometrical Aspects of Non Linear Partial Differential Equations

University Of Texas At Austin, Austin TX

Investigators

Abstract

March 11, 2002. PI: Luis Caffarelli [caffarel@math.utexas.edu], University of Texas, Austin DMS-0140338 Abstract: Our research focus in a series of phenomena that bring together different areas of non linear PDE. For instance, a flame front propagating in a layered medium brings together the interaction of the fast phase transition at the edge of the flame, with the linking of large scales (front speeds) and small scales (possibly random , thin layering). Recent models of front formation, bring together the interaction of optimal allocation,that provides the " variational framework" for front formation (the Monge Ampere equation ) with the issue of how vorticity is transported within, and affects the front. Finally, problems in financial engineering, bring together fully non linear equations ( for instance as "extremals" or intervals of "trust" when only rough bounds on volatility and correlations are available) with issues of phase transition , when trading strategy changes discontinuously at certain values of the parameters,or there are constraints on the range of trading, and homogeneisation, when the underlying space changes randomly. We plan to study several models from different sciences that bring together interactions of complementary non linear phenomena. A good example would be flame propagation in a thinly layered material: Observed at short range, the flame front will "wiggle", moving faster in some of the layers, depending on their composition. From far away, the flame will appear as a homogeneous front. Its speed , though, will depend in a very subtle way from both the nature of the ignition process at the edge of the flame ( free boundary problem) and the properties of the very fine, ( possibly randomly organized, as it usually occurs in nature) thin layering. Another example has to do with the formation of weather fronts: according to some models, globally the front organizes itself trying to " spread its energy" in an optimal way, and within, vorticity ( the "rotational component" of wind) is transported in an interactive way with the front organization. A third example comes from financial engineering when seeking optimal ( or safest) trading strategies, but , as in most cases, only a rough knowledge is available of the way different ways in which different components of the portfolio, or parameters ( interest rates, exchange rates, etc ) interact.This rough or incomplete knowledge, gives rise to non linear strategies, that couple with constraints in the trading range, discontinuous trading strategies, etc.