KUMU PDE Conference Proposal

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

This award will provide support for participants, especially graduate students, junior researchers, women and mathematicians from under-represented groups in the sciences, to attend the regional conference "KUMU Conference in PDE, Dynamical Systems and Applications" to be held at the University of Kansas from April 18-19, 2015, co-organized by faculty from the University of Kansas (KU) and the University of Missouri (MU). Nearly all important physical phenomena are governed by fundamental laws and design principles that directly relate rates of change of some quantity to that of some other quantity. Indeed, given the initial conditions and the physical laws of motion one seeks to predict the future and reconstruct the past. This important observation naturally leads to the idea of a differential equation, thus providing the key to understanding many real-world problems. Differential equations are widely used as models in mathematical physics and have potential applications to many fields including Bose-Einstein condensates, fluid dynamics, pattern formation, gas dynamics and for modeling signals in optical communication networks. This conference will facilitate greater interaction between researchers in differential equations and its related fields from the area close to Kansas and Missouri. Planned as the first of a series of annual meetings, the conference will provide a venue for regional junior and established researchers, as well as graduate students to discuss the recent advances and challenges in their respective fields. In addition, young researchers will be given the opportunity to present their own work and to gain insights into this important subject through interactions with senior experts in the field. The conference website: https://www.math.ku.edu/conferences/2015/KUMUPDE/index.html Complex nonlinear systems abound in science and engineering, and their behavior is often modeled by systems of nonlinear partial differential equations (PDE). Any progress towards understanding the behavior of their solutions is of paramount importance for a variety of practical applications, including fluid flow, flame front propagation and fiber optical communication. Many PDE can be conveniently described as infinite dimensional dynamical systems, allowing for the use of tools and methodologies from dynamical systems theory to make qualitative and quantitative predictions about the solutions of these systems. Objects like invariant manifolds have been a great aid in understanding the behavior of finite-dimensional dynamical systems, but the connections between nonlinear PDE's and dynamical systems is still an area active current research. In the last few decades, collaborations between researchers in these fields, as well as with those working in their applications, have provided tremendous progress in our understanding of the dynamical behavior, stability and robustness of coherent structures in such nonlinear PDE. The main themes of this conference include (i) fluid dynamics, water waves and dispersive PDE's, (ii) existence, dynamics, and stability of nonlinear waves in dissipative systems, and (iii) dynamical systems and 2d-Navier Stokes equations.The techniques used to solve many challenging problems in these broad areas often combine ideas and methodologies from dynamical systems and partial differential equations together with probability theory, spectral and functional analysis, Evans functions, and geometric singular perturbation theory, to name a few.