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Recent advances in nonlinear partial differential equations and applications
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Recent developments in Nonlinear Partial Differential Equations and Applications
Description The articles of this book are written by leading experts in partial differential equations and their applications, who present overviews here of recent advances in this broad area of mathematics. The formation of shocks in fluids, modern numerical computation of turbulence, the breaking of the Einstein equations in a vacuum, the dynamics of defects in crystals, effects due to entropy in hyperbolic conservation laws, the Navier-Stokes and other limits of the Boltzmann equation, occupancy times for Brownian motion in a two dimensional wedge, and new methods of analyzing and solving integrable systems are some of this volume's subjects.
The reader will find an exposition of important advances without a lot of technicalities and with an emphasis on the basic ideas of this field. Product details Format Hardback pages Dimensions Other books in this series. Add to basket. There is already a substantial literature on these topics, both theoretical and numerical, including applications to image processing, semiconductor etching, etc.
Particularly useful has been the level set method of describing the moving interface in terms of an ambient "order parameter", which solves an appropriate nonlinear PDE. Other geometric PDE. The past years have seen a great flowering of geometry, made possible at least in part by methods of nonlinear elliptic PDE, both single equations and systems. I expect this trend to continue, with perhaps more input from the theory of hyperbolic equations.
Dynamical methods in the calculus of variations. The PDE governing many nonequilibrium systems can be approximated by taking time to be discrete, and then solving a minimization problem over each time interval. There remain however profound problems in understanding the limit of the approximations as the time step goes to zero. I believe there is a great subject waiting here to be discovered, some sort of "time-dependent calculus of variations". Kinetic formulations.
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The French PDE school has during the past decade pioneered a fascinating "kinetic" approach to nonlinear transport equations, based upon analogies with the classical passage from the Boltzmann equation to fluid mechanics. The physical procedure is still mathematically unjustified, but some related, and rigorous, procedures provide useful representation formulas for solutions of various nonlinear transport PDE, in terms of functions of more variables "velocities".
Viscosity solutions. The notion of "viscosity solutions" has provided a robust and extremely flexible collection of tools for understanding weak solutions of certain highly nonlinear PDE that satisfy a maximum principle. The biggest successes have been in justifying dynamic programming procedures in control theory, but other applications have included large deviation estimates, interface motions, Hamiltonian dynamics, etc.
Previous page Top of this page Next page. Table of Contents. Summary Article. Statistics as the information science. Statistical issues for databases, the internet, and experimental data. Mathematics in image processing, computer graphics, and computer vision. Future challenges in analysis. Getting inspiration from electrical engineering and computer graphics to develop interesting new mathematics.
Recent Advances in Differential Equations
Research opportunities in nonlinear partial differential equations. Risk assessment for the solutions of partial differential equations. Discrete mathematics for information technology. Random matrix theory, quantum physics, and analytic number theory. Mathematics in materials science. Mathematical biology: analysis at multiple scales. Number Theory and its Connections to Geometry and Analysis.
Revealing hidden values: inverse problems in science and industry. Complex stochastic models for perception and inference. Model theory and tame mathematics. Beyond flatland: the future of space and time. Mathematics in molecular biology and medicine. The year in geometry and topology.