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Abstract

In this course we develop the methodology of “variational modeling” of energy-driven systems. This methodology applies to systems whose evolution (in time) is driven by the decrease of an energy, in a friction-dominated or strongly damped way. Recent developments have shown that a surprisingly large class of evolutionary systems is of this form, even though the energy and the friction mechanism may not be obvious. Examples of these include linear and nonlinear diffusion equations, nonlocal diffusion equations, higher-order parabolic equations, moving-boundary problems, and many others. The course will offer discussions on how to use the mathematical structure as a modeling tool: each choice of an energy and a friction mechanism provide an evolutionary system. The two choices characterize in a remarkably clear way the modeling choices that underlie the resulting differential equations.