Professor Grigory Isaakovich Barenblatt
PROFESSOR IN RESIDENCE, DEPARTMENT OF MATHEMATICS; UNIVERSITY OF CALIFORNIA AT BERKELEY
Scaling, Self-similarity, and the Renormalization Group in Partial Differential Equations
Scaling (self-similar) solutions to the partial differential equations entered the applied mathematics field 200 years ago. Until recently they were treated mostly as "exact special" solutions to some very specific problems--elegant, sometimes useful for qualitative investigations of the models but, in general, very limited in their significance elements of the general theories.
Gradually, it was recognized that the value of these solutions is much more significant: they are the intermediate asymptotics to the solutions to wider classes of problems when the influence of the details of the initial and/or boundary conditions already disappeared, but the solution is still far from its ultimate form. The appearance of computers did not reduce but increased the value of the scaling solutions.
In some cases (in fact, such cases are rather rare) the scaling solutions can be obtained using the dimensional analysis. However, as a rule this is not the case: scaling solutions appear due to the invariance of the problem to an additional group (note,--group, not semigroup), which we identify as the renormalization group.
A survey of these topics will be presented in this lecture; illustrative examples will be used.