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This monograph presents recent and new ideas arising from the study of problems of planar fluid dynamics, and which are interesting from the point of view of geometric function theory and potential theory. In particular, the book is concerned with geometric problems for Hele-Shaw flows. Additionally, Hele-Shaw flows on parameter spaces (e.g., the Teichmüller space) are discussed, and connections with string theory are revealed. Ultimately, the interaction between several branches of complex and potential analysis, and planar fluid mechanics is discussed. The text includes historical remarks concerning people who have contributed to the topic.
This self-contained book assumes a background provided by graduate courses in real and complex analysis, in particular, the theory of conformal mappings and in fluid mechanics.
and Background.- Explicit Strong Solutions.- Weak Solutions and Balayage.- Geometric Properties.- Capacities and Isoperimetric Inequalities.- General Evolution Equations.- Hele-Shaw Evolution and Strings.