Ronald DiPerna

Ronald J. DiPerna (11 February 1947, Somerville, Massachusetts – 8 January 1989, Princeton, New Jersey) was an American mathematician, who worked on nonlinear partial differential equations.

In 1972 DiPerna received from the Courant Institute of Mathematical Sciences his Ph.D. under James Glimm with thesis Global solutions to a class of nonlinear hyperbolic systems. He held academic positions at Brown University, the University of Michigan, the University of Wisconsin, and Duke University, before he became in 1985 a professor at the University of California, Berkeley. He died unexpectedly at age 41 shortly after the end of a sabbatical year as a visiting scholar at the Institute for Advanced Study.

DiPerna was known for his work on nonlinear partial differential equations, especially those that are important in fluid dynamics and the kinetic theory of gases. Probably his best known work is his development and application of the method of compensated compactness. This is a very powerful method for controlling oscillation and thereby proving existence theorems. DiPerna proved existence of weak solutions in the large for the equations of compressible gas dynamics and obtained important results concerning the uniqueness of solutions, their large time behavior, and their local regularity as elements of the appropriate abstract spaces.

In the last part of his career he worked with Pierre-Louis Lions on integro-differential equations in the kinetic theory of gases (Cauchy problem for Boltzmann equations) and the plasma physics generalization (Vlasov equation). He also worked on singularities in compressible flow. DiPerna with Andrew Majda began in 1986 research on the question of the existence of solutions to the Euler equations in two dimensions with initial conditions that are found in the evolution of vortex sheets. DiPerna and Majda introduced the Concentration-Cancellation Method.

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