mathematics

Ronald DiPerna

Ronald DiPerna is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Ronald DiPerna rather than just read about it. In short: Ronald J. DiPerna (11 February 1947 – 8 January 1989) was an American mathematician, who worked on nonlinear partial differential equations.

Key takeaways

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Reference excerpt

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

Ronald Diperna was born in Somerville, Massachusetts in 1947, and did his undergraduate studies at Tufts University before being advised to attend graduate school by Professor George Leger. 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. DiPerna was a Guggenheim Fellow for the academic year 1984–1985 and a Sloan Fellow for the academic year 1978–1979. In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley 1986 and gave a talk Compactness of solutions to nonlinear PDEs. He was married to Maria E. Schonbek, a professor of mathematics at the University of California, Santa Cruz and had a daughter. He died in Princeton, New Jersey in 1989, and in his honor the University of California at Berkeley established the DiPerna Lectures in Applied Mathematics.

Selected publications Diperna, Ronald J. (1973). "Global solutions to a class of nonlinear hyperbolic systems of equations". Comm. Pure Appl. Math. 26: 1–28. doi:10.1002/cpa.3160260102. Diperna, Ronald J. (1973). "Existence in the large for quasilinear hyperbolic conservation laws". Arch. Rational Mech. Anal. 52 (3): 244–257. Bibcode:1973ArRMA..52..244D. doi:10.1007/BF00247735. S2CID 122077205. Uniqueness of solutions to hyperbolic conservation laws, Indiana Univ. Math. J. 28 (1979), 137–188. Diperna, R. J. (1983). "Convergence of approximate solutions to conservation laws". Arch. Rational Mech. Anal. 82 (1): 27–70. Bibcode:1983ArRMA..82...27D. doi:10.1007/BF00251724. hdl:10338.dmlcz/102415. Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), 1–30, Online Diperna, Ronald J. (1985). "Measure-valued solutions to conservation laws". Arch. Rational Mech. Anal. 88 (3): 223–270. Bibcode:1985ArRMA..88..223D. doi:10.1007/BF00752112. S2CID 119711966. Compensated Compactness and general systems of Conservation Laws, Transactions AMS, 292, 1985, 383-420 with Pierre-Louis Lions: Diperna, R. J.; Lions, P. L. (1989). "Global weak solutions of Vlasov-Maxwell systems". Comm. Pure Appl. Math. 42 (6): 729–757. doi:10.1002/cpa.3160420603. with Pierre-Louis Lions: Diperna, R. J.; Lions, P. L. (1989). "On the Cauchy problem for Boltzmann equations: global existence and weak stability". Annals of Mathematics. 130 (2): 321–366. doi:10.2307/1971423. JSTOR 1971423. with Lions: "Ordinary differential equations, Sobolev spaces and transport theory". Inventiones Mathematicae. 98: 511–547. 1989. doi:10.1007/BF01393835. S2CID 123097609.

References

Illustrations

Worked examples

Example 1 — a first encounter with Ronald DiPerna

Start with the simplest possible case. Write down what Ronald DiPerna claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ronald DiPerna before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ronald DiPerna ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ronald DiPerna

In research: Ronald DiPerna appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ronald DiPerna in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ronald DiPerna is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1947 births, 1989 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Ronald DiPerna outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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Frequently asked questions

What is Ronald DiPerna in simple terms?

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

Why does Ronald DiPerna matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ronald DiPerna?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ronald DiPerna.