This is the third volume of a collection of publications of J.-L. Lions. It is devoted mainly to numerical analysis and applications and presents work from the mid-sixties to 2001.
A variety of themes (involving PDEs) is handled: fluid mechanics (with concrete cases of the ocean or the atmosphere, Bingham fluids), elasticity and the theory of shells, coupling, hyperbolic system, finite element methods and domain decomposition, numerical methods for controllability, variable domains. An important idea is to discretize the variational formulation rather than the PDEs themselves.
Most of the publications have already been reviewed (roughly in chronological order): Zbl 0147.12502, Zbl 0191.11601, Zbl 0187.39601, Zbl 0253.73013, Zbl 0241.35035, Zbl 0264.73027, Zbl 0252.35055, Zbl 0294.73060, Zbl 0329.73052, Zbl 0484.76009, Zbl 0631.35051, Zbl 0656.35097, Zbl 0726.76092, Zbl 0746.76019, Zbl 0766.35039, Zbl 0838.93013, Zbl 0838.93014, Zbl 0935.35041, Zbl 0922.35010, Zbl 1013.74050, Zbl 0988.65117, Zbl 1002.65133, Zbl 1010.65038, Zbl 0937.68140, Zbl 0962.93020, Zbl 0960.65103.
The two missing articles are: “Un exemple de problème aux limites couplé parabolique-hyperbolique pour une structure pluri-dimensionelle”. Calcolo 22, No. 1, 7-15 (1985). Here, well-posedness of a coupled system of PDEs, parabolic in a three dimensional domain and hyperbolic in a two dimensional domain with a transmitting point common to the boundaries, is obtained.
“Earth system models and mathematical remarks”. Comput. Methods Appl. Mech. Eng. 89 (1991). Here, important problems of ecology (climatic change, pollution of the soils, distribution of water) are related to the modelling, simulation and control of nonlinear dissipative PDEs. Often initial data are incomplete so that issues of observability are raised (this is also related to sentinels). If humans are to influence the course of events in ecology, then controllability is involved. The ways subsystems interact is of relevance too (the author uses the word “feedback” in this context).
This set of three volumes is of great interest, illustrating once more the incredible productivity of J.-L-Lions and allowing to track his itinerary; the study of PDEs in a functional analytic setting will lead to meaningful interpretations in the control of PDEs and their optimization and will be treated numerically and applied to concrete and important physical situations.