# Algebraic microlocal analysis and nonlinear PDE

Though originating in the study of linear partial differential equations, microlocal analysis has become an invaluable tool in the study of nonlinear pde. Of particular importance has been the application of paraproducts/paradifferential operators and related tools developed by Bony, Coifman, Meyer and others. The theory of paradifferential operators fits nicely within the framework of microlocal analysis pioneered by Calderón, Zygmund, Hörmander, Kohn, Nirenberg, etc. However, there is another rather distinct school of (or approach to) microlocal analysis.

This second school, developed by Sato, Kashiwara, Kawai and others, makes liberal use of tools from algebra as well as the theory of sheaves (hence algebraic microlocal analysis). Additionally, analytic functions (as opposed to $$C^\infty$$ functions) play a much more prominent role in algebraic microlocal analysis. As far as I'm aware (and this isn't terribly far as concerns algebraic microlocal analysis), one can obtain very similar theories of linear pde using either microlocal analysis or algebraic microlocal analysis (though, of course, some differences surely exist). If I'm wrong about this, I'd certainly be interested to hear more. However, I'm not familiar with any work applying algebraic microlocal analysis to the study of nonlinear pde. This leads to my question(s). I have heard that there is at least some work applying algebraic microlocal analysis to nonlinear pde. Could anyone point me towards some references? Additionally, I'd love to hear any thoughts or intuition anyone has about the ability to apply (or the limitations of the applicability of) algebraic microlocal analysis to the study of nonlinear pde.

Edit: So, I was able to locate a couple references that are somewhat relevant to this question. In a set of lecture notes [1] by Pierre Schapira, he indicates that chiral algebras can be used to construct a theory of nonlinear pde in the spirit of $$\mathcal{D}$$-module theory. He gives [2] and [3] as references, which he says sketch the aforementioned theory.

[1] P. Schapira. An Introduction to $$\mathcal{D}$$-Modules. Lecture Notes, https://perso.imj-prg.fr/pierre-schapira/wp-content/uploads/schapira-pub/lectnotes/Dmod.pdf, 2017.

[2] A. Beilinson and V. Drinfeld. Chiral Algebras. Amer. Math. Soc. Coll. Publ., 51, Providence, RI, 2004.

[3] M. Kapranov and Y. Manin. Modules and Morita Theorem for Operads. Amer. J. Math., 123:811-838, 2001.

Edit. I'have added a few other infos taken from the historical notes by Goro Kato and Daniele C. Struppa in [A1], to point out that the application of algebraic microlocal analytic (and thus cohomological) methods to nonlinear (system of) PDEs is nevertheless an old problem.

This is not an answer but perhaps, as an extended comment, it can clarify a bit the status of recent relations between algebraic microlocal analysis and nonlinear PDEs. At the end of his BAMS review [2] (p. 118) of the monograph [1] by François Trèves and Paulo Domingos Cordaro, Pierre Schapira remarks:

However, even if cohomological methods are in many situations a really efficient tool, one should keep in mind that they have their own limits. They rapidly become very hard to manipulate when dealing with precise growth conditions, and so far, don’t apply at all to non-linear problems, contrary to traditional methods which often are suitable with slight modifications. This is a good reason not to reject approaches based on explicit representations of cohomology, and this book is an attempt in this direction.

Despite not being an expert in (algebraic) microlocal analysis, it seems to me that the status of the theory is still the one described by Schapira (who incidentally is one of the leading mathematicians and contributors in that area): algebraic microlocal analysis (in the sense of Kawai, Kashiwara and Schapira) is not (perhaps not yet) able to deal with nonlinear PDEs. However, it is possibly a good idea to ask directly prof. Schapira, since he and his colleague and coauthor Masaki Kashiwara are currently very active in developing this field of research.

While searching for applications of algebraic microlocal analysis, I remembered this remark by Kato and Struppa ([A1], chapter 3, §3.5 p. 180):

Around 1957, Sato began the cohomological study of systems of partial differential equations. The first occasion in which he publicly announced his work on this topic was the series of talks which he delivered in the Spring of 1960 at the Kawada Friday-Seminar just before his departure for the Institute for Advanced Study. During his talks, Sato put emphasis on the importance of cohomological treatments of systems of linear and non-linear partial differential equations. We can say that Sato’s algebraic analysis began in that moment.

Thus, despite applications of the methods from microlocal algebraic analysis to nonlinear PDE currently seem to lack, this possibility was in the mind of Mikio Sato from the very first years of his introduction of the concept of hyperfunction.

References

[1] Cordaro, Paulo D.; Treves, François, Hyperfunctions on hypo-analytic manifolds. (English) Annals of Mathematics Studies, vol. 136, Princeton, NJ: Princeton University Press, pp. xx+377 (1994), ISBN: 0-691-02993-8, MR1311923, Zbl 0817.32001.

[2] P. Schapira, "Book review: Hyperfunctions on hypo-analytic manifolds, by Paulo D. Cordaro and Francois Trèves, Annals of Mathematics Studies, vol. 136, Princeton University Press, Princeton, NJ, 1994, pp. xx+377, ISBN 0-691-92992-X", Bulletin of the American Maththematical Society, vol. 33 (1996) 115-118.