Dear Colleague, mathematician V. Arnold really has written a book on PDE, yet it is really a nice and very professional book in this field, deeply demonstrating its real potential capacity in explaining diverse phenomena, as it was accepted by him. Simultaneously, he was also right that most of "studying some equations in pde of some type satisfying some a priori conditions with some ad hoc constraints" do indeed belong to this scholastical research spam, learning nothing us and imitating simply scientific activity as such...having nothing to do with true nature investigaying. And nothing more!

According to Vladimir Arnold: "Nowadays many are inclined to look disparangingly at this remarkable area of mathematics as an old-fashioned art fo juggling inequalities or as a testing ground for applications of functional analysis". So Arnold is saying that him is better that Sobolev, Gevrey, and Oga Ladyzhenskaya, or even Laurent Schwartz, he also never pay attention that MHD nonlinear equations is not scholastics, if not look at Iter, what about Shafranov equations? etc etc? and no mention Leray-Schauder spaces or Campanato-Morrey Spaces or Besov spaces...and......

Yes, Carlos, you are right that there exists a true PDE area having much to do with investigating the hidden world nature, this is exactly that that had in mind V. Arnold, and exactly that that was studied by Sobolew, Ladyzhenskaya, Schwarz and by many other true mathematical researchers.

Dear Anatolij

I think is important critics and reviews, I was educated under the people of Pedro Calderon, the most prominent mathematician of my native country Argentina. In a very long career Calderon suffered a lot to publish a short list of articles, me I have the same kind of hard times. Actually I work like architect, because the salary of a professor of math is very low after taxes! I cannot understand the the self status of people that actually has not too much contribution in real life!.

One example of that is the book Mechanics of Construction by V. A. Kiseliov, nowadays no one considered important that subject, but a 300 people live of our architects-structures consultancy across Canada, Colombia and Argentina. That kind of reality force me to open this dialogue ...( or critics or reviews?)...A we also a office in Barcelona working with subjects of AI of Gorbatov school.

I do not think I understand the point at issue. If Arnold writes, "Nowadays many are inclined to look disparagingly at this remarkable area of mathematics as an old-fashioned art of juggling inequalities or as a testing ground for applications of functional analysis," he is not necessarily in agreement that the more scholastic papers being disparaged constitute the whole content of PDE theory or, on the other hand, that those (and other) concerns are no longer valuable. Indeed, over the past 60 years I have seen the use of inequalities and functional analysis provide both a deeper understanding of continuum mechanics and development of the computational algorithms which are now so important in making these efficiently applicable. I am now reading in Mordukhovich's recent book on Variational Analysis in connection with generalizing Euler's elastica to flexible rods which twist and contact themselves: very much theory in this simplified case, but with the applied goal of computing the internal forces for a nonlinear model.

Nice contribution!

I have no position in University, I have my work in consultancy mainly in structural engineering concerns, also fluids dynamics, or even simple IT technologist in civil engineering!. I want to point you the book by F. Hartmann and C. Katz "Structural Analysis with Finite Elements" Springer Verlag 2007. You can see almost in all the world the new towers, highways, bridges, dams, water pumping systems etc that are actually builded using variational principles!...and this variational principles are not the studied by the normal boards of mathematics journals. I just want to mention that I have to much people asking me all the time on this concerns, even simple labor people they possed to questions. I also quote the book by, Y-Q Long, S. Cen and Z-F Long; "Advanced Element Method in Structural Engineering" were there even new variational methods applied to thin shells!.

I know the people of the famous "mountain pass lemma" used in what mathematicians widely consireded the significative to mathematical analysis....and unfortunately I never encounter and real and significative application of the "mountain pass lemma"!

An interested article on Arnold written by one of his former students.

https://arxiv.org/pdf/1007.0688.pdf

Arnold had lot of opinions on how mathematics is done. He biggest complain is the "algebraization" of mathematics which he is referring to the formulation of problems in analysis into equivalent problems in algebraic topology - using often sheaf cohomology in order to prove results about analytic/geometric questions - even questions in PDE. It is a common and very powerful method. Among its crowning achievements is the Atliya-Singer Index theorem that ties to elliptic partial differential systems on manifolds to the underlying topology (topological index).

However Arnold on the other hand pushed geometric interpretations of problems and in some cases that was the easiest way to to solve a particular problem - in some cases the "algebraization" was the most efficient path to the solution of a particular problem.

So I am not sure if Arnold's view of how mathematics do mathematics any more correct than any other successful senior mathematical, such as Phil Hartman, Joe Sampson, Jim Ells, John Nash, Lars Hormander, Jean-Pierre Serre, Louis Nirenberg, etc. PDE is a vast area. In order to make progress - the problems needed to be embedded into a larger theory. For example to effectively address PDE's the theory of distributions needed to be developed. The advent of the concept of pseudo differential operators lead to great advances in elliptic operators and systems. Fourier Integral Operators (and oscillatory integrals) have provide tools for non-hypoelliptic problems.

The quest to understand the Navier-Stokes equation has spawned significant generations of what were classical variational problems where a variational functional is considered as a real valued function on an infinite dimensional manifold modeled after a function space (Hilbert or Banach) - which took on the name global analysis. The mountain pass lemma and the Palais-Smale compactness condition (formally known as Smale Condition C) are an important tools in finding the critical points of function on Hilbert/Banach manifolds for rigorous analysis of variational functionals.

Are these abstractions help to someone in the sciences - maybe yes and maybe no. It can take results in pure mathematics years before it has any application at all arises, e.g., number theory to cryptography, algebraic geometry to computer vision. Without the concept of a manifold and the geometry on the manifold developed by a student of Gauss named Bernhard Riemann - Einstein would not had the mathematical structure to model his General Theory and without his good friend and mathematician Mercel Grossman to help him understand Reimannian geometry we would not have had the beautify theory that physics is really about the geometry of space time. That approach is the also at the basis of String Theory as we try to unify gravity and the other forces.

But it gets down to the fact I think that mathematics is at the end of the day not the same as a science. Mathematics can be viewed as the language of science allowing scientific problems to be given in the precise language of mathematics. But the pure mathematician is not a scientist. His interest is not the scientific questions it is exploring and expanding the linguistics of mathematics and that is more often than not independent of any potential application.

To Truman:

In some sence, if we refer mathematics to Science in general, mathematicians are scientists too...😃😃😃