I want to learn and study the theory of Navier - Stokes equations, such as existence and uniqueness in various functional spaces, regularity of the solutions, mild solutons, weak solutions and well posedness. I dont know where to start. Could Any researcher/scientist give suggestions and basic materials?
To study the theory of Navier - Stokes equations I recommend you to use the pdf below if your maths is enough to understand at this level:
https://www.math.iitb.ac.in/~neela/CIMPA/notes/l_14_TIFR_1.pdf
https://www.barnesandnoble.com/w/the-navier-stokes-equations-hermann-sohr/1112801845
The best place to start is the lecture on the N-S equation given at the Clay Institute and the references within
https://www.claymath.org/lectures/navier-stokes-existence-and-smoothness/
and the introductory paper by Charles Fefferman
https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf
While the N-S equation in is a tough nut to crack in general, specific cases have been studied for years with some progress. The swirling flow problem of laminar flow is one such case. Phil Hartman did quite a bit of work on this problem along with his student Lan. See Hartman,
https://www.jstor.org/stable/24890352
and references within.
In
https://core.ac.uk/download/pdf/82146461.pdf
Theorem 2.1 Prevatt extended the general functional analytic tools of Hartman's work on the swirling flow equations to address a more general functional differential equation.
The N-S equation is highly non-linear and has a rich mathematical behavior. Even the specialized cases such as the swirling flow, it seems that specialized methods need to be developed on a case by case basis. That is as of yet a "general theory" have alluded the efforts of some of the best mathematicians.
- Badges
- Science topic
- Similar topics
- Mathematics