What Are the Latest Trends and Challenges in Nonlinear PDEs for Cancer Research?

[Domenico94]

Hi everyone. For people who already saw me in this forum, I know I may seem boring with all these questions about PDE, but I promise this will be the last :D
Anyway, as the title says, which are the main trends of differential equations research, especially nonlinear differential equations(which are widely used in cancer research, just to mention an example)?
Secondly, are those problems most related with finding a particular solution for a differential equation, or are they concerned with the existence and regularity of solutions in a given domain?

 

[jedishrfu]

The major one is solving the Navier-Stokes PDE used in fluid mechanics:

https://en.wikipedia.org/wiki/Navier–Stokes_equations

The Navier-Stokes problem is also a Millenium problem so there's big money behind the solution that you can give away to charity or just refuse like Russian mathematician Grigori Perelman did:

https://en.wikipedia.org/wiki/Grigori_Perelman

and there's these problems from the unsolved list on wikipedia:

Partial differential equations

• Regularity of solutions of Vlasov–Maxwell equations
• Regularity of solutions of Euler equations

https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics

 

[Domenico94]

The major one is solving the Navier-Stokes PDE used in fluid mechanics:

https://en.wikipedia.org/wiki/Navier–Stokes_equations

The Navier-Stokes problem is also a Millenium problem so there's big money behind the solution that you can give away to charity or just refuse like Russian mathematician Grigori Perelman did:

https://en.wikipedia.org/wiki/Grigori_Perelman

and there's these problems from the unsolved list on wikipedia:
https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics

I found Navier Stokes' equations interesting, mainly for the fact that, as far as I know, they've many implications in cancer modelling (Exchange between blood flow and tumor, and interactions between cancer and therapy). The problem is that I only managed to find millenium problems, which I couldn't be able to answer, neither now nor in the future. Are there any "simpler" problems related to Navier Stokes, and which are studied nowadays?
P.S. Yes, I know about Perelman :) He lives with his mother now, or something like that :D

 

[Domenico94]

anyone else?

 

[jedishrfu]

I found this pdf on open problems related to Navier-Stokes:

https://web.math.princeton.edu/~const/2k.pdf

 

[Domenico94]

Thanks a lot!
And what about nonlinearity, instead? I guess there's some research going in that direction as well...