Exploring Nonlinear PDEs: Trends and Challenges in Cancer Research

In summary, the main trends in differential equation research are studying the implications of nonlinearity in cancer modelling, as well as finding solutions to the Navier-Stokes PDE.
  • #1
Domenico94
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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?
 
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  • #2
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

https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics
 
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  • #3
jedishrfu said:
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
 
  • #4
anyone else?
 
  • #6
Thanks a lot!
And what about nonlinearity, instead? I guess there's some research going in that direction as well...
 

1. What are some current open problems in PDE research?

There are many open problems in PDE research, but some of the current ones include the Navier-Stokes equations, the regularity problem for the compressible Navier-Stokes equations, the global well-posedness problem for the 3D Euler equations, and the uniqueness problem for the inverse conductivity problem.

2. Why are these open problems important?

These open problems are important because they have significant applications in various fields such as physics, engineering, and climate modeling. Additionally, solving these problems can provide insight into the behavior of complex systems and lead to the development of new mathematical techniques.

3. What are some approaches that researchers are using to tackle these open problems?

Researchers are using a variety of approaches to tackle these open problems, including numerical simulations, analytical techniques, and new mathematical frameworks. Collaborative efforts between mathematicians and scientists from other disciplines are also becoming increasingly common.

4. How do open problems in PDE research impact society?

The solutions to open problems in PDE research can have a profound impact on society by improving our understanding of physical phenomena, leading to advancements in technology, and aiding in the development of more accurate predictive models. For example, solving the Navier-Stokes equations could lead to more efficient designs for airplanes and cars.

5. Can anyone contribute to the research on open problems in PDE?

Yes, anyone with a strong mathematical background can contribute to the research on open problems in PDE. Many universities have research programs specifically focused on these open problems, and there are also opportunities for collaboration and participation in conferences and workshops. Additionally, advancements in technology have made it easier for researchers to collaborate and share ideas, making it more accessible for individuals to contribute to this field of research.

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