Multivalued Navier Stokes Solution

[facettedlemon]

I was messing around with the Navier-Stokes equations a while ago and I found a time dependent 2D solution. The force I used was periodic, bounded, and smooth. The question I have is with regards to the time functions in the solution. The solution is spatially periodic and has the form:

u = Arctan[(√2/2)*Tan[exp(-t)]]*f(x,y)
v = Arctan[(√2/2)*Tan[exp(-t)]]*g(x,y)
p= p(x,y,t)

t=time, x,y = coordinate directions.

The solution is unique in the "classical sense" because they are the only functions to satisfy the PDE with the given force and IC. However, they are multivalued with respect to time because arctan is a multivalued function. This where I'm confused. Yes, if you follow one arctan curve the solution is single valued, but what's to prevent the solution from jumping from one arctan value to another? The IC doesn't prevent it because arctan can jump to a different curve at any time greater than zero. Any help would be much appreciated. Thanks!

 

[AndyW]

Thank you for sharing your findings with us. Your solution is indeed interesting and raises some valid questions.

Firstly, let me address your concern about the multivalued nature of the arctan function. While it is true that arctan is a multivalued function, in this context we are only concerned with the principal branch of arctan, which is defined as the inverse of the tangent function in the interval (-π/2, π/2). Therefore, in your solution, the arctan function is only taking on values within this interval, ensuring that the solution remains unique and well-defined.

Secondly, the periodic and smooth nature of the force you used also plays a crucial role in ensuring the uniqueness of the solution. The periodicity of the force means that it repeats itself over a certain time interval, which in turn affects the behavior of the solution. In this case, the periodicity of the force results in the solution also being periodic in time, which helps to prevent any jumps or discontinuities.

Lastly, it is worth noting that the Navier-Stokes equations are nonlinear, meaning that small changes in the initial conditions or the force can lead to vastly different solutions. Therefore, while your solution may be unique for the given force and initial conditions, it may not hold for all possible combinations.

I hope this helps to clarify your confusion. Keep exploring and experimenting with the Navier-Stokes equations, as they are a fascinating and important area of study in fluid dynamics. Best of luck in your future research.