Do Navier-Stokes equations have unique solutions for t > 0?

[Guest]

Navier-Stokes equations:

[tex]\rho(\frac{∂\vec{u}}{∂ t} + (\vec{u} \cdot \nabla)\vec{u}) - \mu \nabla \vec{u} + \nabla \rho = \vec{f}[/tex] with [tex]\nabla \cdot \vec{u} = 0[/tex].

Can we prove the existence of unique solutions for all [tex]t > 0[/tex]?

Or can we discover a single initial condition for which our solution breaks down for some [tex]t>0[/tex]?

Relevant Reference Link:

https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations.

[Guest]

Navier-Stokes equations:

1. [tex]\rho(\frac{∂\vec{u}}{∂ t} + (\vec{u} \cdot \nabla)\vec{u}) - \mu \nabla \vec{u} + \nabla p = \vec{f}[/tex] with 2. [tex]\nabla \cdot \vec{u} = 0[/tex].

Can we prove the existence of unique solutions for all [tex]t > 0[/tex]?

Or can we discover a single initial condition for which our solution breaks down for some [tex]t>0[/tex]?

Relevant Reference Link:

https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations.

Moreover, we define [tex]\vec{u} (x_{1 }, x_{2 }, x_{3 }, t) = (u_{1 }(x_{1 }, x_{2 }, x_{3 }, t), u_{2 }(x_{1 }, x_{2 }, x_{3 }, t), u_{3 }(x_{1 }, x_{2 }, x_{3 }, t))[/tex].

From our edited equation one (above) we derive a component equation for each corresponding index, [tex]j \in[/tex] {1, 2, 3}:

1A. [tex]\rho(\frac{∂u_{1 }}{∂ t} + (\sum_{i=1}^{3}u_{1 }\frac{\partial u_{1 } }{\partial x_{i }} )) - \mu (\frac{\partial^{2} u_{1}}{\partial x_{1}^{2}} + \frac{\partial^{2} u_{1}}{\partial x_{2}^{2}} + \frac{\partial^{2} u_{1}}{\partial x_{3}^{2}}) + \frac{\partial p}{\partial x_{1}} = f_{1}[/tex];

1B. [tex]\rho(\frac{∂u_{2 }}{∂ t} + (\sum_{i=1}^{3}u_{2 }\frac{\partial u_{2 } }{\partial x_{i }} )) - \mu (\frac{\partial^{2} u_{2}}{\partial x_{1}^{2}} + \frac{\partial^{2} u_{2}}{\partial x_{2}^{2}} + \frac{\partial^{2} u_{2}}{\partial x_{3}^{2}}) + \frac{\partial p}{\partial x_{2}} = f_{2}[/tex];

1C. [tex]\rho(\frac{∂u_{3 }}{∂ t} + (\sum_{i=1}^{3}u_{3 }\frac{\partial u_{3 } }{\partial x_{i }} )) - \mu (\frac{\partial^{2} u_{3}}{\partial x_{1}^{2}} + \frac{\partial^{2} u_{3}}{\partial x_{2}^{2}} + \frac{\partial^{2} u_{3}}{\partial x_{3}^{2}}) + \frac{\partial p}{\partial x_{1}} = f_{3}[/tex];

And our equation two becomes,

2. [tex]\nabla \cdot \vec{u} = \frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{2}}{\partial x_{2}} + \frac{\partial u_{3}}{\partial x_{3}} = 0[/tex].

Relevant Reference Textbook:

'An Introduction to Ordinary Differential Equations' by Prof. James C. Robinson...'.

[Guest]

Navier-Stokes equations:

[tex]\rho(\frac{∂\vec{u}}{∂ t} + (\vec{u} \cdot \nabla)\vec{u}) - \mu \nabla \vec{u} + \nabla \rho = \vec{f}[/tex] with [tex]\nabla \cdot \vec{u} = 0[/tex].

Can we prove the existence of unique solutions for all [tex]t > 0[/tex]?

Or can we discover a single initial condition for which our solution breaks down for some [tex]t>0[/tex]?

Relevant Reference Link:

https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations.

Relevant Reference Link:

'Researchers develop first mathematical proof for key law of turbulence in fluid mechanics.'

https://phys.org/news/2019-12-mathematical-proof-key-law-turbulence.html.