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

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

Navier-Stokes equations:

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

Can we prove the existence of unique solutions for all $$t > 0$$?

Or can we discover a single initial condition for which our solution breaks down for some $$t>0$$?

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

### Re: Do Navier-Stokes equations have unique solutions for t >

Guest wrote:Navier-Stokes equations:

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

Can we prove the existence of unique solutions for all $$t > 0$$?

Or can we discover a single initial condition for which our solution breaks down for some $$t>0$$?

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

Moreover, we define $$\vec{u} (x_{1 }, x_{2 }, x_{3 }, t) = (u_{1 }(x_{1 }, x_{2 }, x_{3 }, t), u_{2 }(x_{1 }, x_{2 }, x_{3 }, t), u_{3 }(x_{1 }, x_{2 }, x_{3 }, t))$$.

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

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

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

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

And our equation two becomes,

2. $$\nabla \cdot \vec{u} = \frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{2}}{\partial x_{2}} + \frac{\partial u_{3}}{\partial x_{3}} = 0$$.

Relevant Reference Textbook:

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

### Re: Do Navier-Stokes equations have unique solutions for t >

Guest wrote:Navier-Stokes equations:

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

Can we prove the existence of unique solutions for all $$t > 0$$?

Or can we discover a single initial condition for which our solution breaks down for some $$t>0$$?

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

'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.
Guest