Why doesn't the Navier-Stokes equation have a solution?

In summary, the Navier-Stokes equation, which describes the motion of fluid substances, lacks a proven solution for all possible scenarios, particularly in three dimensions. While solutions exist for certain simplified cases, the complexity of turbulent flow and the nonlinear nature of the equations pose significant mathematical challenges. The Clay Mathematics Institute has identified the existence and smoothness of solutions to the Navier-Stokes equation as one of the seven "Millennium Prize Problems," highlighting the ongoing quest in mathematics to resolve these fundamental questions in fluid dynamics.
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Sawawdeh
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Why the navier-stokes equation don't have a solution ?
 
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Sawawdeh said:
Why the navier-stokes equation don't have a solution ?
Because it’s hard enough that so far no one has figured it out. Perhaps no one ever will.

Google for “Millennium prize navier-stokes” for more about what has to be figured out.
 
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The Navier Stokes equations do have solutions for certain specific flows.
 
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If we don't know the solution(s), it does not mean that the equation does not have solutions, does it?
 
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Classically the word solution often refer to a closed form solution, i.e. a "simple" symbolic solution general for large set of initial conditions and parameters, and in that sense we know that there are some (turbulent) flows that cannot have such a solution even if the actual flow dynamics still satisfy the equations.
However, in context of numerical analysis (i.e. in this case computational fluid dynamics) the word solution more imply any possible solutions achievable by numerical means so here it would make sense to say that a specific turbulent flow is a solution to the equations. Since turbulent flows has sensitivity to initial conditions this usually means the numerical solution can only be an approximation that share some statistical measure with the exact solution but also that the two will eventually diverge over time.
 
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@Sawawdeh It's never a surprise when an analytical solution to a problem doesn't exist. We start our Maths and Science education being presented with a number of situations and equations that are soluble analytically and exactly (you have to be encouraged initially) but, once you get into Integral Equations you find that most situations can only be dealt with numerically. In the recent past (pre-digital) people used vast books of tables of integrals to calculate approximate answers for problems.
Then someone discovered Chaos. . . . . .
 
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FAQ: Why doesn't the Navier-Stokes equation have a solution?

What is the Navier-Stokes equation?

The Navier-Stokes equation is a fundamental partial differential equation in fluid dynamics that describes the motion of viscous fluid substances, such as liquids and gases. It accounts for various factors like velocity, pressure, density, and viscosity of the fluid.

Why is the existence and smoothness of the Navier-Stokes equation important?

The existence and smoothness of solutions to the Navier-Stokes equation are crucial because they ensure that the equations accurately describe the behavior of fluids under all conditions. If a solution exists and is smooth, it means the fluid's behavior is predictable and well-defined without singularities or infinite values.

What does it mean for the Navier-Stokes equation to not have a solution?

If the Navier-Stokes equation does not have a solution, it implies that under certain conditions, the equations may break down, leading to undefined or non-physical results. This could mean that the fluid's behavior cannot be predicted accurately, which poses significant challenges for both theoretical and applied fluid dynamics.

What are the main challenges in proving the existence and smoothness of solutions to the Navier-Stokes equation?

The main challenges include dealing with the nonlinear nature of the equation, the complexity of fluid behavior at different scales, and the potential for singularities where quantities like velocity or pressure become infinite. These factors make it extremely difficult to prove whether solutions always exist and remain smooth over time.

Has any progress been made in solving the Navier-Stokes existence and smoothness problem?

Yes, there has been substantial progress in understanding specific cases and developing numerical methods to approximate solutions. However, a general proof for the existence and smoothness of solutions for all conditions remains one of the most significant unsolved problems in mathematics and is one of the seven Millennium Prize Problems, with a reward of one million dollars for a correct proof.

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