The existence of Naville-Stokes equation is very low, and even among the seven difficult problems in the world, it is rarely mentioned. The most important reason is that this problem is really not easy to understand, especially for ordinary people, even the top P/NP problem can be figured out by ordinary people, but the Navier-Stokes equation is difficult to understand, which is why civil science rarely touches on this problem.

This equation was not put forward by one person. 1775, the famous mathematician Euler, yes, is the four kings of mathematics. Now he has mixed fluid mechanics. In his book "General Principles of Fluid Motion", he deduced a set of equations according to the changes of force and momentum of fluid when inviscid fluid is moving.

The equation is as follows: (ax? d？ +bxD+c)y=f(x) (only one of them, and the differential expression of functional extreme conditions, etc. ) is the most important basic equation in inviscid fluid dynamics (ideal fluid mechanics), which refers to the differential equation of motion obtained by applying Newton's second law to inviscid fluid micelles and describes the motion law of ideal fluid. It laid the foundation of ideal fluid mechanics.

Viscous fluid refers to a fluid whose viscosity effect cannot be ignored. The actual fluid in nature is viscous, so the actual fluid is also called viscous fluid, which refers to the property that the active layer between fluid particles generates friction due to relative motion and resists relative motion.

182 1 year, Navid, a famous engineer, popularized Euler's fluid motion equation, and considered intermolecular forces, thus establishing the basic equations of fluid balance and motion. The equation contains only one viscosity constant.

Starting from the continuum model in 1845, Stokes improved his equations of motion of fluid mechanics, and obtained the rectangular coordinate component form of the equations of motion of viscous fluid with two viscosity constants, which was later called Naville-Stokes equation.

Naville-Stokes equation has many expressions.

Before explaining the details of the Naville-Stokes equation, we must first make several assumptions about the fluid. The first is that the fluid is continuous. This emphasizes that it does not involve the formation of internal voids, such as dissolved bubbles, and it does not involve the polymerization of atomized particles. Another necessary assumption is that all involved fields are differentiable, such as pressure p, velocity v, density ρ, temperature q and so on. This equation is derived from the basic principle of conservation of mass, momentum and energy.

In this regard, it is sometimes necessary to consider a finite arbitrary volume, called the control volume, and these principles can be easily applied to it. The finite volume is denoted as ω, and its surface is denoted as ω. The control volume can be fixed in space or can move with the fluid.

It can be said that Naville-Stokes equation is a set of equations describing liquid, air and other fluid substances, which was produced under the impetus of many scientists and engineers. These equations establish the relationship between the change rate (force) of particle momentum of fluid, the change of pressure acting inside the liquid, the dissipation of viscous force (similar to friction) and gravity. These viscous forces come from the interaction between molecules and can tell us how viscous the liquid is. Thus, the Naville-Stokes equation describes the dynamic balance of forces acting on any given liquid area.

In fluid mechanics, there are many equations, but many of them are related to Naville-Stokes equation. Navier-Stokes equation can describe most of the flow field. Of course, it also has its scope of application, which is only applicable to Newtonian fluids.

What is Newtonian fluid? Simply put, it is a fluid in which the shear stress at any point is linearly related to the shear deformation rate. Generally, high viscosity fluids do not satisfy this relationship, which shows that a simple example of Newtonian fluid and non-Newtonian fluid is the well-known siphon phenomenon. Under low viscosity, the suction port must be below the page, but under the high viscosity fluid of non-Newtonian fluid, even if the suction port is higher than the liquid level, siphon can still be carried out because of the high viscosity.

For engineering applications, it is still treated as Newtonian fluid in most cases, or it can be approximated as Newtonian fluid. It can be said that the equation plays a fundamental role in fluid mechanics, but it also plays a decisive role.

The difficult problem involved in this set of equations is how to clarify this set of equations with mathematical theory. Yes, it is even easier to explain Einstein's field equation used to describe strange black holes with mathematical theory than Naville-Stokes equation.

Therefore, the mathematical problems related to the mathematical properties of the solution of Naville-Stokes equation are called the existence and smoothness of the solution of Naville-Stokes equation.

Although Navier-Stokes equation can describe the motion of fluid (liquid or gas) in space. The solution of Naville-Stokes equation can be used in many practical fields. For example, it can be used to simulate weather, ocean currents, water flow in pipes, the movement of stars in galaxies and the airflow around airfoils. They can also be used in the design of aircraft and vehicles, the study of blood circulation, the design of power stations, the analysis of pollution effects and so on.

However, at present, the theoretical research on the solution of Navier-Stokes equation is still insufficient, especially the solution of Navier-Stokes equation often contains turbulence.

Turbulence, also known as turbulence, is a state of fluid flow. When the flow velocity is very small, the fluid flows in layers and does not mix with each other, which is called laminar flow or broken sugar; With the increase of flow velocity, the streamline of fluid begins to oscillate in waves, and the frequency and amplitude of oscillation increase with the increase of flow velocity. This phenomenon is called transitional flow. When the velocity increases to a great extent, the streamline is no longer clearly distinguishable, and many small vortices appear in the flow field, which are called turbulence, also known as turbulence, turbulence or turbulence. (Airplanes are most afraid of turbulence)

Although turbulence is very important in science and engineering, it is disorderly, energy-consuming and diffuse. It is still one of the unsolved physical problems.

In addition, the basic properties of many Naville-Stokes equations have not been proved. Because Naville-Stokes equation depends on differential equation to describe the motion of fluid. Different from algebraic equations, these equations do not seek to establish the relationship between the variables studied (such as velocity and pressure), but seek to establish the relationship between the rate of change or flux of these quantities. In mathematical terms, these rates of change correspond to the derivatives of variables. Among them, the Navier-Stokes equation of an ideal fluid with zero viscosity in the simplest case shows that acceleration (derivative of velocity, or rate of change) is directly proportional to the derivative of internal pressure.

This means that for a given physical problem, at least calculus can be used to solve the Naville-Stokes equation. In fact, only in the simplest case can the known solution be obtained by this method. These situations usually involve non-turbulence in steady state (the flow field does not change with time), in which the viscosity coefficient of the fluid is very large or its velocity is very small (low Reynolds number).

For more complicated cases, such as the lift of global weather systems or wings such as El Nino, the solution of Naville-Stokes equation must be obtained with the help of computers. This scientific field is called computational fluid dynamics.

For example, mathematicians have not proved whether the Navier-Stokes equation has a smooth solution under specific initial conditions in three-dimensional coordinates. It is not proved that if such a solution exists, its kinetic energy has upper and lower limits.

But the problem of Naville-Stokes equation in the Millennium is more difficult. Given an initial velocity field in three-dimensional space-time, there is a vector velocity field and a scalar pressure field, which is the solution of Navier-Stokes equation. The velocity field and pressure field must satisfy the characteristics of smoothness and global definition.

Note that the official expression of each of the seven Millennium mathematical problems in the world except P/NP problem is written by the winner of Fields Prize who has made achievements in this field or this problem, so as to ensure that the problems can be refined and summarized, thus ensuring the rigor of the problems. Because the P/NP problem involves computers, the official statement was written by Stephen Cook, a Turing Prize winner, and the Naville-Stokes equation exists and is smooth. An official statement written by Charles Fei Fuman.

If you can't understand it, it can be simply understood that scientists want to find the general solution of the Naville-Stokes equation, which means that the solution of the equation always exists. In other words, under any initial conditions, can this set of equations describe any fluid at any time in the future?

A group of equations that are difficult to explain by mathematical theory, you still need to prove that the solution of this equation always exists. This has caused many scientists to collapse.

At present, only about one hundred special solutions have been solved. 1934, the mathematician jean leray proved the existence of the weak solution of the so-called Naville-Stokes problem, which satisfies the Naville-Stokes problem on average, but not at every point. Since then, the research on Naville-Stokes problem has stagnated, so it is also called the most difficult mathematical or physical formula. It was not until 80 years later that Tao Zhexuan published an article about the Navier-Stokes problem, "Fine Time Blow-up of an Average Three-dimensional Navier-Stokes Equation". His main purpose was to formalize the supercritical obstacles of the global regularity of the Navier-Stokes equation. Roughly speaking, it is impossible to establish the global regularity of Navier-Stokes equation abstractly. Tao Zhexuan believes that it is not enough to only believe in abstract methods (functional analysis methods based on energy equality such as semigroups) and pure harmonic analysis, and special geometry of NS equation such as vorticity may be needed. This paper is to construct a counterexample similar to NS equation, not the original NS equation.

Imagine, he said, if someone is extremely smart and creates a machine with pure water. It is not composed of rods and gears, but of interacting water currents. Tao said, and drew a shape in the air with his hand like a magician. Imagine that this machine can copy a smaller and faster self, and then this smaller and faster self copies another self, and continue until it reaches infinite speed in a tiny space, thus causing an explosion. Tao said with a smile, he didn't really propose to build such a machine, it was just a thought experiment, just like Einstein's special theory of relativity. But Tao explained that if it can be mathematically proved that nothing can stop this wonderful device from running, it means that water will really explode. In this process, he will also solve the problems of existence and smoothness of Naville-Stokes equation.

In any case, in the process of solving the Naville-Stokes equation, countless new mathematical tools and methods were born, which led the continuous development of mathematics. This is the meaning of these puzzles.