Is the N-S equation problem solved? Juxtaposed with the Riemann Hypothesis, the Millennium Mathematical Puzzle is in sight for victory

This is one of the most famous unsolved problems in mathematics. The new work has been peer-reviewed and the full text is available.

#It’s so exciting, is fluid mechanics ushering in its own superconducting moment?

In recent days, people in mathematics circles have been heatedly discussing that the regular Hamiltonian formula for the Navier-Stokes problem has finally appeared - this unsolved problem in the history of mathematics There may be an answer. In the past, this was even generally considered impossible.

How important is this? The Navier-Stokes equation, like the Riemann Hypothesis, was listed as one of the "Seven Millennium Mathematics Problems" in 2000.

The seven world-class problems are: NP-complete problem, Hodge conjecture, Poincaré conjecture, Riemann hypothesis, Yang-Mills existence and mass gap , Navier-Stokes equation, BSD conjecture. Each of the seven problems has a reward of one million US dollars. In more than 20 years, only the "Poincaré Conjecture" has been solved by the talented Russian mathematician Perelman.

Most of them are familiar, but the "Navier-Stokes equation" (N-S equation) seems to be mentioned less often among them. The reason may be that this problem is too difficult to understand (students who have taken the "Fluid Mechanics" course in college will definitely have an idea). Some even believe that it is the most complex formula in the history of mathematics.

To put it simply, the eighteenth-century mathematician Euler derived the following in "General Principles of Fluid Motion" based on the changes in force and momentum experienced by the fluid when the inviscid fluid moves. A set of equations was produced.

The description of Euler's equation stipulates fluid motion in an idealized world, but there is friction inside the real fluid. Fluids in nature are viscous and are collectively called viscous fluids or real fluids. For example, when we stir honey, we will feel the viscosity effect, and the resistance of the aircraft flying is also largely derived from the viscosity of the air.

Due to the viscosity of actual fluids, our study of fluid motion becomes very complicated.

In the 19th century, French engineer and physicist Claude-Louis Navit and Irish physicist and mathematician George Stokes considered molecules The basic equations of fluid balance and motion are established, and the component forms of motion in rectangular coordinates are described.

This is what later generations called the Navier-Stokes equation. One of the scariest partial differential equations of all time.

^{The Navier-Stokes equations are used to describe fluid substances like liquids and air. These equations relate the rate at which a fluid's particle momentum changes (the force) to the changes in pressure and dissipative viscous forces (analogous to friction) and gravity acting on the interior of the fluid. These viscous forces arise from the interactions of molecules and tell us how viscous a liquid is. In this way, the Navier-Stokes equations describe the dynamic balance of forces acting on any given region of a liquid.}

This is critical for many engineering problems.

If there is a global solution to the Navier-Stokes problem, there will be breakthroughs in many technologies related to fluid mechanics, including but not limited to aerospace, rocket engines, Weather forecasting, pipeline transportation, medical blood flow modeling, and more.

The difficult problem involved with this set of equations is how to explain it using mathematical theory. Even the mathematical theory that explains Einstein's field equations describing exotic black holes is simpler than formulating the Navier-Stokes equations.

The important breakthrough people mentioned comes from the paper "A canonical Hamiltonian formulation of the Navier–Stokes problem", which was published in the top journal in the field of fluid mechanics on April 1 Journal of Fluid Mechanics":

^{Paper link: https://www.cambridge.org/core/journals/journal-of-fluid -mechanics/article/canonical-hamiltonian-formulation-of-the-navierstokes-problem/B3EB9389AE700867A6A3EA63A45E69C6}

This paper proposes a method based on the least squares principle Derivation of a novel Hamiltonian formulation of the isotropic Navier-Stokes problem based on the minimum action principle. The formula uses velocity and pressure as variable field quantities, as well as the canonical conjugate momentum derived from analysis. Based on this, this study constructs a conserved Hamiltonian function H* that satisfies the Hamiltonian canonical equation, and formulates the related Hamiltonian-Jacobian equations for compressible and incompressible flows. This Hamiltonian-Jacobian equation reduces the problem of finding four independent field quantities to finding a single scalar functional among these fields - Hamilton's main functional. Furthermore, Hamilton and Jacobi's transformation theory provides a prescribed method for solving the Navier-Stokes problem: find S*.

If the analytical expression of S * can be obtained, then it will obtain a set of new fields through regular transformation, giving the analytical expressions of the original velocity and pressure fields, These fields will simply be equivalent to their initial values. Failing that, one can only prove that a complete solution to the Hamilton-Jacobian equation exists or does not exist, which would also solve the problem of the existence of the solution.

Could this new research lead to a million-dollar prize? To win, researchers must show that there are solutions to the three-dimensional incompressible Navier-Stokes equations and that, if there are solutions, those solutions are smooth.

Mathematician Terence Tao once thought that this was difficult.

Judging from the current progress, new research has made it easier to solve open problems, and we have taken a big step forward - Navier-Stor The canonical Hamiltonian formulation of the Kers equation may mean that we can bypass the limitations of the standard Lagrangian and reduce the problem to finding a single scalar function.

Perhaps we are not far away from solving the second question of the Millennium Puzzle.

Reference content:

https://www1.grc. nasa.gov/beginners-guide-to-aeronautics/navier-strokes-equation/

https://zhuanlan.zhihu.com/p/263628141

https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

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