Navier–Stokes existence and smoothness
In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.
For the three-dimensional system of equations, and given some initial conditions, mathematicians have neither proved that smooth solutions always exist, nor found any counter-examples.
Since understanding the Navier–Stokes equations is considered to be the first step to understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics.
In physics and engineering, they are a system of equations that model the motion of liquids or non-rarefied gases (in which the mean free path is short enough so that it can be thought of as a continuum mean instead of a collection of particles) using continuum mechanics.
Since the setting of the problem proposed by the Clay Mathematics Institute is in three dimensions, for an incompressible and homogeneous fluid, only that case is considered below.
Since only its gradient appears, the pressure p can be eliminated by taking the curl of both sides of the Navier–Stokes equations.
This means that the equations cannot be solved using traditional linear techniques, and more advanced methods must be used instead.
This nonlinearity allows the equations to describe a wide range of fluid dynamics phenomena, including the formation of shock waves and other complex flow patterns.
This term represents the acceleration of the fluid, and it is a product of the velocity vector v and the gradient operator ∇.
Assuming that the flow is steady (meaning that the velocity and pressure do not vary with time), we can set the time derivative terms equal to zero: We can now consider the flow near the circular obstacle.
This can be seen by considering the continuity equation, which states that the mass flow rate through any surface must be constant.
As a result of these nonlinear effects, the Navier–Stokes equations in this case become difficult to solve, and approximations or numerical methods must be used to find the velocity and pressure fields in the flow.
Consider the case of a two-dimensional fluid flow in a rectangular domain, with a velocity field
We can use a finite element method to solve the Navier–Stokes equation for the velocity field:
To do this, we divide the domain into a series of smaller elements, and represent the velocity field as:
There are many other approaches to solving ordinary differential equations, each with its own advantages and disadvantages.
The choice of approach depends on the specific equation being solved, and the desired accuracy and efficiency of the solution.
In order to rule out the problems at infinity, the Navier–Stokes equations can be set in a periodic framework, which implies that they are no longer working on the whole space
is assumed to be a smooth function as well, and satisfies a very analogous inequality (now the multi-index includes time derivatives as well): For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as
satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector
These equations are typically accompanied by boundary conditions, which describe the behavior of the fluid at the edges of the domain.
The Navier–Stokes equations are nonlinear and highly coupled, making them difficult to solve in general.
This term makes the Navier–Stokes equations highly sensitive to initial conditions, and it is the main reason why the Millennium Prize conjectures are so challenging.
Turbulence is a difficult phenomenon to model and understand, and it adds another layer of complexity to the problem of solving the Navier–Stokes equations.
For example, consider the case of a two-dimensional fluid flow in a rectangular domain, with velocity and pressure fields
For example, using a finite element method, we might represent the velocity and pressure fields as:
Substituting these expressions into the Navier–Stokes equations and applying the finite element method, we can derive a system of ordinary differential equations The functions sought now are periodic in the space variables of period 1.
satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector
In 1934, Jean Leray proved that there are smooth and globally defined solutions to the Navier–Stokes equations under the assumption that the initial velocity
[1] In 2016, Terence Tao published a paper titled "Finite time blowup for an averaged three-dimensional Navier–Stokes equation", in which he formalizes the idea of a "supercriticality barrier" for the global regularity problem for the true Navier–Stokes equations, and claims that his method of proof hints at a possible route to establishing blowup for the true equations.
