Navier–Stokes existence and smoothness

From Wikipedia, the free encyclopedia

 This article needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (April 2008)

The Navier–Stokes equations are one of the pillars of fluid mechanics. These equations describe the motion of a fluid (that is, a liquid or a gas) in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. 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.

Even much more basic properties of the solutions to Navier–Stokes have never been proved. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist, or that if they do exist they have bounded kinetic energy. This is called the Navier–Stokes existence and smoothness problem.

Since understanding the Navier–Stokes equations is considered to be the first step for understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute offered in May 2000 a $1,000,000 prize, not to whomever constructs a theory of turbulence but (more modestly) to the first person providing a hint on the phenomenon of turbulence. In that spirit of ideas, the Clay Institute set a concrete mathematical problem:[1]

Prove or give a counter-example of the following statement:

In three dimensions, given an initial velocity there exist a velocity vector and a scalar pressure, which are smooth and globally defined, that solve the Navier–Stokes equations.

Millennium Prize Problems
P versus NP
The Hodge conjecture
The Poincaré conjecture
The Riemann hypothesis
Yang–Mills existence and mass gap
Navier-Stokes existence and smoothness
The Birch and Swinnerton-Dyer conjecture

Contents

[edit] The Navier–Stokes equations

For mathematics, it is systems of nonlinear differential equations for abstract vector fields of any size. For physics, it is equations that can describe liquids or not rare gases or continuous media, but without effects of capillarity. It is the second law of Newton, where add pressure, gravity, viscous forces and any external force in the volume of gas more then 1 micrometer cubic. Since the setting of the problem proposed by the Clay Mathematics Institute is in three dimensions we will consider only that case.

Let \mathbf{v}(x,t) be is a 3-dimensional vector the velocity of the fluid, and let p(x,t) be the pressure of the fluid. The Navier–Stokes equations are:

\frac{\partial \mathbf{v}}{\partial t} + ( \mathbf{v}\cdot\nabla ) \mathbf{v} = -\nabla p + \nu\triangle \mathbf{v} +\mathbf{f}(x,t)

where ν > 0 is the kinematic viscosity, \mathbf{f}(x,t) the external force, \nabla is the gradient operator and \triangle is the Laplacian operator, which is also denoted by \nabla^2. Note that this is a vector equation, i.e. it has three scalar equations. If we write down the coordinates of the velocity and the external force

\mathbf{v}(x,t)=(\,v_1(x,t),\,v_2(x,t),\,v_3(x,t)\,)\,,\qquad \mathbf{f}(x,t)=(\,f_1(x,t),\,f_2(x,t),\,f_3(x,t)\,)

then for each i = 1,2,3 we have the corresponding scalar Navier–Stokes equation:

\frac{\partial v_i}{\partial t} +\sum_{j=1}^{3}v_j\frac{\partial v_i}{\partial x_j}= -\frac{\partial p}{\partial x_i} + \nu\sum_{j=1}^{3}\frac{\partial^2 v_i}{\partial x_j^2} +f_i(x,t)

The unknowns are the velocity \mathbf{v}(x,t) and the pressure p(x,t). Since in three dimensions we have three equations and four unknowns (three scalar velocities and the pressure), we need a supplementary equation. This extra equation is the continuity equation describing the conservation of mass:

\nabla\cdot \mathbf{v} = 0

Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of "divergence-free" functions. And temperature of continuous medium, density, and viscosity are constants.

[edit] Assumptions on the Navier–Stokes equations

The Navier–Stokes equations are valid under the following assumptions:

1. Conservation of mass, momentum and energy.
2. Newton's second law (F = ma) holds.
3. The fluid is a newtonian fluid: the viscosity can be considered as a constant, and the fluid is isotropic (its properties are the same in all directions).
4. The fluid is incompressible (it cannot be compressed or dilated under external forces).

Assumptions 1 and 2 are the basis of Classical Mechanics, and as such they are standard in any physical problem within the "human scale", which means that we are between the atomic scales (where Quantum mechanics is a more suitable framework) and the large scales of objects such as planets and stars (where General relativity is the adequate theory). Assumption 3, however, is an idealization and does not hold for fluids with complex rheological properties (see Rheology). Assumption 4 is valid for flows with a sufficiently low Mach Number (< 0.3, as a rule of thumb).

There are two different settings for the one-million-dollar-prize Navier–Stokes existence and smoothness problem. The original problem is in the whole space \mathbb{R}^3, which needs extra conditions on the growth behavior of the initial condition and the solutions. In order to rule out the problems at infinity, the Navier–Stokes equations can be set in a periodic framework, which implies that we are no longer working on the whole space \mathbb{R}^3 but in the 3-dimensional torus \mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3. We will treat each case separately.

[edit] Statement of the problem in the whole space

[edit] Hypotheses and growth conditions

The initial condition \mathbf{v}_0(x) is assumed to be a smooth and divergence-free function (see smooth function) such that, for every multi-index α (see multi-index notation) and any K > 0, there exists a constant C = C(α,K) > 0(i.e. this "constant" depends on from alfa and K) such that

\vert \partial^\alpha \mathbf{v_0}(x)\vert\le \frac{C}{(1+\vert x\vert)^K}\qquad for all \qquad x\in\mathbb{R}^3.

The external force \mathbf{f}(x,t) is assumed to be a smooth function as well, and satisfies a very analogous inequality (now the multi-index includes time derivatives as well):

\vert \partial^\alpha \mathbf{f}(x)\vert\le \frac{C}{(1+\vert x\vert + t)^K}\qquad for all \qquad (x,t)\in\mathbb{R}^3\times[0,\infty).

For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as \vert x\vert\to\infty. More precisely, the following assumptions are made:

1. \mathbf{v}(x,t)\in\left[C^\infty(\mathbb{R}^3\times[0,\infty))\right]^3\,,\qquad p(x,t)\in C^\infty(\mathbb{R}^3\times[0,\infty))

2. There exists a constant E\in (0,\infty) such that \int_{\mathbb{R}^3} \vert \mathbf{v}(x,t)\vert dx <E for all t\ge 0\,.

Condition 1 implies that the functions are smooth and globally defined and condition 2 means that the kinetic energy of the solution is globally bounded.

[edit] The million-dollar-prize theorems in the whole space

(A) Existence and smoothness of the Navier–Stokes solutions in \mathbb{R}^3

Let \mathbf{f}(x,t)\equiv 0. For any initial condition \mathbf{v}_0(x) satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector \mathbf{v}(x,t) and a pressure p(x,t) satisfying conditions 1 and 2 above.

(B) Breakdown of the Navier–Stokes solutions in \mathbb{R}^3

There exists an initial condition \mathbf{v}_0(x) and an external force \mathbf{f}(x,t) such that there exists no solutions \mathbf{v}(x,t) and p(x,t) satisfying conditions 1 and 2 above.

[edit] Statement of the periodic problem

[edit] Hypotheses

The functions we seek now are periodic in the space variables of period 1. More precisely, let ei be the unitary vector in the j- direction:

e_1=(1,0,0)\,,\qquad e_2=(0,1,0)\,,\qquad e_3=(0,0,1)

Then \mathbf{v}(x,t) is periodic in the space variables if for any i = 1,2,3 we have that

\mathbf{v}(x+e_i,t)=\mathbf{v}(x,t)\text{ for all } (x,t) \in \mathbb{R}^3\times[0,\infty).

Notice that we are considering the coordinates modulo 1. This allows us to work not on the whole space \mathbb{R}^3 but on the quotient space \mathbb{R}^3/\mathbb{Z}^3, which turns out to be the 3-dimensional torus

\mathbb{T}^3=\{(\theta_1,\theta_2,\theta_3): 0\le \theta_i<2\pi\,,\quad i=1,2,3\}.

We can now state the hypotheses properly. The initial condition \mathbf{v}_0(x) is assumed to be a smooth and divergence-free function and the external force \mathbf{f}(x,t) is assumed to be a smooth function as well. The type of solutions that are physically relevant are those who satisfy these conditions:

3. \mathbf{v}(x,t)\in\left[C^\infty(\mathbb{T}^3\times[0,\infty))\right]^3\,,\qquad p(x,t)\in C^\infty(\mathbb{T}^3\times[0,\infty))

4. There exists a constant E\in (0,\infty) such that \int_{\mathbb{T}^3} \vert \mathbf{v}(x,t)\vert dx <E for all t\ge 0\,.

Just as in the previous case, condition 3 implies that the functions are smooth and globally defined and condition 4 means that the kinetic energy of the solution is globally bounded.

[edit] The periodic million-dollar-prize theorems

(C) Existence and smoothness of the Navier–Stokes solutions in \mathbb{T}^3

Let \mathbf{f}(x,t)\equiv 0. For any initial condition \mathbf{v}_0(x)satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector \mathbf{v}(x,t) and a pressure p(x,t) satisfying conditions 3 and 4 above.

(D) Breakdown of the Navier–Stokes solutions in \mathbb{T}^3

There exists an initial condition \mathbf{v}_0(x) and an external force \mathbf{f}(x,t) such that there exists no solutions \mathbf{v}(x,t) and p(x,t) satisfying conditions 3 and 4 above.

[edit] Partial results

  1. The Navier–Stokes problem in two dimension has already been solved positively since the 60's: there exist smooth and globally defined solutions. [2]
  2. If the initial velocity \mathbf{v}(x,t) is sufficiently small then the statement is true: there are smooth and globally defined solutions to the Navier–Stokes equations.[citation needed]
  3. Given an initial velocity \mathbf{v}_0(x) there exists a finite time T, depending on \mathbf{v}_0(x) such that the Navier–Stokes equations on \mathbb{R}^3\times(0,T) have smooth solutions \mathbf{v}(x,t) and p(x,t). It is not known if the solutions exist beyond that "blowup time" T.[citation needed]
  4. The mathematician Jean Leray in 1934 proved the existence of so called weak solutions to the Navier–Stokes equations, satisfying the equations in mean value, not pointwise.[3]

[edit] References

  1. ^ Official statement of the problem, Clay Mathematics Institute.
  2. ^ O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flows", 2nd edition), Gordon and Breach, 1969.
  3. ^ Leray, J. (1934), “Sur le mouvement d'un liquide visqueux emplissant l'espace”, Acta Mathematica 63: 193–248, DOI 10.1007/BF02547354 

[edit] External links

v • d • e
General subfields within physics