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The Navier–Stokes existence and smoothness problem concerns the
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
properties of solutions to the
Navier–Stokes equations The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
, a system of
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s that describe the motion of a
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
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 In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
, which remains one of the greatest
unsolved problems in physics The following is a list of notable unsolved problems grouped into broad areas of physics. Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomeno ...
, despite its immense importance in science and engineering. Even more basic (and seemingly intuitive) properties of the solutions to Navier–Stokes have never been proven. 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. This is called the ''Navier–Stokes existence and smoothness'' problem. Since understanding the Navier–Stokes equations is considered to be the first step to understanding the elusive phenomenon of
turbulence In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
, the Clay Mathematics Institute in May 2000 made this problem one of its seven
Millennium Prize problems The Millennium Prize Problems are seven well-known complex mathematics, mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem ...
in mathematics. It offered a US$1,000,000 prize to the first person providing a solution for a specific statement of the problem:


The Navier–Stokes equations

In mathematics, the Navier–Stokes equations are a system of
nonlinear In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s for abstract vector fields of any size. 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 In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
is short enough so that it can be thought of as a continuum mean instead of a collection of particles) using
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mec ...
. The equations are a statement of
Newton's second law Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body re ...
, with the forces modeled according to those in a
viscous Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...
Newtonian fluid A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of cha ...
—as the sum of contributions by pressure, viscous stress and an external
body force In physics, a body force is a force that acts throughout the volume of a body.Springer site - Book 'Solid mechanics'preview paragraph 'Body forces'./ref> Forces due to gravity, electric fields and magnetic fields are examples of body forces. Bod ...
. Since the setting of the problem proposed by the Clay Mathematics Institute is in three dimensions, for an
incompressible Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...
and homogeneous fluid, only that case is considered below. Let \mathbf(\boldsymbol,t) be a 3-dimensional vector field, the velocity of the fluid, and let p(\boldsymbol,t) be the pressure of the fluid.More precisely, is the pressure divided by the fluid
density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
, and the density is constant for this incompressible and homogeneous fluid.
The Navier–Stokes equations are: : \frac + ( \mathbf\cdot\nabla ) \mathbf = -\frac\nabla p + \nu\Delta \mathbf +\mathbf(\boldsymbol,t) where \nu>0 is the
kinematic viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
, \mathbf(\boldsymbol,t) the external volumetric force, \nabla is the
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
operator and \displaystyle \Delta is the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
operator, which is also denoted by \nabla\cdot\nabla or \nabla^2. Note that this is a vector equation, i.e. it has three scalar equations. Writing down the coordinates of the velocity and the external force : \mathbf(\boldsymbol,t)=\big(\,v_1(\boldsymbol,t),\,v_2(\boldsymbol,t),\,v_3(\boldsymbol,t)\,\big)\,,\qquad \mathbf(\boldsymbol,t)=\big(\,f_1(\boldsymbol,t),\,f_2(\boldsymbol,t),\,f_3(\boldsymbol,t)\,\big) then for each i=1,2,3 there is the corresponding scalar Navier–Stokes equation: : \frac +\sum_^\fracv_j= -\frac\frac + \nu\sum_^\frac +f_i(\boldsymbol,t). The unknowns are the velocity \mathbf(\boldsymbol,t) and the pressure p(\boldsymbol,t). Since in three dimensions, there are three equations and four unknowns (three scalar velocities and the pressure), then a supplementary equation is needed. This extra equation is the
continuity equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity ...
for
incompressible Incompressible may refer to: * Incompressible flow, in fluid mechanics * incompressible vector field, in mathematics * Incompressible surface, in mathematics * Incompressible string, in computing {{Disambig ...
fluids that describes the
conservation of mass In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter the mass of the system must remain constant over time. The law implies that mass can neith ...
of the fluid: : \nabla\cdot \mathbf = 0. Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of solenoidal ("
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...
-free") functions. For this flow of a homogeneous medium, density and viscosity are constants. Since only its gradient appears, the pressure ''p'' can be eliminated by taking the
curl cURL (pronounced like "curl", ) is a free and open source computer program for transferring data to and from Internet servers. It can download a URL from a web server over HTTP, and supports a variety of other network protocols, URI scheme ...
of both sides of the Navier–Stokes equations. In this case the Navier–Stokes equations reduce to the vorticity-transport equations. The Navier–Stokes equations are
nonlinear In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
, meaning that the terms in the equations do not have a simple linear relationship with each other. 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. One way to understand the nonlinearity of the Navier–Stokes equations is to consider the term (\mathbf\cdot\nabla ) \mathbf in the equations. This term represents the acceleration of the fluid, and it is a product of the velocity vector v and the gradient operator ∇. Because the gradient operator is a linear operator, the term (v · ∇)v is nonlinear in the velocity vector v. This means that the acceleration of the fluid depends on the magnitude and direction of the velocity, as well as the spatial distribution of the velocity within the fluid. Another source of nonlinearity in the Navier–Stokes equations is the pressure term -\frac\nabla p. The pressure in a fluid depends on the density and the gradient of the pressure, and this term is therefore nonlinear in the pressure. To see this more explicitly, consider the case of a circular obstacle of radius R placed in a uniform flow with velocity \mathbf and density \rho. Let \mathbf(\mathbf,t) be the velocity of the fluid at position \mathbf and time t, and let p(\mathbf,t) be the pressure at the same position and time. The Navier–Stokes equations in this case are: : \frac + ( \mathbf\cdot\nabla ) \mathbf = -\frac\nabla p + \nu\Delta \mathbf : \nabla\cdot \mathbf = 0 where \nu is the kinematic viscosity of the fluid. Assuming that the flow is steady (meaning that the velocity and pressure do not vary with time), we can set the
time derivative A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t. Notation A variety of notations are used to denote th ...
terms equal to zero: : ( \mathbf\cdot\nabla ) \mathbf = -\frac\nabla p + \nu\Delta \mathbf : \nabla\cdot \mathbf = 0 We can now consider the flow near the circular obstacle. In this region, the velocity of the fluid will be higher than the uniform flow velocity \mathbf due to the presence of the obstacle. This results in a nonlinear term ( \mathbf\cdot\nabla ) \mathbf in the Navier–Stokes equations that is proportional to the velocity of the fluid. At the same time, the presence of the obstacle will also result in a pressure gradient, with higher pressure near the obstacle and lower pressure farther away. This can be seen by considering the continuity equation, which states that the mass flow rate through any surface must be constant. Since the velocity is higher near the obstacle, the mass flow rate through a surface near the obstacle will be higher than the mass flow rate through a surface farther away from the obstacle. This can be compensated for by a pressure gradient, with higher pressure near the obstacle and lower pressure farther away. 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 \mathbf(x,t) and a pressure field p(x,t). We can use a finite element method to solve the Navier–Stokes equation for the velocity field: \frac + u \frac + v \frac = -\frac \frac + \nu \left( \frac + \frac \right) + f_x(x,y,t) To do this, we divide the domain into a series of smaller elements, and represent the velocity field as: u(x,y,t) = \sum_^N U_i(t) \phi_i(x,y) where N is the number of elements, and \phi_i(x,y) are the shape functions associated with each element. Substituting this expression into the Navier–Stokes equation and applying the finite element method, we can derive a system of ordinary differential equations: \frac = -\frac \sum_^N \left( \frac \right) j\int \phi_j \frac d\Omega + \nu \sum_^N \int_ \left( \frac \right)\phi_j \frac d\Omega + \int f_x \phi_i d\Omega where \Omega is the domain, and the integrals are over the domain. This system of ordinary differential equations can be solved using techniques such as the finite element method or spectral methods. Here, we will use the
finite difference A finite difference is a mathematical expression of the form . Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly d ...
method. To do this, we can divide the time interval
_0, t_f The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> into a series of smaller time steps, and approximate the derivative at each time step using a finite difference formula: \frac \approx -\frac \sum_^N \left( \frac \right)j \int \phi_j \frac d\Omega + \nu \sum_^N \int_ \left( \frac \right)j \phi_j \frac d\Omega + \int f_x \phi_i d\Omega where \Delta t = t_ - t_i is the size of the time step, and U_i and t_i are the values of U_i and t at time step i. Using this approximation, we can iterate through the time steps and compute the value of U_i at each time step. For example, starting at time step i and using the approximation above, we can compute the value of U_i at time step i+1: U_ = U_i + \Delta t \cdot \left(-\frac \sum_^N \left( \frac \right)j \int \phi_j \frac d\Omega + \nu \sum_^N \int_ \left( \frac \right)_j \phi_j \frac d\Omega + \int_ f_x \phi_i d\Omega \right) This process can be repeated until we reach the final time step t_f. 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.


Two settings: unbounded and periodic space

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^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 they are no longer working on the whole space \mathbb^3 but in the 3-dimensional torus \mathbb^3=\mathbb^3/\mathbb^3. Each case will be treated separately.


Statement of the problem in the whole space


Hypotheses and growth conditions

The initial condition \mathbf_0(x) is assumed to be a smooth and divergence-free function (see
smooth function In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...
) such that, for every multi-index \alpha (see
multi-index notation Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices ...
) and any K>0, there exists a constant C=C(\alpha,K)>0 such that : \vert \partial^\alpha \mathbf(x)\vert\le \frac\qquad for all \qquad x\in\mathbb^3. The external force \mathbf(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(x,t)\vert\le \frac\qquad for all \qquad (x,t)\in\mathbb^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: # \mathbf(x,t)\in C^\infty(\mathbb^3\times[0,\infty)),\qquad p(x,t)\in C^\infty(\mathbb^3\times[0,\infty)) # There exists a constant E\in (0,\infty) such that \int_ \vert \mathbf(x,t)\vert^2 \, dx 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.


The Millennium Prize conjectures in the whole space

(A) Existence and smoothness of the Navier–Stokes solutions in \mathbb^3 Let \mathbf(x,t)\equiv 0. For any initial condition \mathbf_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(x,t) and a pressure p(x,t) satisfying conditions 1 and 2 above. (B) Breakdown of the Navier–Stokes solutions in \mathbb^3 There exists an initial condition \mathbf_0(x) and an external force \mathbf(x,t) such that there exists no solutions \mathbf(x,t) and p(x,t) satisfying conditions 1 and 2 above. The Millennium Prize conjectures are two mathematical problems that were chosen by the Clay Mathematics Institute as the most important unsolved problems in mathematics. The first conjecture, which is known as the "smoothness" conjecture, states that there should always exist smooth and globally defined solutions to the Navier–Stokes equations in three-dimensional space. The second conjecture, known as the "breakdown" conjecture, states that there should be at least one set of initial conditions and external forces for which there are no smooth solutions to the Navier–Stokes equations. The Navier–Stokes equations are a set of partial differential equations that describe the motion of fluids. They are given by: \frac + (\mathbf \cdot \nabla) \mathbf = -\frac \nabla p + \nu \nabla^2 \mathbf + \mathbf \nabla \cdot \mathbf = 0 where \mathbf(x,t) is the velocity field of the fluid, p(x,t) is the pressure, \rho is the density, \nu is the kinematic viscosity, and \mathbf(x,t) is an external force. The first equation is known as the momentum equation, and the second equation is known as the continuity equation. These equations are typically accompanied by boundary conditions, which describe the behavior of the fluid at the edges of the domain. For example, in the case of a fluid flowing through a pipe, the boundary conditions might specify that the velocity and pressure are fixed at the walls of the pipe. The Navier–Stokes equations are nonlinear and highly coupled, making them difficult to solve in general. In particular, the difficulty of solving these equations lies in the term (\mathbf \cdot \nabla) \mathbf, which represents the nonlinear advection of the velocity field by itself. 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. In addition to the mathematical challenges of solving the Navier–Stokes equations, there are also many practical challenges in applying these equations to real-world situations. For example, the Navier–Stokes equations are often used to model fluid flows that are turbulent, which means that the fluid is highly chaotic and unpredictable. 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. To solve the Navier–Stokes equations, we need to find a velocity field \mathbf(x,t) and a pressure field p(x,t) that satisfy the equations and the given boundary conditions. This can be done using a variety of numerical techniques, such as finite element methods, spectral methods, or finite difference methods. For example, consider the case of a two-dimensional fluid flow in a rectangular domain, with velocity and pressure fields \mathbf(x,t) and a pressure field p(x,t),respectively. The Navier–Stokes equations can be written as: :\frac + u \frac + v \frac = -\frac \frac + \nu \left( \frac + \frac \right) + f_x(x,y,t) :\frac + u \frac + v \frac = -\frac \frac + \nu \left( \frac + \frac \right) + f_y(x,y,t) :\frac + \frac = 0 :\frac + \frac = 0 where \rho is the density, \nu is the kinematic viscosity, and \mathbf(x,y,t) = (f_x(x,y,t),f_y(x,y,t)) is an external force. The boundary conditions might specify that the velocity is fixed at the walls of the domain, or that the pressure is fixed at certain points. The last identity occurs because the flow is solenoidal. To solve these equations numerically, we can divide the domain into a series of smaller elements, and solve the equations locally within each element. For example, using a finite element method, we might represent the velocity and pressure fields as: u(x,y,t) = \sum_^N U_i(t) \phi_i(x,y) v(x,y,t) = \sum_^N V_i(t) \phi_i(x,y) p(x,y,t) = \sum_^N P_i(t) \phi_i(x,y) where N is the number of elements, and \phi_i(x,y) are the shape functions associated with each element. Substituting these expressions into the Navier–Stokes equations and applying the finite element method, we can derive a system of ordinary differential equations


Statement of the periodic problem


Hypotheses

The functions sought now are periodic in the space variables of period 1. More precisely, let e_i be the unitary vector in the ''i''- direction: : e_1=(1,0,0)\,,\qquad e_2=(0,1,0)\,,\qquad e_3=(0,0,1) Then \mathbf(x,t) is periodic in the space variables if for any i=1,2,3, then: : \mathbf(x+e_i,t)=\mathbf(x,t)\text (x,t) \in \mathbb^3\times mod 1. This allows working not on the whole space \mathbb^3 but on the Quotient space (topology)">quotient space \mathbb^3/\mathbb^3, which turns out to be the 3-dimensional torus: : \mathbb^3=\. Now the hypotheses can be stated properly. The initial condition \mathbf_0(x) is assumed to be a smooth and divergence-free function and the external force \mathbf(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: 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.


The periodic Millennium Prize theorems

(C) Existence and smoothness of the Navier–Stokes solutions in \mathbb^3 Let \mathbf(x,t)\equiv 0. For any initial condition \mathbf_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(x,t) and a pressure p(x,t) satisfying conditions 3 and 4 above. (D) Breakdown of the Navier–Stokes solutions in \mathbb^3 There exists an initial condition \mathbf_0(x) and an external force \mathbf(x,t) such that there exists no solutions \mathbf(x,t) and p(x,t) satisfying conditions 3 and 4 above.


Partial results

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 \mathbf_0(x) is sufficiently small. He also proved the existence of so-called
weak solution In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some prec ...
s to the Navier–Stokes equations, which may not satisfy the equations pointwise but do satisfy them in mean value. In the 1960s, the finite difference method was proven to be convergent for the Navier–Stokes equations and the equations were numerically solved. It was also proven that there are smooth and globally defined solutions to the Navier–Stokes equations in 2 dimensions. It is known that given an initial velocity \mathbf_0(x) there exists a finite "blowup time" ''T'', depending on \mathbf_0(x), such that the Navier–Stokes equations on \mathbb^3\times(0,T) have smooth solutions \mathbf(x,t) and p(x,t). These solutions may, however, not hold for values of t beyond the blowup time. 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.


In popular culture

Unsolved problems have been used to indicate a rare mathematical talent in fiction. The Navier–Stokes problem features in ''The Mathematician's Shiva'' (2014), a book about a prestigious, deceased, fictional mathematician named Rachela Karnokovitch taking the proof to her grave in protest of academia. The movie '' Gifted'' (2017) referenced the Millennium Prize problems and dealt with the potential for a 7-year-old girl and her deceased mathematician mother for solving the Navier–Stokes problem.


See also

*
List of unsolved problems in mathematics Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, Mathematical analysis, analysis, combinatorics, Algebraic geometry, alge ...
*
List of unsolved problems in physics The following is a list of notable unsolved problems grouped into broad areas of physics. Some of the major unsolved problems in physics are theoretical, meaning that existing theories seem incapable of explaining a certain observed phenomenon ...


Notes


References


Further reading

*


External links

* Contributed by: Yakov Sinai
The Clay Mathematics Institute's Navier–Stokes equation prize

Why global regularity for Navier–Stokes is hard
— Possible routes to resolution are scrutinized by Terence Tao.
Navier–Stokes existence and smoothness (Millennium Prize Problem)
A lecture on the problem by Luis Caffarelli. * {{DEFAULTSORT:Navier-Stokes existence and smoothness Fluid dynamics Millennium Prize Problems Partial differential equations Unsolved problems in mathematics Unsolved problems in physics