User:Oleg Schilling/Proposed/Navier-Stokes Equations: Mathematical Properties

• Prof. Roger Temam, Indiana University

• Prof. Ricardo Rosa, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Brasil

For a number of reasons, the Navier-Stokes equations play a central role in different areas of sciences. With some variations depending on the problem, they are a well-accepted model for various phenomena at various scales, from micro-fluid dynamics in the cells (see e.g. Nie, Chen, E, and Robbins (2004)), to classical fluid mechanics (e.g. flows around an airplane, ship, train or car, or flows in a combustion chamber; see Batchelor (1967)), to geophysical flows in the ocean or the atmosphere (see e.g. Pedlosky (1990)), or to the formation of galaxies (see e.g. Larson (1974)).

From a physical point of view, they are, to the best of our knowledge, the only equations in mathematical physics in which the nonlinear term does not come from modeling but is rather introduced by a mathematical reason since the nonlinear term corresponds to the expression of the acceleration in Eulerian coordinates.

This article will be limited to the incompressible Navier-Stokes equations since the compressible case, which is outside the field of expertise of the authors, raise themselves a vast variety of different theoretical questions including, but not limited to, the possible appearance of shocks (discontinuities) and vacuum, the issue of entropy conditions, and the behavior at small viscosity.

Finally, and in relation with the object of this article, these equations are challenging from the mathematical point of view and raise many different questions, being the object of many different studies, the most notorious one being stated as one of the Millennium Prize Problems of the Clay Foundation, namely to determine whether, in space dimension three, the incompressible Navier-Stokes equations are well-posed or not; see below and [1].

Within the framework of the incompressible Navier-Stokes equations, this article will focus on the current state of the theory of existence and uniqueness of solutions in relation with the \(L^2\) theory and the well-posedness problem, and we will address more briefly a few of the many other important recent or not recent issues. In summary we will address the following topics

• The \(L^2\) theory
• Theory in other functional spaces
• Attractors and turbulence
• Vanishing viscosity and relations with the Euler equations

The incompressible Navier-Stokes equations

In their most general form, the equations describing the motion of a fluid in its Eulerian representation are as follows:

\[ \begin{cases} \displaystyle \rho\left(\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot \mathbf{\nabla})\mathbf{u}\right) = \mathbf{\nabla} \cdot \boldsymbol{\sigma} + \mathbf{f}; \\ \displaystyle \frac{\partial \rho}{\partial t} + \mathbf{\nabla}(\rho\mathbf{u}) = 0. \end{cases} \]

All quantities are functions of the space and time variables \(\mathbf{x}=(x_1,x_2,x_3)\) and \(t\ .\) The fluid is expected to fill a region of the three-dimensional space \(\mathbb{R}^3\) which we denote by \(\Omega\ ,\) and which may or may not depend on time. The three-dimensional vector \(\mathbf{u}=\mathbf{u}(\mathbf{x},t)=(u_1,u_2,u_3)\) represents the velocity of an idealized portion of the fluid located at the point \(\mathbf{x}\in \Omega\) at time \(t\ .\) The term \(\mathbf{f}=(f_1,f_2,f_3)\) represents the density of volume forces (such as e.g. gravity forces), while the term \(\boldsymbol{\sigma}=(\sigma_{ij})_{i,j=1}^3\) is the so-called Cauchy stress tensor; the term \(\rho\) is the mass density.

The symbol \(\mathbf{\nabla}\) stands for a vector of partial derivatives \(\mathbf{\nabla} = (\partial/\partial x_1,\partial/\partial x_2, \partial/\partial x_3)\ ,\) so that \[ (\mathbf{u}\cdot\mathbf{\nabla})\mathbf{u} = \left(\sum_{j=1}^3u_j\frac{\partial u_i}{\partial x_j}\right)_{i=1,2,3}, \quad \mathbf{\nabla}\cdot \boldsymbol{\sigma} = \left(\sum_{j=1}^3\frac{\partial \sigma_{ij}}{\partial x_j}\right)_{i=1,2,3}. \]

The expression of the stress tensor in terms of the velocity of the fluid \(\mathbf{u}\) is related to the type of fluid considered, through the so-called constitutive laws of the fluid (stress-strain relations). The most common type of fluids are the Newtonian fluids, in which case the stress-strain relations have the form \[ \sigma_{ij} = \mu \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right) + \left( \lambda \mathbf{\nabla} \cdot \mathbf{u} - p\right)\delta_{ij}, \] where \(\delta_{ij}\) is the Kronecker delta (\(\delta_{ij}=1\) for \(i=j\) and \(0\) for \(i\neq j\)), \(\mu\) is the molecular viscosity and \(p=p(\mathbf{x},t)\) is called the pressure. This embodies both compressible and incompressible Newtonian flows. These equations need to be supplemented with the equation for the conservation of mass and the equation of state.

In the particular case of viscous homogeneous incompressible Newtonian flows, the density \(\rho\) is constant, the conservation of mass reduces to the equation \(\mathbf{\nabla} \cdot \mathbf{u}=0\ ,\) and the Navier-Stokes equations can be written as the closed system \[\tag{1} \begin{cases} \displaystyle \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot \mathbf{\nabla})\mathbf{u} + \mathbf{\nabla} p = \nu \Delta \mathbf{u} + \mathbf{f}, \\ \mathbf{\nabla} \cdot \mathbf{u} = 0. \end{cases} \]

The term \(\nu=\mu/\rho\) is the kinematic viscosity and the term \(p\) now has been renamed and stands for the kinematic pressure (pressure divided by the mass density), while \(\mathbf{f}\) has also been renamed to denote the density of external forces by unit of mass.

Notice that the stress tensor is a linear function of the velocity field and the only non-linear term comes from the advection term \((\mathbf{u}\cdot \mathbf{\nabla})\mathbf{u}\ .\) Here, \[ \gamma(\mathbf{x},t) = \frac{\partial \mathbf{u}(\mathbf{x},t)}{\partial t} + (\mathbf{u}(\mathbf{x},t)\cdot \mathbf{\nabla})\mathbf{u}(\mathbf{x},t) \] is, in the Eulerian representation of a flow, the acceleration of the particle of fluid at \(\mathbf{x}\) at time \(t\ .\)

By considering a typical unit of length \(L\) and a unit of time \(T\ ,\) the Navier-Stokes equations can be transformed into nondimensional form by the change of variables \[ \tilde{\mathbf{x}} = \frac{\mathbf{x}}{L}, \;\; \tilde t = \frac{t}{T}, \;\; \tilde{\mathbf{u}}(\tilde{\mathbf{x}},\tilde t) = \frac{T}{L}\mathbf{u}(L\tilde{\mathbf{x}},T\tilde t), \;\; \tilde p(\tilde{\mathbf{x}},\tilde t) = \frac{T^2}{L^2} p(L\tilde{\mathbf{x}},T\tilde t), \;\; \tilde{\mathbf{f}} = \frac{T^2}{L} \mathbf{f}(L\tilde{\mathbf{x}},T\tilde t). \] The equations then take the form \[\tag{2} \begin{cases} \displaystyle \frac{\partial \tilde{\mathbf{u}}}{\partial \tilde t} + (\tilde{\mathbf{u}}\cdot \mathbf{\nabla})\tilde{\mathbf{u}} + \mathbf{\nabla} \tilde p = \frac{1}{\text{Re}} \Delta \tilde{\mathbf{u}} + \tilde{\mathbf{f}}, \\ \mathbf{\nabla} \cdot \tilde{\mathbf{u}} = 0, \end{cases} \]

where \( \text{Re} = \frac{LU}{\nu} \) is the so-called Reynolds number of the flow (with \(U=L/T\)) and is a measure of the balance between the driving effect of the nonlinear advection term and the damping effect of the viscosity. In order to fix the ideas we will generally work with equation (#NSEclassicalformulation).

Different types of boundary conditions may be associated with equation (#NSEclassicalformulation). The three most common conditions are the following:

• The no-slip boundary condition, when the boundary of the fluid \(\partial \Omega\) is materialized and we require that the particle of the fluid at the boundary has the same velocity as the container, \(\mathbf{u}=\boldsymbol{\varphi}\) on \(\partial \Omega\ .\) For the sake of simplicity, we usually assume that the container is at rest, so that \(\boldsymbol{\varphi}=0\) and we obtain the no-slip (or Dirichlet) boundary condition \(\tag{3} \mathbf{u} = 0 \text{ on } \partial \Omega. \)

• The case of an open boundary for all or part of the boundary of the fluid. On that part we require (for kinematic reasons) that the normal velocity of the fluid \(\mathbf{u}\cdot\mathbf{n}\) is the same as that of the boundary; we also require the tangential stress tensor to be continuous. We will not describe this case in detail since it does not raise any specific mathematical difficulty and may even be simpler in some cases (e.g. at vanishing viscosity, see Masmoudi and Rousset (2010)).
• The space-periodic case, which is of interest for mathematical purposes and for certain studies on turbulence and for flows away from the boundary. In this case, we consider \(L_1, L_2, L_3 > 0\ ,\) call \(\Omega=(0,L_1)\times (0,L_2)\times (0,L_3)\) the period, and assume that

\[\tag{4} \mathbf{u} \text{ and } p \text{ are periodic with period } L_i \text{ in the direction } x_i, \;i=1,2,3. \]

The equations are usually also supplemented with an initial condition on the velocity, \[\tag{5} \mathbf{u}(\mathbf{x},0)= \mathbf{u}_0(\mathbf{x}), \]

where \(\mathbf{u}_0\) is a given velocity field. The pressure does not need an initial condition since it can actually be determined, at each time \(t\ ,\) as a functional of \(\mathbf{u}(\cdot,t)\ .\) In the language of geophysical fluid dynamics, this means that \(\mathbf{u}\) is a prognostic variable (for which we prescribe an initial value \(\mathbf{u}_0\)), while \(p=\mathcal{N}(\mathbf{u})\) is a diagnostic variable. The following observation is common in computational fluid dynamics: Taking the divergence of the first equation (#NSEclassicalformulation) and using the incompressibility condition \(\boldsymbol{\nabla}\cdot\mathbf{u}=0\ ,\) we find \[\tag{6} \Delta p = \boldsymbol{\nabla} \cdot \mathbf{f} - \boldsymbol{\nabla}\cdot((\mathbf{u}\cdot\boldsymbol{\nabla})\mathbf{u}) = \boldsymbol{\nabla}\cdot\mathbf{f} - \sum_{i,j=1}^3 \frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i}. \]

Furthermore, in the no-slip case equation (#noslipcondition) we take the scalar product of the first equation (#NSEclassicalformulation) with the outward unit normal \(\mathbf{n}=(n_1,n_2,n_3)\) on \(\partial \Omega\) and we find \[\tag{7} \frac{\partial p}{\partial \mathbf{n}} = \left(\mathbf{f} - \nu\Delta \mathbf{u}\right)\cdot\mathbf{n}. \]

The equations equation (#pequation)- equation (#pbc) form a Neumann problem, which allows to determine \(p\) as a function of \(\mathbf{u}\ .\) In the case of equation (#periodiccondition) we replace equation (#pbc) by the periodicity of \(p\ .\) In both cases, the necessary compatibility conditions are satisfied for suitable types of \(\mathbf{f}\ ,\) and \(p\) is defined up to an additive constant.

The linear part of the Navier-Stokes equations, without the advection term, is called the Stokes equations, and the associated stationary problem is called the Stokes problem: \[\tag{8} \begin{cases} - \nu \Delta \mathbf{u} + \boldsymbol{\nabla} p = \mathbf{f}, \\ \boldsymbol{\nabla} \cdot \mathbf{u} = 0. \end{cases} \]

The Stokes problem equation (#Stokesproblem) may be associated with any of the boundary conditions above.

Two-dimensional flows are also of interest. In two space dimensions, we consider flows of the form \(\mathbf{u}(\mathbf{x},t) = (u_1(x_1,x_2,t),u_2(x_1,x_2,t),0)\) on a cylindrical domain of the form \(\Omega=\Omega'\times \mathbb{R}\ ,\) where \(\Omega'\subset\mathbb{R}^2\) is called the section of the domain. In this case, the equations reduce to a system of equations for the two-dimensional vector field \(\mathbf{u}'(\mathbf{x}',t) = (u_1(x_1,x_2,t), u_2(x_1,x_2,t))\ ,\) with \(\mathbf{x}'=(x_1,x_2)\in \Omega'\ .\)

The Leray weak formulation and the \(L^2\) theory

The modern mathematical theory of the Navier-Stokes equations started with the pioneering works of J. Leray (1933, 1934a,b), with the introduction of the concept of weak formulation and the first fundamental results of global existence and local uniqueness of solutions. In the late 1930s J. Leray wrote also a pioneering work concerning the existence of stationary (time independent) solutions of equation (#NSEclassicalformulation) which lead to the classical Leray-Schauder fixed point theorem in topology. We will not address this question here and will focus on the time-dependent case.

The weak formulation consists in multiplying the Navier-Stokes equations with appropriate (smooth) test functions, integrating the result over the domain \(\Omega\ ,\) and using integration by parts to reduce the number of derivatives in the velocity field necessary for the equations to make sense.

Since we are considering the incompressible case it is natural to consider test functions which are smooth divergence-free vector fields on \(\Omega\) satisfying the corresponding boundary conditions. In the no-slip case, one considers the space of test functions \[ \mathcal{V} = \{ \mathbf{v} \in \mathcal{C}_{\text{c}}^\infty(\Omega;\mathbb{R}^3); \;\boldsymbol{\nabla} \cdot \mathbf{v} = 0 \}, \] where \(\mathcal{C}_{\text{c}}(\Omega;\mathbb{R}^3)\) denotes the space of infinitely-differentiable vector functions from \(\Omega\) to \(\mathbb{R}^3\) with compact support included in \(\Omega\ .\)

Multiplying the incompressible Navier-Stokes equations by such a test function \(\mathbf{v}\) and integrating the result over \(\Omega\) one arrives at the equation \[ \frac{d}{dt} \int_\Omega \mathbf{u}\cdot\mathbf{v} \;d\mathbf{x} + \int_\Omega ((\mathbf{u}\cdot\boldsymbol{\nabla})\mathbf{u})\cdot\mathbf{v} \;d\mathbf{x} + \int_\Omega (\boldsymbol{\nabla} p)\cdot\mathbf{v} \;d\mathbf{x} = \nu\int_\Omega (\Delta \mathbf{u})\cdot\mathbf{v}\;d\mathbf{x} + \int_\Omega \mathbf{f}\cdot\mathbf{v} \;d\mathbf{x}. \] Integrating by parts the terms with the pressure and the Laplacian, and using the boundary and divergence-free conditions on \(\mathbf{v}\ ,\) one arrives at \[\tag{9} \frac{d}{dt} \int_\Omega \mathbf{u}\cdot\mathbf{v} \;d\mathbf{x} + \int_\Omega ((\mathbf{u}\cdot\boldsymbol{\nabla})\mathbf{u})\cdot\mathbf{v} \;d\mathbf{x} = - \nu\int_\Omega (\mathbf{\nabla}\mathbf{u})\cdot(\boldsymbol{\nabla}\mathbf{v})\;d\mathbf{x} + \int_\Omega \mathbf{f}\cdot\mathbf{v} \;d\mathbf{x}. \]

Now, considering test functions in time of the form \(\varphi\in \mathcal{C}_{\text{c}}^\infty((0,T),\mathbb{R})\ ,\) for given \(T>0\ ,\) we may also multiply the equation above by \(\varphi\ ,\) integrate in time, and use integration by parts in the first term to obtain the equation

\[\tag{10} \begin{align} \int_0^T \int_\Omega (\mathbf{u}\cdot\mathbf{v})\varphi' \;d\mathbf{x} \;dt + \int_0^T\int_\Omega (((\mathbf{u}\cdot\boldsymbol{\nabla})\mathbf{u})\cdot\mathbf{v})\varphi \;d\mathbf{x} \;dt = - \nu \int_0^T \int_\Omega ((\boldsymbol{\nabla}\mathbf{u})\cdot(\boldsymbol{\nabla}\mathbf{v})) \varphi \;d\mathbf{x} \;dt + \int_0^T \int_\Omega (\mathbf{f}\cdot\mathbf{v})\varphi \;d\mathbf{x} \;dt, \end{align} \]

where \(\varphi'\) denotes the time derivative of \(\varphi\ .\)

The weak formulation consists in finding a divergence-free vector field \(\mathbf{u}=\mathbf{u}(\mathbf{x},t)\) satisfying the corresponding initial and boundary conditions and the equation above for every pair of test functions \(\mathbf{v}=\mathbf{v}(\mathbf{x})\) and \(\varphi=\varphi(t)\ .\)

This is one of the weak forms of the Navier-Stokes equations, as introduced by J. Leray. Note that the pressure \(p\) does not appear anymore in equation (#NSEweakformulation) (nor in (#differentialNSEweakformulation)).

The \(L^2\) framework

For the equation (#NSEweakformulation) to make sense one needs some kind of regularity on the function \(\mathbf{u}=\mathbf{u}(\mathbf{x},t)\ .\) This regularity amounts to assuming that the solution belongs to some functional space; it is "necessary" for each of the terms in equation (#differentialNSEweakformulation), equation (#NSEweakformulation) to make sense. For instance, one can further integrate by parts the terms involving the space-derivatives of \(\mathbf{u}\) and ask that the components \(u_i\) and the products \(u_iu_j\) are locally integrable. Many other more regular functional spaces have been considered and in fact a number of different theories have been developed depending on that. We shall detail here the natural \(L^2\) framework and mention later on other frameworks such as \(L^p\) and Besov spaces. We consider only the no-slip case for simplicity.

The \(L^2\) framework starts with two basic spaces obtained as the completion of the space \(\mathcal{V}\) of test functions with respect to the \(L^2(\Omega)^3\) and the \(H^1(\Omega)^3\) norms, and denoted by \(H\) and \(V\ ,\) respectively. In the no-slip case and if the domain \(\Omega\) is regular these spaces can be characterized as \[ H = \{ \mathbf{u}\in L^2(\Omega)^3; \;\boldsymbol{\nabla}\cdot\mathbf{u} = 0 \text{ in } \Omega, \;\mathbf{u}\cdot \mathbf{n} = 0 \text{ on } \partial\Omega \}, \] where \(\mathbf{n}\) denotes the outward unit normal to the boundary \(\partial \Omega\ ,\) and \[ V = \{ \mathbf{u}\in H_0^1(\Omega)^3; \;\boldsymbol{\nabla}\cdot\mathbf{u} = 0 \}. \] These are Hilbert spaces with inner product given respectively by \[ (\!(\mathbf{u},\mathbf{v})\!)_H = \int_\Omega \mathbf{u}(\mathbf{x})\cdot\mathbf{v}(\mathbf{x}) \;d\mathbf{x} = \sum_{j=1}^3 \int_\Omega u_j(\mathbf{x})v_j(\mathbf{x}) \;d\mathbf{x}, \] and \[ (\!(\mathbf{u},\mathbf{v})\!)_V = \sum_{i,j=1}^3 \int_\Omega \frac{\partial u_i}{\partial x_j}\frac{\partial v_i}{\partial x_j} \;d\mathbf{x}. \] and associated norms \(\|\mathbf{u}\|_H = \sqrt{(\!(\mathbf{u},\mathbf{u})\!)_H}\ ,\) \(\|\mathbf{u}\|_V = \sqrt{(\!(\mathbf{u},\mathbf{u})\!)_V}\ .\)

Notice that these are natural spaces since the square of the norm in \(H\) is directly related to the total kinetic energy of the flow (divided by \(2/\rho\)) and the square of the norm in \(V\) is the total enstrophy (idem), which can be interpreted as the total "energy of rotation".

In the context of the \(L^2\) theory, a weak solution on a time interval \((0,T)\ ,\) \(T>0\ ,\) is considered to be a velocity field \(\mathbf{u}\) which is (essentially) bounded from \((0,T)\) into \(H\) and square integrable from \((0,T)\) into \(V\ ,\) \(\mathbf{u}\in L^\infty(0,T;H)\cap L^2(0,T;V)\) and which satisfies the weak formulation equation (#NSEweakformulation). Strong solutions are more regular, being continuous from \([0,T]\) into \(V\ ,\) and satisfying equation (#NSEclassicalformulation) in a suitable sense.

Formally, assuming the solution is sufficiently regular, we may take the scalar product of the Navier-Stokes equations with the velocity field \( \mathbf{u}\) and integrate over the domain \(\Omega\) to obtain an equation for the total kinetic energy of the flow (up to the density factor), in the form

\[ \frac{1}{2}\frac{d}{dt} \int_\Omega |\mathbf{u}|^2 \;d\mathbf{x} + \nu \sum_{i,j=1}^3 \int_\Omega \left|\frac{\partial u_i}{\partial x_j}\right|^2 \;d\mathbf{x} = \int_\Omega \mathbf{f}\cdot\mathbf{u} \;d\mathbf{x}. \]

The nonlinear term vanishes due to the boundary and divergence-free conditions, reflecting the fact that this term is not responsible for the production of energy; its role is to redistribute the energy among the different scales of motion.

However, due to the lack of sufficient regularity of the weak solutions it is not known whether in general a weak solution satisfy some form of energy equation or even inequality. Nevertheless, it is possible to obtain the existence of weak solutions which satisfies in addition the following form of energy inequality in the distribution sense:

\[\tag{11} \frac{1}{2}\frac{d}{dt} \int_\Omega |\mathbf{u}|^2 \;d\mathbf{x} + \nu \sum_{i,j=1}^3 \int_\Omega \left|\frac{\partial u_i}{\partial x_j}\right|^2 \;d\mathbf{x} \leq \int_\Omega \mathbf{f}\cdot\mathbf{u} \;d\mathbf{x}. \]

It is also possible to prove the existence of weak solutions satisfying a form of the energy inequality which is local in space. Such weak solutions are called suitable weak solutions, and were introduced by Caffarelli, Kohn and Nirenberg (1982) (see the discussion below concerning the dimension of the singularity set).

Main results of the \(L^2\) theory (dimensions \(2\) and \(3\))

For the three-dimensional case, Leray (1934b) proved the global existence of weak solutions satisfying the energy inequality (#LerayHopfenergyinequality) and the local existence and uniqueness of strong solutions in the whole-space case \(\Omega=\mathbb{R}^3\ .\) The method used by Leray was by mollifying the nonlinear term, replacing the first \(\mathbf{u}\) in the inertial term \((\mathbf{u}\cdot\boldsymbol{\nabla})\mathbf{u}\) by a mollification of \(\mathbf{u}\ .\) Then, Hopf (1951), constructing approximate solutions by the Galerkin method, proved the global existence of weak solutions satisfying the energy inequality (#LerayHopfenergyinequality) for flows in a bounded domain in \(\mathbb{R}^3\) with no-slip boundary conditions.

As for the two-dimensional case, Leray (1933,1934a)) proved first the existence and uniqueness of regular solutions for all time in the whole space case, and then for some interval of time in the bounded domain case (bounded section for the cylinder). Then, Ladyzhenskaya (1958,1959) proved the existence and uniqueness of strong solutions on a bounded domain for all time and Lions and Prodi (1959) proved the uniqueness of weak solutions for all time.

The current status of the \(L^2\) theory for existence, uniqueness, regularity and continuous dependence on the data is essentially as follows:

• 3D case: Given a domain in \(\mathbb{R}^3\ ,\) a forcing term, and an initial condition, there exists at least one Leray-Hopf weak solution associated with these data and defined for all time, i.e. on the time interval \([0,\infty)\ .\) Moreover, upon further regularity on the initial condition (and some mild extra conditions on the forcing term), there exists a time \(T^*>0\ ,\) depending on the data, such that there exists a unique strong solution of the Navier-Stokes equations on the interval \((0,T^*)\ .\) The strong solution also depends continuously on the data, as long as the necessary regularity on the data holds. Furthermore, higher order regularity of the strong solution depends on further regularity on the data (see e.g. Temam (1983)).
• 2D case: In this case, the theory is more complete, with existence and uniqueness of weak solutions for all time, with the solution depending continuously on the data, and, moreover, with the weak solution being a strong solution provided the data is sufficiently regular, and as regular as the data allow.

In the three-dimensional case, the global existence or non-existence of a unique solution strong is still an open problem and has been recognized as a major open problem as illustrated by the fact that it is one of the seven Millennium Prize Problems of the Clay Foundation [2]. A few partial results exist, such as global existence of strong solutions for small data or for highly oscillating data, and eventual regularity in the absence of a forcing term (i.e. any weak solution on the interval \([0,\infty)\) becomes a strong solution on an interval \([T,\infty)\) for \(T\) sufficiently large) (see Leray (1933)).

The results mentioned above are now classical results of the \(L^2\) theory of the Navier-Stokes equations and can be found in many books, such as Ladyzhenskaya (1963), J.-L. Lions (1969), Temam (1974, 1983), Constantin and Foias (1988), P.-L. Lions (1996), Sohr (2001), and von Wahl (1985).

Other theories and results

Existence, uniqueness, and regularity results in other functional spaces

The Navier-Stokes equations have been considered in a number of functional spaces other than \(L^2\ .\) An important earlier result based on \(L^p\) spaces is a condition for interior regularity and uniqueness known as Serrin's condition, in which a Leray-Hopf weak solution is a smooth, strong solution and is uniquely determined by the initial condition provided it belongs to the space \(L^s(0,T;L^q(\Omega)^3)\ ,\) for some \(3/q+2/s\leq 1\ ,\) with \(3\leq q \leq \infty\ ,\) \(2\leq s \leq \infty\ .\) The uniqueness result was proved by Serrin (1963) in the case \(3<q\leq \infty\ ,\) while the regularity was proved by Serrin (1962) under the non-strict condition \(3/q+2/s <1\ .\) The regularity condition was subsequently extended to the non-strict inequality \(3/q+2/s\leq 1\) and the uniqueness and regularity conditions were extended to some particular critical cases (such as \(q=3\ ,\) \(s=\infty\ ,\) assuming e.g. continuity in time with values in \(L^3(\Omega)\)) by Fabes, Jones, and Riviere (1972), Masuda (1984), Sohr and von Wahl (1984), von Wahl (1986), Giga (1986), P.-L. Lions and Masmoudi (1998), and Furioli, Lemarié-Rieusset, and Terraneo (2000); the latter one being credited as the first work to prove uniqueness of mild solutions in \(\mathcal{C}([0,T], L^3(\mathbb{R}^3)\) without any additional hypothesis. The full critical case \(L^\infty(0,T;L^3(\Omega)^3)\) was settled by Kozono and Sohr (1996), in regards to uniqueness, and by Escauriaza, Seregin, and Šverák (2003), in regards to regularity.

Many of these extensions on Serrin's results were given through the concept of mild solution. The definition and existence of mild solutions in the context of the Navier-Stokes equations, akin to that of the classical theory of linear semigroup theory, were initially given by Fujita and Kato (1964), in \(L^2\)-based spaces, obtaining local solutions in \(\mathcal{C}([0,T];H^{1/2}(\Omega)^3)\ ,\) for divergence-free initial conditions in \(H^{1/2}(\Omega)^3\ .\) A mild solution satisfies an integral equation related to the variation of constants formula, of the form \[ \mathbf{u}(t) = e^{-t A}\mathbf{u}_0 + \int_0^t e^{-t A} G(\mathbf{u}(s)) \;ds, \] where \(A\) is the Stokes operator, i.e. the linear operator associated with the Stokes problem (the stationary linear part of the Navier-Stokes equations) and \(G(\mathbf{u})\) is associated with the nonlinear term. One looks for a solution by solving a fixed point problem, in a suitable space, on a short interval of time.

This notion of mild solution was then adapted to address divergence-free initial conditions in \(L^3(\Omega)^3\) by Weissler (1980), in the half-space case \(\Omega=\mathbb{R}^3_+\ ,\) local in time, and by Kato (1984), in the whole space \(\Omega=\mathbb{R}^3\ ,\) with also global existence, uniqueness and decay results for small data. This notion was subsequently exploited to reduce even further the regularity required on the initial condition for the existence and uniqueness of a solution, as given, for instance, by Cannone (1995) and Planchon (1996), with initial conditions in the Besov space \(\dot B_q^{-1+3/q),\infty}\ ,\) with \(3<q<\infty\ ,\) among other spaces; Meyer (1997), in the Morey-Campanato space \(M^2(\mathbb{R}^3)\) and in some homogeneous Besov spaces; Furioli, Lemarié-Rieusset, and Terraneo (2000) (see also the book Lemarié-Rieusset (2002)), in the Besov space \(\dot B_2^{1/2,\infty}\ ;\) Koch and Tataru (2001), in the space \(\text{BMO}^{-1}\) of bounded mean oscillations; and Bourgain and Pavlović (2010), in the Besov space \(\dot B_\infty^{-1,\infty}\ .\) Many of these spaces do not compare with each other, but the Besov space in the latter result, namely \(\dot B_\infty^{-1,\infty}\ ,\) contains all the previous ones. Continuous dependence on the initial condition has been obtained in many of these spaces, however Bourgain and Pavlović (2008) showed that the set of mild solutions do not depend continuously on the initial data in the space \(\dot B_\infty^{-1,\infty}\ .\)

Another issue of great interest is the study of the singularity set of weak solutions, which is the set of points in space-time in which the velocity field is singular in the sense of not being essentially bounded on any space-time neighborhood of the point. Outside the singularity set, the velocity field is smooth. One is interested in estimating how large or small is the singularity set, and one way of doing this is through estimating its fractal or Hausdorff dimension. Scheffer (1980) proved that the singularity set has Hausdorff dimension less than \(5/3\ .\) The currently sharpest result is due to Caffarelli, Kohn and Nirenberg (1982), who introduced the notion of suitable weak solution, which is essentially a weak solution of the Navier-Stokes equations which satisfies a local (in space and time) version of the energy inequality. They proved the existence of suitable weak solutions for all times and proved that the (parabolic) Hausdorff dimension of the singular set of a suitable weak solution is less than or equal to \(1\ .\) Scheffer (1976) also studied the set of singular times, which can be defined as the set of times for which the velocity field at those times does not belong to the space \(V\ ,\) and proved that this set has Hausdorff dimension at most \(1/2\ .\)

Attractors and turbulence

Another issue of major theoretical and practical interest is the understanding of the dynamics of the flow of a fluid. A fundamental mathematical object related, in particular, to the long-time dynamics and the asymptotic behavior of the solutions is the global attractor. Global attractors may be defined for most autonomous dissipative evolution equations and, in particular, for the two-dimensional incompressible Navier-Stokes equations, and, in a slightly modified sense, also to the three-dimensional equations. In essence, a global attractor is the smallest subset of the phase space of the system which attracts all the solutions of the system, uniformly with respect to bounded sets of initial conditions. Extensions of this notion to non-autonomous equations give rise to the so-called uniform attractors and pull-back attractors (Chepyzhov and Vishik (2002), Caraballo, Lukaszewicz, and Real (2006)).

Besides investigating the existence of such an object, many works have been devoted to proving that the global attractor has finite dimension and to estimating its dimension from above and below. The dimension of the attractor is somehow related to the notion of degrees of freedom of the motion, and one of the fundamental achievements has been to proving upper bound estimates which agree, and hence corroborate, the estimates of the number of degrees of freedom of a turbulent flow obtained heuristically in the conventional theory of turbulence.

The existence of the global attractor for the dynamical system generated by the autonomous two-dimensional Navier-Stokes equations appears in works of Ladyzhenskaya (1973,1975). The finite-dimensionality, in terms of Hausdorff and fractal dimensions, of this global attractor has been proved by Foias and Temam (1979). In this latter work, the authors also proved that, in the three-dimensional case, the subsets \(X\) of the phase space \(H\) which are bounded in the norm of \(V\) and invariant for the flow (i.e. such that any weak solution \(\mathbf{u}\) on the time interval \([0,\infty)\) with \(\mathbf{u}(0)\in X\) is such that \(\mathbf{u}(t)\in X\) for all \(t\geq 0\)) have finite Hausdorff and fractal dimensions.

An important characteristic non-dimensional quantity associated somehow with the strength of the forcing term is the Grashof number. This number may be defined in a number of ways, depending on the geometry of the domain and on the regularity of the driving term. In the two-dimensional case with a time-independent forcing term \(\mathbf{f}\) belonging to the space \(H\ ,\) one such form is given by \[ \text{Gr}=\frac{L^{3/2}}{\nu^2} \left(\int_\Omega |\mathbf{f}(\mathbf{x})| \;d\mathbf{x}\right)^{1/2}, \] where \(L\) is a suitable characteristic length associated with the spatial domain \(\Omega\) occupied by the fluid. The upper bound originally obtained by Foias and Temam (1979) for the dimension of the global attractor in the two-dimensional case was exponential in \(\text{Gr}\ .\) One expects that the higher the Grashof number, the more complicated the flow, and, hence, the higher the dimension of the global attractor. Although that appears to be so in general situations, Marchioro (1986) has showed an example where the attractor reduces to a single globally stable fixed point (and hence with dimension zero) for a family of forcing terms with arbitrarily large Grashof numbers.

A number of subsequent results aimed towards improving the upper bounds for the dimension of the attractor down to a polynomial dependence of lower and lower degree on \(\text{Gr}\ .\) Then, Temam (1986) obtained, for the no-slip case, an upper bound for the dimension of the attractor of the order of \(\text{Gr}\ .\) For the space-periodic case, Constantin, Foias, and Temam (1988) obtained an upper bound of the order of \(\text{Gr}^{2/3}(1+\ln\text{Gr})^{1/3}\ .\) This later estimate is nearly optimal, in view of a result by Liu (1994) saying that, in that case, for any given Grashof number \(\text{Gr}\ ,\) there exist forcing terms such that the dimension of the global attractor is bounded from below by \(\text{Gr}^{2/3}\ .\)

The first result yielding a lower bound for the dimension of the attractor was given earlier by Babin and Vishik (1985,1986), in the space-periodic case in an elongated domain of the form \((0,1)\times (0,L)\) and in which the forcing term had a specific form, associated with a uniform pressure gradient along the channel. The authors showed that the dimension of the global attractor in this case is bounded from below by a quantity proportional to \(L\ .\) Then, Ziane (1997), using the general framework introduced by Constantin, Foias, and Temam (1985), proved that in this geometry the dimension of the attractor is also bounded from above by a quantity proportional to \(L\ ,\) showing that at least in this geometry the theory of Constantin, Foias, and Temam (1985) is sharp.

For the three-dimensional periodic case, an upper-bound for the dimension of an invariant set \(X\ \) bounded in \(V\ \) in the sense described earlier in this section was given by Constantin, Foias, and Temam (1985) to be of the order of \((L/L_\epsilon)^3\ ,\) where \(L\) is the period in space and \(L_\epsilon\) is a quantity related to the Kolmogorov dissipation length. More precisely, \(L_\epsilon=(\nu^3/\epsilon)^{1/4}\ ,\) where \[ \epsilon = \nu \limsup_{t\rightarrow \infty} \sup_{\mathbf{u}_0\in X} \frac{1}{|\Omega|t} \sum_{i,j=1}^3 \int_0^t\int_\Omega |\frac{\partial u_i}{\partial x_j}| \;d\mathbf{x}\;d s, \] and where \(\mathbf{u}=(u_1,u_2,u_3)\ \) is the solution starting at \(\mathbf{u}_0\ \) at time \(t=0\ ,\) which exists globally in time and is uniquely defined by \(\mathbf{u}_0\) thanks to the assumption of \(X\ \) being bounded in \(V\ \) and invariant. This upper-bound estimate is of the order of the estimate of the number of degrees of freedom of a turbulent flow as argued by Landau and Lifschitz (1987) on heuristic grounds.

Similarly, another upper bound estimate given by Constantin, Foias, and Temam (1988) for the dimension of the global attractor in the two-dimensional periodic case is of the order of \((L/L_\eta)^2(1+\ln(L/L_\eta))^{2/3}\ ,\) where \(L\) is the period in space and \(L_\eta\) is a quantity associated with the Kraichnan dissipation length. This upper-bound estimate is nearly the same as the one obtained on heuristic grounds in the two-dimensional case.

Since in the three-dimensional case it is not known whether the system is globally well-posed or not, it is not known whether it generates a semigroup of operators, so that the global attractor cannot be defined in the classical sense. Nevertheless, a weak global attractor has been defined by Foias and Temam (1987) as an invariant bounded set in the phase space \(H\) which attracts all global weak solutions in the weak topology. This set was proved to exist but it is not known, however, whether it is bounded in \(V\) and, hence, whether it has finite dimension. In space dimension three, if the invariant set or attractor \(X\) has bounded norm in \(V\) (bounded enstrophy), then \(X\) has finite dimension with an upper bound on the dimension depending on the Grashof number and on the diameter of \(X\) on \(V\ .\)

Vanishing viscosity limit and the Euler equations

An outstanding problem in mathematical physics (in mathematics and physics) is the behavior of the Navier-Stokes equations when the (kinematic) viscosity \(\nu\) converges to \(0\ .\) We consider the equations in the form equation (#NSEclassicalformulation); then the behavior depends on the associated boundary condition.

The simplest case is that of periodic boundary condition as in equation (#periodiccondition). Denoting \((\mathbf{u}^\nu,p^\nu)\) the solution of equation (#NSEclassicalformulation), equation (#periodiccondition), equation (#initialcondition), one can prove that \((\mathbf{u}^\nu,p^\nu)\) converges to the solution \((\mathbf{u}^0,p^0)\) of the incompressible Euler equations \[\tag{12} \begin{cases} \displaystyle \frac{\partial \mathbf{u}^0}{\partial t} + (\mathbf{u}^0\cdot \boldsymbol{\nabla})\mathbf{u}^0 + \boldsymbol{\nabla} p^0 = \mathbf{f}, \\ \boldsymbol{\nabla} \cdot \mathbf{u}^0 = 0, \end{cases} \]

with the same boundary and initial conditions equation (#periodiccondition), equation (#initialcondition). This convergence assumes mild regularity properties for \(\mathbf{f}\) and \(\mathbf{u}_0\ ,\) and is valid for all time in space dimension \(2\ ,\) and for the time interval of regularity \((0,T^*)\) in space dimension \(3\ ;\) see Kato (1972).

The situation is again manageable if we replace equation (#periodiccondition) by the second set of boundary conditions stated between equation (#noslipcondition) and equation (#periodiccondition). J.-L. Lions (1969) proves in this case the convergence of \((\mathbf{u}^\nu,p^\nu)\) to \((\mathbf{u}^0,p^0)\) for which the boundary condition is \[ \mathbf{u}^0\cdot \mathbf{n} = 0 \text{ on } \partial \Omega. \] The associated boundary layers have been studied by several authors; see Gie and Kelliher (2011) and the references therein.

Finally, the case of no-slip boundary condition is considered for simplicity in two separated typical cases, namely the case in which \(\boldsymbol{\varphi}\cdot \mathbf{n}\) has a constant sign on each connected component of the boundary \(\partial \Omega\) and the case in which \(\boldsymbol{\varphi}=0\) as in equation (#noslipcondition). The case in which \(\boldsymbol{\varphi}\cdot\mathbf{n}=U_0>0\ ,\) for some positive scalar \(U_0\ ,\) has been studied by Temam and Wang (2002), who determined the boundary layer equation (analogue of the Prandtl equation but a simple solvable linear equation in this case) and could prove the convergence of \((\mathbf{u}^\nu,p^\nu)\) to \((\mathbf{u}^0,p^0)\ .\) Further results on the boundary layer at higher orders were established by Hamouda and Temam (2007).

When \(\boldsymbol{\varphi}\cdot\mathbf{n}\neq 0\) on \(\partial \Omega\ ,\) the boundary \(\partial \Omega\) is noncharacteristic. In the characteristic case, when e.g. \(\boldsymbol{\varphi}=0\ ,\) the problem remains an outstanding one. On the mathematical side, no convergence result has been proved even in the two-dimensional case for which we know the existence and uniqueness for all time of the solutions of the Navier-Stokes equations and of the Euler equations. In fact, certain authors question the convergence of \(\mathbf{u}^\nu\) to \(\mathbf{u}^0\ .\) From the physical point of view the problem is also an outstanding one, in relation with the theory of the turbulent boundary layer. However, due to the needs of aeronautics and other classical fluid mechanics problems, a body of engineering results has been developed, starting in the 1930s, based on the concept of Prandtl boundary layer theory (see e.g. Batchelor (1967) and Schlichting (1979)). Note that no comprehensive mathematical theory of the Prandtl equations is available and this is of course one of the road blocks for studying the behavior of \((\mathbf{u}^\nu,p^\nu)\) as \(\nu\rightarrow 0\ .\) Indeed, there is no satisfactory theory of existence and uniqueness of solutions for the Prandtl equation. Note however the result of existence and uniqueness of solutions in the analytic case (data and solutions are analytic); see e.g. Cannone, Lombardo and Sammartino (2001), and Gargano, Sammartino, and Sciacca (2009).

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