Finite-time blowup for a Navier–Stokes model equation for the self-amplification of strain

The incompressible Navier–Stokes equation is one of the fundamental equations of fluid mechanics. Although it is over 150 years old, much about its solutions, including the global existence of smooth solutions, remains unknown. The Navier–Stokes equation is given by

	$\displaystyle\partial_{t}u-\Delta u+(u\cdot\nabla)u+\nabla p$	$\displaystyle=0$		(1.1)
	$\displaystyle\nabla\cdot u$	$\displaystyle=0,$		(1.2)

where $u\in\mathbb{R}^{3}$ is the velocity and $p$ is the pressure. The first equation is a statement of Newton’s second law, $F=ma$, where $\partial_{t}u+(u\cdot\nabla)u$ gives the acceleration in the Lagrangian frame, $\Delta u$ describes the viscous forces due to the internal friction of the fluid, and $-\nabla p$ describes the force due to the pressure. The second equation, the divergence free constraint, comes from the conservation of mass. We will note that $p$ is not an independently evolving function, but is determined entirely by $u$ by convolution with the Poisson kernel,

$$p=(-\Delta)^{-1}\sum_{i,j=1}^{3}\frac{\partial u_{j}}{\partial x_{i}}\frac{\partial u_{i}}{\partial x_{j}}.$$

(1.3)

It is possible to state the incompressible Navier–Stokes equation without giving any reference to pressure at all by making use of the Helmholtz projection onto the space of divergence free vector fields, yielding the equation

$$\partial_{t}u-\Delta u+P_{df}\nabla\cdot\left(u\otimes u\right)=0.$$

(1.4)

Note that we have used the fact that $\nabla\cdot(u\otimes u)=(u\cdot\nabla)u,$ because $\nabla\cdot u=0$, and the fact that the Helmholtz decomposition implies that $P_{df}(\nabla p)=0$.

The first major advances towards a rigorous mathematical understanding of the Navier–Stokes equation came in the seminal paper by Leray [Leray]. For all initial data $u^{0}\in L^{2}_{df},$ Leray proved the global-in-time existence of weak solutions, in the sense of integrating against smooth test functions, satisfying the energy inequality, which states that for all $t>0$,

$$\frac{1}{2}\|u(t)\|_{L^{2}}^{2}+\int_{0}^{t}\|u(\tau)\|_{\dot{H}^{1}}^{2}\mathop{}\!\mathrm{d}\tau\leq\frac{1}{2}\left\|u^{0}\right\|_{L^{2}}^{2}.$$

(1.5)

Unfortunately, while such solutions are well suited to study in that the sense that global-in-time existence is guaranteed for all finite-energy initial data, they are not known to be either smooth or unique, leaving major problems for the well-posedness theory.

The lack of a uniqueness and regularity theory for Leray weak solutions led Fujita and Kato to develop the notion of mild solutions, which satisfy (1.4) in the sense of Duhamel’s formula. Unlike Leray’s weak solutions, mild solutions must be both smooth and unique. Kato and Fujita proved the local-in-time existence, uniqueness, and smoothness of mild solutions for initial data in $\dot{H}^{1}$, with the time of existence bounded below uniformly in the $\dot{H}^{1}$ norm [KatoFujita].

We will note that because mild solutions are smooth and unique, the initial value problem for mild solutions of the Navier–Stokes equation is locally well-posed in $\dot{H}^{1}$—and also in a number of larger spaces; however, it is not known to be globally well-posed. Whether the Navier–Stokes equation has global smooth solutions or admits smooth solutions that blowup in finite-time is one of the biggest open problems in PDEs and one of the “Millennium Problems” put forward by the Clay Mathematics Institute [Clay].

The main difficulty is that the only bounds that are available on the growth of solutions are the bounds in $L^{\infty}_{t}L^{2}_{x}$ and $L^{2}_{t}\dot{H}^{1}_{x}$ due to the energy equality, and these bounds are not enough to guarantee the global existence of smooth solutions because the energy equality is super-critical with respect to the invariant rescaling of the Navier–Stokes equation. The solution set of the Navier–Stokes equation is preserved under the rescaling,

$$u^{\lambda}(x,t)=\lambda u(\lambda x,\lambda^{2}t),$$

(1.7)

for all $\lambda>0.$ This means that is not enough to control the $L^{\infty}_{t}L^{2}_{x}$ or $L^{2}_{t}\dot{H}^{1}_{x}$ norms of $u,$ which are supercritical in terms of scaling; in order to guarantee global regularity, we need to control a scale critical norm. Ladyzhenskaya [Ladyzhenskaya], Prodi [Prodi], and Serrin [Serrin] independently proved a family of scale critical regularity criteria, which state that if $T_{max}<+\infty,$ and $\frac{2}{p}+\frac{3}{q}=1,$ with $3<q\leq+\infty$ then

$$\int_{0}^{T_{max}}\|u(t)\|_{L^{q}}^{p}\mathop{}\!\mathrm{d}t=+\infty.$$

(1.8)

Escauriaza, Seregin and Sv̌erák [L3] extended this result to the endpoint case $q=3$. They proved that if $T_{max}<+\infty,$ then

$$\limsup_{t\to T_{max}}\|u(t)\|_{L^{3}}=+\infty.$$

(1.9)

Recently, Tao further extended this regularity criterion giving a quantitative lower bound on the rate of the blowup of the $L^{3}$ norm [TaoL3]. This result is very slightly supercritical—in fact triple logarithmically—with respect to scaling, and is the first supercritical regularity criterion for the Navier–Stokes equation.

Two crucially important objects for the study of the Navier–Stokes equation are the strain, which is the symmetric gradient of the velocity, $S=\nabla_{sym}u,$ with $S_{ij}=\frac{1}{2}\left(\partial_{i}u_{j}+\partial_{j}u_{i}\right),$ and the vorticity, which is a vector that represents the anti-symmetric part of the velocity, and is given by $\omega=\nabla\times u.$ Physically, the strain describes how a parcel of the fluid is deformed, while the vorticity describes how a parcel of the fluid is rotated.

Taking the curl of (1.1), we find the evolution equation for $\omega$ is given by

$$\partial_{t}\omega-\Delta\omega+(u\cdot\nabla)\omega-S\omega=0.$$

(1.10)

Taking the symmetric gradient of (1.1), we find the evolution equation for $S$ is given by,

$$\partial_{t}S-\Delta S+(u\cdot\nabla)S+S^{2}+\frac{1}{4}\omega\otimes\omega-\frac{1}{4}|\omega|^{2}I_{3}+\operatorname*{Hess}(p)=0.$$

(1.11)

We will note that the vorticity equation is invariant under the rescaling,

$$\omega^{\lambda}(x,t)=\lambda^{2}\omega(\lambda x,\lambda^{2}t),$$

(1.12)

and the strain equation is invariant under the rescaling

$$S^{\lambda}(x,t)=\lambda^{2}S(\lambda x,\lambda^{2}t).$$

(1.13)

The extra factor of $\lambda$ comes from the fact that both $\omega$ and $S$ scale like $\nabla u.$

The vorticity has been studied fairly exhaustively for its role in the dynamics of the Navier–Stokes equation. For instance the Beale-Kato-Majda regularity criterion [BKM], which holds for smooth solutions of both the Euler and Navier–Stokes equations, states that if $T_{max}<+\infty,$ then

$$\int_{0}^{T_{max}}\|\omega(\cdot,t)\|_{L^{\infty}}\mathop{}\!\mathrm{d}t=+\infty.$$

(1.14)

Chae and Choe proved a regularity criterion on two components of vorticity that has a geometric significance, guaranteeing that blowup must be fully three dimensional [ChaeVort]. They showed that if a smooth solution of the Navier–Stokes equation blows up in finite-time $T_{max}<+\infty$, then for all $\frac{3}{2}<q<+\infty,\frac{2}{p}+\frac{3}{q}=2$,

$$\int_{0}^{T_{max}}\|e_{3}\times\omega(\cdot,t)\|_{L^{q}}^{p}\mathop{}\!\mathrm{d}t=+\infty.$$

(1.15)

The fixed direction condition in this regularity criterion was recently loosened by the author in [MillerVort]. In another key result involving vorticity, Constantin and Fefferman proved that the direction of the vorticity must vary rapidly in regions where the vorticity is large if there is finite-time blowup [VortDirection]. There are many other results involving vorticity, far too many to list here.

The strain equation has been investigated much less thoroughly, but can provide some insights that do not follow as clearly from the vorticity equation. We will refer to the evolution equation for $S$ in (1.11) as the Navier–Stokes strain equation. This equation is an evolution equation on the constraint space $L^{2}_{st},$ the space of strain matrices, which replaces the divergence free constraint for the Navier–Stokes and vorticity equations. We define $L^{2}_{st}$ as follows.

The role of this constraint space in the evolution equation (1.11) was examined by the author in [MillerStrain]. One geometric restriction on the matrices $S\in L^{2}_{st}$ is that they must be trace free, because

$$\operatorname*{tr}(S)=\nabla\cdot u=0.$$

(1.17)

Furthermore, in that paper, the author proved that Hessians and scalar multiples of the identity matrix must be in the orthogonal compliment of $L^{2}_{st}.$

For sufficiently smooth solutions to the Navier–Stokes strain equation $\frac{1}{4}|\omega|^{2},\operatorname*{Hess}(p)\in L^{2},$ so we can conclude that the terms $\frac{1}{4}|\omega|^{2}I_{3}$ and $\operatorname*{Hess}(p)$ are orthogonal to the constraint space, $\frac{1}{4}|\omega|^{2}I_{3},\operatorname*{Hess}(p)\in\left(L^{2}_{st}\right)^{\perp}.$ This means that the Navier–Stokes strain equation can be expressed in terms of the projection onto $L^{2}_{st}$ as

$$\partial_{t}S-\Delta S+P_{st}\left((u\cdot\nabla)S+S^{2}+\frac{1}{4}\omega\otimes\omega\right)=0.$$

(1.20)

This is analogous to defining the Navier–Stokes equation without any reference to $\nabla p$ by using the Helmholtz projection onto the space of divergence free vector fields in (1.4). We will use (1.20) to define mild solutions to the Navier–Stokes strain equation in section 3.

It is not actually necessary to separately prove the existence of mild solutions to the strain equation, as it is straightforward to reduce this problem to the existence of mild solutions of the Navier–Stokes equation. The author proved the equivalence of these formulations in [MillerStrain].

The strain evolution equation is extremely useful, because it allows us to prove a simplified identity for enstrophy growth, which can equivalently be defined in terms of the square of $L^{2}$ norm of $S,\omega,$ or $\nabla u$ based on an isometry proven by the author in [MillerStrain].

Enstrophy is a very important quantity because Theorem 1.1 states that a smooth solution of the Navier–Stokes equation must exist locally in time for initial data in $u^{0}\in\dot{H}^{1}.$ This implies that enstrophy controls regularity, because as long as enstrophy remains bounded on some time interval, a smooth solution can be continued to some later time.

The standard estimate for enstrophy growth is given in terms of nonlocal interaction of the vorticity and the strain:

$$\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\frac{1}{2}\|\omega(t)\|_{L^{2}}^{2}=-\|\omega\|_{\dot{H}^{1}}^{2}+\left<S,\omega\otimes\omega\right>.$$

(1.31)

This is a nonlocal identity because $S$ can be determined in terms of $\omega$ by a nonlocal, zeroth order pseudo-differential operator, with $S=\nabla_{sym}\nabla\times(-\Delta)^{-1}\omega.$ Using the isometry in Proposition 1.5, and the evolution equations for both the strain and the vorticity, this identity can be drastically simplified, with the nonlocal term replaced by a term involving only the determinant of $S.$

This identity was first proven by Neustupa and Penel in [NeustupaPenel1, NeustupaPenel2]. The analogous result without the dissipation term $-2\|S\|_{\dot{H}^{1}}^{2}$ was later proven independently by Chae in the context of smooth solutions of the Euler equation in [ChaeStrain] using similar methods to Neustupa and Penel. This identity was also proven using the evolution equation for the strain, a different approach to that of Neustupa and Penel, by the author in [MillerStrain]. The identity in Proposition 1.9 directly implies a family of scale-invariant regularity criteria in terms of the positive part of the middle eigenvalue of $S.$

This regularity criterion was first proven by Neustupa and Penel in [NeustupaPenel1, NeustupaPenel2, NeustupaPenel3]. It was also proven independently by the author in [MillerStrain]. Note that because $\operatorname*{tr}(S)=0,$ this regularity criterion significantly restricts the geometry of any finite-time blowup for the Navier–Stokes equation: any blowup must be driven by unbounded planar stretching and axial compression, with the strain having two positive eigenvalues, and one very negative eigenvalue.

There are many other conditional regularity results, which guarantee the regularity of solutions as long as some scale critical quantity remains finite, including regularity criteria involving the derivative in just one direction $\partial_{3}u$ [Kukavica], and involving just one velocity direction $u_{3}$ [CheminOneDirection, CheminOneDirection2]. For a more thorough, but by no means exhaustive, treatment of regularity criteria for the Navier–Stokes equation see Chapter 11 in [NS21].

In this paper, we will take the opposite approach. We will prove finite-time blowup for solutions of the Navier–Stokes equation with a fairly broad set of initial data, assuming that a certain scale invariant quantity related to the structure of the nonlinearity remains small. We will do this first by considering a model equation for the Navier–Stokes strain equation and proving finite-time blowup for solutions of this this model equation, and then by viewing the actual Navier–Stokes strain equation as a perturbation of the model equation.

In order to do this, we will drop the advection and the vorticity terms from the evolution equation (1.20) entirely, along with a piece of the $S^{2}$ term so that the enstrophy growth identity in Proposition 1.9 still holds. We will show that

$$\left<S,\omega\otimes\omega\right>=-4\int\det(S)=-\frac{4}{3}\left<S^{2},S\right>,$$

(1.36)

and

$$\left<(u\cdot\nabla)S,S\right>=0,$$

(1.37)

and therefore

$$\left<P_{st}\left((u\cdot\nabla)S+\frac{1}{3}S^{2}+\frac{1}{4}\omega\otimes\omega\right),S\right>=0.$$

(1.38)

Using this identity, we can rewrite the full Navier–Stokes strain equation as

$$\partial_{t}S-\Delta S+\frac{2}{3}P_{st}\left(S^{2}\right)+P_{st}\left((u\cdot\nabla)S+\frac{1}{3}S^{2}+\frac{1}{4}\omega\otimes\omega\right)=0,$$

(1.39)

Dropping the term $P_{st}\left((u\cdot\nabla)S+\frac{1}{3}S^{2}+\frac{1}{4}\omega\otimes\omega\right)$ from the evolution equation, our strain model equation will be given by

$$\partial_{t}S-\Delta S+\frac{2}{3}P_{st}\left(S^{2}\right)=0.$$

(1.40)

We will refer to (1.40) as the strain self-amplification model equation, because it isolates the interaction of the strain with itself, discarding the nonlocal interaction with the vorticity and the effects of advection. In the model equation, we are dropping a combination of terms that are orthogonal to $S$ in $L^{2},$ while keeping the two terms that contribute to the evolution in time of the $L^{2}$ norm to first order. We will also show that for solutions of the strain self-amplification model equation

$$\frac{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}t}\|S(t)\|_{L^{2}}^{2}=-2\|S\|_{\dot{H}^{1}}^{2}-4\int\det(S),$$

(1.41)

so the strain self-amplification model equation does in fact have the same identity for enstrophy growth as the Navier–Stokes equation, and consequently has a regularity criterion for $\lambda_{2}^{+}$ in the critical Lebesgue spaces $L^{p}_{t}L^{q}_{x}$ entirely analogous to the regularity criterion for the Navier–Stokes equation in Theorem 1.10.

Solutions of this model equation blowup in finite-time for a fairly wide range of initial conditions.

Because we chose our strain self-amplification model equation (1.40) by dropping some terms from the full strain equation, we can prove a new conditional blowup result for the full Navier–Stokes equation, by viewing the actual strain equation as a perturbation of the strain self-amplification model equation.

We cannot show that there are any solutions of Navier–Stokes equation which satisfy the perturbative condition in Theorem 1.14 up until $T^{*}$. If we could, then this would solve the Navier–Stokes regularity problem by implying the existence of finite-time blowup. We can, however, use scaling arguments to prove that this condition is satisfied for short times for some solutions of the Navier–Stokes equation.

In section 2, we will discuss the relationship between our results and previous results for simplified model equations for Navier–Stokes. In section 3, we will define a number of the spaces used in our analysis and give precise definitions of mild solutions. In section 4, we will develop the local well-posedness theory for the strain self-amplification model equation, including proving global well-posedness for small initial data, and scale critical regularity criteria in terms of $\lambda_{2}^{+}$ and in terms of two vorticity components. In section 5, we will prove Theorem 1.11, demonstrating the existence of finite-time blowup for solutions of the strain self-amplification model equation, and will prove a number of properties about the set of initial data satisfying the hypothesis of this theorem. Finally in section 6, we will prove Theorem 1.14, the conditional blowup result for the full Navier–Stokes equation when a perturbative condition is satisfied by the history of the solution, and further show that this perturbative condition is satisfied for short times for some solutions of the Navier–Stokes equation.

There are a number of previous results that prove blowup for simplified model equations for Navier–Stokes with the hope of elucidating possibilities of extending this to the full Navier–Stokes equation. Montgomery-Smith introduced a scalar toy model equation, replacing the first order pseudo-differential operator $P_{df}\nabla\cdot$ by $-(-\Delta)^{\frac{1}{2}},$ and replacing the quadratic term $u\otimes u$ by $u^{2},$ giving the scalar equation

$$\partial_{t}u-\Delta u-(-\Delta)^{\frac{1}{2}}\left(u^{2}\right)=0,$$

(2.1)

and proved the existence of finite-time blowup solutions for this equation [MontgomerySmith]. This blowup result was extended by Gallagher and Paicu to a model equation on the space of divergence free vector fields by adjusting the Fourier symbol of the first order pseudo differential operator [GallagherToy]. However, while Gallagher and Paicu’s model equation is an evolution equation on natural constraint space, the space of divergence free vector fields, neither of these model equations respects the energy equality, and so both are still quite far from the actual fluid equations. They are nonetheless important in that they establish that it is not possible to prove global regularity for the Navier–Stokes equation using heat semi-group methods alone.

Tao improved on these earlier blowup results by introducing a Fourier space averaged Navier–Stokes model equation [TaoModel]. His model equation is given by

$$\partial_{t}u-\Delta u+\tilde{B}(u,u)=0,$$

(2.2)

where $\tilde{B}(u,u)$ is a Fourier space averaged version of $P_{df}\nabla\cdot(u\otimes u)$. This equation is an improvement over the previous results because $\tilde{B}$ is constructed so that

$$\left<\tilde{B}(u,u),u\right>=0,$$

(2.3)

so Tao’s model equation (2.2) respects the energy equality, with for all $0<t<T_{max}$,

$$\frac{1}{2}\|u(t)\|_{L^{2}}^{2}+\int_{0}^{t}\|u(\tau)\|_{\dot{H}^{1}}^{2}\mathop{}\!\mathrm{d}\tau=\frac{1}{2}\left\|u^{0}\right\|_{L^{2}}^{2},$$

(2.4)

while also exhibiting finite-time blowup. The operator $\tilde{B}$ also has some of the same harmonic analysis bounds as those found for full Navier–Stokes equation, in particular

$$\|\tilde{B}(u,u)\|_{L^{2}}\leq C\|u\|_{L^{4}}\|\nabla u\|_{L^{4}}.$$

(2.5)

The fact that there are finite-time blowup solutions to Tao’s model equation shows that if there is global regularity for solutions of the Navier–Stokes equation with arbitrary smooth initial data, the proof will require more than the energy equality and the standard harmonic analysis techniques. New a priori bounds are needed. We will note in particular that the bound in (2.5) implies that Tao’s model equation respects the Ladyzhenskaya-Prodi-Serrin regularity criterion, that is if $T_{max}<+\infty$ for a solution $u$ of (2.2), then for all $\frac{2}{p}+\frac{3}{q}=1,3<q\leq+\infty,$

$$\int_{0}^{T_{max}}\|u\|_{L^{q}}^{p}=+\infty.$$

(2.6)

While the Tao model equation respects the energy equality and some of the structure of the velocity equation, it does not respect the structure of the vorticity or strain equations. In particular, Tao’s model does not respect—or at least has not been shown to respect—the identity for enstrophy growth in Proposition 1.9, the regularity criterion on $\lambda_{2}^{+}$ in Theorem 1.10, or the regularity criterion on two components of the vorticity in (1.15). The finite-time blowup result for the strain self-amplification model equation is an advance on Tao’s model equation if the Navier–Stokes regularity problem is considered from the point of view of enstrophy growth. The model equation considered here, unlike Tao’s model equation, does not respect the energy equality; however, from a mathematical point of view, the energy equality is less fundamental to the Navier–Stokes regularity problem than the identity for enstrophy growth, because energy does not control regularity. Blowup for the Navier–Stokes equation in finite-time is equivalent to the blowup of enstrophy in finite-time, so mathematically it is very significant that we are able to show blowup for an evolution equation on $L^{2}_{st}$ that respects the identity for enstrophy growth in Proposition 1.9. In summary, Tao’s model equation reflects more of the structure of the velocity formulation of the Navier–Stokes regularity problem, while the strain self-amplification model equation reflects more of the structure of both the strain and vorticity formulations of the regularity problem.

The strain self-amplification model equation is the first model equation of possible Navier–Stokes blowup that respects regularity criteria for the Navier–Stokes equation based not just on size, but on geometric structure as well. It is straightforward to show that the strain self-amplification, Montgomery-Smith, Gallagher-Paicu, and Tao model equations all respect the Ladyzhenskaya-Prodi-Serrin regularity criterion on the size of $u$, but the strain self-amplification model equation also respects the regularity criterion on $\lambda_{2}^{+}$ in Theorem 1.10. This implies as a corollary that the strain self-amplification model equation must respect the regularity criterion on two vorticity components proven by Chae and Choe as well. This suggests it captures significantly more of the geometry of potential Navier–Stokes blowup than any of the previous model equations, at least as far as deformation and vorticity are concerned.

Theorem 1.14 shows that the local part of the nonlinearity of the strain evolution equation tends to lead to finite-time blowup for a wide range of initial conditions, so there must be finite-time blowup for the Navier–Stokes equation similar to the blowup for the model equation for the self-amplification of strain unless the vorticity and advection terms act to deplete this nonlinearity and prevent blowup. This is consistent with a number of previous works for model equations related to the Navier–Stokes and Euler equations that suggest that advection plays a regularizing role. For instance, there are a number of previous works on the Constantin-Lax-Majda [ConstantinLaxMajda] and De Gregorio [DeGregorio] 1D models for the vorticity equation, which showed that advection may have a regularizing effect [DiegoCordoba, ElgindiAdvection, Sverak1DModels]. Theorem 1.11, which states that finite-time blowup occurs for a wide range of initial data for the strain self-amplification model equation, extends the analysis of the regularizing role of advection from 1D models that do not respect the structure of the constraint space, to a 3D model that does respect the structure of the constraint space.

There is also previous research on model equations for the axisymmetric Navier–Stokes and Euler equations which preserve more of the structures of three dimensional fluid mechanics than the Constantin-Lax-Majda or De Gregorio models. These model equations also show that advection plays a regularizing role [HouZhen, HouLiuWang]. Furthermore, there has been research on the possible role of advection in the depletion of nonlinearity related to its interaction with the pressure in the growth of subcritical $L^{q}$ norms of $u$ [TranYu]. Theorem 1.14 is entirely novel, however, because it is the first perturbative, finite-time blowup result related to the possible role of nonlinear depletion by advection. The previous results were either heuristic or numerical; in contrast, Theorem 1.14 provides a quantitative condition guaranteeing blowup as long as the terms which could potentially deplete the nonlinear self-amplification of strain are small enough relative to strain self-amplification.

Finally, we should mention that very recently, Elgindi [ElgindiFiniteTimeEuler] and Elgindi, Ghoul, and Masmoudi [ElgindiGhoulMasmoudi] proved finite-time blowup for a class of $C^{1,\alpha}\left(\mathbb{R}^{3}\right)$ solutions of the Euler equation that conserve energy. While the question of blowup for smooth solutions to the Euler equation remains open, this represents an enormous step forward in providing an example of classical solutions to the Euler equation that blowup in finite-time. The blowup solutions of the Euler equation constructed in [ElgindiFiniteTimeEuler, ElgindiGhoulMasmoudi] are axisymetric and swirl-free, and are closely related to an example of finite-time blowup that we will construct for the strain self-amplification model equation. We will discuss this further in section 5.

We begin by defining the homogeneous and inhomogeneous Hilbert spaces.

Note that when referring to $H^{s}\left(\mathbb{R}^{3}\right),\dot{H}^{s}\left(\mathbb{R}^{3}\right),\text{or}\>L^{p}\left(\mathbb{R}^{3}\right),$ the $\mathbb{R}^{3}$ will often be omitted for brevity’s sake. All Hilbert and Lebesgue norms are taken over $\mathbb{R}^{3}$ unless otherwise specified. Furthermore, $\mathbb{S}^{3\times 3}$ will refer to the space of three by three symmetric matrices,

$$\mathbb{S}^{3\times 3}=\left\{\left(\begin{array}[]{ccc}a&d&e\\ d&b&f\\ e&f&c\end{array}\right):a,b,c,d,e,f\in\mathbb{R}\right\}.$$

(3.3)

We now define the subspaces of divergence free vector fields and strain matrices in Hilbert spaces.

We will also define axisymmetric, swirl-free vector fields and strain matrices.