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This coming fall quarter, I am teaching a class on topics in the mathematical theory of incompressible fluid equations, focusing particularly on the incompressible Euler and Navier-Stokes equations. These two equations are by no means the only equations used to model fluids, but I will focus on these two equations in this course to narrow the focus down to something manageable. I have not fully decided on the choice of topics to cover in this course, but I would probably begin with some core topics such as local well-posedness theory and blowup criteria, conservation laws, and construction of weak solutions, then move on to some topics such as boundary layers and the Prandtl equations, the Euler-Poincare-Arnold interpretation of the Euler equations as an infinite dimensional geodesic flow, and some discussion of the Onsager conjecture. I will probably also continue to more advanced and recent topics in the winter quarter.

In this initial set of notes, we begin by reviewing the physical derivation of the Euler and Navier-Stokes equations from the first principles of Newtonian mechanics, and specifically from Newton’s famous three laws of motion. Strictly speaking, this derivation is not needed for the mathematical analysis of these equations, which can be viewed if one wishes as an arbitrarily chosen system of partial differential equations without any physical motivation; however, I feel that the derivation sheds some insight and intuition on these equations, and is also worth knowing on purely intellectual grounds regardless of its mathematical consequences. I also find it instructive to actually see the journey from Newton’s law

$\displaystyle F = ma$

to the seemingly rather different-looking law

$\displaystyle \partial_t u + (u \cdot \nabla) u = -\nabla p + \nu \Delta u$

$\displaystyle \nabla \cdot u = 0$

for incompressible Navier-Stokes (or, if one drops the viscosity term ${\nu \Delta u}$, the Euler equations).

Our discussion in this set of notes is physical rather than mathematical, and so we will not be working at mathematical levels of rigour and precision. In particular we will be fairly casual about interchanging summations, limits, and integrals, we will manipulate approximate identities ${X \approx Y}$ as if they were exact identities (e.g., by differentiating both sides of the approximate identity), and we will not attempt to verify any regularity or convergence hypotheses in the expressions being manipulated. (The same holds for the exercises in this text, which also do not need to be justified at mathematical levels of rigour.) Of course, once we resume the mathematical portion of this course in subsequent notes, such issues will be an important focus of careful attention. This is a basic division of labour in mathematical modeling: non-rigorous heuristic reasoning is used to derive a mathematical model from physical (or other “real-life”) principles, but once a precise model is obtained, the analysis of that model should be completely rigorous if at all possible (even if this requires applying the model to regimes which do not correspond to the original physical motivation of that model). See the discussion by John Ball quoted at the end of these slides of Gero Friesecke for an expansion of these points.

Note: our treatment here will differ slightly from that presented in many fluid mechanics texts, in that it will emphasise first-principles derivations from many-particle systems, rather than relying on bulk laws of physics, such as the laws of thermodynamics, which we will not cover here. (However, the derivations from bulk laws tend to be more robust, in that they are not as reliant on assumptions about the particular interactions between particles. In particular, the physical hypotheses we assume in this post are probably quite a bit stronger than the minimal assumptions needed to justify the Euler or Navier-Stokes equations, which can hold even in situations in which one or more of the hypotheses assumed here break down.)

I’ve just uploaded to the arXiv my paper Finite time blowup for a supercritical defocusing nonlinear Schrödinger system, submitted to Analysis and PDE. This paper is an analogue of a recent paper of mine in which I constructed a supercritical defocusing nonlinear wave (NLW) system ${-\partial_{tt} u + \Delta u = (\nabla F)(u)}$ which exhibited smooth solutions that developed singularities in finite time. Here, we achieve essentially the same conclusion for the (inhomogeneous) supercritical defocusing nonlinear Schrödinger (NLS) equation

$\displaystyle i \partial_t u + \Delta u = (\nabla F)(u) + G \ \ \ \ \ (1)$

where ${u: {\bf R} \times {\bf R}^d \rightarrow {\bf C}^m}$ is now a system of scalar fields, ${F: {\bf C}^m \rightarrow {\bf R}}$ is a potential which is strictly positive and homogeneous of degree ${p+1}$ (and invariant under phase rotations ${u \mapsto e^{i\theta} u}$), and ${G: {\bf R} \times {\bf R}^d \rightarrow {\bf C}^m}$ is a smooth compactly supported forcing term, needed for technical reasons.

To oversimplify somewhat, the equation (1) is known to be globally regular in the energy-subcritical case when ${d \leq 2}$, or when ${d \geq 3}$ and ${p < 1+\frac{4}{d-2}}$; global regularity is also known (but is significantly more difficult to establish) in the energy-critical case when ${d \geq 3}$ and ${p = 1 +\frac{4}{d-2}}$. (This is an oversimplification for a number of reasons, in particular in higher dimensions one only knows global well-posedness instead of global regularity. See this previous post for some exploration of this issue in the context of nonlinear wave equations.) The main result of this paper is to show that global regularity can break down in the remaining energy-supercritical case when ${d \geq 3}$ and ${p > 1 + \frac{4}{d-2}}$, at least when the target dimension ${m}$ is allowed to be sufficiently large depending on the spatial dimension ${d}$ (I did not try to achieve the optimal value of ${m}$ here, but the argument gives a value of ${m}$ that grows quadratically in ${d}$). Unfortunately, this result does not directly impact the most interesting case of the defocusing scalar NLS equation

$\displaystyle i \partial_t u + \Delta u = |u|^{p-1} u \ \ \ \ \ (2)$

in which ${m=1}$; however it does establish a rigorous barrier to any attempt to prove global regularity for the scalar NLS equation, in that such an attempt needs to crucially use some property of the scalar NLS that is not shared by the more general systems in (1). For instance, any approach that is primarily based on the conservation laws of mass, momentum, and energy (which are common to both (1) and (2)) will not be sufficient to establish global regularity of supercritical defocusing scalar NLS.

The method of proof in this paper is broadly similar to that in the previous paper for NLW, but with a number of additional technical complications. Both proofs begin by reducing matters to constructing a discretely self-similar solution. In the case of NLW, this solution lived on a forward light cone ${\{ (t,x): |x| \leq t \}}$ and obeyed a self-similarity

$\displaystyle u(2t, 2x) = 2^{-\frac{2}{p-1}} u(t,x).$

The ability to restrict to a light cone arose from the finite speed of propagation properties of NLW. For NLS, the solution will instead live on the domain

$\displaystyle H_d := ([0,+\infty) \times {\bf R}^d) \backslash \{(0,0)\}$

and obey a parabolic self-similarity

$\displaystyle u(4t, 2x) = 2^{-\frac{2}{p-1}} u(t,x)$

and solve the homogeneous version ${G=0}$ of (1). (The inhomogeneity ${G}$ emerges when one truncates the self-similar solution so that the initial data is compactly supported in space.) A key technical point is that ${u}$ has to be smooth everywhere in ${H_d}$, including the boundary component ${\{ (0,x): x \in {\bf R}^d \backslash \{0\}\}}$. This unfortunately rules out many of the existing constructions of self-similar solutions, which typically will have some sort of singularity at the spatial origin.

The remaining steps of the argument can broadly be described as quantifier elimination: one systematically eliminates each of the degrees of freedom of the problem in turn by locating the necessary and sufficient conditions required of the remaining degrees of freedom in order for the constraints of a particular degree of freedom to be satisfiable. The first such degree of freedom to eliminate is the potential function ${F}$. The task here is to determine what constraints must exist on a putative solution ${u}$ in order for there to exist a (positive, homogeneous, smooth away from origin) potential ${F}$ obeying the homogeneous NLS equation

$\displaystyle i \partial_t u + \Delta u = (\nabla F)(u).$

Firstly, the requirement that ${F}$ be homogeneous implies the Euler identity

$\displaystyle \langle (\nabla F)(u), u \rangle = (p+1) F(u)$

(where ${\langle,\rangle}$ denotes the standard real inner product on ${{\bf C}^m}$), while the requirement that ${F}$ be phase invariant similarly yields the variant identity

$\displaystyle \langle (\nabla F)(u), iu \rangle = 0,$

so if one defines the potential energy field to be ${V = F(u)}$, we obtain from the chain rule the equations

$\displaystyle \langle i \partial_t u + \Delta u, u \rangle = (p+1) V$

$\displaystyle \langle i \partial_t u + \Delta u, iu \rangle = 0$

$\displaystyle \langle i \partial_t u + \Delta u, \partial_t u \rangle = \partial_t V$

$\displaystyle \langle i \partial_t u + \Delta u, \partial_{x_j} u \rangle = \partial_{x_j} V.$

Conversely, it turns out (roughly speaking) that if one can locate fields ${u}$ and ${V}$ obeying the above equations (as well as some other technical regularity and non-degeneracy conditions), then one can find an ${F}$ with all the required properties. The first of these equations can be thought of as a definition of the potential energy field ${V}$, and the other three equations are basically disguised versions of the conservation laws of mass, energy, and momentum respectively. The construction of ${F}$ relies on a classical extension theorem of Seeley that is a relative of the Whitney extension theorem.

Now that the potential ${F}$ is eliminated, the next degree of freedom to eliminate is the solution field ${u}$. One can observe that the above equations involving ${u}$ and ${V}$ can be expressed instead in terms of ${V}$ and the Gram-type matrix ${G[u,u]}$ of ${u}$, which is a ${(2d+4) \times (2d+4)}$ matrix consisting of the inner products ${\langle D_1 u, D_2 u \rangle}$ where ${D_1,D_2}$ range amongst the ${2d+4}$ differential operators

$\displaystyle D_1,D_2 \in \{ 1, i, \partial_t, i\partial_t, \partial_{x_1},\dots,\partial_{x_d}, i\partial_{x_1}, \dots, i\partial_{x_d}\}.$

To eliminate ${u}$, one thus needs to answer the question of what properties are required of a ${(2d+4) \times (2d+4)}$ matrix ${G}$ for it to be the Gram-type matrix ${G = G[u,u]}$ of a field ${u}$. Amongst some obvious necessary conditions are that ${G}$ needs to be symmetric and positive semi-definite; there are also additional constraints coming from identities such as

$\displaystyle \partial_t \langle u, u \rangle = 2 \langle u, \partial_t u \rangle$

$\displaystyle \langle i u, \partial_t u \rangle = - \langle u, i \partial_t u \rangle$

and

$\displaystyle \partial_{x_j} \langle iu, \partial_{x_k} u \rangle - \partial_{x_k} \langle iu, \partial_{x_j} u \rangle = 2 \langle i \partial_{x_j} u, \partial_{x_k} u \rangle.$

Ideally one would like a theorem that asserts (for ${m}$ large enough) that as long as ${G}$ obeys all of the “obvious” constraints, then there exists a suitably non-degenerate map ${u}$ such that ${G = G[u,u]}$. In the case of NLW, the analogous claim was basically a consequence of the Nash embedding theorem (which can be viewed as a theorem about the solvability of the system of equations ${\langle \partial_{x_j} u, \partial_{x_k} u \rangle = g_{jk}}$ for a given positive definite symmetric set of fields ${g_{jk}}$). However, the presence of the complex structure in the NLS case poses some significant technical challenges (note for instance that the naive complex version of the Nash embedding theorem is false, due to obstructions such as Liouville’s theorem that prevent a compact complex manifold from being embeddable holomorphically in ${{\bf C}^m}$). Nevertheless, by adapting the proof of the Nash embedding theorem (in particular, the simplified proof of Gunther that avoids the need to use the Nash-Moser iteration scheme) we were able to obtain a partial complex analogue of the Nash embedding theorem that sufficed for our application; it required an artificial additional “curl-free” hypothesis on the Gram-type matrix ${G[u,u]}$, but fortunately this hypothesis ends up being automatic in our construction. Also, this version of the Nash embedding theorem is unable to prescribe the component ${\langle \partial_t u, \partial_t u \rangle}$ of the Gram-type matrix ${G[u,u]}$, but fortunately this component is not used in any of the conservation laws and so the loss of this component does not cause any difficulty.

After applying the above-mentioned Nash-embedding theorem, the task is now to locate a matrix ${G}$ obeying all the hypotheses of that theorem, as well as the conservation laws for mass, momentum, and energy (after defining the potential energy field ${V}$ in terms of ${G}$). This is quite a lot of fields and constraints, but one can cut down significantly on the degrees of freedom by requiring that ${G}$ is spherically symmetric (in a tensorial sense) and also continuously self-similar (not just discretely self-similar). Note that this hypothesis is weaker than the assertion that the original field ${u}$ is spherically symmetric and continuously self-similar; indeed we do not know if non-trivial solutions of this type actually exist. These symmetry hypotheses reduce the number of independent components of the ${(2d+4) \times (2d+4)}$ matrix ${G}$ to just six: ${g_{1,1}, g_{1,i\partial_t}, g_{1,i\partial_r}, g_{\partial_r, \partial_r}, g_{\partial_\omega, \partial_\omega}, g_{\partial_r, \partial_t}}$, which now take as their domain the ${1+1}$-dimensional space

$\displaystyle H_1 := ([0,+\infty) \times {\bf R}) \backslash \{(0,0)\}.$

One now has to construct these six fields, together with a potential energy field ${v}$, that obey a number of constraints, notably some positive definiteness constraints as well as the aforementioned conservation laws for mass, momentum, and energy.

The field ${g_{1,i\partial_t}}$ only arises in the equation for the potential ${v}$ (coming from Euler’s identity) and can easily be eliminated. Similarly, the field ${g_{\partial_r,\partial_t}}$ only makes an appearance in the current of the energy conservation law, and so can also be easily eliminated so long as the total energy is conserved. But in the energy-supercritical case, the total energy is infinite, and so it is relatively easy to eliminate the field ${g_{\partial_r, \partial_t}}$ from the problem also. This leaves us with the task of constructing just five fields ${g_{1,1}, g_{1,i\partial_r}, g_{\partial_r,\partial_r}, g_{\partial_\omega,\partial_\omega}, v}$ obeying a number of positivity conditions, symmetry conditions, regularity conditions, and conservation laws for mass and momentum.

The potential field ${v}$ can effectively be absorbed into the angular stress field ${g_{\partial_\omega,\partial_\omega}}$ (after placing an appropriate counterbalancing term in the radial stress field ${g_{\partial_r, \partial_r}}$ so as not to disrupt the conservation laws), so we can also eliminate this field. The angular stress field ${g_{\partial_\omega, \partial_\omega}}$ is then only constrained through the momentum conservation law and a requirement of positivity; one can then eliminate this field by converting the momentum conservation law from an equality to an inequality. Finally, the radial stress field ${g_{\partial_r, \partial_r}}$ is also only constrained through a positive definiteness constraint and the momentum conservation inequality, so it can also be eliminated from the problem after some further modification of the momentum conservation inequality.

The task then reduces to locating just two fields ${g_{1,1}, g_{1,i\partial_r}}$ that obey a mass conservation law

$\displaystyle \partial_t g_{1,1} = 2 \left(\partial_r + \frac{d-1}{r} \right) g_{1,i\partial r}$

together with an additional inequality that is the remnant of the momentum conservation law. One can solve for the mass conservation law in terms of a single scalar field ${W}$ using the ansatz

$\displaystyle g_{1,1} = 2 r^{1-d} \partial_r (r^d W)$

$\displaystyle g_{1,i\partial_r} = r^{1-d} \partial_t (r^d W)$

so the problem has finally been simplified to the task of locating a single scalar field ${W}$ with some scaling and homogeneity properties that obeys a certain differential inequality relating to momentum conservation. This turns out to be possible by explicitly writing down a specific scalar field ${W}$ using some asymptotic parameters and cutoff functions.

Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an a posteriori fashion once the identities are in hand, without the need to rigorously justify the manipulations used to come up with these identities).

It is a remarkable fact in the theory of differential equations that many of the ordinary and partial differential equations that are of interest (particularly in geometric PDE, or PDE arising from mathematical physics) admit a variational formulation; thus, a collection ${\Phi: \Omega \rightarrow M}$ of one or more fields on a domain ${\Omega}$ taking values in a space ${M}$ will solve the differential equation of interest if and only if ${\Phi}$ is a critical point to the functional

$\displaystyle J[\Phi] := \int_\Omega L( x, \Phi(x), D\Phi(x) )\ dx \ \ \ \ \ (1)$

involving the fields ${\Phi}$ and their first derivatives ${D\Phi}$, where the Lagrangian ${L: \Sigma \rightarrow {\bf R}}$ is a function on the vector bundle ${\Sigma}$ over ${\Omega \times M}$ consisting of triples ${(x, q, \dot q)}$ with ${x \in \Omega}$, ${q \in M}$, and ${\dot q: T_x \Omega \rightarrow T_q M}$ a linear transformation; we also usually keep the boundary data of ${\Phi}$ fixed in case ${\Omega}$ has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on ${\Phi}$ and ${D\Phi}$, which require the machinery of Lagrange multipliers to deal with, but which will only serve as a distraction for the current discussion.) It is common to use local coordinates to parameterise ${\Omega}$ as ${{\bf R}^d}$ and ${M}$ as ${{\bf R}^n}$, in which case ${\Sigma}$ can be viewed locally as a function on ${{\bf R}^d \times {\bf R}^n \times {\bf R}^{dn}}$.

Example 1 (Geodesic flow) Take ${\Omega = [0,1]}$ and ${M = (M,g)}$ to be a Riemannian manifold, which we will write locally in coordinates as ${{\bf R}^n}$ with metric ${g_{ij}(q)}$ for ${i,j=1,\dots,n}$. A geodesic ${\gamma: [0,1] \rightarrow M}$ is then a critical point (keeping ${\gamma(0),\gamma(1)}$ fixed) of the energy functional

$\displaystyle J[\gamma] := \frac{1}{2} \int_0^1 g_{\gamma(t)}( D\gamma(t), D\gamma(t) )\ dt$

or in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

$\displaystyle J[\gamma] = \frac{1}{2} \int_0^1 g_{ij}(\gamma(t)) \dot \gamma^i(t) \dot \gamma^j(t)\ dt.$

As discussed in this previous post, both the Euler equations for rigid body motion, and the Euler equations for incompressible inviscid flow, can be interpreted as geodesic flow (though in the latter case, one has to work really formally, as the manifold ${M}$ is now infinite dimensional).

More generally, if ${\Omega = (\Omega,h)}$ is itself a Riemannian manifold, which we write locally in coordinates as ${{\bf R}^d}$ with metric ${h_{ab}(x)}$ for ${a,b=1,\dots,d}$, then a harmonic map ${\Phi: \Omega \rightarrow M}$ is a critical point of the energy functional

$\displaystyle J[\Phi] := \frac{1}{2} \int_\Omega h(x) \otimes g_{\gamma(x)}( D\gamma(x), D\gamma(x) )\ dh(x)$

or in coordinates (again ignoring coordinate patch issues)

$\displaystyle J[\Phi] = \frac{1}{2} \int_{{\bf R}^d} h_{ab}(x) g_{ij}(\Phi(x)) (\partial_a \Phi^i(x)) (\partial_b \Phi^j(x))\ \sqrt{\det(h(x))}\ dx.$

If we replace the Riemannian manifold ${\Omega}$ by a Lorentzian manifold, such as Minkowski space ${{\bf R}^{1+3}}$, then the notion of a harmonic map is replaced by that of a wave map, which generalises the scalar wave equation (which corresponds to the case ${M={\bf R}}$).

Example 2 (${N}$-particle interactions) Take ${\Omega = {\bf R}}$ and ${M = {\bf R}^3 \otimes {\bf R}^N}$; then a function ${\Phi: \Omega \rightarrow M}$ can be interpreted as a collection of ${N}$ trajectories ${q_1,\dots,q_N: {\bf R} \rightarrow {\bf R}^3}$ in space, which we give a physical interpretation as the trajectories of ${N}$ particles. If we assign each particle a positive mass ${m_1,\dots,m_N > 0}$, and also introduce a potential energy function ${V: M \rightarrow {\bf R}}$, then it turns out that Newton’s laws of motion ${F=ma}$ in this context (with the force ${F_i}$ on the ${i^{th}}$ particle being given by the conservative force ${-\nabla_{q_i} V}$) are equivalent to the trajectories ${q_1,\dots,q_N}$ being a critical point of the action functional

$\displaystyle J[\Phi] := \int_{\bf R} \sum_{i=1}^N \frac{1}{2} m_i |\dot q_i(t)|^2 - V( q_1(t),\dots,q_N(t) )\ dt.$

Formally, if ${\Phi = \Phi_0}$ is a critical point of a functional ${J[\Phi]}$, this means that

$\displaystyle \frac{d}{ds} J[ \Phi[s] ]|_{s=0} = 0$

whenever ${s \mapsto \Phi[s]}$ is a (smooth) deformation with ${\Phi[0]=\Phi_0}$ (and with ${\Phi[s]}$ respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

$\displaystyle \int_\Omega \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}\ dx = 0. \ \ \ \ \ (2)$

Write ${\delta \Phi := \frac{d}{ds} \Phi[s]|_{s=0}}$ for the infinitesimal deformation of ${\Phi_0}$. By the chain rule, ${\frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}}$ can be expressed in terms of ${x, \Phi_0(x), \delta \Phi(x), D\Phi_0(x), D \delta \Phi(x)}$. In coordinates, we have

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \delta \Phi^i(x) L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) \ \ \ \ \ (3)$

$\displaystyle + \partial_{x^a} \delta \Phi^i(x) L_{\partial_{x^a} q^i} (x,\Phi_0(x), D\Phi_0(x)),$

where we parameterise ${\Sigma}$ by ${x, (q^i)_{i=1,\dots,n}, (\partial_{x^a} q^i)_{a=1,\dots,d; i=1,\dots,n}}$, and we use subscripts on ${L}$ to denote partial derivatives in the various coefficients. (One can of course work in a coordinate-free manner here if one really wants to, but the notation becomes a little cumbersome due to the need to carefully split up the tangent space of ${\Sigma}$, and we will not do so here.) Thus we can view (2) as an integral identity that asserts the vanishing of a certain integral, whose integrand involves ${x, \Phi_0(x), \delta \Phi(x), D\Phi_0(x), D \delta \Phi(x)}$, where ${\delta \Phi}$ vanishes at the boundary but is otherwise unconstrained.

A general rule of thumb in PDE and calculus of variations is that whenever one has an integral identity of the form ${\int_\Omega F(x)\ dx = 0}$ for some class of functions ${F}$ that vanishes on the boundary, then there must be an associated differential identity ${F = \hbox{div} X}$ that justifies this integral identity through Stokes’ theorem. This rule of thumb helps explain why integration by parts is used so frequently in PDE to justify integral identities. The rule of thumb can fail when one is dealing with “global” or “cohomologically non-trivial” integral identities of a topological nature, such as the Gauss-Bonnet or Kazhdan-Warner identities, but is quite reliable for “local” or “cohomologically trivial” identities, such as those arising from calculus of variations.

In any case, if we apply this rule to (2), we expect that the integrand ${\frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}}$ should be expressible as a spatial divergence. This is indeed the case:

Proposition 1 (Formal) Let ${\Phi = \Phi_0}$ be a critical point of the functional ${J[\Phi]}$ defined in (1). Then for any deformation ${s \mapsto \Phi[s]}$ with ${\Phi[0] = \Phi_0}$, we have

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \hbox{div} X \ \ \ \ \ (4)$

where ${X}$ is the vector field that is expressible in coordinates as

$\displaystyle X^a := \delta \Phi^i(x) L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)). \ \ \ \ \ (5)$

Proof: Comparing (4) with (3), we see that the claim is equivalent to the Euler-Lagrange equation

$\displaystyle L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) - \partial_{x^a} L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)) = 0. \ \ \ \ \ (6)$

The same computation, together with an integration by parts, shows that (2) may be rewritten as

$\displaystyle \int_\Omega ( L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) - \partial_{x^a} L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)) ) \delta \Phi^i(x)\ dx = 0.$

Since ${\delta \Phi^i(x)}$ is unconstrained on the interior of ${\Omega}$, the claim (6) follows (at a formal level, at least). $\Box$

Many variational problems also enjoy one-parameter continuous symmetries: given any field ${\Phi_0}$ (not necessarily a critical point), one can place that field in a one-parameter family ${s \mapsto \Phi[s]}$ with ${\Phi[0] = \Phi_0}$, such that

$\displaystyle J[ \Phi[s] ] = J[ \Phi[0] ]$

for all ${s}$; in particular,

$\displaystyle \frac{d}{ds} J[ \Phi[s] ]|_{s=0} = 0,$

which can be written as (2) as before. Applying the previous rule of thumb, we thus expect another divergence identity

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \hbox{div} Y \ \ \ \ \ (7)$

whenever ${s \mapsto \Phi[s]}$ arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain ${\Omega}$ is the Euclidean space ${{\bf R}^d}$, and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable ${x}$, thus

$\displaystyle L( x, \Phi(x), D\Phi(x) ) = L( \Phi(x), D\Phi(x) ), \ \ \ \ \ (8)$

then we obtain ${d}$ translation symmetries

$\displaystyle \Phi[s](x) := \Phi(x - s e^a )$

for ${a=1,\dots,d}$, where ${e^1,\dots,e^d}$ is the standard basis for ${{\bf R}^d}$. For a fixed ${a}$, the left-hand side of (7) then becomes

$\displaystyle \frac{d}{ds} L( \Phi(x-se^a), D\Phi(x-se^a) )|_{s=0} = -\partial_{x^a} [ L( \Phi(x), D\Phi(x) ) ]$

$\displaystyle = \hbox{div} Y$

where ${Y(x) = - L(\Phi(x), D\Phi(x)) e^a}$. Another common type of symmetry is a pointwise symmetry, in which

$\displaystyle L( x, \Phi[s](x), D\Phi[s](x) ) = L( x, \Phi[0](x), D\Phi[0](x) ) \ \ \ \ \ (9)$

for all ${x}$, in which case (7) clearly holds with ${Y=0}$.

If we subtract (4) from (7), we obtain the celebrated theorem of Noether linking symmetries with conservation laws:

Theorem 2 (Noether’s theorem) Suppose that ${\Phi_0}$ is a critical point of the functional (1), and let ${\Phi[s]}$ be a one-parameter continuous symmetry with ${\Phi[0] = \Phi_0}$. Let ${X}$ be the vector field in (5), and let ${Y}$ be the vector field in (7). Then we have the pointwise conservation law

$\displaystyle \hbox{div}(X-Y) = 0.$

In particular, for one-dimensional variational problems, in which ${\Omega \subset {\bf R}}$, we have the conservation law ${(X-Y)(t) = (X-Y)(0)}$ for all ${t \in \Omega}$ (assuming of course that ${\Omega}$ is connected and contains ${0}$).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if ${\Omega \subset {\bf R}}$ and the Lagrangian has no explicit time dependence, thus

$\displaystyle L(t, \Phi(t), \dot \Phi(t)) = L(\Phi(t), \dot \Phi(t)),$

then by using the time translation symmetry ${\Phi[s](t) := \Phi(t-s)}$, we have

$\displaystyle Y(t) = - L( \Phi(t), \dot\Phi(t) )$

as discussed previously, whereas we have ${\delta \Phi(t) = - \dot \Phi(t)}$, and hence by (5)

$\displaystyle X(t) := - \dot \Phi^i(x) L_{\dot q^i}(\Phi(t), \dot \Phi(t)),$

and so Noether’s theorem gives conservation of the Hamiltonian

$\displaystyle H(t) := \dot \Phi^i(x) L_{\dot q^i}(\Phi(t), \dot \Phi(t))- L(\Phi(t), \dot \Phi(t)). \ \ \ \ \ (10)$

For instance, for geodesic flow, the Hamiltonian works out to be

$\displaystyle H(t) = \frac{1}{2} g_{ij}(\gamma(t)) \dot \gamma^i(t) \dot \gamma^j(t),$

so we see that the speed of the geodesic is conserved over time.

For pointwise symmetries (9), ${Y}$ vanishes, and so Noether’s theorem simplifies to ${\hbox{div} X = 0}$; in the one-dimensional case ${\Omega \subset {\bf R}}$, we thus see from (5) that the quantity

$\displaystyle \delta \Phi^i(t) L_{\dot q^i}(t,\Phi_0(t), \dot \Phi_0(t)) \ \ \ \ \ (11)$

is conserved in time. For instance, for the ${N}$-particle system in Example 2, if we have the translation invariance

$\displaystyle V( q_1 + h, \dots, q_N + h ) = V( q_1, \dots, q_N )$

for all ${q_1,\dots,q_N,h \in {\bf R}^3}$, then we have the pointwise translation symmetry

$\displaystyle q_i[s](t) := q_i(t) + s e^j$

for all ${i=1,\dots,N}$, ${s \in{\bf R}}$ and some ${j=1,\dots,3}$, in which case ${\dot q_i(t) = e^j}$, and the conserved quantity (11) becomes

$\displaystyle \sum_{i=1}^n m_i \dot q_i^j(t);$

as ${j=1,\dots,3}$ was arbitrary, this establishes conservation of the total momentum

$\displaystyle \sum_{i=1}^n m_i \dot q_i(t).$

Similarly, if we have the rotation invariance

$\displaystyle V( R q_1, \dots, Rq_N ) = V( q_1, \dots, q_N )$

for any ${q_1,\dots,q_N \in {\bf R}^3}$ and ${R \in SO(3)}$, then we have the pointwise rotation symmetry

$\displaystyle q_i[s](t) := \exp( s A ) q_i(t)$

for any skew-symmetric real ${3 \times 3}$ matrix ${A}$, in which case ${\dot q_i(t) = A q_i(t)}$, and the conserved quantity (11) becomes

$\displaystyle \sum_{i=1}^n m_i \langle A q_i(t), \dot q_i(t) \rangle;$

since ${A}$ is an arbitrary skew-symmetric matrix, this establishes conservation of the total angular momentum

$\displaystyle \sum_{i=1}^n m_i q_i(t) \wedge \dot q_i(t).$

Below the fold, I will describe how Noether’s theorem can be used to locate all of the conserved quantities for the Euler equations of inviscid fluid flow, discussed in this previous post, by interpreting that flow as geodesic flow in an infinite dimensional manifold.

The Euler equations for incompressible inviscid fluids may be written as

$\displaystyle \partial_t u + (u \cdot \nabla) u = -\nabla p$

$\displaystyle \nabla \cdot u = 0$

where ${u: [0,T] \times {\bf R}^n \rightarrow {\bf R}^n}$ is the velocity field, and ${p: [0,T] \times {\bf R}^n \rightarrow {\bf R}}$ is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that ${u}$ is bounded. We will take the dimension ${n}$ to be at least two, with the three-dimensional case ${n=3}$ being of course especially interesting.

The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post, one potential route to establishing finite time blowup for the latter equations when ${n=3}$ is to be able to construct “computers” solving the Euler equations, which generate smaller replicas of themselves in a noise-tolerant manner (as the viscosity term in the Navier-Stokes equation is to be viewed as perturbative noise).

Perhaps the most prominent obstacles to this route are the conservation laws for the Euler equations, which limit the types of final states that a putative computer could reach from a given initial state. Most famously, we have the conservation of energy

$\displaystyle \int_{{\bf R}^n} |u|^2\ dx \ \ \ \ \ (1)$

(assuming sufficient decay of the velocity field at infinity); thus for instance it would not be possible for a computer to generate a replica of itself which had greater total energy than the initial computer. This by itself is not a fatal obstruction (in this paper of mine, I constructed such a “computer” for an averaged Euler equation that still obeyed energy conservation). However, there are other conservation laws also, for instance in three dimensions one also has conservation of helicity

$\displaystyle \int_{{\bf R}^3} u \cdot (\nabla \times u)\ dx \ \ \ \ \ (2)$

and (formally, at least) one has conservation of momentum

$\displaystyle \int_{{\bf R}^3} u\ dx$

and angular momentum

$\displaystyle \int_{{\bf R}^3} x \times u\ dx$

(although, as we shall discuss below, due to the slow decay of ${u}$ at infinity, these integrals have to either be interpreted in a principal value sense, or else replaced with their vorticity-based formulations, namely impulse and moment of impulse). Total vorticity

$\displaystyle \int_{{\bf R}^3} \nabla \times u\ dx$

is also conserved, although it turns out in three dimensions that this quantity vanishes when one assumes sufficient decay at infinity. Then there are the pointwise conservation laws: the vorticity and the volume form are both transported by the fluid flow, while the velocity field (when viewed as a covector) is transported up to a gradient; among other things, this gives the transport of vortex lines as well as Kelvin’s circulation theorem, and can also be used to deduce the helicity conservation law mentioned above. In my opinion, none of these laws actually prohibits a self-replicating computer from existing within the laws of ideal fluid flow, but they do significantly complicate the task of actually designing such a computer, or of the basic “gates” that such a computer would consist of.

Below the fold I would like to record and derive all the conservation laws mentioned above, which to my knowledge essentially form the complete set of known conserved quantities for the Euler equations. The material here (although not the notation) is drawn from this text of Majda and Bertozzi.

Today I’d like to discuss (in the Tricks Wiki format) a fundamental trick in “soft” analysis, sometimes known as the “limiting argument” or “epsilon regularisation argument”.

Title: Give yourself an epsilon of room.

Quick description: You want to prove some statement $S_0$ about some object $x_0$ (which could be a number, a point, a function, a set, etc.).  To do so, pick a small $\varepsilon > 0$, and first prove a weaker statement $S_\varepsilon$ (which allows for “losses” which go to zero as $\varepsilon \to 0$) about some perturbed object $x_\varepsilon$.  Then, take limits $\varepsilon \to 0$.  Provided that the dependency and continuity of the weaker conclusion $S_\varepsilon$ on $\varepsilon$ are sufficiently controlled, and $x_\varepsilon$ is converging to $x_0$ in an appropriately strong sense, you will recover the original statement.

One can of course play a similar game when proving a statement $S_\infty$ about some object $X_\infty$, by first proving a weaker statement $S_N$ on some approximation $X_N$ to $X_\infty$ for some large parameter N, and then send $N \to \infty$ at the end.

General discussion: Here are some typical examples of a target statement $S_0$, and the approximating statements $S_\varepsilon$ that would converge to $S$:

 $S_0$ $S_\varepsilon$ $f(x_0) = g(x_0)$ $f(x_\varepsilon) = g(x_\varepsilon) + o(1)$ $f(x_0) \leq g(x_0)$ $f(x_\varepsilon) \leq g(x_\varepsilon) + o(1)$ $f(x_0) > 0$ $f(x_\varepsilon) \geq c - o(1)$ for some $c>0$ independent of $\varepsilon$ $f(x_0)$ is finite $f(x_\varepsilon)$ is bounded uniformly in $\varepsilon$ $f(x_0) \geq f(x)$ for all $x \in X$ (i.e. $x_0$ maximises f) $f(x_\varepsilon) \geq f(x)-o(1)$ for all $x \in X$ (i.e. $x_\varepsilon$ nearly maximises f) $f_n(x_0)$ converges as $n \to \infty$ $f_n(x_\varepsilon)$ fluctuates by at most o(1) for sufficiently large n $f_0$ is a measurable function $f_\varepsilon$ is a measurable function converging pointwise to $f_0$ $f_0$ is a continuous function $f_\varepsilon$ is an equicontinuous family of functions converging pointwise to $f_0$ OR $f_\varepsilon$ is continuous and converges (locally) uniformly to $f_0$ The event $E_0$ holds almost surely The event $E_\varepsilon$ holds with probability 1-o(1) The statement $P_0(x)$ holds for almost every x The statement $P_\varepsilon(x)$ holds for x outside of a set of measure o(1)

Of course, to justify the convergence of $S_\varepsilon$ to $S_0$, it is necessary that $x_\varepsilon$ converge to $x_0$ (or $f_\varepsilon$ converge to $f_0$, etc.) in a suitably strong sense. (But for the purposes of proving just upper bounds, such as $f(x_0) \leq M$, one can often get by with quite weak forms of convergence, thanks to tools such as Fatou’s lemma or the weak closure of the unit ball.)  Similarly, we need some continuity (or at least semi-continuity) hypotheses on the functions f, g appearing above.

It is also necessary in many cases that the control $S_\varepsilon$ on the approximating object $x_\varepsilon$ is somehow “uniform in $\varepsilon$“, although for “$\sigma$-closed” conclusions, such as measurability, this is not required. [It is important to note that it is only the final conclusion $S_\varepsilon$ on $x_\varepsilon$ that needs to have this uniformity in $\varepsilon$; one is permitted to have some intermediate stages in the derivation of $S_\varepsilon$ that depend on $\varepsilon$ in a non-uniform manner, so long as these non-uniformities cancel out or otherwise disappear at the end of the argument.]

By giving oneself an epsilon of room, one can evade a lot of familiar issues in soft analysis.  For instance, by replacing “rough”, “infinite-complexity”, “continuous”,  “global”, or otherwise “infinitary” objects $x_0$ with “smooth”, “finite-complexity”, “discrete”, “local”, or otherwise “finitary” approximants $x_\varepsilon$, one can finesse most issues regarding the justification of various formal operations (e.g. exchanging limits, sums, derivatives, and integrals).  [It is important to be aware, though, that any quantitative measure on how smooth, discrete, finite, etc. $x_\varepsilon$ should be expected to degrade in the limit $\varepsilon \to 0$, and so one should take extreme caution in using such quantitative measures to derive estimates that are uniform in $\varepsilon$.]  Similarly, issues such as whether the supremum $M := \sup \{ f(x): x \in X \}$ of a function on a set is actually attained by some maximiser $x_0$ become moot if one is willing to settle instead for an almost-maximiser $x_\varepsilon$, e.g. one which comes within an epsilon of that supremum M (or which is larger than $1/\varepsilon$, if M turns out to be infinite).  Last, but not least, one can use the epsilon room to avoid degenerate solutions, for instance by perturbing a non-negative function to be strictly positive, perturbing a non-strictly monotone function to be strictly monotone, and so forth.

To summarise: one can view the epsilon regularisation argument as a “loan” in which one borrows an epsilon here and there in order to be able to ignore soft analysis difficulties, and can temporarily be able to utilise estimates which are non-uniform in epsilon, but at the end of the day one needs to “pay back” the loan by establishing a final “hard analysis” estimate which is uniform in epsilon (or whose error terms decay to zero as epsilon goes to zero).

A variant: It may seem that the epsilon regularisation trick is useless if one is already in “hard analysis” situations when all objects are already “finitary”, and all formal computations easily justified.  However, there is an important variant of this trick which applies in this case: namely, instead of sending the epsilon parameter to zero, choose epsilon to be a sufficiently small (but not infinitesimally small) quantity, depending on other parameters in the problem, so that one can eventually neglect various error terms and to obtain a useful bound at the end of the day.  (For instance, any result proven using the Szemerédi regularity lemma is likely to be of this type.)  Since one is not sending epsilon to zero, not every term in the final bound needs to be uniform in epsilon, though for quantitative applications one still would like the dependencies on such parameters to be as favourable as possible.

Prerequisites: Graduate real analysis.  (Actually, this isn’t so much a prerequisite as it is a corequisite: the limiting argument plays a central role in many fundamental results in real analysis.)  Some examples also require some exposure to PDE.