You are currently browsing the tag archive for the ‘conservation laws’ tag.

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

to the seemingly rather different-looking law

for incompressible Navier-Stokes (or, if one drops the viscosity term , 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 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.)

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 of one or more fields on a domain taking values in a space will solve the differential equation of interest if and only if is a critical point to the functional

involving the fields and their first derivatives , where the Lagrangian is a function on the vector bundle over consisting of triples with , , and a linear transformation; we also usually keep the boundary data of fixed in case has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on and , 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 as and as , in which case can be viewed locally as a function on .

Example 1 (Geodesic flow)Take and to be a Riemannian manifold, which we will write locally in coordinates as with metric for . A geodesic is then a critical point (keeping fixed) of the energy functionalor in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

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

reallyformally, as the manifold is now infinite dimensional).More generally, if is itself a Riemannian manifold, which we write locally in coordinates as with metric for , then a harmonic map is a critical point of the energy functional

or in coordinates (again ignoring coordinate patch issues)

If we replace the Riemannian manifold by a Lorentzian manifold, such as Minkowski space , 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 ).

Example 2 (-particle interactions)Take and ; then a function can be interpreted as a collection of trajectories in space, which we give a physical interpretation as the trajectories of particles. If we assign each particle a positive mass , and also introduce a potential energy function , then it turns out that Newton’s laws of motion in this context (with the force on the particle being given by the conservative force ) are equivalent to the trajectories being a critical point of the action functional

Formally, if is a critical point of a functional , this means that

whenever is a (smooth) deformation with (and with respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

Write for the infinitesimal deformation of . By the chain rule, can be expressed in terms of . In coordinates, we have

where we parameterise by , and we use subscripts on 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 , 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 , where 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 for some class of functions that vanishes on the boundary, then there must be an associated differential identity 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 should be expressible as a spatial divergence. This is indeed the case:

Proposition 1(Formal) Let be a critical point of the functional defined in (1). Then for any deformation with , we havewhere is the vector field that is expressible in coordinates as

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

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

Since is unconstrained on the interior of , the claim (6) follows (at a formal level, at least).

Many variational problems also enjoy one-parameter continuous *symmetries*: given any field (not necessarily a critical point), one can place that field in a one-parameter family with , such that

for all ; in particular,

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

whenever arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain is the Euclidean space , and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable , thus

then we obtain translation symmetries

for , where is the standard basis for . For a fixed , the left-hand side of (7) then becomes

where . Another common type of symmetry is a *pointwise* symmetry, in which

for all , in which case (7) clearly holds with .

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 is a critical point of the functional (1), and let be a one-parameter continuous symmetry with . Let be the vector field in (5), and let be the vector field in (7). Then we have the pointwise conservation law

In particular, for one-dimensional variational problems, in which , we have the conservation law for all (assuming of course that is connected and contains ).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if and the Lagrangian has no explicit time dependence, thus

then by using the time translation symmetry , we have

as discussed previously, whereas we have , and hence by (5)

and so Noether’s theorem gives conservation of the *Hamiltonian*

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

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

For pointwise symmetries (9), vanishes, and so Noether’s theorem simplifies to ; in the one-dimensional case , we thus see from (5) that the quantity

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

for all , then we have the pointwise translation symmetry

for all , and some , in which case , and the conserved quantity (11) becomes

as was arbitrary, this establishes conservation of the *total momentum*

Similarly, if we have the rotation invariance

for any and , then we have the pointwise rotation symmetry

for any skew-symmetric real matrix , in which case , and the conserved quantity (11) becomes

since is an arbitrary skew-symmetric matrix, this establishes conservation of the *total angular momentum*

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

where is the velocity field, and is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that is bounded. We will take the dimension to be at least two, with the three-dimensional case 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 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

(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

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

and angular momentum

(although, as we shall discuss below, due to the slow decay of 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

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

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

**General discussion: **Here are some typical examples of a target statement , and the approximating statements that would converge to :

for some independent of | |

is finite | is bounded uniformly in |

for all (i.e. maximises f) | for all (i.e. nearly maximises f) |

converges as | fluctuates by at most o(1) for sufficiently large n |

is a measurable function | is a measurable function converging pointwise to |

is a continuous function | is an equicontinuous family of functions converging pointwise to OR is continuous and converges (locally) uniformly to |

The event holds almost surely | The event holds with probability 1-o(1) |

The statement holds for almost every x | The statement holds for x outside of a set of measure o(1) |

Of course, to justify the convergence of to , it is necessary that converge to (or converge to , etc.) in a suitably strong sense. (But for the purposes of proving just *upper* bounds, such as , 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 on the approximating object is somehow “uniform in “, although for “-closed” conclusions, such as measurability, this is not required. [It is important to note that it is only the *final* conclusion on that needs to have this uniformity in ; one is permitted to have some intermediate stages in the derivation of that depend on 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 with “smooth”, “finite-complexity”, “discrete”, “local”, or otherwise “finitary” approximants , 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. should be expected to degrade in the limit , and so one should take extreme caution in using such quantitative measures to derive estimates that are uniform in .] Similarly, issues such as whether the supremum of a function on a set is actually attained by some maximiser become moot if one is willing to settle instead for an almost-maximiser , e.g. one which comes within an epsilon of that supremum M (or which is larger than , 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.

## Recent Comments