Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws
I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First, I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfi...
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| description | I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First, I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Ampère type.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 047, 24 pages
Geometry and Conservation Laws for a Class
of Second-Order Parabolic Equations II:
Conservation Laws
Benjamin B. MCMILLAN
University of Adelaide, Adelaide, South Australia
E-mail: benjamin.mcmillan@adelaide.edu.au
Received March 17, 2020, in final form April 27, 2021; Published online May 11, 2021
https://doi.org/10.3842/SIGMA.2021.047
Abstract. I consider the existence and structure of conservation laws for the general
class of evolutionary scalar second-order differential equations with parabolic symbol. First
I calculate the linearized characteristic cohomology for such equations. This provides
an auxiliary differential equation satisfied by the conservation laws of a given parabolic
equation. This is used to show that conservation laws for any evolutionary parabolic equation
depend on at most second derivatives of solutions. As a corollary, it is shown that the
only evolutionary parabolic equations with at least one non-trivial conservation law are
of Monge–Ampère type.
Key words: conservation laws; parabolic symbol PDEs; Monge–Ampère equations; characte-
ristic cohomology of exterior differential systems
2020 Mathematics Subject Classification: 35L65; 58A15; 35K10; 35K55; 35K96
1 Introduction
The celebrated contributions of Noether began the systematic study of conservation laws for
non-linear partial differential equations. More recently, Vinogradov introduced cohomological
tools to the calculation of conservation laws, with the C-spectral sequence of [7] and [8]. Much
work has been developed from the C-spectral sequence by Vinogradov and subsequent authors,
but particularly relevant here is the work of Bryant and Griffiths in [3], where they translate the
C-spectral sequence to the setting of exterior differential systems. Their approach is particularly
amenable to considerations of the coordinate invariant geometry of a given class of differential
equation.
In a follow-up paper [4], Bryant and Griffiths applied their technique to the class of non-linear
second-order scalar parabolic equations for one unknown function of 2 variables. They defined the
class of 7-dimensional exterior differential systems that are locally equivalent to such parabolic
equations and showed that, in this case, conservation laws are in bijection with functions satisfying
an auxiliary differential equation on the 7-manifold. Their 7-manifold is essentially the space
of 2-jets of solutions, so an immediate consequence of their theorem is that any conservation law
in this context depends on at most second derivatives of solutions. This should be contrasted
with other classes of differential equation, of which there are many well known examples whose
conservation laws depend on arbitrarily many derivatives of solutions – one famous example
being the KdV equation. Bryant and Griffiths also showed that a 2-variable parabolic equation
with at least one non-trivial conservation law is necessarily of Monge–Ampère type.
There are new phenomena for parabolic equations in three or more variables. For example,
it is no longer the case that every parabolic equation can be put into evolutionary form. In her
thesis [5], Clelland studied the exterior differential systems corresponding to evolutionary parabolic
equations in 3 variables. She showed that in this case too, conservation laws depend on at most
mailto:benjamin.mcmillan@adelaide.edu.au
https://doi.org/10.3842/SIGMA.2021.047
2 B.B. McMillan
second derivatives of solutions and only Monge–Ampère equations have non-trivial conservation
laws.
In this paper, I extend these results to evolutionary parabolic equations in any number
of variables. I show that for any evolutionary parabolic equation, conservation laws are functionals
depending on at most second derivatives of solutions. As a corollary, if an evolutionary parabolic
equation has at least one non-trivial conservation law, then it is necessarily Monge–Ampère
(Theorem 8.2).
The jump from 3 to all degrees of freedom relies on the author’s recent progress on the
geometry of parabolic differential equations. In [6] (henceforth, Part I), I introduce the general
class of exterior differential systems that model second-order parabolic differential equations.
This puts the class of parabolic equations in a more geometric form, which I then study using
Cartan’s method of equivalence, introducing several local invariants of interest.
The local invariants derived from the equivalence problem – and their geometric interpretation –
are crucial to the calculations in this paper. Indeed, the extended Goursat invariants introduced
in Part I allow for a simplification of the structure equations of evolutionary parabolic type
exterior differential systems. This reduction makes the calculations done here significantly
more tractable. The Monge–Ampère invariants, also introduced in Part I, provide a geometric
characterization of Monge–Ampère parabolic equations. To wit, the existence of a non-trivial
conservation law on an evolutionary parabolic equation allows one to show that a component of
the Monge–Ampère invariants vanishes identically, which is enough to show that the equation is
Monge–Ampère.
To be clear, this strategy is the same as was applied by the previous authors in dimensions 2
and 3. The difficulty is that the complexity of the equivalence problem grows quickly; the
number of local invariants is of order 4 in dimension. This complexity can be mitigated in higher
dimensions with a careful application of representation theory. Indeed, the local invariants take
values in various representations of SO(n), so that we may consider the invariants in families.
From this perspective, the Goursat and the Monge–Ampère invariants are treated the same
in all dimensions, in much the same way as the Riemannian curvature tensor of Riemannian
geometry is.
To expand on this point, in any Cartan equivalence problem there is an associated Lie group
that controls the local invariants. The Lie group G associated to the geometry of parabolic
equations is somewhat complicated to write down, but it contains a copy of SO(n). (Here n+ 1
is the number of independent variables in the given parabolic equation.) The local invariants of
a parabolic equation are equivariant functions from a G-structure to a particular representation
of G. The local invariants can then be categorized according to which subrepresentations they
take values in, and thus treated as unitary objects. It is worth noting that in 2 and 3 dimensions
the representation theory is obscured simply by low rank considerations, and there it is simpler
to treat each scalar invariant separately.
The first several sections of this paper are more or less expository, devoted to recalling known
results and tools, but with an emphasis on parabolic equations. In Section 2, I recall the geometric
background on parabolic systems that will be applicable to conservation laws. The material in that
section is derived from results in [6]. In particular, I recall parabolic systems, the exterior differ-
ential systems that are locally equivalent to parabolic equations. Also of note are the geometric
characterizations of evolutionary parabolic equations and of Monge–Ampère parabolic equations.
In Section 3, I work out the structure equations for the infinite prolongation of a parabolic
system (M0, I0). This is an infinite-dimensional replacement of M0 that has the same solutions
as M0, but sees information about arbitrarily many derivatives. This technical step is necessary
to deal with the (a priori) possibility that conservation laws can depend on high derivatives
of solutions. To experts, there will be no surprises here, just a direct calculation depending
on the symbol type of a parabolic system.
Conservation Laws for a Class of Second-Order Parabolic Equations II 3
Sections 4, 5, and 6 recall tools of [3], but with an emphasis on parabolic equations. Effort
has been made throughout to present the material in such way that they may provide a useful
guide for other examples.
In Section 7, these tools are used to prove Theorem 7.1, which describes the canonical form that
any conservation law takes, and its dependence on a single function on the infinite prolongation
of M0. This function satisfies an auxiliary differential equation on the infinite prolongation,
so that in principle, one needs to solve an infinite-dimensional PDE to find conservation laws.
In Section 8, I prove Theorem 8.1, which states that the defining function of a conservation
law is in fact defined on M0, instead of on the infinite prolongation. Since points of M0 are the
2-jets of solutions, this result can be restated as follows.
Theorem 1.1. Any conservation law for any evolutionary parabolic equation depends on at most
second derivatives of solutions.
This means in particular that the classification problem of finding all conservation laws for
a parabolic system is a finite-dimensional problem, in distinct contrast to other symbol classes.
Using the reduction of conservation laws to M0 it is then fairly simple to prove Corollary 8.2,
which can be paraphrased as follows.
Theorem 1.2. If an evolutionary parabolic equation has at least one non-trivial conservation
law, then it is of Monge–Ampère type in the neighborhood where the conservation law does not
vanish.
2 Background
I begin by briefly recalling some geometric results that will be needed here. For more details,
and proofs, see [6].
In coordinates, a scalar, (n+1)-variable, second-order weakly parabolic equation is a differential
equation for a single unknown function u of n+ 1 variables x0, . . . , xn, the differential equation
taking the form
F
(
xa, u,
∂u
∂xa
,
∂2u
∂xa∂xb
)
= 0, a, b = 0, . . . , n, (2.1)
subject to the constraint that the linearization at the 2-jet of any solution has parabolic symbol.
Here and throughout, “space-time” indices a, b, . . . will range from 0 to n, while “spatial”
indices i, j, . . . will range from 1 to n. The Einstein summation convention will be used without
further comment. In fact, because all of the vector spaces that will be associated to a parabolic
equation have a well defined spatial trace, it will be convenient to apply the spatial Einstein
summation convention, so that, for example,
∂2u
∂xi∂xi
:=
n∑
i=1
∂2u
∂xi∂xi
.
The distinction between trace and spatial trace will be made by using (respectively) repeated
space-time indices a, b, . . . and spatial indices i, j, . . . .
A parabolic equation is evolutionary if there is a choice of coordinates so that it is in evolu-
tionary form
∂u
∂x0
= G
(
xa, u,
∂u
∂xi
,
∂2u
∂xi∂xj
)
.
4 B.B. McMillan
It should be noted that a generic weakly parabolic equation cannot be put into evolutionary form
for any change of coordinates, even locally. Indeed, the extended Goursat invariants provide the
obstruction to finding such coordinates. The Goursat invariants in turn provide the geometric
condition that characterizes evolutionary parabolic equations. This condition is described below,
and is crucial to the calculations to follow.
The following definition describes the exterior differential systems that are locally equivalent
to scalar second-order weakly parabolic equations.
Definition 2.1. A weakly parabolic system in n+ 1 variables is a (2n+ 2 + (n+ 1)(n+ 2)/2)-
dimensional1 exterior differential system (M0, I0) such that any point has a neighborhood
equipped with a spanning set of 1-forms
θ∅, θa, ωa, πab = πba
that satisfy:
1. The forms θ∅, θa generate I0 as a differential ideal.
2. The structure equations
dθ∅ ≡ −θa ∧ωa
(
mod θ∅
)
,
dθa ≡ −πab ∧ωb
(
mod θ∅, θb
)
.
3. The parabolic symbol relation (see comment above about spatial Einstein summation)
πii ≡ 0
(
mod θ∅, θa, ω
a
)
.
Remark 2.2. There is a natural association from weakly parabolic equations to parabolic
systems. This association is such that the (graphs of) solutions of the parabolic equation are
in bijection with the solution submanifolds of the corresponding parabolic system. For the
reader’s convenience, I briefly recall the essential facts here, but refer to Example 1.4 of [6] for
more details.
The space of 2-jets J = J2
(
Rn+1,R
)
is isomorphic as a manifold to Rn+1×R×Rn+1×S
(
Rn+1
)
.
A choice of coordinate xa on Rn+1 and u on R determines jet coordinates xa, u, pa, pab = pba
on J , where the pa correspond to the first derivatives of u with respect to xa and pab to the
second derivatives. These coordinates may be used to define a coframing of J ,
θ̂∅ := du− padxa, θ̂a := dpa − pabdxb, dxa, dpab. (2.2)
The coframing here depends on the choice of coordinates, but the following structure equations
hold for any such choice,
dθ̂∅ ≡ −dpa ∧dxa
(
mod θ̂∅
)
,
dθ̂a ≡ dpab ∧ dxb
(
mod θ̂∅, θ̂b
)
,
and these structure equations are reflected in condition 2 of Definition 2.1.
Now, a second order differential equation such as equation (2.1) defines, in a clear way,
a function F (xa, u, pa, pab) on J , and assuming F is sufficiently non-degenerate, the zero set
M0 = F−1(0) is a manifold of codimension 1. If Σ ↪→M0 is a submanifold for which the ideal
I0 =
{
θ̂∅, θ̂b
}
pulls back to zero and the pullback of dx0∧ · · · ∧dxn is nowhere vanishing, then Σ
1This is 1 less than the dimension of J2(Rn+1,R), corresponding to the fact that a parabolic equation is defined
by a single equation on J2(Rn+1,R).
Conservation Laws for a Class of Second-Order Parabolic Equations II 5
is locally the 2-jet graph of a solution to the differential equation F . Such Σ are called solution
submanifolds of the exterior differential system (M0, I0), and it is straightforward to see that the
2-jet graph of a solution to the differential equation is a solution submanifold.
The coframing (2.2) pulls back to M0, and the equation dF = 0 determines a single relation
between the forms of the coframing. In particular, for F a weakly parabolic equation, there
exists at every point of M0 a change of coframe from (2.2) to one satisfying the conditions
of Definition 2.1.
Conversely, every small enough neighborhood in a parabolic system arises in this way from
a weakly parabolic differential equation. See [2, Theorem 5.10] or the proof of Theorem 5.3 in [6]
for details. On the other hand, there are parabolic systems which don’t have a global embedding
into J2, such as the parabolic system modelling mean curvature flow, taken up in Example 1.6
of [6].
Remark 2.3. Every spanning set of 1-forms as in the definition is adapted to the parabolic
symbol type of the system. As such, they are called parabolic (extended) coframings. It is worth
noting that they are not strictly coframings, because of the parabolic symbol relation. Nonethless,
an extended parabolic coframing may be used to define a coframing: for any point x ∈M0, the
extended coframing defines a linear injection
TxM −−−→ R⊕Rn+1⊕
(
Rn+1
)∨ ⊕ S2 Rn+1 .
By construction, post-composing this with the quotient map annihilating the spatial trace element
in S2 Rn+1, we get an isomorphism of TxM with a canonical vector space – a legimate coframe.
The issue then is that this quotient space does not have a geometrically natural choice of basis.
This unnatural choice can be avoided by using extended parabolic coframings.
Parabolic coframes can be used to define a G-structure in the standard manner, and it is
in this context that the geometric invariants are developed.
Remark 2.4. It follows from the structure equations of any parabolic coframing that the
1-form θ∅ is uniquely defined up to rescaling by a function. As a consequence, the Cartan system
of θ∅ is well defined independent of a choice of parabolic coframing. It can be directly computed
from the structure equations that Cartan system of θ∅ is the Frobenius ideal
J = {θ∅, θa, ωa}.
This ideal will help to define vertical derivatives of functions in Section 6.
Evolutionary equations are geometrically privileged in the class of all parabolic equations,
and more can be said of them. Indeed, there is geometric test for evolutionarity: it was shown
in Theorem 5.4 of [6] that a given parabolic equation can be put in evolutionary form if and only
if its associated parabolic system has a choice of parabolic coframing for which
dω0 ≡ 0
(
mod ω0
)
.
This in turn is equivalent to the refined structure equations
dθi ≡ −πia ∧ωa
(
mod θ∅, θj
)
(2.3)
for i, j = 1, . . . , n. Furthermore, because the sub-principal symbol of an evolutionary parabolic is
non-vanishing, the coframing may be chosen so that
πii = θ0. (2.4)
6 B.B. McMillan
In fact, an evolutionary parabolic system has parabolic coframings satisfying the structure
equations
dθ∅ ≡ −θa ∧ωa
(
mod θ∅
)
,
dθi ≡ −πia ∧ωa
(
mod θ∅, θi
)
,
dθ0 ≡ −π0a ∧ω
a
(
mod θ∅, θa
)
,
dω0 ≡ 0
(
mod ω0
)
,
dωi ≡ 0
(
mod θ∅, θa, ω
a
)
.
Furthermore, from d2θi = 0, one readily computes that
dπij ≡ 0
(
mod θ∅, θa, ω
a, πkl
)
,
dπi0 ≡ 0
(
mod θ∅, θa, ω
a, πkl, πk0
)
.
Monge–Ampère parabolic equations are even more geometrically privileged than evolutionary
parabolics, and have evident deep connections with conservation laws. An evolutionary parabolic
equation is Monge–Ampère if it is of the form
∂u
∂x0
=
∑
I,J⊆{1,...,n}
|I|=|J |
AI,J
(
xa, u,
∂u
∂xi
)
HI,J ,
where the AI,J are arbitrary smooth functions and HI,J is the row I, column J minor sub-
determinant of the Hessian of u. Monge–Ampère equations are quantifiably less complex than
the generic case: their corresponding exterior differential systems have a natural de-prolongation
to an exterior differential system of dimension only 2n+ 3.
Definition 2.5. A quasi parabolic Monge–Ampère system in n+ 1 variables is a (2n+ 3)-dimen-
sional exterior differential system (M−1, I−1) such that I−1 is locally generated by a 1-form θ∅
and an (n+ 1)-form Υ satisfying:
1. θ∅ is maximally non-integrable:
θ∅ ∧ (dθ∅)n+1 6= 0.
2. At each point of M−1,
Υ 6≡ 0
(
mod θ∅,dθ∅
)
.
3. There is (locally) a 1-form ω0 on M−1 independent of θ∅ and so that
ω0 ∧Υ ≡ 0
(
mod θ∅,dθ∅
)
,
and any other such 1-form is a linear combination of θ∅ and ω0.
In [6], I developed the local invariants that provide an effective test for a parabolic system
to have a Monge–Ampère deprolongation. Briefly, for an evolutionary parabolic system M0
and any evolutionary parabolic coframing, there is a co(n)-valued 1-form
(
βji
)
=
(
δji βtr + β̊ji
)
(so β̊ji = −β̊ij), a symmetric traceless matrix valued 1-form
(
ξji
)
, and a 1-form κ∅ so that
dθi ≡ −βji ∧ θj − ξ
j
i ∧ θj − πia ∧ω
a
(
mod θ∅,Λ
2 I0
)
,
dωi ≡ −
(
δijκ∅ − βij
)
∧ωj + ξij ∧ω
j
(
mod θ∅, θa, ω
0
)
. (2.5)
Conservation Laws for a Class of Second-Order Parabolic Equations II 7
There are functions V jkl
i on M0 so that
ξji ≡ V
jkl
i πkl
(
mod θ∅, θa, ω
a
)
.
The functions V jkl
i generically take values in the co(n) representation S2
(
S2
0 Rn
)
, but a coframe
reduction may be made absorbing the ik-trace, so that V jkl
i takes values in S2
0
(
S2
0 Rn
)
. These
functions are the secondary2 Monge–Ampère invariants, which determine whether M0 comes
from a parabolic equation of Monge–Ampère type. The following is part of Theorem 4.3 in [6].
Theorem 2.6. For a parabolic system (M0, I0) the following are equivalent:
1. M0 is locally EDS equivalent to a neighborhood in the prolongation of a quasi-parabolic
Monge–Ampère system.
2. The Monge–Ampère invariants V jkl
i of M0 take values in
b2 := ker
(
S2
0
(
S2
0 Rn
)
−−−→ S3
0 Rn⊗Rn
)
.
It is worth noting that
S2
0
(
S2
0 Rn
)
= b2 ⊕ S4
0 Rn,
so it suffices for the totally symmetric component of the secondary Monge–Ampère invariant
to vanish. For the parabolic system M0 associated to a parabolic equation, condition (1) is
equivalent to the parabolic equation being of Monge–Ampère type. On the other hand, once the
conservation law theory has been worked out here, condition (2) will be immediately obvious for
any parabolic system with at least one non-trivial conservation law.
3 The prolongation of (M0,I0)
A priori, the conservation laws of a parabolic equation may depend on arbitrarily many derivatives
of solutions. In order to prove that this is not the case for evolutionary parabolic equations, it
will be necessary to work on the infinite prolongation of (M0, I0), whose definition I recall here.
Let Jr
(
Rn+1,R
)
denote the manifold of r-jets of functions from Rn+1 to R. A choice of linear
coordinates x0, . . . , xn on the domain Rn+1 and coordinate p∅ on the codomain R induces natural
coordinates on Jr
(
Rn+1,R
)
– for the symmetric multi-index I in {0, . . . , n}, the coordinate pI
corresponds to the derivative ∂|I|p∅
∂xI
.
The jet manifolds may be arranged in a tower, where each Jr+1
(
Rn+1,R
)
is a bundle over
Jr
(
Rn+1,R
)
with fiber isomorphic to the symmetric product Sr+1
(
Rn+1
)
. Furthermore, each
jet space Jr
(
Rn+1,R
)
has a natural exterior differential system structure, with the differential
ideal C(r) generated by the 1-forms3
θ̂I = dpI − pIadxa, |I| < r.
Let (J∞
(
Rn+1,R
)
, C(∞)) denote the inverse limit of this tower. On J∞
(
Rn+1,R
)
there are
natural differential operators, the total derivatives, defined by the formal sum
Da = ∂xa +
∑
I
pIa∂pI .
2As the name suggests, there are primary Monge–Ampère invariants. They vanish automatically for evolutionary
parabolics.
3Here and throughout, Ia will denote the symmetric multi-index obtained from I by appending a.
8 B.B. McMillan
These can be used to describe the operation of prolongation. For example, consider a second-order
differential equation
F
(
xa, p∅,
∂p∅
∂xa
,
∂p∅
∂xa∂xb
)
= 0
for the unknown scalar function p∅ of n+ 1 variables, which defines a function
F (xa, p∅, pa, pab)
on J∞
(
Rn+1,R
)
that clearly factors through J2
(
Rn+1,R
)
. The first prolongation of this
equation is, by definition, the system of n+ 2 functions
F (1) : J3
(
Rn+1,R
)
−−−→ R×Rn+1,
where F (1) = F × (DaF ). This process may be repeated inductively to define the prolonged
systems F (2), F (3), . . . on to F (∞). It is important to note that solutions of F are in bijection
with those of F (r) for each r, so no information is lost at any step.
Now consider a parabolic system (M0, I0), as defined above. Locally on M0 there is a parabolic
second-order equation F and an embedding so that the diagram of EDS maps
(M0, I0)
ι
↪−−−−→
(
J2
(
Rn+1,R
)
, C(2)
) F−−−−→ (R, {0})
is exact, in the sense that M0 = F−1(0) and I0 = ι∗ C(2).
The exterior differential system prolongations of M0 (see [2, 3] for the intrinsic definition and
more details) can be seen to fit into the diagram
...
...
...
(
M
(2)
0 , I(2)
0
) (
J4
(
Rn+1,R
)
, C(4)
) (
R1+n+1+(n+1
2 ), {0}
)
(
M
(1)
0 , I(1)
0
) (
J3
(
Rn+1,R
)
, C(3)
) (
R1+n+1, {0}
)
(M0, I0)
(
J2
(
Rn+1,R
)
, C(2)
)
(R, {0})
F (2)
F (1)
F
It is a key point that the left column may be calculated globally, without reference to local
embeddings into jet space. In particular, infinite prolongations exist even for exterior differential
systems that have no global embedding into jet bundles, such as the mean curvature flow.
Furthermore, any (n+ 1)-dimensional solution submanifold in M0 has a unique lift to M
(r)
0 for
each r, so the solution manifolds of M0 are in bijection with those of M
(r)
0 for each r.
Let (M, I) be the infinite prolongation of M0, the inverse limit of the prolongation tower.
More precisely, the manifold M is given by the inverse limit of underlying manifolds,
M = lim←−M
(r)
0 ,
and the ideal I is given by
I =
∞⋃
r=0
I(r) .
Conservation Laws for a Class of Second-Order Parabolic Equations II 9
The manifold M is infinite-dimensional, but this will not cause any technical difficulty in prac-
tice, because any conservation law factors through some finite level. In particular, we are
concerned with conservation laws of finite type, which are represented by certain differential forms
pulled back from a finite prolongation of M0. Accordingly, it will suffice to consider finite-type
functions on M , those that can be expressed as the pullback of a function on M
(r)
0 for some r.
The space of finite-type smooth functions is given by
C∞(M) =
∞⋃
r=0
C∞
(
M
(r)
0
)
and the finite-type differential forms by
Ω∗(M) =
∞⋃
r=0
Ω∗
(
M
(r)
0
)
.
It is worth noting that a function A ∈ C∞
(
M
(r)
0
)
is one which, when restricted to the r-jet
graph of a solution u(x) to the differential equation F , results in a functional of x, of u, and
of the derivatives of u up to order r + 2. For example, a point of M0 is the 2-jet of a solution,
and a function on M0 is a functional depending on at most second derivatives of solutions.
Now for some linear algebra preliminaries, which will be used to describe the structure
of (M, I). Fix a vector space W and subspace W ′ of respective dimensions n + 1 and n, and
basis ea on W so that
W ′ = R{e1, . . . , en} ⊂W = R{e0, . . . , en}.
Let
W ′′ = W/W ′ ∼= R{e0}.
This decomposition of W induces one on the symmetric powers of W , so that
SrW ∼=
⊕
s+t=r
Ss(W ′)⊗ St(W ′′).
For notation, elements will be indexed with regards to this splitting, so that Sr(W ) has basis
given by the elements
eI,t := ei1◦ · · · ◦eis◦(e0)t
for I = (i1 . . . is) as I ranges over symmetric multi-indices of length |I| up to r, and t = r − |I|.
It will be convenient to relabel the adapted coframing ωa, θ∅, θa, πab of M0 so that it is
consistent with the basis of S2W :
θ∅ 7−−→ θ∅,0, πij 7−−→ πij,0,
θi 7−−→ θi,0, πi0 7−−→ πi,1,
θ0 7−−→ θ∅,1, π00 7−−→ π∅,2.
The total symmetric product S• := S•(W ) of W is an algebra, with multiplication given
by the action of W :
eieI,t = eIi,t,
e0eI,t = eIi,t+1.
10 B.B. McMillan
Note that this is nothing but a free polynomial algebra with a special indeterminate picked out.
There is also an action of W∨ on S•, which is essentially the directional derivative. In the given
basis,
eieI,t := #(i, I)eI\i,t, e0eI,t+1 := (t+ 1)eI,t,
where #(i, I) is the number of times i appears in I. These can be used to define the spatial trace
operator on S•, given by
tr =
n∑
i=1
eiei.
Now, a parabolic system (M0, I0) is a linear Pfaffian system, and the general theory of such
(see [2, Chapter IV]) determines the associated tableaux
K ∼= S2
0(W ′)⊕ (W ′ ⊗W ′′)⊕ S2(W ′′) ⊂W ⊗W,
where S2
0(W ′) is the traceless component of S2(W ′). The r-fold prolongation of K is by definition
K(r) = (K ⊗ SrW ) ∩
(
W ⊗ Sr+1W
)
,
which is readily computed to be
K(r) ∼=
2+r⊕
s=0
Ss0(W ′)⊗ S2+r−s(W ′′).
Each K(r) is naturally identified with the kernel of the spatial trace
tr : Sr+2(W )
eiei−−−−−−→ Sr(W ).
This linear algebra determines the structure equations in the following proposition, whose
proof is standard.
Proposition 3.1 (principal structure equations). Near any point of the infinite prolonga-
tion (M, I) of a parabolic system, there is a spanning set of 1-forms
ωa, θI,t,
where I ranges over symmetric multi-indices and t over non-negative integers, so that
1. I = {θI,t},
2. I(r) = {θI,t : |I|+ t ≤ r + 1},
3. for any θI,t, the principal structure equations
dθI,t ≡ −θIi,t ∧ωi − θI,t+1 ∧ω
0
(
mod θJ,s : |J |+ s ≤ |I|+ t
)
(3.1)
hold,
4. for each r ≥ 0, the pullback to M of Ω1
(
M
(r)
0
)
is locally spanned by ωa and the 1-forms
of I(r+1), and
5. for each fixed I and t, the 1-forms in I are subject to the relation
n∑
i=1
θIii,t ≡ 0
(
mod θJ,s for which |J |+ s < |I|+ t+ 2
)
.
Conservation Laws for a Class of Second-Order Parabolic Equations II 11
Definition 3.2. Any choice of ωa, θI,t as in Proposition 3.1 is called a parabolic coframing of M .
Remark 3.3. As in Remark 2.3, a parabolic coframing only defines a coframing of M after
composing with the appropriate quotient. Nonetheless, the bundle of all parabolic coframes
can be used to define a G(∞)-structure, where G(∞) is the infinite group prolongation of G.
Essentially by construction, G(∞) acts stably on the filtration of Ω∗(M) by the ideals I(r).
The theorem has the following corollary.
Corollary 3.4. A function A ∈ C∞(M) factors through C∞
(
M
(r)
0
)
if and only if
dA ≡ 0
(
mod ωa, I(r+1)
)
.
Proof. The function A is finite type, so factors through C∞
(
M
(R)
0
)
for sufficiently large R. Since
the fibers of M
(R)
0 are connected, A factors through C∞
(
M
(r)
0
)
if and only if dA ∈ Ω1
(
M
(r)
0
)
,
which by the theorem holds if and only if
dA ≡ 0
(
mod ωa, I(r+1)
)
. �
4 The characteristic cohomology
With (M, I) the infinitely prolonged parabolic system described in the last section, let Ω
∗
be
chain complex defined by the exact sequence
0 −−→ I∗ −−→ Ω∗(M) −−→ Ω
∗ −−→ 0.
Because I is a differential ideal, the differential of Ω∗(M) descends to a differential dh on
the quotient Ω
∗
, which is therefore a graded commutative differential algebra. This cdga is
intimately connected with the behavior of solutions to M . The next definition follows Bryant
and Griffiths [3].
Definition 4.1. The characteristic cohomology H
q
of an infinitely prolonged parabolic sys-
tem (M, I) is the cohomology of the complex Ω
∗
,
H
q
= Hq
(
Ω
∗
, dh
)
.
Due to the symbol of (M, I), it follows from the results in [3] that H
n
and H
n+1
are the only
nontrivial characteristic cohomology groups. Furthermore, H
n
is naturally identified as the space
of conservation laws of (M, I):
Definition 4.2. The space of conservation laws for a parabolic system is given by the degree n
characteristic cohomology,
C = H
n
(M).
The cohomology of the differential operator dh is difficult to compute directly. However, the
characteristic cohomology fits into a spectral sequence. In turn, the first page of this spectral
sequence can be computed using a second spectral sequence. This second spectral sequence is
quite amenable to calculations, as the differential of it first page is linear over functions, allowing
a pointwise computation determined entirely by the symbol of (M, I).
To define the first spectral sequence, consider the filtration of Ω∗(M) defined recursively by
F0 = Ω∗(M),
Fp+1 = I ∧ Fp .
12 B.B. McMillan
Each level of the filtration is graded, with graded components
Fp,q = Fp ∩Ωp+q(M).
Then the bi-graded associated graded spaces are given by
Grp,q := Fp,q /Fp+1,q−1 .
I is formally Frobenius, so, for each p ≥ 0, the exterior derivative descends to a well defined
operator dh. These complexes comprise page 0 of the filtration spectral sequence, so that
Ep,q0 = Grp,q.
It follows immediately from the definition that
E0,q
0 = Ω
q
,
and thus
E0,q
1 = H
q
.
On the other hand,
E∞ ⇒ H∗(M),
so for any contractible neighborhood in M the spectral sequence converges to the homology
of a point. As we are concerned here with local conservation laws, assume henceforth that M is
contractible, restricting attention to a neighborhood if necessary.
It follows from the two-line theorem of Vinogradov [1], that for parabolic equations,
Ep,q1 = 0 for q < n,
so the E1 page is
0 H
n+1
E1,n+1
1 E2,n+1
1 · · · ,
0 H
n
E1,n
1 E2,n
1 · · · ,
0 0 0 0 · · · .
dv dv dv
dv dv dv
Note that as an immediate consequence, the bottom row is exact, so that H
n
is isomorphic to
the kernel of dv in E1,n
1 . This motivates the following definition, from [3].
Definition 4.3. The space of differentiated conservation laws is
C = ker
(
E1,n
1
dv−−−→ E2,n
1
)
.
The operator dv provides an isomorphism between C and C , and the latter space may be
computed in two steps: First one computes E1,n
1 , and then one computes the kernel of dv. It is
the second spectral sequence that helps to compute E1,n
1 , which is explained in the next section.
Conservation Laws for a Class of Second-Order Parabolic Equations II 13
5 The weight filtrations
Bryant and Griffiths introduced a second filtration on Ep,q1 when p ≥ 1, the principal weight
filtration, that linearizes the calculation of each Ep,q1 .
Definition 5.1. A weight function is a function wt: Ω∗(M) → Z satisfying the following
properties:
1. wt(f) = 0 for f ∈ C∞(M).
2. wt(α∧β) = wt(α) + wt(β) for α, β ∈ Ω∗(M).
3. wt(α+ β) = max(wt(α),wt(β)).
Example 5.2 (principal weight filtration). Fix any parabolic coframing of M and consider the
weight function pwt on Ω∗(M) uniquely specified by
pwt(ωa) = −1,
pwt(θI,t) = |I|+ t.
The weight function pwt descends to Grp,∗ for each p > 0. As argued in [3], pwt on Grp,∗ is
independent of the choice of parabolic coframing. See also Remark 3.3.
For each integer k and each p ≥ 1, define
Fpk =
{
α ∈ Fp : pwt(α) ≤ k
}
/Fp+1,
the subspace in Grp,∗ of forms with principal weight less than k. The grading by form degree
descends to Fpk, with graded components
Fp,qk =
{
α ∈ Fp,q : pwt(α) ≤ k
}
/Fp+1 .
The principal weight filtration on Grp,∗ is defined by the sequence
· · · ⊂ Fpk ⊂ Fpk+1 ⊂ · · · ⊂ Grp,∗ .
It follows immediately from equations (3.1) that this filtration is stable under dh.
For each fixed p > 0 there is a spectral sequence E associated to this filtration, with E0 page
given by
E
q,k
0 = Fp,qk /Fp,qk−1 .
This spectral sequence converges to Ep,∗1 .
In calculations, it will be convenient to abuse notation slightly, using Fpk again to denote the
preimage of Fpk under the mapping Fp → Grp,∗, that is,
Fpk =
{
α ∈ Fp : pwt(α) ≤ k
}
+ Fp+1
and
Fp,qk =
{
α ∈ Fp,q : pwt(α) ≤ k
}
+ Fp+1 .
Then for any element [Φ] ∈ Eq,k0 represented by a form Φ, the choice of Φ is uniquely defined
modulo Fpk−1. For example, if dhΦ is any representative of dh[Φ] ∈ Eq+1,k
0 , then
dhΦ ≡ dΦ
(
mod Fpk−1
)
.
The distinct usages occur in different contexts, and should not cause confusion.
14 B.B. McMillan
Any element of E
q,k
0 can be represented by a linear combination of (p+ q)-forms
θI1,t1 ∧ · · · ∧ θIp,tp ∧ωa1 ∧ · · · ∧ωaq
of principal weight exactly k – explicitly, those for which
|I1|+ t1 + · · ·+ |Ip|+ tp − q = k.
The exterior derivative dh is C∞(M)-linear on E
∗,k
0 , and treats the forms ωa as constants.
Indeed, by R-linearity of dh, it suffices to check that
dh(fθI1,t1 ∧ · · · ∧ θIp,tp ∧ωa1 ∧ · · · ∧ωaq)
≡ f dh(θI1,t1 ∧ · · · ∧ θIp,tp) ∧ωa1 ∧ · · · ∧ωaq
(
mod Fpk−1
)
,
but this follows from the observation that dh strictly decreases weight for functions and for
each ωa.
Corollary 3.4 can be restated in terms of the principal weight filtration:
Corollary 5.3. A function A ∈ C∞(M) factors through C∞
(
M
(r)
0
)
if and only if
dA ≡ 0
(
mod ωa,F1
r+2
)
.
Example 5.4 (sub-principal weight filtration). For parabolic systems, the principal weight
filtration doesn’t see “lower order” information, such as the sub-principal symbol. So, it will also
be necessary to introduce the sub-principal weight filtration, which sees further into the structure
of (M, I). The proof of Theorem 8.1 relies heavily on the sub-principal weight filtration.
Unlike the previous case, the sub-principal weight filtration on Grp,∗ will necessarily depend
on a choice of parabolic coframing. However, it will momentarily be shown that there are more
refined choices of parabolic coframing and that all such coframings define the same sub-principal
weight filtration.
Relative to a parabolic coframing, the sub-principal weight function is the unique weight
function wt such that
wt
(
ωi
)
= −1,
wt
(
ω0
)
= −2,
wt(θI,t) = |I|+ 2t.
For each integer k and each p ≥ 1, define the subspace
Fpk =
{
α ∈ Fp : wt(α) ≤ k
}
/Fp+1
of Grp,∗. The sub-principal weight filtration on Grp,∗ is defined by the sequence
· · · ⊂ Fpk ⊂ F
p
k+1 ⊂ · · · ⊂ Grp,∗ .
As with the principal weight filtration, it will be useful to have the notation
Fpk =
{
α ∈ Fp : wt(α) ≤ k
}
+ Fp+1
and
Fp,qk =
{
α ∈ Fp,q : wt(α) ≤ k
}
+ Fp+1 .
For a generic choice of parabolic coframing, the induced filtration is not automatically dh-stable.
So make the following definition.
Conservation Laws for a Class of Second-Order Parabolic Equations II 15
Definition 5.5. A parabolic coframing ωa, θI,t is a refined parabolic coframing of M if it satisfies
the refined structure equations
dθI,t ≡ −θIi,t ∧ωi − θI,t+1 ∧ω
0
(
mod F1
|I|+2t−1
)
. (5.1)
Before turning to the proof that refined parabolic coframings exist for evolutionary parabolic
systems, I state several useful properties of such coframings. Only the proof of the second
property relies on the parabolic system being evolutionary.
Proposition 5.6. Suppose given a refined parabolic coframing ωa, θI,t on an evolutionary
parabolic system and corresponding sub-principal weight filtration F .
1. The ideal FpS is dh-stable for each p > 0 and all S.
2. On FpS /F
p
S−1, the operator dh is C∞(M)-linear and treats the ωa as constants.
3. The higher symbol relations
θIii,t ≡ θI,t+1
(
mod F1
I+2+2t−1
)
hold for all I and t.
4. Any other choice of refined parabolic coframing gives the same sub-principal weight filtration.
Proof. 1. It suffices to check this for elements in FpS of the form
fθI1,t1 ∧ · · · ∧ θIp,tp ∧ωa1 ∧ · · · ∧ωaq .
The claim is clear from the structure equations (5.1) and from the fact that dh does not increase
weight for functions and the ωa.
2. This follows because dh strictly decreases sub-principal weight of functions and the ωa.
In particular, because the system is assumed evolutionary,
dhω
0 ≡ 0
(
mod ω0,F1
)
.
3. Equation (2.4), pulled back to M , gives
θii,0 = θ∅,1,
providing the base case for induction. If
θIii,t ≡ θI,t+1
(
mod F1
|I|+2+2t−1
)
holds, then applying d to both sides,
−θIjii,t ∧ωj − θIii,t+1 ∧ω
0 ≡ −θIj,t+1 ∧ω
j − θI,t+2 ∧ω
0
(
mod F1
|I|+2+2t−1
)
,
so an application of Cartan’s lemma gives
θIjii,t − θIj,t+1 ≡ 0
(
mod ωa,F1
|I|+2+2t
)
,
θIii,t+1 − θI,t+2 ≡ 0
(
mod ωa,F1
|I|+3+2t
)
.
But the left hand side is contained in I, so cannot have nonzero ωa terms, and thus
θIjii,t − θIj,t+1 ≡ 0
(
mod F1
|I|+2+2t
)
,
θIii,t+1 − θI,t+2 ≡ 0
(
mod F1
|I|+3+2t
)
.
16 B.B. McMillan
4. Let ω̃a, θ̃I,t be a second choice of refined parabolic coframing, with associated sub-principal
weight filtration F̃ . It suffices to show that for each weight N and sub-principal weight S,
F1,0
N ∩F
1,0
S = F1,0
N ∩F̃
1,0
S .
For N = 0, there is nothing to show, as F1,0
0 is spanned by θ∅,0 which is a multiple of θ̃∅,0.
For each weight N > 0,
F1,0
N ⊂ F
1,0
2N ,
for if θI,t is such that |I|+ t ≤ N , then |I|+ 2t ≤ 2N . Likewise,
F1,0
N ⊂ F̃
1,0
2N .
Thus, for each N ,
F1,0
N ∩F
1,0
2N = F1,0
N = F1,0
N ∩F̃
1,0
2N .
So, suppose N is the first weight for which the claim fails, and then S < 2N the largest
sub-principal weight such that
F1,0
N ∩F
1,0
S 6= F1,0
N ∩ F̃
1,0
S .
Then there are some θ̃Ii,t ∈ F1,0
N ∩ F̃
1,0
S and functions AJ,sIi,t for which
θ̃Ii,t ≡
∑
|J |+s≤N
|J |+2s>S
AJ,sIi,tθJ,s 6≡ 0
(
mod F1,0
N ∩F
1,0
S
)
and thus for θ̃I,t ∈ F1,0
N−1 ∩F̃
1,0
S−1 = F1,0
N−1 ∩F
1,0
S−1,
dθ̃I,t ≡ −θ̃Ii,t ∧ωi ≡
∑
AJ,sIi,tθJ,s ∧ω
i
(
mod ω0,F1,1
N−2 ∩F
1,1
S−2
)
.
The right hand side is not in F1,1
N−1 ∩F
1,1
S−1. However, F1
N−1 ∩F1
S−1 is dh-stable, so dθ̃I,t is
in F1,1
N−1 ∩F
1,1
S−1. This contradiction proves the statement. �
Proposition 5.7. Let (M, I) be the infinite prolongation of an evolutionary parabolic system.
There exist refined parabolic coframings.
Proof. It will suffice to inductively prove the following statement for each N > 0: there is
a parabolic coframing such that for each θI,t with pwt(θI,t) ≤ N ,
dθI,t ≡ −θIi,t ∧ωi − θI,t+1 ∧ω
0
(
mod F1
|I|+t−1 ∩F
1
|I|+2t−1
)
. (5.2)
The base case N = 1 follows immediately from the pullback to M of the refined structure
equations (2.3) for an evolutionary parabolic system.
Suppose the induction has been carried out up to N−1. Then, for each sub-principal weight k,
one has
d
(
F1,1
N−2 ∩F
1,1
k
)
⊂ F1,2
N−2 ∩F
1,2
k .
This follows because, modulo F2, any element of F1,1
N−2 ∩F
1,1
k is a linear combination of terms
θI,t∧ω
a such that pwt(θI,t) ≤ N − 1.
Conservation Laws for a Class of Second-Order Parabolic Equations II 17
It follows from the principal structure equations of Proposition 3.1 and from
F1,1
N−1 =
{
. . . , θ∅,N ∧ω
i
}
⊂ F1,1
2N−1
that
dθ0,N ≡ −θi,N ∧ωi − θ0,N+1 ∧ω
0
(
mod F1
N−1
)
,
≡ −θi,N ∧ωi − θ0,N+1 ∧ω
0
(
mod F1
N−1 ∩F1
2N−1
)
which provides the base case for a second induction.
Suppose then that (5.2) holds for all θJ,s with
pwt(θJ,s) = N, wt(θJ,s) > N + S.
Consider θIi,t of weights
pwt(θIi,t) = N, wt(θIi,t) = N + S.
From the principal structure equations, there are functions so that
dθIi,t ≡ −θIij,t ∧ωj − θIi,t+1 ∧ω
0 −
∑
G
(J,s)
(Ii,t),aθJ,s ∧ω
a
(
mod F1
N−1 ∩F1
N+S−1
)
,
the sum over indices J, s, a such that
pwt(θJ,s ∧ω
a) ≤ N − 1 and wt(θJ,s ∧ω
a) ≥ N + S.
From the first induction hypothesis,
dθI,t ≡ −θIi,t ∧ωi − θI,t+1 ∧ω
0
(
mod F1,1
N−2 ∩F
1,1
N+S−2
)
and thus (using now the second induction)
0 = d2θI,t ≡
∑
G
(J,s)
(Ii,t),aθJ,s ∧ω
a ∧ωi
(
mod F1,2
N−2 ∩F
1,2
N+S−2
)
,
so that
G
(J,s)
(Ii,t),j = G
(J,s)
(Ij,t),i and G
(J,s)
(Ii,t),0 = 0.
From this it follows that one may make the coframe modification
θIij,t 7−−−→ θIij,t −G(J,s)
(Ii,t),jθJ,s,
which absorbs all of the GJ,sIi,t,j while preserving the principal structure equations and the previous
induction steps. �
6 Horizontal and vertical derivatives
The differentials dh and dv in the characteristic spectral sequence have natural geometric
interpretations, which I describe here. They are essentially the same operators as defined in [3].
By definition, the operator dh is the restriction of d to the associated graded Grp,∗. More expli-
citly, given an equivalence class [α] ∈ Grp,∗ represented by α ∈ Fp, the horizontal derivative dh[α]
is represented by dα ∈ Fp, so that
dhα ≡ dα
(
mod Fp+1
)
.
18 B.B. McMillan
For any element [ϕ] ∈ F0 /F1, the operator dh is horizontal with respect to any solution
manifold Σ:
(dhϕ)|Σ = d(ϕ|Σ).
In particular, a function A ∈ C∞(M) is a functional on solution manifolds Σ, from which we
may define new functionals DaA so that
dhA ≡ (DaA)ωa
(
mod F1
)
.
Then from
(dhA)|Σ = d(A|Σ)
it follows that the operators Da are locally the restriction to M of the total derivatives Da
defined on J∞
(
Rn+1,R
)
.
Now consider the vertical derivative dv. Given a form α ∈ Fp such that dhα ≡ 0, the
derivative dα is an element of Fp+1. As such, define the vertical derivative of α by
dvα ≡ dα
(
mod Fp+2
)
.
Then, the horizontal derivative of this vanishes,
dhdvα ≡ dhdα ≡ d2α = 0
(
mod Fp+2
)
.
This last observation will be useful, because it is often easier to compute dhdv than dh, as the
operator dv aids one in filtering by weights.
It will be useful for calculations to extend the operator dv to all functions. This is accomplished
in an invariant manner using the Cartan system J = {θ∅,0, θa, ωa} =
{
ωa,F1,0
1
}
defined
in Remark 2.4, so that
dvA ≡ dA
(
mod J
)
.
The following lemma shows how to compute the vertical derivative of the directional total
derivatives of a function.
Lemma 6.1. Let M be the infinite prolongation of an evolutionary parabolic system. For any
function A ∈ C∞(M) with w = wt(dA), there are functions AI,s, not all zero, for which
dA ≡ (DaA)ωa +
∑
|I|+2s=w
AI,sθI,s
(
mod F1
w−1
)
.
Then
dv(DiA) ≡
∑
AI,sθIi,s
(
mod J ,F1
2,F1
w
)
,
dv(D0A) ≡
∑
AI,sθI,s+1
(
mod J ,F1
2,F1
w+1
)
.
Furthermore, if w ≥ 4, then
dv(DiA) ≡
∑
AI,sθIi,s
(
mod J ,F1
w
)
,
dv(D0A) ≡
∑
AI,sθI,s+1
(
mod J ,F1
w+1
)
.
Conservation Laws for a Class of Second-Order Parabolic Equations II 19
Proof. From the structure equations for ωi and ω0,
dωa ≡ 0
(
mod Λ2J ,F1
1
)
.
Then
0 = d2A ≡ d
(
(DaA)ωa +
∑
|I|+2s=w
AI,sθI,s
) (
mod F1
w−1
)
,
≡ dv(DaA)ωa −
∑
|I|+2s=w
AI,s
(
θIi,sω
i + θI,s+1ω
0
) (
mod Λ2J ,F1
1,F1
w−1
)
,
so that the result follows from an application of Cartan’s lemma. The last claim follows from the
observation that F1,0
2 ⊆ F1,0
4 . �
The previous lemma holds with principal weight replacing sub-principal weight. The proof is,
mutatis mutandis, the same, so omitted.
Lemma 6.2. Let M be the infinite prolongation of a parabolic system. For any function
A ∈ C∞(M), with N = pwt(dA) ≥ 2, there are functions AI,s, not all zero, for which
dA ≡ (DaA)ωa +
∑
|I|+s=N
AI,sθI,s
(
mod F1
N−1
)
.
Then
dv(DiA) ≡
∑
AI,sθIi,s
(
mod J ,F1
N
)
,
dv(D0A) ≡
∑
AI,sθI,s+1
(
mod J ,F1
N
)
.
Corollary 6.3. If f ∈ C∞
(
M
(r)
0
)
, then Daf ∈ C∞
(
M
(r+1)
0
)
.
Proof. For a function f ∈ C∞
(
M
(r)
0
)
, one has by Corollary 5.3,
dvf ≡ 0
(
mod J ,F1
r+2
)
,
and thus by the previous lemma,
dvDaf ≡ 0
(
mod J ,F1
r+3
)
,
so that Daf ∈ C∞
(
M
(r+1)
0
)
. �
7 Conservation laws of parabolic systems
The first step in calculating the space of conservation laws for a parabolic system is to com-
pute E1,n
1 . The statement of the following Theorem uses the omitted index notation, wherein for
an anti-symmetric index set I, the (n+ 1− |I|)-form ω(I) is defined by
ω(I) = ±
∏
a∈{0,...,n}−I
ωa,
with sign so that(∏
a∈I
ωa
)
∧ω(I) = +ω(∅) := ω0 ∧ · · · ∧ωn.
20 B.B. McMillan
Theorem 7.1. For an evolutionary parabolic system (M, I) with parabolic coframing as in Pro-
position 3.1, there are functions ai and an (n+ 1)-form
Υ = θi,0 ∧ω(i) − θ∅,0 ∧ω(0) + aiθ∅,0 ∧ω(i)
so that any element of E1,n
1 is represented by a form
Φ ≡ AΥ− (DiA)θ∅,0 ∧ω(i)
(
mod F2
)
,
where A is a function in C∞(M). The function A satisfies a differential constraint determined
by dhΦ ≡ 0.
The functions ai are in C∞
(
M
(1)
0
)
and are determined by the local invariants of M0. There
are functions W I,t ranging over |I|+ t = 2 so that
dvai ≡ V jkl
i θjkl,0 +W I,tθIi,t
(
mod J ,F2
2
)
.
Proof. It suffices to understand the principal weight spectral sequence E
∗,k
∗ (with p = 1), which
converges to E1,∗
1 . From the general theory of characteristic spectral sequences, the zeroth
page E0 is isomorphic to the Spencer complex of the tableaux K. This Spencer complex in turn
calculates the minimal free resolution of the symbol module associated to K. See [3], as well as
Chapter VIII of [2]. The parabolic symbol module has exactly 1 relation, and thus no relations
between relations, so the following part of E0 contains the only terms that don’t immediately
degenerate,
· · · E
n+1,−n+1
0 E
n+1,−n
0 E
n+1,−n−1
0
· · · E
n,−n+1
0 E
n,−n
0 0
· · · E
n−1,−n+1
0 0 0
dh dh
dh
Since dh is a C∞(M)-linear vector bundle map between these spaces, it suffices to compute
pointwise. Localizing at each point of M results in the following diagram of vector spaces,
R{θij,0∧ω(∅), θi,1∧ω(∅), θ∅,2∧ω(∅)} R{θi,0∧ω(∅), θ∅,1∧ω(∅)} R{θ∅,0∧ω(∅)}
R{θi,0∧ω(b), θ∅,1∧ω(b)} R{θ∅,0∧ω(a)} 0
R{θ∅,0∧ω(ab)} 0 0
dh dh
dh
Then it is clear that the E1 page is
0 0 C∞(M){θ0,0∧ω(∅)}
C∞(M){θi,0∧ω(i)} 0 0
0 0 0
(7.1)
Conservation Laws for a Class of Second-Order Parabolic Equations II 21
Now there is enough degeneracy to see that the E2 page is given by
0 0 C∞(M){θ0,0∧ω(∅)}
C∞(M){θi,0∧ω(i)} 0 0
0 0 0
δ
After this page the spectral sequence degenerates, so
E1,n
1 = ker(δ).
The operator δ determines a linear differential operator on functions A ∈ C∞(M), giving the
differential constraint for A to correspond to an element E1,n
1 .
In more concrete language, the calculation just done may be unwound as follows. For any
element Φ ∈ E1,n
1 , there are functions A and Aa in C∞(M) so that
Φ ≡ Aθi,0 ∧ω(i) +A0 θ∅,0 ∧ω(0) +Ai θ∅,0 ∧ω(i)
(
mod F2
)
and Φ is dh-closed. At highest sub-principal weight, one finds
0 = dhΦ ≡ −(A+A0)θ0,1 ∧ω(∅)
(
mod θ∅,0,F1
−n−1
)
,
so that A0 = −A. Then, with no restriction on weights,
0 = dhΦ ≡ Adh
(
θi,0 ∧ω(i) − θ∅,0 ∧ω(0)
)
− (DiA+Ai)θi,0 ∧ω(∅)
(
mod θ∅,0,F
2
)
.
Since
dh
(
θi,0 ∧ω(i) − θ∅,0 ∧ω(0)
)
≡ 0
(
mod θ∅,0, θi,0,F
2
)
,
there are functions a∅, ai ∈ C∞
(
M
)
so that
dh
(
θi,0 ∧ω(i) − θ∅,0 ∧ω(0)
)
≡ (a∅θ∅,0 + aiθi,0) ∧ω(∅)
(
mod F2
)
,
and it is clear that
Ai = −DiA+Aai.
This establishes that
Φ ≡ AΥ− (DiA)θ∅,0 ∧ω(i)
(
mod F2
)
.
Recall the structure equations (2.5), which can be restated in the language of weights as
dθi,0 ≡ −βji ∧ θj,0 − ξ
j
i ∧ θj,0 − θij,0 ∧ω
j − θi,1 ∧ω0
(
mod F2
2
)
,
dω0 ≡ β0
0 ∧ω
0
(
mod ω0
)
,
dωi ≡ −(δijκ∅ − βij) ∧ωj + ξij ∧ω
j
(
mod ω0,F1
0
)
,
with
ξji ≡ V
jkl
i θkl,0
(
mod J
)
22 B.B. McMillan
and
κ∅ ≡ β0
0 ≡ β
j
i ≡ 0
(
mod J ,F1
2
)
.
Using these, one computes
d(θi,0 ∧ω(i) − θ∅,0 ∧ω(0)) ≡ (a∅θ∅,0 + aiθi,0) ∧ω(∅) − 2V jkl
i θkl,0 ∧ θj,0 ∧ω(i)
+
(
β0
0 − κ∅ + (n− 1)βtr
)
∧ θi,0 ∧ω(i)
(
mod F2
2−n
)
.
Since β0
0 − κ∅ + (n− 1)βtr ∈ F1
2, there are functions W I,t for |I|+ t = 2 so that
β0
0 − κ∅ + (n− 1)βtr ≡ −
∑
W I,tθI,t
(
mod F1
1
)
.
Then
0 = d2(θi,0 ∧ω(i) − θ∅,0 ∧ω(0))
≡ (dai) ∧ θi,0 ∧ω(∅) − 2V ikl
j θjkl,0 ∧ θi ∧ω(∅) −
∑
W I,tθIi,t ∧ θi,0 ∧ω(∅)
(
mod θ∅,0,F
2
2−n
)
so that, up to an application of Cartan’s lemma,
dai ≡ 2V jkl
i θjkl,0 +
∑
W I,tθIi,t
(
mod J ,F1
2
)
. �
Remark 7.2. It will be useful below to note that in the course of the proof, it was shown that
dΥ ≡ dhΥ− 2V jkl
i (θkl,0 ∧ θj,0 − θjkl,0 ∧ θ∅,0) ∧ω(i)
−W I,t(θI,t ∧ θi,0 − θIi,t ∧ θ∅,0) ∧ω(i)
(
mod F2
2−n
)
.
8 Conservation laws of strongly parabolic systems
Finally, with the tools developed, it is not difficult to prove the following theorem.
Theorem 8.1. Let (M, I) be the infinite prolongation of an evolutionary parabolic system
(M0, I0) and Φ a differentiated conservation law on (M, I). The defining function A of Φ factors
through C∞
(
M0
)
.
Proof. By Proposition 7.1, a conservation law Φ has a defining function A ∈ C∞(M) so that
the I-linear part of Φ is given by
Φ ≡ AΥ− (DiA)θ∅,0 ∧ω(i)
(
mod F2
)
.
Since Φ is closed, dhΦ ≡ 0 and dvΦ is defined. Calculating directly,
dvΦ ≡ (dA) ∧Υ− d(DiA) ∧ θ∅,0 ∧ω(i)
(
mod F2
2−n,F2
2−n
)
. (8.1)
Here it is worth noting that only I-quadratic terms need be considered (the I-linear terms cancel
each other), and that the I-quadratic terms of dΥ were computed in Remark 7.2.
Now, using Corollary 5.3, it suffices to demonstrate that
pwt(dA) ≤ 2.
This is done by first bounding the sub-principal weight of dA. To this end, let
w = wt(dA),
Conservation Laws for a Class of Second-Order Parabolic Equations II 23
and N the largest integer for which
dvA 6≡ 0
(
mod J ,F1
N−1,F1
w−1
)
.
Then, for S = w −N , there are functions AI,S , BJ,S+1 so that
dvA ≡
∑
|I|+S=N
|I|+2S=w
AI,SθI,S +
∑
|J |+S+1=N−1
|J |+2(S+1)=w
BJ,S+1θJ,S+1
(
mod J ,F1
N−2,F1
w−1
)
.
(If |I| = N − S = 0, 1 then the second sum is vacuous, so let the BJ,S+1 = 0 in that case.) From
Lemma 6.1, if N ≥ 3
dv(DiA) ≡
∑
AI,SθIi,S +
∑
BJ,S+1θJi,S+1
(
mod J ,F1
N−1,F1
w
)
.
Suppose that either w ≥ 5 (in which case N ≥ 3 is automatic) or w = 4 and N ≥ 3, and plug
into equation (8.1),
dvΦ ≡ AI,S
(
θI,S ∧ θi,0 ∧ω(i) − θI,S ∧ θ∅,0 ∧ω(0) − θIi,S ∧ θ∅,0 ∧ω(i)
)
+BJ,S+1
(
θJ,S+1 ∧ θi,0 ∧ω(i) − θJ,S+1 ∧ θ∅,0 ∧ω(0) − θJi,S+1 ∧ θ∅,0 ∧ω(i)
)(
mod F2
N−n−1,F2
w−n−1
)
.
One then computes immediately
0 ≡ dhdvΦ ≡ −2AI,SθI,S+1 ∧ θ∅,0 ∧ω(∅)
(
mod F2
N−n−1,F2
w−n−1
)
,
so that each AI,S vanishes, contradicting the maximality of N .
It has just been shown that
dvA ≡ 0
(
mod J ,F1
2,F1
3
)
.
As such, there are functions Ajkl,0, Bj,1, and B∅,2 so that
dvA ≡ Ajkl,0θjkl,0 +Bj,1θj,1 +B∅,2θ∅,2
(
mod J ,F1
1,F1
2
)
.
The proof of Lemma 6.1 applies here. Indeed, from
0 = d2A ≡ (dv(DiA)−Ajkl,0θjkli,0 −Bj,1θji,1 −B∅,2θi,2) ∧ωi
(
mod ω0,Λ2J ,F1
1,F1
2
)
it is an application of Cartan’s lemma to obtain
dv(DiA) ≡ Ajkl,0θijkl,0 +Bj,1θji,1 +B∅,2θi,2
(
mod J ,F1
2,F1
3
)
.
Plugging into equation (8.1),
dvΦ ≡ Ajkl,0
(
θjkl,0 ∧ θi,0 ∧ω(i) − θjkl,0 ∧ θ∅,0 ∧ω(0) − θijkl,0 ∧ θ∅,0 ∧ω(i)
)
+Bj,1
(
θj,1 ∧ θi,0 ∧ω(i) − θj,1 ∧ θ∅,0 ∧ω(0) − θij,1 ∧ θ∅,0 ∧ω(i)
)
+B∅,2
(
θ∅,2 ∧ θi,0 ∧ω(i) − θ∅,2 ∧ θ∅,0 ∧ω(0) − θi,2 ∧ θ∅,0 ∧ω(i)
) (
mod F2
2−n,F2
2−n
)
and thus
0 ≡ dhdvΦ ≡ −2Aijk,0θijk,1 ∧ θ∅,0 ∧ω(∅)
(
mod F2
2−n,F2
2−n
)
.
It follows that each Ajkl,0 vanishes, which proves the theorem. �
24 B.B. McMillan
With the theorem just proven, it is not difficult to prove the following theorem.
Theorem 8.2. Given an evolutionary parabolic system M0 with a non-trivial conservation law Φ
on its infinite prolongation M , in any neighborhood of M0 where the defining function of Φ is
not zero, M0 has a Monge–Ampère deprolongation.
Proof. Suppose M0 has a non-trivial conservation law
Φ ≡ AΥ− (DiA)θ∅,0 ∧ω(i)
(
mod F2
)
.
By Theorem 8.1, A is a function on M0. Consequently, in any neighborhood where A is non-zero,
there are parabolic cofframings that reduce A to 1. For example, the change of coframing that
fixes the ωa but transforms the other coframing terms as
θ∅,0 7−−→ (1/A)θ∅,0, θi,0 7−−→ (1/A)θi,0, θ∅,1 7−−→ (1/A)θ∅,1,
πij,0 7−−→ (1/A)πij,0, πi,1 7−−→ (1/A)πi,1, π∅,2 7−−→ (1/A)π∅,2
preserves the evolutionary parabolic structure of (M0, I0), and after this reduction,
Φ ≡ Υ
(
mod F2
)
.
By Remark 7.2,
dvΦ ≡ −2V jkl
i (θkl,0 ∧ θj,0 − θjkl,0 ∧ θ∅,0) ∧ω(i)
−W I,t(θI,t ∧ θi,0 − θIi,t ∧ θ∅,0) ∧ω(i)
(
mod F2
2−n
)
.
Then, recalling that V jkl
i has values in S2
0(S2
0 Rn),
0 = dhdvΦ ≡ −2V jkl
i θijkl,0 ∧ θ∅,0 ∧ω(∅)
(
mod F2
2−n
)
,
and the S4
0 Rn component of the Monge–Ampère invariant vanishes identically. �
Acknowledgements
This material is based upon work supported by the National Science Foundation under Grants
No. DGE-1106400 and 74341.2010, as well as the Australian Research Council, Discovery Program
DP190102360.
References
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khov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and conservation laws for differential equations
of mathematical physics, Translations of Mathematical Monographs, Vol. 182, Amer. Math. Soc., Providence,
RI, 1999.
[2] Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems,
Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991.
[3] Bryant R.L., Griffiths P.A., Characteristic cohomology of differential systems. I. General theory, J. Amer.
Math. Soc. 8 (1995), 507–596.
[4] Bryant R.L., Griffiths P.A., Characteristic cohomology of differential systems. II. Conservation laws for
a class of parabolic equations, Duke Math. J. 78 (1995), 531–676.
[5] Clelland J.N., Geometry of conservation laws for a class of parabolic partial differential equations, Selecta
Math. (N.S.) 3 (1997), 1–77.
[6] McMillan B.B., Geometry and conservation laws for a class of second-order parabolic equations I: Geometry,
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[7] Vinogradov A.M., The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear
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https://doi.org/10.1090/mmono/182
https://doi.org/10.1007/978-1-4613-9714-4
https://doi.org/10.2307/2152923
https://doi.org/10.2307/2152923
https://doi.org/10.1215/S0012-7094-95-07824-7
https://doi.org/10.1007/s000290050005
https://doi.org/10.1007/s000290050005
https://doi.org/10.1016/j.geomphys.2020.103824
https://arxiv.org/abs/1810.00458
https://doi.org/10.1016/0022-247X(84)90071-4
https://doi.org/10.1016/0022-247X(84)90072-6
1 Introduction
2 Background
3 The prolongation of (M0, I0)
4 The characteristic cohomology
5 The weight filtrations
6 Horizontal and vertical derivatives
7 Conservation laws of parabolic systems
8 Conservation laws of strongly parabolic systems
References
|
| id | nasplib_isofts_kiev_ua-123456789-211302 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T11:23:54Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | McMillan, Benjamin B. 2025-12-29T11:06:08Z 2021 Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws. Benjamin B. McMillan. SIGMA 17 (2021), 047, 24 pages 1815-0659 2020 Mathematics Subject Classification: 35L65; 58A15; 35K10; 35K55; 35K96 arXiv:1810.02346 https://nasplib.isofts.kiev.ua/handle/123456789/211302 https://doi.org/10.3842/SIGMA.2021.047 I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First, I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservation laws of a given parabolic equation. This is used to show that conservation laws for any evolutionary parabolic equation depend on at most second derivatives of solutions. As a corollary, it is shown that the only evolutionary parabolic equations with at least one non-trivial conservation law are of Monge-Ampère type. This material is based upon work supported by the National Science Foundation under Grant No. DGE-1106400 and 74341.2010, as well as the Australian Research Council, Discovery Program DP190102360. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws Article published earlier |
| spellingShingle | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws McMillan, Benjamin B. |
| title | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_full | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_fullStr | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_full_unstemmed | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_short | Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws |
| title_sort | geometry and conservation laws for a class of second-order parabolic equations ii: conservation laws |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211302 |
| work_keys_str_mv | AT mcmillanbenjaminb geometryandconservationlawsforaclassofsecondorderparabolicequationsiiconservationlaws |