No-go theorem on reduction operators of linear second-order parabolic equations
The problem on description of the reduction operators, i.e. the operators of nonclassical (conditional) symmetry, of an arbitrary second-order linear parabolic partial differential equation in two independent variables proves to be equivalent, in some sense, to solving the initial equation.
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552966681001984 |
|---|---|
| author | Popovych, R. Попович, Р. |
| author_facet | Popovych, R. Попович, Р. |
| author_institution_txt_mv | [
{
"author": "R. Popovych",
"institution": null
}
] |
| author_sort | Popovych, R. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
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| datestamp_date | 2020-08-13T10:42:30Z |
| description | The problem on description of the reduction operators, i.e. the operators of nonclassical (conditional) symmetry, of an arbitrary second-order linear parabolic partial differential equation in two independent variables proves to be equivalent, in some sense, to solving the initial equation. |
| first_indexed | 2026-08-04T01:07:50Z |
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| fulltext |
Збiрник праць Iнституту математики НАН України 2006, т.3, N 2, 231–238
УДК 517.912:512.816
No-go theorem
on reduction operators of linear
second-order parabolic equations
R.O. POPOVYCH
Institute of Mathematics of NAS of Ukraine, Kyiv
Fakultät für Mathematik, Universität Wien
E-mail: rop@imath.kiev.ua
Доведено, що проблема опису всiх операторiв редукцiї, тобто опера-
торiв некласичної (умовної) симетрiї, довiльного лiнiйного рiвняння з
частинними похiдними другого порядку з двома незалежними змiнни-
ми еквiвалентна в певному сенсi розв’язанню вихiдного рiвняння.
The problem on description of the reduction operators, i.e. the operators
of nonclassical (conditional) symmetry, of an arbitrary second-order linear
parabolic partial differential equation in two independent variables proves
to be equivalent, in some sense, to solving the initial equation.
1. Introduction. The notion of nonclassical symmetry (called also
Q-conditional or, simply, conditional symmetry) was introduced in [1]
by the example of the one-dimensional linear heat equation and a par-
tial class of operators. A precise and rigorous definition was suggested
later (see e.g. [4,5,14]). In contrast to classical Lie symmetry, the system
of determining equations on the coefficients of conditional symmetry
operators of the heat equation was found to be nonlinear and less over-
determined. First this system was investigated in [12] in detail, where the
system was partially linearized and its Lie symmetries were found. The
problem on conditional symmetries of the heat equation was completely
solved in [3]. Namely, in the both arising cases the maximal Lie invari-
ance algebras of the determining equations were calculated and the
determining equations were reduced to the initial equation with nonlocal
transformations. Results of [3] were in [2,6,7] extended to a class of linear
transfer equations which generalize the heat equation. Thus, for these
equations the “no-go” theorems on reduction of determining equations
for coefficients of conditional symmetry operators to the initial equations
232 R.O. Popovych
were proved in detail and wide multi-parametric families of exact solu-
tions were constructed with non-Lie reductions. It was observed in [13]
that the proof of the theorem from [3] on linearization of determini-
ng equations in case of conditional symmetry operators with vanishing
coefficients of ∂t are extended to the class of one-dimensional evolution
equations. This theorem was also generalized to multi-dimensional evolu-
tion equations [8] and even systems of such equations [11].
In this paper we investigate the class of second-order linear parabolic
partial differential equation in two independent variables, which have the
general form
Lu = ut − a2(t, x)uxx − a1(t, x)ux − a0(t, x)u = 0, (1)
where the coefficients ai, i = 1, 2, 3, are (real or complex) analytic func-
tions of t and x, a2 6= 0. The “no-go” theorems on reduction of determi-
ning equations for coefficients of conditional symmetry operators to the
initial equations are proved for class (1). All possible reductions to ordi-
nary differential equations are described.
Conditional invariance of a differential equation with respect to a vec-
tor field is equivalent to that any ansatz associated with the vector
field reduces the equation to a differential equation with the lesser by
1 number of independent variables [14]. That is why, below we use the
shorter and more natural term “reduction operator” instead of “operator
of conditional symmetry” and say that an operator reduces a differential
equation in case the equation is reduced by the associated ansatz.
2. Determining equations for reduction operators. Prelimi-
nary description of reduction operators of equations (1) is given by the
following theorem.
Theorem 1. Any reduction operator of equation (1) is equivalent to ei-
ther an operator ∂t + g1(t, x)∂x + (g2(t, x)u+ g3(t, x))∂u with the coeffi-
cients g1, g2 and g3 satisfying the system
L̃g1 +Ha1 + a1
xg
1 + 2a2g2
x + a1
t = 0,
L̃g2 −Ha0 − a0
xg
1 − a0
t = 0, (2)
L̃g3 − a0g3 = 0,
where L̃ = ∂t − a2∂xx − a1∂x + H and H = 2g1
x − (a2
xg
1 + a2
t )/a
2, or
an operator ∂x + η(t, x, u)∂u, where the function η is a solution of the
Reduction operators of linear second-order parabolic equations 233
equation
ηt = a2(ηxx + 2ηηxu + η2ηuu) + a2
x(ηx + ηηu)
+ (a1η)x + a0(η − uηu) + a0
xu. (3)
Proof. In the case of two independent variables t and x, reduction ope-
rators are written as Q = τ(t, x, u)∂t + ξ(t, x, u)∂x + η(t, x, u)∂u, where
(τ, ξ) 6= (0, 0). The conditional invariance criterion for equation (1) and
the operator Q has the form [4]
Q(2)Lu
∣∣
Lu=0, Q[u]=0, DtQ[u]=0, DxQ[u]=0
= 0, (4)
where Q(2) is the standard second prolongation of Q, Q[u] = η−τut−ξux
is its characteristic, Dt and Dx denote the total differentiation operators
with respect to t and x. All equalities hold true as algebraic relations
in the second-order jet space J (2) over the space of the independent
variables (t, x) and the dependent variable u.
Since (1) is an evolution equation, there are two principally different
cases of finding Q: τ 6= 0 and τ = 0.
If τ 6= 0 we can assume τ = 1 up to the usual equivalence of reduction
operators. There is only one unconstrained variable in (4). We choose the
derivative ux as such variable and express the other derivatives being
in Q(2)Lu via (t, x, u) and ux on the constrained set of J (2):
ut = η − ξux, uxx =
η − ξux − a1ux − a0
a2
.
Splitting in the obtained equation with respect to ux results in the
determining equations for coefficients ξ and η which imply ξu = 0,
ηuu = 0, i.e. ξ = g1(t, x), η = g2(t, x)u + g3(t, x). Further splitting
with respect to u leads to system (2).
The condition τ = 0 gives ξ 6= 0 since (τ, ξ) 6= (0, 0). Therefore, wi-
thout loss of generality we can put ξ = 1 in view of the usual equivalence
of reduction operators. All derivatives being in Q(2)Lu are expressed on
the constrained set of J (2) via the variables (t, x, u):
ux = η, uxx = ηx + ηηu, ut = a2(ηx + ηηu) + a1η + a0u.
After substituting these expressions to the equation Q(2)Lu = 0, we
obtain equation (3).
234 R.O. Popovych
Note 1. We can essentially simplify and order investigation of reduction
operators, additionally taking into account Lie symmetry transformati-
ons in case of a single equation [9] and transformations from the equi-
valence group or the whole set of admissible transformations in case of
a class of equations [10]. Up to the equivalence relation generated by
the equivalence group of class (1) on the set of pairs “(an equation of
form (1), its reduction operator)”, it is enough to investigate only the
subclass of equations (1) with a2 = 1, a1 = 0.
3. No-go theorems. There is a connection of system (2) and equati-
on (3) with initial equation (1) via non-point transformations.
Theorem 2. Nonlinear coupled system (2) is reduced by the transfor-
mation
g1 = −a2 v
1v2
xx − v1
xxv
2
v1v2
x − v1
xv
2
− a1, g2 = −a2 v
1
xv
2
xx − v1
xxv
2
x
v1v2
x − v1
xv
2
+ a0,
g3 =
a2
v1v2
x − v1
xv
2
∣∣∣∣∣∣
v1 v1
x v1
xx
v2 v2
x v2
xx
v3 v3
x v3
xx
∣∣∣∣∣∣
(5)
to the uncouple linear system of three copies vit−a2vixx−a1vix−a0vi = 0
of equation (1) for the functions vi = vi(t, x), and the functions v1 and
v2 being linearly independent. Hereafter i = 1, 2, 3.
Note 2. Let W (ϕ1, . . . , ϕn) = det(∂l−1ϕk/∂xl−1) nk,l=1 be the Wronski-
an of the functions ϕk = ϕk(t, x), k = 1, n, with respect to the variable x.
Then transformation (5) can be rewritten as
g1 = −a2 (W (v1, v2))x
W (v1, v2)
− a1, g2 = −a2W (v1
x, v
2
x)
W (v1, v2)
+ a0,
g3 = a2W (v1, v2, v3)
W (v1, v2)
.
Proof. For any tuple of functions gi = gi(t, x) there exists functions vi
determined by (5). Really, relations (5) can be rewritten in the form
Q̂v1 = 0, Q̂v2 = 0, Q̂v3 = g3, (6)
where Q̂ = Q̄−L = a2∂xx+(a1+g1)∂x+a0−g2. (Here Q̄ = ∂t+g
1∂x−g2
and L = ∂t − a2(t, x)∂xx − a1(t, x)∂x − a0(t, x) are linear differential
Reduction operators of linear second-order parabolic equations 235
operators acting in the space of functions of (t, x). Q̄ is associated wi-
th the operator Q and L is taken from equation (1).) System (6) with
respect to vi is a system of three uncoupled second-order linear ordi-
nary differential equations with the independent variable x, and t being
treated as a parameter. We can take any fundamental tuple of solutions
of the equation Q̂v = 0 as (v1, v2) and any partial solution of the equati-
on Q̂v = g3 as v3. The functions vi are determined by (6) ambiguously,
namely up to the transformations
ṽp = ϕpq(t)vq, ṽ3 = v3 + ψq(t)vq or
vp = ϕ̃pq(t)ṽq, v3 = ṽ3 + ψ̃q(t)ṽq,
where ϕpq and ψq are arbitrary functions of t, |ϕpq| 6= 0, (ϕ̃pq) = (ϕpq)−1,
ψ̃q = −ψpϕ̃pq. Hereafter p, q = 1, 2 and the summation over the repeated
indices is implied.
Let (v1, v2, v2) be a fixed solution of (6), where the parameter-functi-
ons gi = gi(t, x) satisfy system (2). We will show that the functions ϕpq
and ψq (or ϕ̃pq and ψ̃q) can be chosen in such way that the functions
ṽi = 0 will satisfy equation (1), i.e. Lṽi = 0.
The left parts of equations (2) can be rewritten with representati-
on (5) as
R1 = a2 v
2
v1
R̄1 + a2 v
1
v2
R̄2, R2 = a2 v
2
x
v1
R̄1 + a2 v
1
x
v2
R̄2, R3 = Q̂Lv3,
where
R̄1 =
(
W (v1, Lv1)
W (v1, v2)
)
x
=
(
(Lv1/v1)x
(v2/v1)x
)
x
,
R̄2 =
(
W (v2, Lv2)
W (v2, v1)
)
x
=
(
(Lv2/v2)x
(v1/v2)x
)
x
.
Two first equations R1 = R2 = 0 of (2) is a linear system of algeb-
raic equations with respect to the values R̄1 and R̄2 with the non-vani-
shing determinant (a2)2W (v1, v2)/(v1v2). Therefore, its unique solution
is R̄1 = R̄2 = 0. Integration the latter equations and the equation R3 =
Q̂Lv3 = 0 with respect to x leads to the conclusion that Lvi = ζip(t)vp,
where ζip are functions of t. If the functions ϕ̃pq and ψ̃q being in the
“ambiguity” transformations satisfy the system of ODEs ϕ̃pqt = ζpq
′
ϕ̃q
′q,
ψ̃qt = ζ3q′ϕ̃q
′q then Lṽi = 0.
236 R.O. Popovych
And vice versa, if the functions vi = vi(t, x) satisfy the equation
Lv = 0, and the functions v1 and v2 being linearly independent, then
the corresponding expressions R1, R2 and R3 vanish identically. It means
that the functions gi = gi(t, x) determined by (5) give a solution of
system (2).
Theorem 3. Nonlinear equations (3) is reduced by composition of the
nonlocal substitution η = −Φx/Φu, where Φ is a function of (t, x, u), and
the hodograph transformation
the new independent variables: t̃ = t, x̃ = x, κ = Φ,
the new dependent variable: ũ = u (7)
to the equation Lũ = ũt̃ − a2(t̃, x̃)ũx̃x̃ − a1(t̃, x̃)ũx̃ − a0(t̃, x̃)ũ = 0, where
κ plays the role of a parameter.
We do not adduced the proof of theorem 3 since it has already proved
for both some equations [3] or subclasses [6] from class (1) and much more
general classes of evolution equations [8, 11,13].
4. Ansatzes and solutions. Inverting of theorem 3 gives the followi-
ng true statement [8, 11].
Theorem 4. For any one-parametric family of solutions of equation (1)
there exists an operator ∂x + η(t, x, u)∂u which reduces equation (1) and
with respect to which the family of solutions is invariant.
An ansatz associated with the operator ∂x+η∂u, where η = −Φx/Φu
and the function Φ = Φ(t, x, u) is obtained from the one-parametric
solution u = f(t, x,κ) of equation (1) with the inverse hodograph trans-
formation to (7), has the form u = f(t, x, ϕ(ω)), ω = t. Here ω and ϕ are
the new (“invariant”) independent and dependent variables correspon-
dingly. Substitution of the ansatz to equation (1) implies the reduced
equation ϕω = 0, i.e. ϕ = C = const. As a result, we obtain the obvious
one-parametric solution u = f(t, x, C).
Theorems 3 and 4 can be united to the single statement.
Theorem 5. For any equation of form (1), there exists one-to-one
correspondence between one-parametric families of its solutions and re-
duction operators with zero coefficients of ∂t. Namely, each operator of
such kind corresponds to the family of solutions which are invariant
Reduction operators of linear second-order parabolic equations 237
with respect to this operator. The problems of construction of all one-
parametric families of solutions of equation (1) and complete descripti-
on of its reduction operators with zero coefficients of ∂t are completely
equivalent.
An ansatz associated with the operator ∂t + g1∂x + (g2u+ g3)∂u, has
the form u = v1ϕ(ω) + v3, ω = v2/v1. Here ω and ϕ again denote the
new (“invariant”) independent and dependent variables. The functions
vi = vi(t, x) are solutions of equation (1), which are connected with the
reduction operator coefficients gi = gi(t, x) via transformation (5), and
v1 and v2 being linearly independent. Integration of the corresponding
reduced equation ϕωω = 0 gives ϕ = C2ω + C1, where C1 and C2 are
arbitrary constants. After substituting the expression for ϕ to the ansatz,
we obtain the two-parametric solution
u = C1v
1 + C2v
2 + v3. (8)
And vice versa, given a two-parametric solution of form (8), the coeffi-
cients g1, g2 and g3 are unambiguously repaired by formula (5).
Supposed triviality of the above ansatzes and reduced equations is
connected with usage of the special representations for the solutions of
the determining equations. Under this approach difficulties in constructi-
on of ansatzes and integration of reduced equations are replaced by dif-
ficulties in obtaining of the representations for coefficients of reduction
operators.
4. Conclusion. The “no-go” results of this paper can be extended
with investigation of Lie symmetries and Lie reductions of determini-
ng equations (2) and (3). Indeed, the maximal Lie invariance algebras
of (2) and (3) are isomorphic to the maximal Lie invariance algebras of
equation (1). This result is well known for the linear heat equation [3].
Let us note also that the term “no-go” has to be treated only as
impossibility of exhaustive solving of the problem. At the same time,
imposing additional constraints on the coefficients, one can construct
a number of particular examples of reduction operators and then apply
them to finding exact solutions of the initial equation. Since the determi-
ning equation has more independent variables and, therefore, more free-
dom degrees, it is more convenient often to guess a simple solution or a
simple ansatz for the determining equation, which can give a parametric
set of complicated solutions of the initial equation. (It is similar to si-
tuation with Lie symmetries of first-order ordinary differential equa-
238 R.O. Popovych
tions.) This approach was applied to interesting subclass of equations (1),
which arises under symmetry reduction of the Navier–Stokes equati-
ons [2,6,7], and allowed to construct series of multi-parametric solutions.
The research of the author was supported by the Austrian Science
Fund (FWF), Lise Meitner project M923-N13.
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|
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| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:07:50Z |
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| spelling | oai:trim.imath.kiev.ua:article-4672020-08-13T10:42:30Z No-go theorem on reduction operators of linear second-order parabolic equations Заборонна теорема щодо редукції операторів лінійних параболічних рівнянь другого порядку Popovych, R. Попович, Р. The problem on description of the reduction operators, i.e. the operators of nonclassical (conditional) symmetry, of an arbitrary second-order linear parabolic partial differential equation in two independent variables proves to be equivalent, in some sense, to solving the initial equation. Доведено, що проблема опису всіх операторів редукції, тобто операторів некласичної (умовної) симетрії, довільного лінійного рівняння з частинними похідними другого порядку з двома незалежними змінними еквівалентна в певному сенсі розв'язанню вихідного рівняння. Інститут математики НАН України 2006-11-14 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/467 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 3 No. 2 (2006): Symmetry and Integrability of Equations of Mathematical Physics (Dedicated to the 70-th Anniversary of Professor W.I. Fushchych); 231-238 Сборник Трудов Института математики НАН Украины; Том 3 № 2 (2006): Симетрія та інтегровність рівнянь математичної фізики (До 70-річчя від дня народження Вільгельма Ілліча Фущича); 231-238 Збірник Праць Інституту математики НАН України; Том 3 № 2 (2006): Симетрія та інтегровність рівнянь математичної фізики (До 70-річчя від дня народження Вільгельма Ілліча Фущича); 231-238 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/467/463 Авторське право (c) 2020 Р. Попович http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Popovych, R. Попович, Р. No-go theorem on reduction operators of linear second-order parabolic equations |
| title | No-go theorem on reduction operators of linear second-order parabolic equations |
| title_alt | Заборонна теорема щодо редукції операторів лінійних параболічних рівнянь другого порядку |
| title_full | No-go theorem on reduction operators of linear second-order parabolic equations |
| title_fullStr | No-go theorem on reduction operators of linear second-order parabolic equations |
| title_full_unstemmed | No-go theorem on reduction operators of linear second-order parabolic equations |
| title_short | No-go theorem on reduction operators of linear second-order parabolic equations |
| title_sort | no-go theorem on reduction operators of linear second-order parabolic equations |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/467 |
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