Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics

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2 Invariant Subspaces: Mathematics in 1D 93

Here, we use the notation C = (C

, ..., C

)

∈ IR

for constant vectors, and

C(t) = (C

(t), ..., C

(t))

for sufﬁciently smooth vector-functions. For all the

equations, we are looking for solutions in the standard form

u(x , t) =



i=1

(t) f

(x ), (2.153)

where the functional set {f

(x )} is linearly independent.

2.8.1 Invariant and partially invariant subspaces

• Invariant subspaces. Consider the equation

l[u] = F[u]. (2.154)

Here, l is a linear operator that depends on t and includes only derivatives in t, while

F is a nonlinear operator that depends on x and includes only derivatives in x.A

linear subspace is





i=1

(x ) : C = (C

, ..., C

)

∈ IR



. (2.155)

If W

is invariant with respect to F,i.e.,F[W

] ⊆ W

,or,whichisthesame,





i=1

(x )





i=1



[C] f

(x ) for any C,

where 

[C] = 

, ..., C

) are some algebraic functions, then (2.154) possesses

solutions (2.153) with the coefﬁcients satisfying the dynamical system

l[C

(t)] = 

[C(t)], i = 1, ..., n. (2.156)

This is the most frequently occurring instance throughout this text.

•Partially invariant subspaces. For the subspace (2.155), there exists an (invariant)

set M ⊂ W

that is deﬁned by a system of algebraic equations

[C] = 0fork = 1, ..., r,

such that F[M] ⊂ W

. Solutions (2.153) are determined by



l[C

(t)] = 

[C(t)], i = 1, ..., n,

[C(t)] = 0, k = 1, ..., r.

Example 2.53 The operator

F[u] = u

+ (u

)

+ u

− 1

admits the invariant subspace W

= L{1, cos x, sin x}. It is checked that, for any

constants {C

F[C

+ C

cosx + C

sin x] = C

+ C

− 1

+(2C

− 1)C

cos x + (2C

− 1)C

sin x ∈ W

The subspace W

= L{cosx, sin x} is partially invariant with respect to F:

F[C

cosx + C

sin x] = C

+ C

− 1 − C

cos x − C

sin x ∈ W

iff C

+ C

= 1.

94 Exact Solutions and Invariant Subspaces

Example 2.54 For the operator

F[u] = u



1 −



)

+ 2



1 +



there exists the partially invariant subspace W

= L{1, x

F[C

+ C

] = 2C

+ 2C

2(C

+1)

∈ W

iff C

=−1.

2.8.2 Invariant and partially invariant modules

• Invariant modules. The equation takes the form

∗

[u] = 0, (2.157)

where F

∗

is a nonlinear operator that depends on x and t and includes derivatives in

these variables. The set of linear combinations





i=1

(t) f

(x )



(2.158)

is a module, [373, Ch. III]. If W

is invariant with respect to F

∗

, F

∗

] ⊆ W

,or

∗





i=1

(t) f

(x )





i=1



[C(t)] f

(x ) for any C(t),

where 

[C(t)] = 

(t, C(t), C



(t), ...) are some differential operators, then, for

solutions (2.153) of equation (2.157), there occurs the system



[C(t)] = 0, i = 1, ..., n. (2.159)

If the module W

is invariant with respect to operator F

∗

,thePDE

∂

∂t

= F

∗

[u],

where q is greater than the maximal order of the time derivatives that are available

in F

∗

, possesses solutions (2.153) with coefﬁcients that solve the DS

(q)

(t) = 

[C(t)], i = 1, ..., n.

• Partially invariant modules. If there exists a set M ⊂ W

that is deﬁned by

[C(t)] = 0fork = 1, ..., r,

such that F[M] ⊂ W

, then, instead of (2.159), solutions (2.153) are obtained from

the system





[C(t)] = 0, i = 1, ..., n,

[C(t)] = 0, k = 1, ..., r,

with 

[C] = 

(t, C(t), C



(t), ...) and χ

[C] = χ

(t, C(t), C



(t), ...) being dif-

ferential operators.

Example 2.55 For the operator

F[u] = u

−

there exists the invariant module W

= L{1, x , x

, x

F[C

+ C

x + C

+ C

] = C



− 3C







− C



− 9C









− 5C



− 6C









− 3C





∈ W

2 Invariant Subspaces: Mathematics in 1D 95

Example 2.56 For the operator

F[u] = u

− u

, (2.160)

there exists the partially invariant module W

= L{1, e

, xe

F[C

+ C

] =−(C

+ 2C



−C



+ (C

+ C



∈ W

iff (C

+ C



= 0.

2.8.3 Further generalizations

• As above, consider the equation (2.157) and the module (2.158). Let there exist

another module





i=1

(t)

(x )



with basis {

(x )} and expansion coefﬁcients {

(t)}.IfF[W

] ⊆

,i.e.,forany

C(t), there exist differential operators 

[C(t)] ≡ 

(t, C(t), C



(t), ...), such that

∗





i=1

(t) f

(x )





i=1



[C(t)]

(x ),

then, for solutions (2.153), there occurs the system



[C(t)] = 0, i = 1, ..., m.

• If there exists a set M ⊂ W

that is deﬁned by

[C(t)] = 0fork = 1, ..., r,

such that F[M] ⊂

, then, for the coefﬁcients of solutions (2.153), there occurs

the system





[C(t)] = 0, i = 1, ..., m,

[C(t)] = 0, k = 1, ..., r.

• Another straightforward generalization is achieved when the bases of modules W

and

are functions of both independentvariables x and t,i.e.,





i=1

(t) f

(x , t)



and





i=1

(t)

(x , t)



The variables x and t can also be vectors, x = (x

, ..., x

) and t = (t

, ..., t

Example 2.57 For the operator (2.160), there exist the modules

= L{e

, e

−x

, e

} and

= L{1, e

, e

}, such that

F[C

+ C

−x

+ C

]

=−2(C

)



−



3(C

)



+ C



+ 9C





3(C

)



− C



− 9C



∈

Example 2.58 For the operator

F[u] = u

xxt

+ uu

xxx

− u

= L{x, cos(γ (t)x ), sin(γ (t)x)} and

= L{cos(γ (t)x), sin(γ (t)x), x cos(γ (t)x), x sin(γ (t)x)},

(see Example 1.37), we have

96 Exact Solutions and Invariant Subspaces

where γ(t) is a smooth function. In this case,

F[C

x + C

cos(γ x ) + C

sin(γ x )]



− (C

)



cos(γ x ) +



− (C

)



sin(γ x )

+γ

(γ



+ γ C

) x [C

sin(γ x ) − C

cos(γ x)] ∈

Notice that, at the same time, W

can be considered as a partially invariant module:

F[W

] ⊆ W

iff C

=−



Remarks and comments on the literature

§2.1.We follow [541, 544]. A survey on some results connected with applications of invariant

subspaces to evolution PDEs is given in [546]. Extended surveys on this and other methods of

reduction of differential equations can be found in [127, 312, 445].

§2.2.A general scheme for main proofs is explained in [543]; see [544, 545] for the statements

of the results and discussion.

§ 2.3–2.4. Concerning these results, see [545].

§2.6.We follow [547]. A detailed analysis of the structure of the set of polynomial operators,

preserving polynomial subspaces, based on the notion of deﬁciency, is fulﬁlled in [257]. The

description of translation-invariant quadratic operators presented in Section 2.6.4 is similar to

that given in [257, Sect. 6].

Open problems

• Given an arbitrary subspace (2.1), describe the set



n−1

of translation-invariant

operators preserving W

. [In Section 2.6.3, this is done for polynomial subspaces.

The whole set of operators (2.3) preserving W

is given by Theorem 2.1.]

• Perform a general description of the set of translation-invariant homogeneous

polynomialoperators of a given degree p ≥ 3 that preserve the polynomialsubspace

(2.96). [For quadratic operators, see Theorem 2.35 and [257, Sect. 6].]

• The same problem for quadratic and higher-degree operators preserving sub-

spaces of trigonometric and exponential types. [For quadratic operators with expo-

sitions 1.29–1.32, 1.41, 1.44, and 1.45.]

• Prove that a nonlinear differential operator (2.3) preserving a subspace of the

maximal dimension is necessarily quadratic. [In Sections 2.3 and 2.4 this was proved

for operators of the ﬁrst and second order.]

• Perform a full description of second-order operators (2.3) preserving invariant

subspaces of the maximal dimension. [For translation-invariant operators, see Sec-

tion 2.4.]

• Perform the previous description for operators of orders greater than two.

• See Conjecture 2.1 in Section 2.6.1.

nential subspaces, see Example 1.49; other particular cases are considered in Propo-

CHAPTER 3

Parabolic Equations in One Dimension: Thin

Film, Kuramoto-Sivashinsky, and Magma

Models

In the following three chapters, we use invariant subspaces for various higher-order opera-

tors and PDEs in one space dimension. More attention is now paid to models of physical

importance and evolution properties of the exact solutions. In the current chapter, we will deal

mainly with quasilinear parabolic PDEs, including the well-known thin ﬁlm and Kuramoto–

Sivashinsky models. We also consider the lesser known magma equation, generating an in-

teresting family of formally pseudo-parabolic models. These classes of PDEs are associated

with different areas of mechanics and physics. Using the exact solutions makes it possible to

detect new and unusual properties of nonlinear models. Mathematical existence, uniqueness,

and regularity theory of many higher-order PDEs to be studied is still a ways from being com-

plete. Their solutions on ﬁnite-dimensional invariant subspaces yield an exceptional oppor-

tunity for describing key evolution aspects of such problems, including singularity formation

phenomena of blow-up and extinction. Whenever possible and reasonable, we discuss some

mathematical aspects of these nonlinear problems slightly more rigorously, and state, when it

seems necessary and interesting,

OPEN PROBLEMS

For most 1D higher-order ordinary differential operators, the results on invariant subspaces

can be extracted from the Main Theorem in Section 2.1 and from other results of Chapter 2.

Nevertheless, dealing with simple polynomial or trigonometric subspaces for typical fourth or

higher-order operators, sometimes we include direct simple calculations which help to reveal

some additional features of the associated PDEs of parabolic type. Such an easy geometric

analysis often gives straightforward extensions of the models, admitting invariant subspaces,

and thus avoiding the use of technical manipulations via the Main Theorem.

3.1 Thin ﬁlm models and solutions on polynomial subspaces

We ﬁrst deal with a class of quasilinear fourth and higher-order operators, which,

since the 1980s, began to play a determining role in the theory of nonlinear degen-

erate parabolic PDEs. It is not an exaggeration to say that nowadays higher-order

nonlinear degenerate thin ﬁlm equations are of the same fundamental importance as

the second-order porous medium equations used in the 1950s-80s.

3.1.1 Typical quasilinear models from thin ﬁlm theory

Fourth-order thin ﬁlm equations. Thin ﬁlm ﬂows correspond to the case where the

size of the ﬂow domain in one direction is essentially smaller in comparison with that

in the other directions. This enables to derive simpliﬁed models from the Navier–

98 Exact Solutions and Invariant Subspaces

Stokes equations. Namely, coupling to boundary conditions on the ﬂuid-substrate

interface and ﬂuid surface yields a single scalar equation for the ﬂuid height. The

quasilinear parabolic fourth-order thin ﬁlm equation (TFE)

=−



xxx



, with a ﬁxed exponent n = 0, (3.1)

is key in thin ﬁlm theory. For n = 0, this is the well-known linear bi-harmonic

equation

=−u

xxxx

. (3.2)

The connection and certain similarities between (3.1) and (3.2) will be important in

what follows.

The quasilinear TFE (3.1) with n = 3, as well as

)

=−3



xxx



, (3.3)

arise from Reynolds’ equation for Stokes’ ﬂow

= (u

)

, or u

=∇

· (u

∇

p)(n = 3). (3.4)

This describes slow sliding or spreading of thin Newtonian liquid drops (ﬁlm) over

a horizontal surface in the presence of high surface tension. Here, u = u(x , t) ≥ 0

denotes the height of the droplet, and p = p(x , t) stands for the pressure of the

ﬂuid. For n = 3, this is precisely Reynolds’ equation from lubrication theory. It also

applies to the case n = 2, where the ﬂuid is able to “slip” over the solid.

The case n = 1 corresponds to ﬂows in a porous medium or in a Hele–Shaw

cell which is formed by two immiscible ﬂuids that are separated by a thin interface

with thickness 2u. The dependence of p upon u (and its spatial derivatives) is of

importance in establishing the force driving the spreading of the droplet. The fourth-

order TFE (3.1) corresponds to capillary driven ﬂow of a thin droplet of thickness

u(x , t), so that, under suitable rescaling, the pressure is

p =−u

. (3.5)

Equation (3.3) correspondsto steady capillary-gravity driven ﬂows down a substrate

at an angle α to the horizontal in a special (x , y)-geometry.

∗

Illustrating a typical derivation of thin ﬁlm models, consider a Hele–Shaw ﬂow

between two parallel plates separated by a ﬁxed distance h

. This is governedby two

PDEs,



conservation of mass: u

+ (uv)

= 0, and

Darcy’slaw: v =−

12µ

(3.6)

where v is the average velocity of the ﬂuid in the ﬁlm, µ is the ﬂuid viscosity, and p

is the pressure. If the ﬂow is governedby the surface tension, then

=−γκ ≡−γ u

(cf. (3.5)), where γ is the surface tension and κ is the curvatureof the surface.Substi-

tuting p

into the second equation in (3.6) and the resulting v into the ﬁrst equation

∗

As usual, we put extra references on this subject in the Remarks.

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 99

yields the PDE

γ h

12µ

(uu

xxx

)

= 0.

This is (3.1) with n = 1, where the constant is scaled out. The case n = 2represents

Navier-slip-dominated Stokes ﬂows of thin ﬁlm. Then the analog of (3.3) has the

form

)

=−2



xxx



Taking into account other monomial differential terms occurring after differenti-

ating in the fourth-order operators in the above models, let us introduce generalized

TFEs comprising a family of homogeneous operators

= F[u] ≡−



xxx

+ βu

n−1

+ γ u

n−2

)



, (3.7)

where β and γ are constants. These PDEs share the scaling and homogenuity prop-

erties of the standard TFE, which is important for our invariant subspace analysis.

More models containing other terms and operators appear in lubrication theory,

where the long-wave unstable lubrication equations take the form

=−



xxx



−





, with n > 0, m > 0. (3.8)

In general, the lubrication approximation that describes the motion of long-wave

unstable thin ﬁlms deals with equations of the same type (3.4) for which the homo-

geneous nonlinearity u

(with typically n = 3) is replaced by a perturbed function

+ βu

with β>0andk ∈ (0, 3). This corresponds to classes of slip boundary

conditions, while the abovecase, β = 0, reﬂects the no-slip conditions.Furthermore,

instead of (3.5), the pressure p in the thin ﬁlm is given by

p =−u

+ G(u),

where the term G(u) describes forces exerted on the ﬁlm, e.g., hydrostatic body

forces or intermolecular forces due to van der Waals interaction. Such models lead

to a more general class of the fourth-order parabolic PDEs, such as

=−( f (u)u

xxx

)

− (g(u)u

)

(3.9)

with given functions f and g, or to extended models containing other terms. As was

mentioned, a typical nonlinearity in the main higher-orderterm is

f (u) = αu

+ βu

with an extra exponent k > 0 and positive constants α and β. Such PDEs also occur

as stable and unstable Cahn–Hilliard models in phase transition, ﬂuid interfaces,

and rupture of thin ﬁlms. They can describe thin jets in Hele–Shaw cells and ﬂuid

droplets hanging from a ceiling. In the semilinear case, n = 0, these equations are of

themodiﬁedKuramoto–Sivashinskytypethat describeﬂame propagationandseveral

other phenomena, such as solidiﬁcation of a hyper-cooled melt.

More lower-order terms enter the Benney equation that describes the nonlinear

dynamics of the interface of 2D liquid ﬁlms ﬂowing on a ﬁxed inclined plane,

)

+ ε



−

cotθ h



+ % h

xxx



= 0, (3.10)

100 Exact Solutions and Invariant Subspaces

where R is the unit-order Reynolds number of the ﬂow driven by gravity, σ is the

rescaled Weber number (related to surface tension σ), θ is the angle of plane inclina-

tion to the horizontal,and ε =

& 1, with d being the average thickness of the ﬁlm

and λ the wavelength of the characteristic interfacial disturbances.

Another class of TFEs in 1D includes a two-parameter family of models, such as

=−



)

xxx



+ (lower-order terms), (3.11)

where n and s are some ﬁxed exponents. We will deal with some of these PDEs

which can be reduced to quadratic or cubic representations.

Concerning multi-dimensional TFEs with non-power nonlinearities, as a typical

example, consider the PDE

+∇·



−Gh

BMh

2P(1+Bh)



∇h



+ S ∇·(h

∇h) = 0 (3.12)

that describes, in the dimensionless form, the dynamics of a ﬁlm in IR

subject to the

actions of thermocapillary, capillary, and gravity forces. Here, G, M, P, B,andS

are the gravity, Marangoni, Prandtl, Biot, and inverse capillary numbers respectively.

Another important related fourth-order model with similar operators in IR

or IR

is the Cahn–Hilliard equation applied in the study of phase separations in cooling

binary solutions, such as alloys, glasses, and polymer mixtures, where the pattern

formation phenomena occur,

=−∇·( f (u)∇u + g(u)∇u) in IR

× IR

Here, f (u) and g(u) are given coefﬁcients which are typically quadratic or cubic

functions. Here, u measures the difference in mass fractions of the two components

of the alloy.

The 1D TFE (3.1) and other similar models exhibit some exceptional mathemat-

ical properties. In particular, some free-boundary problems (FBPs) for (3.1) admit

nonnegative solutions, which is a natural and desirable property for thin ﬁlm ap-

plications. Recall that, typically, u(x, t) measures the height of the ﬁlm and is not

intended to change sign (at least, signiﬁcantly). Mathematical theory of such fourth

and higher-order TFEs has been originated in the pioneering work by Bernis and

Friedman [44], who developed new ideas, techniques, and initiated further system-

atic mathematical study of weak solutions of the higher-order quasilinear degenerate

parabolic PDEs. Further references are put into the Remarks.

Sixth-order thin ﬁlm equations. Thin ﬁlm theory generates other higher-ordernon-

linear PDEs. For instance, another pressure dependence in (3.4) given by

p = u

xxxx

corresponds to the case where an elastic plate covers the droplet surface. This leads

to the sixth-order parabolic equation



xxxxx



+ (lower-order terms). (3.13)

According to Landau–Lifshitz [372], m =−u

is the bending moment on the over-

laying plate, and % = u

xxx

is the shearing force. As in the fourth-order case, n = 3

3 Thin Film, Kuramoto-Sivashinsky, and Magma Models 101

corresponds to Newtonian viscous ﬂuids that do not slip along the substrate, while

n = 2 is associated with the Navier-slip dominated case. In addition, n = 3arisesin

a model of the oxidation of silicon in semiconductor devices. The parameter n = 1

corresponds to thin ﬁlm ﬂows in a Hele–Shaw cell. The linear counterpart of this

TFE for n = 0isthetri-harmonic equation

= u

xxxxxx

Setting n = 1 gives the leading quadratic operator to be considered later on,

= (uu

xxxxx

)

+ (lower-order terms).

On free-boundary conditions. For one-dimensional TFEs, the free-boundary con-

ditions at the interface x = s(t),whereu(s(t), t) = 0, are of principal importance.

For the fourth-order PDE (3.1), the most physically relevant are the zero contact

angle and zero-ﬂux (mass conservation) conditions, i.e.,

u = u

= u

xxx

= 0atx = s(t). (3.14)

These make sense also for n = 0, i.e., for the bi-harmonic equation (3.2). In general,

forthe fourth-orderTFEs, these three conditionson the aprioriunknownfree bound-

ary (together with standard Dirichlet or Neumann ones at ﬁxed boundary points and

given initial data) are expected to be sufﬁcient to specify a solution. A rigorousproof

of uniqueness (to be discussed later) is always a difﬁcult mathematical problem, re-

maining open for many FBPs.

Other types of free-boundary conditions can also be considered, e.g., those of the

one-phase Florin type

= S[u]atx = s(t),

where the operator S may depend on other derivatives of u at the interface. For the

heat equation, u

= u

, the case S[u] = constant gives Florin’s boundarycondition

which has been known since the 1950s, [190]. See [454] for details.

In both cases, it is important to derive the dynamic interface equation representing

the dependence of the interface speed s



(t) on the corresponding interface “slopes”

that are determined in terms of the derivatives u

, u

,... at x = s(t). For instance,

for the zero-ﬂux Stefan FBP with conditions

u = u

xxx

= 0andu

= S[u]atx = s(t),

for sufﬁciently smooth solutions, we have, by differentiating, that

u(s(t), t) = u



+ u

= 0,

yielding the following formal Stefan-type free-boundary condition, which is the in-

terface equation:



=−

≡−

S[u]

xxx

)



u=0

Here, the right-hand side may contain an indeterminacy (if, say, S[u] = 0onthein-

terface). Existence of such a limit as x → s(t) is very difﬁcult to justify for general

weak solutions. Using particular explicit solutions, we reveal the structure of dy-

namic interface equations and discuss related mathematical aspects. In some cases,

we will need more general free-boundaryconditions for some exact solutions.

102 Exact Solutions and Invariant Subspaces

For the sixth-order TFEs, such as (3.13), a standard FBP setting includes four

free-boundary conditions

u = u

= u

xxxxx

= 0atx = s(t), (3.15)

meaning zero-height, zero contact angle, zero-curvature, and zero-ﬂux. Some other

conditionsof Stefan–Florin’s type can also be introduced to treat solutions on invari-

ant subspaces.

2mth-order thin ﬁlm equations in IR and IR

. The 2mth-order1D TFEs for n = 1

take the form

= (−1)

m+1



2m−1



+ (lower-order terms), (3.16)

where D

∂

∂x

, and, as usual, by lower-order terms, we mean operators of lower

differential order. The sign multiplier (−1)

m+1

in front of the higher-order operator

guarantees that, for any m = 1, 2, ..., the PDE is of parabolic type in the positivity

domain{u > 0}of the solution.This ensures local existenceand regularityproperties

of strictly positive classical solutions that sometimes can be extended to compactly

supported solutions by using special types of regularization; see Remarks. A full

mathematicaltheoryof such PDEs concerningexistence,uniqueness,and differential

properties of weak, strong, or maximal solutions of the Cauchy problem, or FBPs is

still not fully justiﬁed, especially for higher orders where m ≥ 3.

All of the above models admit a natural formulation in the N-dimensional geom-

etry. For instance, (3.16) in IR

× IR

reads

= (−1)

m+1

∇·(u∇

m−1

u) + (lower-order terms).

Free-boundary conditions should also be adapted to this 2mth-order, N-dimensional

case and should include m+1 conditions on the free-boundarysurface.

3.1.2 Quadratic models: polynomial subspaces

As usual, for construction of solutions on linear invariant subspaces, we need to

rewrite the PDE in a form having principle algebraically homogeneous polynomial

operators,or, even better, including a quadraticor a cubic one. The thin ﬁlm operator

F in (3.1) is already quadratic for n = 1 and is cubic for n = 2. Some other thin ﬁlm-

type models can be reduced to quadratic PDEs. For instance, consider the quasilinear

fourth-order TFE of the type (3.11), with n = 1,

=−(v(v

)

xxx

)

Setting u = v

yields a homogeneous quadratic operator in the PDE

=−s



xxxx

xxx



Let us next introduce a more general class of quadratic operators in the TFE

= F[u] ≡−uu

xxxx

+ βu

xxx

+ γ(u

)

. (3.17)