Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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the gravity field. For a container consisting of a
semi-infinite vertical cylinder, closed at the bottom,
one obtains
div Tu ¼
0
g
u þ g 1 cos !ðÞþ ½32
where ! is the angle between the upward directed
surface normal and the vertical axis, and is to be
determined by a volume constraint. Athanassenas
and Finn proved that for a general smooth domain
, prescribed , and prescribed fluid mass M subject
to the restriction
M <
0
jj
=g ½33
there exists exactly one solution of [32] achieving
the boundary data .
The condition [33] is necessary for existence with
the prescribed mass.
The methods used for this theorem do not permit
regularity conditions to be relaxed to allow domains
with corner points. An approximation procedure
yields an existence theorem for such cases, however
the uniqueness proof then fails; it can be replaced by
a weaker result, estimating the difference between
two eventual solutions: Let u, v, be solutions of [32]
in a piecewise smooth domain , and suppose
Tu Tv on =@ except at the corner points,
where no data are prescribed. Then
u v þ =
0
½34
throughout .
Note that in this result, no growth condition is
imposed at the corner points. It can happen that
both u and v are unbounded at a corner point;
nevertheless, [34] holds uniformly over .
The solutions of [32] emulate many of the
characteristics of solutions of [16]. Notably, there is
again a dichotomy of behavior, depending on open-
ing angle 2 at a corner point, with all solutions
either bounded, or unbounded with growth like 1=r.
The Equations II
If in addition to taking account of the change of density
with height, one accounts for the energy change due to
expansion or contraction of volume elements with
changing density, one is led to the equation
div Tu ¼
0
p
0
e
gu
1ðÞ
þ g 1 cos !ðÞþ ½35
Here the changes from the incompressible case are
much more significant than for [32]. In order to
ensure stable behavior of solutions, it seems appro-
priate to impose the condition
0
>p
0
. The general
existence theorem above can no longer be expected;
it is possible to give explicit examples of analytic
domains, and constant data , for which no solution
of the problem exists. Thus, even in a larg e down-
ward gravity field, the solutions can emulate the
behavior of solutions of [18]. That can happen,
however, only for data exceeding =2. The
condition [33] is again necessary for existence.
For eqn [34], cannot be eliminated by addition
of a constant to the solution, and its determination
creates a new level of difficulty toward solution of
the physical existence question. Athanassenas and
Finn proved unique existence of solutions of [35],
[17] for a capillary tube of general smooth section
dipped into an infinite liquid bath (which corre-
sponds to =0), when 0 = 2. If >=2 then
solutions do not always exist; it can happen that the
surface moves down to the bottom of the tube,
regardless of the depth of immersion. Under a
hypothesis of radial symmetry, Finn and Luli were
able to prove the existence of solutions with
prescribed mass in a semi-infinite cylinder closed at
the bottom, in the rang e 0 <, and uniqueness
if 0 =2. Note that in this case, values >
=2 are not excluded. For large enough mass , the
surface will always cover the base of the tube.
Closing Remarks
This brief survey is intended only as a general
indication of the current state of the theory; much
material of interest could not be included. Nor have
we addressed hysteresis effects on contact angle.
Detailed references to the material discussed and also
to further information can be found in the articles
listed below. More recent publications can be located
by following links in MathSciNet or Zentralblatt.
Acknowledgmnt
I owe a special debt of thanks to my colleague
Paul Concus, who read the material in detail and
provided many effectual suggestions, leading to a
much-improved exposition.
See also: Compressible Flows: Mathematical Theory;
Interfaces and Multicomponent Fluids; Newtonian Fluids
and Thermohydraulics.
Further Reading
References for text material and for further reading are cited in
the expository articles:
Finn R (2002a) Milan Journal of Mathematics 70: 1–23.
Finn R (2002b) Mathematical Intelligencer 24: 21–33.
Capillary Surfaces 445
Cartan Model see Equivariant Cohomology and the Cartan Model
Cauchy Problem for Burgers-Type Equations
G M Henkin, Universite
´
P.-M. Curie, Paris VI,
Paris, France
ª 2006 Elsevier Ltd. All rights reserved.
Burgers Type Equations
We consider here tw o types of equations: the scalar
partial differential equations (PDEs) of the form
@f
@t
þ ’ðf Þ
@f
@x
¼ "
@
2
f
@x
2
;">0 ½1
f = f ( x, t ), x 2 R, t 2 R
þ
, and the scalar differe nce–
differential equations of the form
@F
@t
þ ’ðFÞ
Fðx; tÞF ðx "; tÞ
"
¼ 0;">0 ½2
F = F(x, t), x 2 R, t 2 R
þ
.
Equation [1] for t he case of linear f 7!’(f )
was ca lled as Burgers equation by Hopf (1950),
who justified this by the remark: ‘‘equation was
first
@f
@t
þ f
@f
@x
¼ "
@
2
f
@x
2
introduced by J. M. Bur gers (1940) as a simplest
model to the differential equations of fluid flow’’. In
fact, eqn [1] for linear ’(f ) was introduced earlier in
1915 by Bateman. Equation [1] for general ’(f )
appeared later in very different models, for example,
in the model for displacement of oil by water, in a
model of road traffic, etc.
For ’(f ) = a þ b f , Hopf and Cole have studied
[1] basing on the substitution
f ¼
1
b
a þ "
@g
@x
g
reducing [1] to the heat equation
@y
@t
¼ "
@
2
g
@x
2
This transformation (often called as the Hopf–
Cole transform) appeared for the first time in 1906
in the book of Forsyth ‘‘Theory of differential
equations.’’
Equation [2] first appeared for ’(F) = a þ b F,
" = 1, x = n 2 Z, in Levi, Ragnisco, Bruchi (1983) as
a semidiscrete equation reducible to the linear
equation
dG
n
ðtÞ
dt
¼ aðG
n1
ðtÞG
n
ðtÞÞ
by the substitution
Fðn; tÞ¼
a
b
G
n
ðtÞG
n1
ðtÞ
G
n
ðtÞ
Equation [2] for general ’(F) w as introduced by
Henkin, Polterovich (1991) for the description of a
Schumpeterian evolution of industry. For any ">0,
one can consider [2] as t he family of differ en ce–
differential equations, depending on the parameter
= {x="} 2 [0, 1), w here {x="} denotes the frac-
tional part of x=". For physical applications of [1]
(see Gelfand (1959), Landan and Lifschitz (1968),
Lax (1973)), the inviscid case (" = þ0) is the most
interesting. But, for some special physical models
and for some social and biological applications (see
Henkin, Polterovich (1991), Serre (1999)), the
interesting case concerns eqn [2] with " = 1and
x 2 Z.
The results considered in this article concern
mainly the Cauchy problem for eqns [1] and [2]
with initial data f(x, 0), F(x, 0) satisfying the
conditions
f ðx; 0Þ!
; x !1
Z
0
1
jf ðx; 0Þ
jdx
þ
Z
1
0
j
þ
f ðx; 0Þjdx < 1
½3
and correspondingly
Fðk" þ "Þ!
; k !1
X
0
k¼1
jFðk" þ "; 0Þ
j
þ
X
1
k¼0
j
þ
Fðk" þ "; 0Þj < 1
½4
where
þ
, 2 [0, 1) and the mapping
7!{F(k" þ ",0)
sgn k
, k 2 Z} 2 l
1
is smooth.
446 Cauchy Problem for Burgers-Type Equations
The standard classical questions concerning
Cauchy problems [1], [3] and [2], [4],namely
those relating to existence , unicity, regularity, and
conservation laws a re well established (see Oleinik
(1959),andSerre ( 1999)). This section formulates
only those which are essential for the study
of asymptotic behavior of solutions f( x, t)and
F(x, t), when t !1or " ! 0, and of the relation
between vanishing viscosity and difference scheme
approximations for inviscid Burgers type
equations.
One can see that asymptotic behavior of solutions
of [2], [4] when " !þ0 is not the same as the
asymptotic behavior of [1], [3] when " !þ0, in
spite of fact that in the limiting case " = þ0 both [1]
and [2] look identical. It can be explained by the
fact that eqn [2] can be interpreted as a semidiscrete
approximation of the nonconservative (nonphysical)
equation
@F
@t
þ ’ðFÞ
@F
@x
¼
"
2
’ðFÞ
@
2
F
@x
2
However, the problem [2], [4] can be naturally
transformed into conservative (physical) initial pro-
blem. Indeed, the substitution
f ¼
Z
F
0
dy
’ðyÞ
(under condition of integrability of 1=’(y)) trans-
forms [2] into the equation
@f ðx; tÞ
@t
þ
ðf ðx; tÞÞ ðf ð x "; tÞÞ
"
¼ 0 ½5
where
0
(f ) = ’(F). Equation [5] is the so-called
monotone one-sided semidiscrete approximation of
conservative viscous equation,
@f
@t
þ ’ðFÞ
@f
@x
¼
"
2
@
@x
’ðFÞ
@f
@x
½6
where
f ðx; 0Þ!
Z
0
dy
’ðyÞ
; x !1
The results of finite-difference approximations
for nonlinear conservation laws (see A. Harten,
J. Hyman, P. Lax (1976)) explain both the similarity
of behavior of [6] and [5] as well as some difference
in the behavior of [1] and [2].
For further exposition the following assum ption is
useful:
Assumption 1 Let ’ in [1], [2] be a positive and
continuously differentiable function on the interval
[
,
þ
]. Let ’
0
have only isolated zeros.
From references one can deduce the following gene-
ral properties of Cauchy problems [1], [3] and [2], [4].
Theorem 0 Under Assumption 1, we have:
(i) There exists a unique (weak) solution f(x, t), x 2
R, t 2 R
þ
of the problem [1], [3]; this solution is
necessarily smooth for t > 0; besides, it satisfies
the following conservation laws for t > 0:
f ðx; tÞ!
; x !1
f ðx; tÞ!
þ
; x !þ1
d
dt
Z
1
0
ð
þ
f ðx; tÞÞ
dx
Z
0
1
ðf ðx; tÞ
Þdx
¼
Z
þ
’ðyÞdy
Moreover, if the initial value f(x,0) is nonde-
creasing as a function of x, then solution f(x, t)
is nondecreasing as a funct ion of x for all t 0.
(ii) There ex ists a unique solution F (x, t) x 2 R, t 2
R
þ
of the proble m [2], [4]; this solution is
smooth for t > 0; besides, it satisfies the follow-
ing conservation laws for t > 0 and 2 [0, 1):
Fðk" þ "; tÞ!
; k !1
Fðk" þ "; tÞ!
þ
; k !þ1
d
dt
X
1
k¼1
Z
þ
Fðk"þ";tÞ
dy
’ðyÞ
X
0
k¼1
Z
Fðk"þ";tÞ
dy
’ðyÞ
"#
¼
þ
Moreover, if for some 2 [0, 1) the F(k" þ ",0) is
nondecreasing as a function of k 2 Z then solution
F(k " þ ", t) is also nondecreasing as a function of
k 2 Z for all t 0 and the same .
Gelfand’s Problem and Iljin–Oleinik
Theorem
The main results considered in this article are related
to the following problem, formulated explicitly by
Gelfand (1959): to find the asymptotic ( t !1)ofthe
solution f (x, t)oftheeqn [1] with the initial condition
f ðx; 0Þ¼
; if x > x
f
0
ðxÞ; if x 2½x
; x
þ
½7
where
þ
.
Gelfand found a solution to this problem for the
inviscid case " = þ0 with initial conditions
f (x,0)=
if x < 0, and f (x,0)=
þ
if x 0 (see
below), and remarked that it would be interesting to
prove that the main term of the asymptotic (t !1)
of f (x, t ) satisfying [1], [7] coincides with the
solution of [1], [7] for " = þ0.
Cauchy Problem for Burgers-Type Equations 447
Gelfand’s problem admits natural extension for
eqn [2] with the initial conditions
Fðx; 0Þ¼
; if x > x
Fðx; 0Þ¼F
0
ðxÞ; if x 2½x
; x
þ
½8
Let us introduce, for u 2 [
,
þ
], the function
(u) =
R
u
’(y)dy. Let the function
ˆ
(u), u 2
[
,
þ
], be upper bound of the convex set
fðu; vÞ: v ðuÞ; u 2½
;
þ
g
By Assumption 1, the set s = {u 2 [
,
þ
]:
(u) <
ˆ
(u)} is the finite union of intervals,
s=(
,
0
) [(
1
,
1
) [(
L
,
þ
), where
=
0
0
1
<
1
L
L
=
þ
.
Let us define the function
ˆ
f(x, t)by
^
f ðx; tÞ¼
; if x <’ð
Þt
ð
^
0
Þ
ð1Þ
x=tðÞ; if ’ð
Þt x ’ð
þ
Þt
þ
; if x >’ð
þ
Þt
8
<
:
where in the case
ˆ
0
(u)
l
, u 2 (
l
,
l
), l = 0,
1, ..., L; also, by definition, (
ˆ
0
)
(1)
(
l
) = [
l
,
l
].
Theorem 1 (Gelfand) The solution f (x, t) of the
problem [1], [7] for the case " = þ0 and initial
conditions f (x,0)=
, if x > 0, has the explicit
form: f (x, t) =
ˆ
f(x, t).
The analogous statement is valid also for the
problem [2], [8] if, in the construction above, one
takes
ðuÞ¼
Z
u
0
dy
’ðyÞ
instead of (u), u 2 [
,
þ
].
The Gelfand problem for [1], [3] and [1], [7] with
monotonic ’(f ) was solved by Iljin and Oleinik
(1960). In the case
=
þ
, the solution of this
problem follows from an earlier work of Lax (1957).
For the case of linear ’(f ), the solution of this problem
follows from an earlier work of Hopf (1950).
For semidiscre te initial problems [2], [4] and [2],
[8], the analog of the asymptotic results of Hopf and
Iljin–Oleinik ha ve been obtained and applied by
Henkin and Polterovich (1991).
The case of increasing ’(f ) has been studied in
detail. In this case, for both initial problems [1], [3]
and [2], [4], there is uniform convergence of solutions
f (x, t)andF(x, t) to the so-called rarefaction profile
gx=tðÞ¼
; x >’ð
Þt
’
ð1Þ
x=tðÞ; x 2½’ð
Þt;’ð
þ
Þt
t !1(see Iljin and Oleinik (1960) and Henkin
and Polterovich (1991)). More precise result i n
this case about convergence t o the so-called
N-wave has been obtained by Dafermos (1977)
and Liu (1978).
For the case of a general ’(f ), in particular, for
the case of nonincr easing ’(f ), we need the notion
of shock profile. Following Serre (1999), three
definitions can be introduced.
Definition The initial problem [1], [3] (correspond-
ingly, [2], [4]) admits (
,
þ
)-shock profile (
<
þ
)
if there exists a traveling-wave solution of this equation,
that is, of the form f =
˜
f(x ct) (correspondingly,
F =
˜
F(x Ct)), such that
˜
f(x) !
when x !1
(correspondingly,
˜
F(x) !
when x !1).
From the results of Gelfand (1959) and Oleinik
(1959), it follows that initial problem [1], [3] admits
(
,
þ
)-shock profile iff
c ¼
1
þ
Z
þ
’ðyÞdy
<
1
u
Z
u
’ðyÞdy; 8u 2ð
;
þ
Þ½9
From the results of Henkin and Polterovich
(1991) and Belenky (1990), it follows that initial
problem [2], [4] admits (
,
þ
)-shock profile iff
1
C
¼
1
þ
Z
þ
dy
’ðyÞ
>
1
u u
Z
u
dy
’ðyÞ
; 8u 2ð
;
þ
Þ½10
In the case " = þ0, the equality in [9] and [10] is
called the Rankine–Hugoniot condition, the inequal-
ity in [9] and [10] is called the entropy condition (or
the Gelfand–Oleinik condition).
Definition For initial problem [1], [3] (correspond-
ingly, [2], [4]) admitting (
,
þ
)-shock profile and
for " = þ0, we will call by shock waves the weak
solutions of [1], [3] (corr espondingly, [2], [5], [4]) of
the form
~
f
ðx ctÞ¼
; if x ct
~
F
ðx CtÞ¼
; if x Ct
where c, C satisfy Rankine–Hugoniot and entropy
conditions [9], [10].
Definition The (
,
þ
)-shock profile for [1] (cor-
respondingly, for [2]) is called strict if in addition to
[9], [10] we have the Lax (1954) condition:
’ð
þ
Þ < c <’ð
Þ½11
and correspondingly
’ð
þ
Þ < C <’ð
Þ½12
448 Cauchy Problem for Burgers-Type Equations
The (
,
þ
)-shock profile for [1] or [2] is called
semicharacteristic if one of the inequalities in [11] or
[12] is strict and the other is an equality. This profile
is called characteristic if both inequalities in [11] or
[12] are equalities.
One can check (Iljin and Oleinik 1960, Henkin and
Polterovich 1991)thatifinadditiontoAssumption 1
the function ’ on [
,
þ
] is nonconstant and
nonincreasing then eqn [1] (correspondingly, [2])
admits a strict (
,
þ
)-shock profile.
The main result of Iljin–Oleinik (19 60) for eqn [1]
and analogous statement of Henkin and Polterovich
(1991) for eqn [2] can be presented as follows.
Theorem 2
(i) Let the initial problem [1], [3] admit a strict
(
,
þ
)-shock profile
˜
f. Let f (x, t), x 2 R, t 2
R
þ
, be a solution of [1], [3]. Then there exists
d
0
2 R
sup
x2R
jf ðx; tÞ
~
f ðx ct d
0
Þj ! 0; t !1 ½13
The value of d
0
is determined uniquely by relation
Z
1
1
ff ðx; 0Þ
~
f ðx d
0
Þgdx ¼ 0
(ii) Let the initial problem [2], [4] admit a strict
(
,
þ
)-shock profile
˜
F. Let F(x, t), x 2 R, t 2
R
þ
be a solution of [2], [4]. Then there exists
continuous function D
0
(), 2 [0, 1), such that
sup
x2R
jF ðx; tÞ
~
Fðx Ct D
0
ðfx="gÞj ! 0;
t !1
½14
The function D
0
(), 2 [0, 1], is determined
uniquely from relation
X
1
k¼1
fðFðn; 0ÞÞ ð
~
Fðn D
0
ÞÞg ¼ 0
where
ðFÞ¼
Z
A
F
dy
’ðyÞ
; F < A;
~
F < A
(iii) If in conditions (i) and (ii), we take " = þ0 then
there exist d
0
, D
0
such that 8>0, we have
sup
xctþd
0
þ
j
þ
f ðx; tÞj
þ sup
xctþd
0
j
f ðx; tÞj ! 0; t !1
sup
xCtþD
0
þ
j
þ
Fðx; tÞj
þ sup
xCtþD
0
j
Fðx; tÞj ! 0 ; t !1
½15
The values of d
0
and D
0
are determined by
Z
d
0
1
ðf ðx; 0Þ
Þdx þ
Z
1
d
0
ðf ðx; 0Þ
þ
Þdx ¼ 0
Z
D
0
1
ðFðx; 0Þ
Þdx þ
Z
1
D
0
ðFðx; 0Þ
þ
Þd x ¼ 0
Remarks
(i) The statements of Theorem 2 give a positive
answer to Gelfand’s question for the case of
initial problem [1], [3] and [2], [4], admitting
strict shock profiles.
(ii) For linear ’(f ) = a þ bf , a > 0, a þ b
þ
> 0,
b < 0, the traveling waves
˜
f,
˜
F for [1], [3] and
[2], [4] can be found explicitly:
~
f ¼
þ
þ
1 þ expfpðx ctÞg
c ¼ a þ
b
2
ð
þ
þ
Þ; p ¼
ð
þ
Þb
2"
~
F ¼
þ
þ
1 þ expfPðx CtÞg
C ¼ b
ln
a þ b
þ
a þ b
; P ¼
"
ln
a þ b
a þ b
þ
where
b ¼ð þ b
Þ 1
a þ b
þ
a þ b
(iii) For initial problems [1], [7] and [2], [8],
þ
>
, the asymptotic convergence statements
[13]–[15] admit the precise asymptotic esti-
mates (see Iljin and Oleinik (1960) for [1], [7]:
sup
x2R
jf ðx; tÞ
~
f ðx ct d
0
Þj ¼ Oðe
t
Þ
>0;"> 0
½16
sup
x2R
jF ðx; tÞ
~
Fðx Ct D
0
ðfx="gÞÞj ¼ Oðe
t
Þ
>0;">0 ½17
f ðx; tÞ¼
for x > ðct þ d
0
Þ
t t
0
;"¼þ0
Fðx; tÞ¼
for x > ðCt þ D
0
Þ
t t
0
;"¼þ0
½18
Theorem 2(i) is proved basing on the following
idea. Let f satisfy the initial problem [1], [3] and let
Cauchy Problem for Burgers-Type Equations 449
˜
f(x ct þ d
0
)be(
,
þ
)-shock profile for [1],
satisfying condition [13]. Put
ðx; tÞ¼
Z
x
1
ff ðy; tÞ
~
f ðy ct d
0
Þgdy
The function (x, t) satisfies the nonlinear parabolic
equation
@
@t
þ ’ð
~
f þð1 Þf Þ
@
@x
¼ "
@
2
@x
2
where (x, t) is some smooth function of (x, t) with
values in [0, 1].
Besides, by conservation law of Theorem 0(i),we
have ( x, t ) ! 0, x !1, 8t 0.
Estimates basing on maximum principle and
appropriate comparison statements give that
(x, t) ) 0, x 2 R, t !1. It implies that
f ðx; tÞ
~
f ðx ct d
0
Þ)0; x 2 R; t !1
Theorem 2(ii) is proved in a similar way. Let F(n,
t) satisfy the initial problem [2], [4] with x = n 2
Z, " = 1, = {x} = 0, and let
˜
F(n Ct D
0
)be
(
,
þ
)-shock profile for [2], satisfying condition
[14]. Put
ðn; tÞ¼
X
n
1
fðFðn; tÞÞ ð
~
Fðn Ct D
0
ÞÞg
Then function (n, t) satisfies the semidiscrete
parabolic equation
dðn; tÞ
dt
¼’ð
ð1Þ
ððFÞ
þð1 Þð
~
FÞÞÞððn 1; tÞðn; tÞÞ
where (n, t) is some function with values in [0, 1].
Besides, by conservation law of Theorem 0(ii),we
have
ðn; tÞ!0; n !1; 8 t 0
Estimates, basing on generalized maximum prin-
ciple and comparison statem ents, give that
(n, t) ) 0, n 2 Z, t !1. It implies that
Fðn; tÞ
~
Fðn Ct þ D
0
Þ)0; n 2 Z; t !1
Remark For the cases of nonstrict shock profiles
(characteristic or semicharacteristic) the statements
of Theorem 2 are not valid. The reason is that,
under initial conditions [3], [4] for any d
0
and D
0
,
we have
Z
1
1
ff ðx; oÞ
~
f ðx d
0
Þgdx ¼1
and, correspondingly,
X
1
1
fð
~
Fðk" þ " D
0
ÞðF ðk" þ "; 0ÞÞg ¼ 1
So, the crucial argument, related to conservation
law, does not hold.
One can extend the important Theorems 2(i), 2(ii)
for the case of nonstrict shock profiles in two different
ways: by changing conditions of these theorems or by
changing conclusions of these theorems.
The first method (started by Mei, Matsumura, and
Nishihara in 1994) was completed by the following
L
1
-asymptotic stability result (Serre 2004).
Theorem 3 (Freistu¨ hler–Serre). Let eqns [1], [2]
admit (
,
þ
)-shock profiles and
˜
f,
˜
F – the corre-
sponding train-wave solutions of [1], [2]. Let
f (x, t), F(n, t), x 2 R, n 2 Z, t 2 R
þ
be solutions of
eqns [1], [2] with such initial conditions that
Z
1
1
jf ðx; 0Þ
~
f ðxÞjdx < 1
X
1
1
jF ðn; 0Þ
~
FðnÞj < 1
Then
Z
1
1
jf ðx; tÞ
~
f ðx ct d
0
Þjdx ! 0
and, correspondingly,
X
1
1
jFðn; tÞ
~
Fðn Ct D
0
Þj ! 0; t !1
where consta nts d
0
and D
0
are calculated from the
same relations as in Theorem 2.
Remark For the inviscid case " = þ0, the state-
ment of Theorem 3 is still v alid for equations
admitting strict shock profiles, but generally is not
valid for equations admitt ing only nonstrict shock
profiles (see Serre (2004)).
The second method permits, keeping initial con-
ditions [3], [4], to localize the positions of viscous
shock waves for generalized Burgers equations
(see the next section).
Asymptotic Behavior of Solutions of
Generalized Burgers Equations
The main current interest and the main difficulty in
the study of Gelfand’s problem for generalized
Burgers equations consist in the following question
formulated explicitly for initial problem [1], [3] by
Liu et al. (1998): ‘‘In the Cauchy problem there is
450 Cauchy Problem for Burgers-Type Equations
the question of determining the location of viscous
shock waves’’. A similar question and related
conjecture were formulated by Henkin and Potter-
ovich (1999) for the initial problem [2], [4].
For solving this problem, it is important to solve it
first for the Burgers type equations admitting
nonstrict shock profiles.
Theorem 4 (Henkin–Shananin–Tumanov).
(i) Let the initial problem [1], [3] admit the nonstrict
(
,
þ
)-shock profile [9] and
˜
f(x ct) be a
corresponding traveling-wave solution. Let
’
0
ð
Þ 6¼ 0; if ’ð
Þ¼c
’
0
ð
þ
Þ 6¼ 0; if ’ð
þ
Þ¼c
Let f (x, t) be a solution of [1], [3]. Then there
exist constants
0
and d
0
such that
sup
x2R
jf ðx; tÞ
~
f ðx ct
0
ln t d
0
Þj ! 0; t !1
where
ð
þ
Þ
0
¼
1=’
0
ð
þ
Þ; if ’ð
Þ > c ¼ ’ð
þ
Þ
1=’
0
ð
Þ; if ’ð
Þ¼c >’ð
þ
Þ
1=’
0
ð
Þ1=’
0
ð
þ
Þ; if ’ð
Þ¼c ¼ ’ð
þ
Þ
8
>
<
>
:
(ii) Let the initial problem [2], [4] with = 1 admit the
nonstrict (
,
þ
)-shock profile [10] and
˜
F(n Ct)
be a corresponding traveling-wave solution. Let
’
0
ð
Þ 6¼ 0; if ’ð
Þ¼C
’
0
ð
þ
Þ 6¼ 0; if ’ð
þ
Þ¼C
Let F( n, t ) be a solution of [2], [4]. Let
Fðn; 0Þ¼
def
Fðn; 0ÞFðn 1; 0Þ0
Then there exist constants
0
and D
0
such that
sup
n2Z
jFðn; tÞ
~
Fðn Ct
0
ln t D
0
Þj ! 0;
t !1
where
ð
þ
Þ
0
¼
C=ð2’
0
ð
þ
ÞÞ; if ’ð
Þ > C ¼ ’ð
þ
Þ
C=ð2’
0
ð
ÞÞ; if ’ð
Þ¼C >’ð
þ
Þ
ðC=2Þ½1=’
0
ð
þ
Þ
þ1=’
0
ð
Þ; if ’ð
Þ¼C ¼ ’ð
þ
Þ
8
>
>
>
<
>
>
>
:
Remarks
(i) One could think that nonstrict shock profiles
as in Theorem 4 can appear only in exceptional
cases. But Proposition 2 and Theorem 5 below
show, on the contrary, that characteristic shock
profiles and, as a consequence, the behavior of
initial problems [1], [3] and [2], [4] as in Theorem
4 are rather a rule than an exception.
(ii) The statement of Theorem 4(i) (and also of
Theorem 5(i)) below) disprove the Gelfand hope
that the main term of asymptotic (t !1)of
f (x, t), satisfying [1], [7], coincides with the
solution of [1], [7] for = þ0withthesame
initial condition. Indeed, in conditions of Theorem
4, we have ’(
) = c or ’(
þ
) = c,but’
0
(
) 6¼
’
0
(
þ
); then for any >0 the traveling wave
˜
f(x ct
0
ln t d
0
)for[1], [3], concentrated
near the point x
(t) = ct þ
0
ln t þ d
0
,moves
away (t !1) from the shockwave for [1], [7] for
= þ0, concentrated near the point x
0
(t) = ct þ
o(lnt), where o(lnt)= ln t ! 0, t !1.
(iii) Theorem 4 (and also Theorem 5 below) also
illustrate another interesting phenomenon: for
the case ’
0
(
) 6¼ ’
0
(
þ
), one has asymptotic
convergence of the solution of [1], [3] (corre-
spondingly of [2], [4]) to the traveling
wave
˜
f(x ct
0
ln t d
0
) (correspondingly
˜
F(x Ct
0
ln t D
0
)), which does not
satisfy eqn [1] or correspondingly eqn [2]. Such
a phenomenon was first discovered by Liu and
Yu (1997) in the special boundary-value pro-
blem for the classi cal Burgers equations, if
u(x, t) satisfies the follow ing conditions:
if u
t
þu u
x
¼ u
xx
; uð0; tÞ¼1; uð1; tÞ¼1;
uðx; 0Þ¼th
x
2
; then
juðx; tÞþth
1
2
ðx lnð1 þtÞÞj ! 0; t !1; x 0
Theorem 4 is proved in basing on the following
idea. Let f (x, t) satisfy [1], [3] and F(n, t) satisfy [2],
[4]. Let
˜
f(x ct) be the traveling wave for [1], [3]
and
˜
F(n Ct) be the trave ling wave for [2], [4].
Suppose that ’(
) > c = C = ’(
þ
). Let d
A
(t) and
D
A
(t), A > 0 be functions such that
Z
ctþA
ffiffi
t
p
ctA
ffiffi
t
p
ff ðx; tÞ
~
f ðx ct d
A
ðtÞÞgdx ¼ 0 ½19
and, correspondingly,
X
½CtþA
ffiffi
t
p
k¼½CtA
ffiffi
t
p
fðFðk; tÞÞð
~
Fðk Ct D
A
ðtÞÞÞg
þðCt þA
ffiffi
t
p
½Ct þA
ffiffi
t
p
ÞððFðCt þA
ffiffi
t
p
þ1; tÞÞ
ð
~
Fð½Ct þA
ffiffi
t
p
þ1 Ct þD
A
ðtÞÞ ¼ 0
½20
Cauchy Problem for Burgers-Type Equations 451
The relations [9], [20] can be called ‘‘localized
conservation law.’’ The proof contains two difficult
parts.
The first part consists in proving that for A > 2
ffiffiffi
c
p
(correspondingly, A > 2
ffiffiffiffi
C
p
) the following asymp-
totics are valid:
d
A
ðtÞ¼
ln t
ð
þ
Þ’
0
ð
þ
Þ
þ d
0
þ oð1Þ; t !1
D
A
ðtÞ¼
C ln t
2ð
þ
Þ’
0
ð
þ
Þ
þ D
0
þ oð1Þ; t !1
½21
where d
0
, D
0
are independent of A.
The second part gives the following convergence
statements:
sup
x2½ctA
ffiffi
t
p
;ctþA
ffiffi
t
p
Z
x
ctA
ffiffi
t
p
ff ðy; tÞ
~
f ðy ct d
A
ðtÞÞg
dy
! 0; t !1
sup
x2½CtA
ffiffi
t
p
;CtþA
ffiffi
t
p
X
n
k¼½CtA
ffiffi
t
p
fðFðk; tÞÞ
ð
~
Fðk Ct D
A
ðtÞÞÞg
! 0; t !1
The precise a priori estimates of local solutions of
[1], [2] play an important role in the proof. An
example of such an estimate, also useful for further
results, is given below.
Proposition 1 Let, in eqn [2], C = ’(0) > 0, =
1, 0 ’
0
(0) <
0
,
¯
x
def
=
(x Ct)=
ffiffiffiffiffiffi
Ct
p
. Let the func-
tion F(x, t), defined in the domain
0
= {(x, t): a
1
<
¯
x < a
2
}, a
2
> 0, satisfy eqn [2],
Fðx; tÞ¼
def
Fðx; tÞF ðx 1; tÞ0
jFðx; tÞj
ffiffiffiffiffiffi
Ct
p
; ðx; tÞ2
0
; t t
0
Then
Fðx; tÞ
B
Ct
; ðx; tÞ2
0
; t t
0
where
B ¼ B
0
a
2
þ
1
d
þ
0
C
1 þ lnð1 þ a
2
ÞðÞ
d ¼ minð
x a
1
; a
2
xÞ
and B
0
is an absolute constant.
It is interesting to compare a priori estimate of
Proposition 1 with some similar (but less precise)
estimates in the theory of classical quasilinear
parabolic equations (Ladyzhenskaya et al. 1968).
We will formulate now the genera l conjecture
concerning asymptotic behavior of solutions of
initial problems [1], [3] and [2], [4] and some
partial results which confirm this conjecture. To
simplify formulation we admit the following.
Assumption 2 Let
ˆ
(u)and
ˆ
(u) be upper bounds of
the convex hulls for the graphs of
ðuÞ¼
Z
u
’ðyÞdy
and
ðuÞ¼
Z
u
dy
’ðyÞ
respectively, with u 2 [
,
þ
]. We suppose that
s ¼fu 2½
;
þ
: ðuÞ <
^
ðuÞg
¼ð
;
0
Þ[ð
1
;
1
Þ[ð
L
;
þ
Þ
where
¼
0
<
0
<
1
<
1
< <
L
<
L
¼
þ
or, correspondingly,
S ¼fu 2½
;
þ
:ðuÞ <
^
ðuÞg
¼ð
; b
0
Þ[ða
1
; b
1
Þ[ða
M
;
þ
Þ
where
¼ a
0
< b
0
< a
1
< b
1
< < a
M
< b
M
¼
þ
In addition, we suppose that ’
0
(
l
) 6¼0, ’
0
(
l
) 6¼
0, l = 0, 1, ... , L or, correspondingly, ’
0
(a
m
) 6¼
0, ’
0
(b
m
) 6¼0, m=0, 1,... ,M.
Proposition 2 (Weinberger 1990, Henkin and
Polterovich 1999). Under Assumptions 1, 2, one has:
(i) If u 2 [
,
þ
] n sand, correspondingly, u 2
[
,
þ
] n S, then following functions are well
defined:
g
l
x
t
¼
l
; if x <’ð
l
Þt
’
ð1Þ
x=tðÞ; if ’ð
l
Þt x
’ð
lþ1
Þt
lþ1
; if x >’ð
lþ1
Þt;
l ¼ 0; 1; ...; L
8
>
>
>
<
>
>
>
:
and, correspondingly,
G
l
x
t
¼
b
m
; if x <’ðb
m
Þt
’
ð1Þ
x=tðÞ; if ’ðb
m
Þt x
’ða
mþ1
Þt
a
m
; if x >’ða
mþ1
Þt;
m ¼ 0; 1; ...; M
8
>
>
>
<
>
>
>
:
(ii) For any interval (
l
,
l
) s and, correspond-
ingly,(a
m
, b
m
) S there exist traveling waves
˜
f
l
(x c
l
t) for [1] with overfall (
l
,
l
) and,
452 Cauchy Problem for Burgers-Type Equations
correspondingly,
˜
F
m
(x C
m
t ) for [2] with over-
fall (a
m
, b
m
), where
c
l
¼
1
l
l
Z
l
l
’ðyÞdy
c
l
¼ ’ð
l
Þ; l ¼ 0; ...; L 1
c
l
¼ ’ð
l
Þ; l ¼ 1; ...; L
and, correspondingly,
C
1
m
¼
1
b
m
a
m
Z
b
m
a
m
dy
’ðyÞ
C
m
¼ ’ðb
m
Þ; m ¼ 0; ...; M 1
C
m
¼ ’ða
m
Þ; m ¼ 1; ...; M
Conjecture (Henk in and Polterovich 1994, 1999,
Henkin and Shananin 2004). Let
~
f ðx; t;
0
; ...;
L
Þ
¼
X
L
l¼0
~
f
l
ðx c
l
t "
l
ðtÞÞ þ
X
L1
l¼0
g
l
x
t
X
L1
l¼0
l
X
L
l¼1
l
; L 1
~
Fðn"; t;
0
; ...;
M
Þ
¼
X
M
m¼0
~
F
m
ðn" C
m
t "
m
ðtÞÞ þ
X
M1
m¼0
G
m
n"
t
X
M1
m¼0
b
m
X
M
m¼1
a
m
; M 1
Then under Assumptions 1, 2, the following state-
ments are valid:
(i) For any solution f (x, t), x 2 R, t 2 R
þ
,ofini-
tial problem [1], [3], there exist shift-functions
l
(t):
l
ln t þ Oð1Þ
l
ðtÞ
þ
l
ln t þ Oð1Þ
0
l
þ
l
< 1; l ¼ 0; 1; ...; L
such that
sup
x2R
jf ðx ; tÞ
~
f ðx; t;
0
;
1
; ...;
L
Þj ! 0;
t !1
(ii) Moreover, in (i) one can take
l
¼
þ
l
¼
"
ð
l
l
Þ
1
’
0
ð
l
Þ
; if l ¼ 0 < L;’ð
0
Þ6¼ ’ð
0
Þ
1
’
0
ð
l
Þ
1
’
0
ð
l
Þ
; if 0 < l < L
1
’
0
ð
l
Þ
; if l ¼ L > 0;’ð
L
Þ6¼ ’ð
L
Þ
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
(iii) For any solution F(n", t), n 2 Z, t 2 R
þ
,ofinitial
problem [2], [4], there exist shift-functions
m
(t):
m
ln t þ Oð1Þ
m
ðtÞ
þ
m
ln t þ Oð1Þ
0
m
þ
m
< 1; l ¼ 0; 1; ...; L
such that
sup
n2Z
jF ðn"; tÞ
~
Fðn"; t;
0
;
1
; ...;
M
Þj ! 0;
t !1
(iv) Moreover, in (iii) one can take
m
¼
þ
m
¼
C
m
ðb
m
a
m
Þ
1
’
0
ðb
m
Þ
; if m ¼0 < M;’ða
0
Þ6¼’ðb
0
Þ
1
’
0
ða
m
Þ
1
’
0
ðb
m
Þ
; if 0 < m < M
1
’
0
ða
m
Þ
; if m ¼M > 0;’ða
M
Þ6¼’ðb
M
Þ
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
The mai n result confirming formulated conjec-
tures is the following.
Theorem 5 (Henkin and Shananin). Conjecture
(i) for L = 1 and corresponding conjecture (iii) for
M = 1 are true, that is,for solution of initial problem
[1], [3] there exist shift functions
l
(t) = O (ln t) such
that for t !1we have
f ðx; tÞ7!
~
f
0
ðx c
0
t "
0
ðtÞÞ ; if x c
0
t
’
ð1Þ
ðx=tÞ; if c
0
t x c
1
t
~
f
1
ðx c
1
t "
1
ðtÞÞ ; if x c
1
t
8
<
:
and for solution of initial problem [2], [4] there exist
shift functions
m
(t) = O(ln t) such that for t !1
we have
Fðn"; tÞ7!
~
F
0
ðn" C
0
t "
0
ðtÞÞ ; if n" C
0
t
’
ð1Þ
ðn"=t Þ; if C
0
t n"
C
1
t
~
F
1
ðn" C
1
t "
1
ðtÞÞ ; if n" C
1
t
8
>
>
<
>
>
:
The proof of Theorem 5 is of the same nature as
the proof of Theorem 4.
Remarks
(i) The proof of stronger Conjectures (ii) and (iv)
for L = 1orM = 1 are in preparation.
(ii) The numerical results, Rykova and Spivak (pre-
print, 2004), confirm conjecture (iii) for M = 2.
(iii) The results of Weinberger (1990) and Henkin
and Polterovich (1999) confirm convergence
statements of Conjectures (i), (iii) for all L and
M, but only on the intervals of rarefaction
Cauchy Problem for Burgers-Type Equations 453
profiles: x 2 [’(
l
)t, ’(
l þ1
)t] or, correspond-
ingly, x 2 [’(b
m
)t, ’(a
m þ1
)t], t > 0.
The problem of finding asymptotics (t !1)of
solutions of (viscous) conservation laws has been
posed originally not only for generalized Burgers
equations but also for systems of conservation laws in
one spatial variable (see Gelfand (1959)). In this
direction many important results on existence and
asymptotic stability of viscous shock profiles (con-
tinuous and discrete) have been obtained and applied
(see Benzoni-Gavage (2004), Lax (1973), Serre
(1999), Zumbrun and Howard (1998) and references
therein). The results of type of Theorems 4,5 have not
yet been obtained for systems of conservation laws.
It is also very interesting to study asymptotic
behavior of scalar (viscous) conservation laws in
several spatial variables (continuous or discrete),
basing on the asymptotic properties of Burgers type
equations. In this direction there have been several
important results and problems (see Bauman and
Phillips (1986), Henkin and Polterovich (1991),
Hoff and Zumbrun (2000), Serre (1999),
Weinberger (1990), and references therein).
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some points of the statistical theory of free turbulence. Proc.
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e 83: 1457–1500.
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dimensional viscous shock fronts. Indiana University Mathe-
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þ uu
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= u
xx
.
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