A singular perturbation nonlinear boundary value problem and the 𝐸-condition for a scalar conservation law
Authors: Jiang Jie , Wang Xue Kong 
Published: December 2016

This paper deals with the singular perturbation boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace Subscript v left-parenthesis negative normal infinity right-parenthesis equals upper A comma v left-parenthesis plus normal infinity right-parenthesis equals upper B semicolon epsilon greater-than-or-equal-to 0 comma upper A greater-than upper B Superscript epsilon left-parenthesis k left-parenthesis v left-parenthesis s right-parenthesis right-parenthesis v prime left-parenthesis s right-parenthesis right-parenthesis Super Superscript prime Superscript plus left-parenthesis s g left-parenthesis v left-parenthesis s right-parenthesis right-parenthesis minus phi left-parenthesis v left-parenthesis s right-parenthesis right-parenthesis right-parenthesis v prime left-parenthesis s right-parenthesis plus f left-parenthesis v left-parenthesis s right-parenthesis right-parenthesis equals 0 normal i normal n upper R comma">

{

v

(

−
∞

)

=
A
,

v

(

+
∞

)

=
B
;

ε
≥
0
,

A
>
B

ε

(

k

(

v

(
s
)

)

v
′

(
s
)

)

′

+

(

s
g

(

v

(
s
)

)

−
φ

(

v

(
s
)

)

)

v
′

(
s
)

+
f

(

v

(
s
)

)

=
0

i
n

R
,

\left \{ {_{v\left ( { - \infty } \right ) = A, \qquad v\left ( { + \infty } \right ) = B; \qquad \varepsilon \ge 0, \qquad A > B}^{\varepsilon {{\left ( {k\left ( {v\left ( s \right )} \right )v’\left ( s \right )} \right )}’} + \left ( {sg\left ( {v\left ( s \right )} \right ) - \varphi \left ( {v\left ( s \right )} \right )} \right )v’\left ( s \right ) + f\left ( {v\left ( s \right )} \right ) = 0 \qquad \mathrm {in}\,R,}} \right .

\]
whose solution

v
ε

(
s
)

{v_\varepsilon }\left ( s \right )

is constructed by the aid of the solution

w
ε

(
t
)

{w_\varepsilon }\left ( t \right )

to the two-point boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row minus left-parenthesis StartFraction w prime left-parenthesis t right-parenthesis minus phi left-parenthesis t right-parenthesis plus epsilon f left-parenthesis t right-parenthesis k left-parenthesis t right-parenthesis slash w left-parenthesis t right-parenthesis Over g left-parenthesis t right-parenthesis EndFraction right-parenthesis Superscript prime Baseline equals StartFraction epsilon k left-parenthesis t right-parenthesis Over w left-parenthesis t right-parenthesis EndFraction normal i normal n left-parenthesis upper A comma upper B right-parenthesis comma 2nd Row w left-parenthesis upper A right-parenthesis equals 0 comma w left-parenthesis upper B right-parenthesis equals 0 period EndLayout">

{

−

(

w
′

(
t
)

−
φ

(
t
)

+
ε
f

(
t
)

k

(
t
)

/

(
t
)

w

(
t
)

w

(
t
)

g

(
t
)

)

′

=

ε
k

(
t
)

w

(
t
)

i
n

(

A
,
B

)

,

w

(
A
)

=
0
,

w

(
B
)

=
0.

\left \{ {\begin {array}{*{20}{c}} { - {{\left ( {\frac {{w’\left ( t \right ) - \varphi \left ( t \right ) + \varepsilon f\left ( t \right )k{{\left ( t \right )} \left / {\vphantom {{\left ( t \right )} {w\left ( t \right )}}} \right . {w\left ( t \right )}}}}{{g\left ( t \right )}}} \right )}’} = \frac {{\varepsilon k\left ( t \right )}}{{w\left ( t \right )}} \;\mathrm {in}\; \left ( {A, B} \right ),}\\ {w\left ( A \right ) = 0, \qquad w\left ( B \right ) = 0.} \end {array}} \right .

\]
The restrictions on

φ

(
t
)

\varphi \left ( t \right )

,

g

(
t
)

g\left ( t \right )

,

k

(
t
)

k\left ( t \right )

, and

f

(
t
)

f\left ( t \right )

not only ensure that the two-point boundary value problem has a solution

w
ε

(
t
)

{w_\varepsilon }\left ( t \right )

but also guarantee that as

ε
\varepsilon

tends to zero the solution

w
ε

(
s
)

{w_\varepsilon }\left ( s \right )

pointwise converges to
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v 0 left-parenthesis s right-parenthesis equals upper A plus left-parenthesis upper B minus upper A right-parenthesis upper H left-parenthesis s minus StartFraction normal upper Phi left-parenthesis upper B right-parenthesis Over upper G left-parenthesis upper B right-parenthesis EndFraction right-parenthesis comma s element-of upper R">

v
0

(
s
)

=
A
+

(

B
−
A

)

H

(

s
−

Φ

(
B
)

G

(
B
)

)

,

s
∈
R

{v_0}\left ( s \right ) = A + \left ( {B - A} \right )H\left ( {s - \frac {{\Phi \left ( B \right )}}{{G\left ( B \right )}}} \right ), \qquad s \in R

\]
, the solution to the reduced problem, where

H

(
s
)

H\left ( s \right )

is the multiple-valued Heaviside function,

G

(
t
)

=:

∫
A

t

g

(
s
)

d
s

G\left ( t \right ) =:\int _A^{t} g\left ( s \right ) ds

, and

Φ

(
t
)

=:

∫

A

t

φ

(
s
)

d
s

\Phi \left ( t \right ) =:\int _{A}^{t} \varphi \left ( s \right ) ds

. Moreover, the function

u
ε

(

x
,
t

)

=:

v
ε

(

x

/

x
t

t

)

{u_\varepsilon }\left ( {x, t} \right ) = :{v_\varepsilon }\left ( {{x \left / {\vphantom {x t}} \right . t}} \right )

, as a solution to the Riemann problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row StartFraction partial-differential upper G left-parenthesis u right-parenthesis Over partial-differential t EndFraction plus StartFraction partial-differential normal upper Phi left-parenthesis u right-parenthesis Over partial-differential x EndFraction equals StartFraction f left-parenthesis u right-parenthesis Over t EndFraction plus epsilon t StartFraction partial-differential Over partial-differential x EndFraction left-parenthesis k left-parenthesis u right-parenthesis StartFraction partial-differential u Over partial-differential x EndFraction right-parenthesis comma x element-of upper R comma t greater-than 0 comma 2nd Row u left-parenthesis x comma 0 right-parenthesis equals upper A plus left-parenthesis upper B minus upper A right-parenthesis upper H left-parenthesis x right-parenthesis comma f o r x element-of upper R EndLayout">

{

∂
G

(
u
)

∂
t

+

∂
Φ

(
u
)

∂
x

=

f

(
u
)

t

+
ε
t

∂

∂
x

(

k

(
u
)

∂
u

∂
x

)

,

x
∈
R
,

t
>
0
,

u

(

x
,
0

)

=
A
+

(

B
−
A

)

H

(
x
)

,

f
o
r
x
∈
R

\left \{ {\begin {array}{*{20}{c}} {\frac {{\partial G\left ( u \right )}}{{\partial t}} + \frac {{\partial \Phi \left ( u \right )}}{{\partial x}} = \frac {{f\left ( u \right )}}{t} + \varepsilon t\frac {\partial }{{\partial x}}\left ( {k\left ( u \right )\frac {{\partial u}}{{\partial x}}} \right ), \qquad x \in R, \qquad t > 0,}\\ {u\left ( {x, 0} \right ) = A + \left ( {B - A} \right )H\left ( x \right ), \qquad for x \in R} \end {array}} \right .

\]
pointwise converges to

u
0

(

x
,
t

)

=:

v
0

(

x

/

x
t

t

)

{u_0}\left ( {x, t} \right ) = :{v_0}\left ( {{x \left / {\vphantom {x t}} \right . t}} \right )

, the discontinuous solution of the Riemann problem for the scalar conservation law

(

ε
=
0

)

\left ( {\varepsilon = 0} \right )

. Obviously,

u
0

(

x
,
t

)

{u_0}\left ( {x, t} \right )

satisfies the classical Rankine-Hugoniot condition on the line of discontinuity

x
=

t
Φ

(
B
)

/

(
B
)

g

(
B
)

g

(
B
)

x = \\ t\Phi {{\left ( B \right )} \left / {\vphantom {{\left ( B \right )} {g\left ( B \right )}}} \right . {g\left ( B \right )}}

, and the restriction on

Φ

(
u
)

\Phi \left ( u \right )

and

G

(
u
)

G\left ( u \right )

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi left-parenthesis u right-parenthesis minus upper G left-parenthesis u right-parenthesis normal upper Phi left-parenthesis upper B right-parenthesis slash upper G left-parenthesis upper B right-parenthesis greater-than-or-equal-to 0 o n left-bracket upper A comma upper B right-bracket comma">

Φ

(
u
)

−
G

(
u
)

Φ

(
B
)

/

(
B
)

G

G

(
B
)

≥
0

o
n

[

A
,
B

]

,

\Phi \left ( u \right ) - G\left ( u \right )\Phi {{\left ( B \right )} \left / {\vphantom {{\left ( B \right )} G}} \right . G}\left ( B \right ) \ge 0 \qquad on\left [ {A, B} \right ],

\]
is exactly the E-Condition proposed first by Oleinik. The technical arguments, which involve only the use of the Schauder Fixed Point Theorem and integral representations, are elementary.