The Riemann problem in gas dynamics
Authors: Smith Randolph G. 
Published: June 2010

We consider the Riemann problem (R.P.) for the

3

×

3

3\, \times \, 3

system of gas dynamics equations in a single space variable. We assume that the specific internal energy

e
=
e
(
v
,

s
)

e = e(v,\,s)

(s = specific entropy, v = specific volume) satisfies the usual hypotheses,

p
v

>

0
,

p

v
v

>

0
,

p
s

>

0

(
p

=

−

e
v

=

{p_v}\, > \,0,\,{p_{vv}}\, > \,0,\,{p_s}\, > \,0\,(p\, = \, - \,{e_v}\, =

pressure); we also assume some reasonable hypotheses about the asymptotic behavior of e. We call functions e satisfying these hypotheses energy functions Theorem 1. For any initial data
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper U Subscript l Baseline comma upper U Subscript r Baseline right-parenthesis left-parenthesis upper U Subscript l Baseline equals left-parenthesis v Subscript l Baseline comma p Subscript l Baseline comma u Subscript l Baseline right-parenthesis">

(

U
l

,

U
r

)

(

U
l

=

(

v
l

,

p
l

,

u
l

)

({U_l},\,{U_r})\,({U_l}\, = \,({v_l},\,{p_l},\,{u_l})

,

U
r

=

(

v
r

,

p
r

,

u
r

)

{U_r}\, = \,({v_r},\,{p_r},\,{u_r})

, u = flow velocity), the R. P. has a solution. We introduce two conditions:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-parenthesis upper I right-parenthesis right-parenthesis EndLabel StartLayout 1st Row 1st Column StartFraction partial-differential Over partial-differential v EndFraction p left-parenthesis v comma e right-parenthesis less-than-or-slanted-equals StartFraction p squared Over 2 e EndFraction 2nd Column a m p semicolon left-parenthesis v comma e greater-than 0 right-parenthesis comma EndLayout EndLayout">

(

(I)

)

∂

∂
v

p
(
v
,

e
)
⩽

p
2

2
e

a
m
p
;

(
v
,

e

>

0
)
,

\begin {array}{*{20}{c}}\tag {$\text {(I)}$} {\frac {\partial } {{\partial v}}\,p(v,\,e) \leqslant \frac {{{p^2}}} {{2e}}} & {(v,\,e\, > \,0),} \\ \end {array}

\]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-parenthesis II right-parenthesis right-parenthesis EndLabel StartLayout 1st Row 1st Column StartFraction partial-differential Over partial-differential v EndFraction e left-parenthesis v comma p right-parenthesis greater-than-or-slanted-equals minus StartFraction p Over 2 EndFraction 2nd Column a m p semicolon left-parenthesis v comma p greater-than 0 right-parenthesis period EndLayout EndLayout">

(

(II)

)

∂

∂
v

e
(
v
,

p
)

⩾
−

p
2

a
m
p
;

(
v
,

p

>

0
)
.

\begin {array}{*{20}{c}}\tag {$\text {(II)}$} {\frac {\partial }{{\partial v}}\,e(v,\,p)\, \geqslant - \frac {p} {2}} & {(v,\,p\, > \,0).} \\ \end {array}

\]
Theorem 2. (I) is necessary and sufficient for uniqueness of solutions of the R. P. Nonuniqueness persists under small perturbations of the initial data. (I) is implied by the known condition
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis left-parenthesis asterisk right-parenthesis right-parenthesis EndLabel StartFraction partial-differential Over partial-differential v EndFraction e left-parenthesis v comma p right-parenthesis greater-than 0 left-parenthesis v comma p greater-than 0 right-parenthesis comma EndLayout">

(

(
∗
)

)

∂

∂
v

e
(
v
,
p
)
>
0

(
v
,
p
>
0
)
,

\begin{equation}\tag {$(\ast )$} {\frac {\partial } {{\partial v}}e(v,p) > 0} \qquad (v,p > 0), \end{equation}

which holds for all usual gases. (I) implies (II). We construct energy functions e that violate (II), that satisfy (II) but violate (I), and that satisfy (I) but violate (*). In all solutions considered, the shocks satisfy the entropy condition and the Lax shock conditions.