Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 78, pp. 1–12.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

EXTENDING LAGRANGIAN TRANSFORMATIONS TO

NONCONVEX SCALAR CONSERVATION LAWS

PRERONA DUTTA

Abstract. This article studies a method of finding Lagrangian transforma-

tions, in the form of particle paths, for all scalar conservation laws having a
smooth flux. These are found using the notion of weak diffeomorphisms. More

precisely, from any given scalar conservation law, we derive a Temple system

having one linearly degenerate and one genuinely nonlinear family. We modify
the system to make it strictly hyperbolic and prove an existence result for it.

Finally we establish that entropy admissible weak solutions to this system are

equivalent to those of the scalar equation. This method also determines the
associated weak diffeomorphism.

1. Introduction

Reformulating systems of hyperbolic conservation laws in a Lagrangian form
provides an equivalent description of the problem by means of particle paths. For
smooth solutions, this means that the system is described as the flow in time of a
diffeomorphism of space. For hyperbolic conservation laws, the solutions are gen-
erally not smooth. This article extends recent work which shows that the erstwhile
diffeomorphisms then become invertible bi-Lipschitz mappings, referred to as weak
diffeomorphisms, that also satisfy systems of conservation laws. For systems which
describe fluid flow, construction of particle paths in the form of weak diffeomor-
phisms follows naturally from the associated velocity field. However, the notion of
a particle path can be extended to scalar equations and to systems which do not
include velocity fields explicitly.

Research in this direction was pioneered by Arnold in 1966. He described the dy-
namics of an ideal fluid in [1], as a geodesic flow on the group of volume-preserving
diffeomorphisms of a fixed domain equipped with the metric defined by the kinetic
energy. Thereafter in [6], the work of Ebin and Marsden extended this program.
In this article we determine how to find weak diffeomorphisms for any scalar con-
servation law with a smooth flux, not necessarily convex. The aim of our study is
to gain a deeper insight into conservation laws and to propel further investigation
into the prospects of using Lagrangian transformations to answer important open
questions in this area.

Previous work in [9] illustrated a procedure to find weak diffeomorphisms for
scalar conservation laws in one space variable, that works only when the flux is

2020 Mathematics Subject Classification. 35L65, 35L40, 35L45.

Key words and phrases. Scalar conservation law; Temple system; weak diffeomorphism.
©2022. This work is licensed under a CC BY 4.0 license.
Submitted October 24, 2022. Published November 21, 2022.

1



2 P. DUTTA EJDE-2022/78

convex. Holmes, Keyfitz and Tiglay established in [9] that under the standard
notions of admissibility, weak solutions of the original equation correspond to weak
solutions of the transformed system. For classical solutions, the authors of [9] used a
method similar to that in [6] and for weak solutions, their approach was to consider
particle paths as curves through the space of absolutely continuous and invertible
isomorphisms, which they called weak diffeomorphisms.

A different notion of Lagrangian variables was used by Bianchini and Marconi
in [2]. They used generalized characteristics to define their notion of Lagrangian
representation and applied it to prove results on the structure of bounded entropy
admissible weak solutions to scalar conservation laws. The Lagrangian representa-
tion in this case was not related to diffeomorphisms. In fact, the construction in [2]
was based on wavefront tracking. Thereafter, to find an approximation scheme and
compactness estimates, the authors of [2] utilized the transport collapse method
first introduced by Brenier in [3].

Our definition of a Lagrangian transformation is different from the perspective
in [2]. Our goal is to extend the idea of particle paths for scalar conservation laws
to weak solutions. In this case, as characteristics may intersect, unlike [2] we can
no longer consider particle paths to be characteristic curves. With this in mind,
given a scalar conservation law

ρt + f(ρ)x = 0 , ρ(x, 0) = ρ0(x) , x ∈ R (1.1)

since an abstract scalar conservation law may not describe the flow of a substance,
we define u := f(ρ)/ρ and rewrite (1.1) with u defining the flow speed as

ρt + (uρ)x = 0 , ρ(x, 0) = ρ0(x) , x ∈ R . (1.2)

Given a particle at x at t = 0, define γ(x, t) as the position it would reach at time
t if it travels with velocity u at every time. Thus γ is defined by the equation

γ̇ = u ◦ γ or γt(x, t) = u(γ(x, t), t) with γ(x, 0) = x . (1.3)

Our objective is to establish that for each fixed t, γ is a diffeomorphism for smooth
solutions and generalizes to a weak diffeomorphism in the absence of classical solu-
tions. To accomplish this, we manipulate the conservation law and define a mapping
γ that satisfies a Hamilton-Jacobi equation. From this, as done in [9], we derive
a Temple system with one linearly degenerate and one genuinely nonlinear family.
The system obtained in [9] was strictly hyperbolic owing to the convexity assump-
tion on the flux. Without the convexity of f , the resulting Temple system would
not be strictly hyperbolic. There are no relevant existence results for non-strictly
hyperbolic Temple systems as yet. Our achievement in this paper is to convert
the problem (1.1) into an equivalent problem whose diffeomorphism equation is a
strictly hyperbolic Temple system. Furthermore, we slightly extend a prior result
on existence of global large data solutions for strictly hyperbolic Temple systems.
In [11] this was proved by assuming both families to be genuinely nonlinear. In the
current paper we develop an existence result along the lines of [11], with suitable
modifications to account for one family being linearly degenerate. Finally we re-
cover γ from the Temple system and show the equivalence of solutions of the system
with those of the original equation.

This article is organized as follows. In Section 2, we show that a scalar conser-
vation law with bounded initial data can be transformed to an equivalent Cauchy
problem for a convex conservation law. Then we state the existence result (proved



EJDE-2022/78 LAGRANGIAN TRANSFORMS FOR SCALAR CONSERVATION LAWS 3

in Appendix A) for strictly hyperbolic Temple systems having one linearly degen-
erate and one genuinely nonlinear family. Finally we prove that weak solutions of
the derived Temple system are equivalent to those of the original scalar equation.
We refer to [4, 5, 14] for definitions and results related to conservation laws that are
used throughout the rest of this article, specifically pg. 81-85 in [4] for definitions
of admissibility conditions and entropy admissible weak solutions.

2. Weak diffeomorphisms for scalar conservation law

Given a scalar conservation law of the form (1.1) where f is a smooth flux and
ρ0 ∈ I = [a, b] for any a, b ∈ R, we rewrite it as

ρt + (uρ)x = 0 , ρ(x, 0) = ρ0(x) , x ∈ R , (2.1)

where we define u to be

u ≡ F (ρ) =
f(ρ)

ρ
. (2.2)

The weak diffeomorphism corresponding to (2.1) is defined by

γ̇ = u ◦ γ or γt(x, t) = u(γ(x, t), t) with γ(x, 0) = x . (2.3)

We show that for each fixed t, γ defines a diffeomorphism of x for smooth solutions
and generalizes to a weak diffeomorphism when classical solutions no longer exist.
This was achieved for a special case in [9] where the authors assumed that f was
convex and ρ, F, F ′ > 0, to deduce an equation for γ using the invertibility of F .
Motivated by [9], in this paper we provide a construction for weak diffeomorphisms
for a Cauchy problem with bounded data and a smooth flux, not necessarily convex.

2.1. Transforming the problem. Given a scalar conservation law

ρt + f(ρ)x = 0 , ρ(x, 0) = ρ0(x) , x ∈ R (2.4)

where f is any smooth flux, assume that ρ0 ∈ I = [a, b] for some a, b ∈ R.
It is clear from (2.2) that for F to be well-defined, ρ must be non-zero. Also, ρ

must maintain the same sign over I so that the Riemann problem for the derived
Temple system has a solution. Therefore if I contains 0, we use the following
translation.

Given ρ0 ∈ I = [a, b] with a < 0 < b, choose a constant L ∈ R such that L > a
and define σ := ρ+ L. Then (2.4) becomes

σt + f(σ − L)x = 0 , σ(x, 0) = σ0(x) , x ∈ R (2.5)

where σ0 ∈ Ĩ with Ĩ = I + L ⊂ R+.
From (2.5), we now define the flow speed as

u ≡ F (σ) =
f(σ − L)

σ
. (2.6)

The equation for γ was derived in [9] by assuming that F is invertible. In that case,
from (2.6) taking σ = b(u) where b = F−1, (2.5) becomes

b′(u)ut + ub′(u)ux + b(u)ux = 0 . (2.7)

Differentiating (2.3) with respect to t and substituting (2.7),

γtt = −
( b(u)

b′(u)

)
ux ◦ γ . (2.8)



4 P. DUTTA EJDE-2022/78

Differentiating (2.3) with respect to x,

ux =
(
γt ◦ γ−1

)
x

=
(γxt
γx

)
◦ γ−1 . (2.9)

Using (2.9) to eliminate ux from (2.8) we obtain(b′(γt)
b(γt)

)
γtt = −γxt

γx
. (2.10)

Integrating (2.10) from time 0 to t and using γt(x, 0) = F (σ0(x)), γx(x, 0) = 1
results in

ln
b(γt)

σ0
= − ln γx (2.11)

which yields

b(γt) =
σ0
γx

. (2.12)

Since b = F−1, (2.12) gives a Hamilton-Jacobi type equation

γt = F
(σ0
γx

)
(2.13)

which can be replaced by an equivalent conservation law system, with η = γx:

ηt − F
(v
η

)
x

= 0 , η(x, 0) = 1 ,

vt = 0 , v(x, 0) = σ0(x) .
(2.14)

Proving that solutions of system (2.14) are equivalent to those of the original scalar
equation (2.4) means showing that given an admissible weak solution (η, v) to (2.14),
one can recover the admissible weak solution to (2.4) with the corresponding data.

System (2.14) has the structure of a Temple system, for which the shock and
rarefaction waves coincide in state space. In quasilinear form, (2.14) becomes(

η
v

)
t

+

( v
η2F

′( vη ) − 1
ηF
′( vη )

0 0

)(
η
v

)
x

=

(
0
0

)
(2.15)

with eigenvalues λ1 = 0 and λ2 = v
η2F

′( vη ). The family corresponding to the

eigenvalue λ1 = 0 is linearly degenerate, while the family corresponding to the
eigenvalue λ2 is genuinely nonlinear. From (2.6),

F ′(σ) =
σf ′(σ − L)− f(σ − L)

σ2
.

Note that a system of the form (2.14) associated with the scalar equation (2.4)
via γ is strictly hyperbolic only when F is invertible. Existence of global large
data solutions for strictly hyperbolic 2 × 2 Temple systems was proved in [11]. If

F ′(σ) = 0 for some σ ∈ Ĩ, we modify the conservation law in Lemma 2.1 to obtain
an equation equivalent to (2.4) such that the associated Temple system is strictly
hyperbolic. The new equation and the corresponding system establish our results.
Our process replaces F with a velocity function G where G′ > 0.

Lemma 2.1. Given f in (2.5) such that F ′(σ) = 0 for some σ ∈ Ĩ, define g(σ) =
f(σ − L) +K and G by

G(σ) =
g(σ)

σ



EJDE-2022/78 LAGRANGIAN TRANSFORMS FOR SCALAR CONSERVATION LAWS 5

where K ∈ R is a constant. If K < minσ∈Ĩ{σf ′(σ − L) − f(σ − L)}, then G′ > 0

and G is invertible for σ ∈ Ĩ. Moreover, the system replacing F in (2.14) with G
is strictly hyperbolic.

Proof. Here G′ is given by

G′(σ) =
σf ′(σ − L)− f(σ − L)−K

σ2

for σ ∈ Ĩ. Thus G′ > 0 if σf ′(σ − L) − f(σ − L) > K. We choose any K <
minσ∈Ĩ{σf ′(σ − L)− f(σ − L)}.

Now for the modified system

ηt −G(
v

η
)x = 0 , η(x, 0) = 1 ,

vt = 0 , v(x, 0) = σ0(x) ,
(2.16)

we obtain λ2 = v
η2G

′( vη ) > 0. Thus (2.16) is strictly hyperbolic. �

Note: For ρ0 ∈ [a, b], if a > 0, then L = 0 and σ = ρ+ 0 = ρ. If b < 0, then define
σ = −ρ.

2.2. Main results. In the previous section we transformed the Cauchy problem
(2.4) for a smooth flux f into a strictly hyperbolic Temple system. The modified
scalar equation is

σt + g(σ)x = 0 , σ(x, 0) = σ0(x) , x ∈ R , (2.17)

where g is obtained from f via Lemma 2.1 and σ is obtained from ρ so that σ > 0
in Ĩ. Proposition 2.6 shows that (2.17) is equivalent to the original scalar equation
(2.4).

To prove the existence of global large data solutions for strictly hyperbolic Tem-
ple systems in [11], Leveque and Temple assumed both families to be genuinely
nonlinear and used Godunov’s method. Here we extend their result to when one
family is linearly degenerate.

Theorem 2.2. Consider the Cauchy problem

ηt −G(
v

η
)x = 0 , η(x, 0) = 1 ,

vt = 0 , v(x, 0) = σ0(x)
(2.18)

where σ0 ∈ Ĩ has bounded total variation, G ∈ C2 and G′ > 0 for σ ∈ Ĩ. Then
(2.18) has a unique admissible weak solution (η, v) ∈

(
C(0,∞), BV 2

)
for t > 0.

The proof is given in Appendix A and proceeds as in [11], with modifications to
account for the linearly degenerate family in system (2.18). As a direct consequence
of Theorem 2.2, we have the following corollary.

Corollary 2.3. Let (η, v) ∈
(
C(0,∞), BV 2

)
be the admissible weak solution to the

system (2.18). The distributional solution γ to

γx(x, t) = η(x, t), γt(x, t) = G
(v(x, t)

η(x, t)

)
, γ(x, 0) = x (2.19)

is well-defined, absolutely continuous and invertible and γ−1 is absolutely continu-
ous.



6 P. DUTTA EJDE-2022/78

Proof. For a fixed t, γ is an antiderivative of a strictly positive function of bounded
variation. Therefore it follows using the Fundamental Theorem of Calculus for
Lebesgue Integrals [7, Theorem 3.35] that γ is an absolutely continuous function of
x and is invertible.

The function γ is strictly increasing in x and differentiable almost everywhere by
the same theorem. Thus the inverse in x of γ is continuous and strictly increasing.
Let X be the set of points at which γ has a finite, positive derivative and let Y
be the collection of the remaining points. So Y is a set of measure zero. As γ is
absolutely continuous, it satisfies the Lusin N-property [13, Lemma 7.25], implying
that γ(Y ) is also a set of measure zero. Let A = γ(X) and B = γ(Y ). We know that
γ−1 has finite positive derivative at every point in A and B is a set of measure zero.
Thus γ−1 maps any measure zero subset of A to a set of measure zero and also maps
every subset of B to a set of measure zero, i.e., a subset of Y. Therefore γ−1 satisfies
the Lusin N-property as well, which shows that it is absolutely continuous. �

This leads to our main result, where we prove that admissible weak solutions to
equation (2.17) can be recovered from those of system (2.18).

Theorem 2.4. Define

σ :=
v(γ−1(x, t), t)

η(γ−1(x, t), t)
(2.20)

where (η, v) is the admissible weak solution to (2.18) and γ is defined in Corollary
2.3. Then σ is an admissible weak solution to (2.17).

Proof. With γ−1 defined in Corollary 2.3, construct σ as in (2.20). Let ϕ(x, t) be
a test function, i.e. ϕ ∈ C1 and has compact support. Recalling g(σ) = σG(σ), we
have

I :=

∫∫ (
ϕtσ + ϕxg(σ)

)
dx dt

=

∫∫ [
ϕt ·

(
(
v

η
) ◦ γ−1

)
+ ϕx ·

((v
η
G(
v

η
)
)
◦ γ−1

)]
dx dt .

Using the change of variable (x, t) 7→ (γ(x, t), t)

I =

∫∫ [
(ϕt ◦ γ)(

v

η
) + (ϕx ◦ γ)

(v
η
G(
v

η
)
)]
η dx dt . (2.21)

Now,

∂

∂t
(ϕ ◦ γ) = (ϕx ◦ γ) · γt + (ϕt ◦ γ) ,

∂

∂x
(ϕ ◦ γ) = (ϕx ◦ γ) · γx .

(2.22)

Let ϕ̃ = ϕ ◦ γ. From (2.22) we obtain

ϕt ◦ γ = ϕ̃t −
ϕ̃x
γx
γt and ϕx ◦ γ =

ϕ̃x
γx

.



EJDE-2022/78 LAGRANGIAN TRANSFORMS FOR SCALAR CONSERVATION LAWS 7

Then (2.21) becomes

I =

∫∫ [
ϕ̃tv + ϕ̃x

(v
η
G(
v

η
)− v

η
G(
v

η
)
)]
dx dt

=

∫∫
ϕ̃tv dx dt

= −
∫
ϕ̃(x, 0)σ0(x) dx .

(2.23)

Thus σ as defined in (2.20) using the weak solution to system (2.18), satisfies the
definition of a weak solution to equation (2.17).

Next we establish that admissibility conditions for the system [4, Section 4.4]
correspond to admissibility conditions for the scalar conservation law. In order to
do so, we prove that given a convex extension for (2.17), we can recover a convex
extension for (2.18) and vice versa.

Definition 2.5. A pair of C1 functions (E ,Q) : R → R is an entropy-entropy flux
pair for (2.17) if

Q′(σ) = E ′(σ) · g′(σ) (2.24)

at every σ where E ,Q and g are differentiable. In particular, if E is convex and
smooth, then the pair (E ,Q) is called a convex extension.

A convex extension (Ē , Q̄) of (2.18) [4, Definition 4.4] satisfies the equation
Ēt + Q̄x = 0 for smooth variables. From this, using the chain rule and (2.18) we
obtain

Q̄η = Ēη ·
v

η2
G′(

v

η
) and Q̄v = −Ēη ·

1

η
G′(

v

η
) .

Equating second order mixed partial derivatives yields Ēηη + v
η Ēηv = 0 and solving

this partial differential equation we obtain

Ē(
v

η
) = η · S(

v

η
),

Q̄(
v

η
) = −S(

v

η
)G(

v

η
) +

∫ v/η

S′(x)G(x) dx

(2.25)

where Ē is convex if S′′ > 0.
Now we return to the proof of Theorem 2.4. Suppose that (E1,Q1) is a convex

extension for the scalar equation (2.17), i.e., Q′1(σ) = E ′1(σ) · g′(σ). Then σ is an
admissible solution to (2.17), if for any test function ψ,

J :=

∫∫ [
ψtE1 + ψxQ1

]
dx dt ≥ 0 . (2.26)

Now we use a change of variable (x, t) 7→ (γ(x, t), t) and rewrite J as

J =

∫∫ [
(ψt ◦ γ) · E1(

v

η
) + (ψx ◦ γ) · Q1(

v

η
)
]
η dx dt. (2.27)

Assuming (2.26), applying (2.22) to (2.27) and using the notation ψ̃ = ψ ◦ γ, we
proceed as in (2.23) to deduce that

J :=

∫∫ [
ψ̃tE2(η, v) + ψ̃xQ2(η, v)

]
dx dt ≥ 0 , (2.28)

where E2(η, v) = ηE1( vη ) and Q2(η, v) = Q1( vη )−G( vη )·E1( vη ). Thus E2 is of the form

obtained in (2.25) and (E2)t + (Q2)x = 0 whenever (η, v) is a classical solution to



8 P. DUTTA EJDE-2022/78

the system (2.18). Furthermore, as E1 is convex, it follows by a direct computation
that the Hessian matrix of E2 is positive semi-definite. For the reverse implication,
consider a convex entropy Ē for the system (2.18) as defined in (2.25). Then the
convex function S in (2.25) is a convex entropy for (2.17). This completes the
proof. �

In conclusion, we observe that ρ is an admissible weak solution to the Cauchy
problem for the scalar conservation law (2.4) where ρ = σ − L and σ is given by
(2.20).

Proposition 2.6. Equations (2.4) and (2.17) are equivalent in the sense that en-
tropy admissible weak solutions of (2.17) are equivalent to those of (2.4).

Proof. Equation (2.17) is obtained from (2.4) via Lemma 2.1. Given a convex
extension (E1,Q1) for (2.4), it satisfies Q′1(ρ) = E ′1(ρ) · f ′(ρ) for ρ ∈ I. As ρ is an
admissible weak solution to (2.4), for any test function ϕ,∫∫

[ϕtE1(ρ) + ϕxQ1(ρ)] dx dt ≥ 0

which means ∫∫
[ϕtE1(σ − L) + ϕxQ1(σ − L)] dx dt ≥ 0 .

Taking E2(σ) = E1(σ − L) and Q2(σ) = Q1(σ − L) yields∫∫
[ϕtE2(σ) + ϕxQ2(σ)] dx dt ≥ 0

for any test function ϕ, where Q′2(σ) = E ′2(σ) · g′(σ) and σ is an admissible weak
solution to (2.17). Therefore (E2,Q2) is a convex extension for (2.17) obtained from
a convex extension of (2.4) and the reverse follows similarly. This implies entropy
admissible weak solutions of (2.4) and (2.17) are equivalent. �

Appendix A

In this appendix, we provide details for the proof of Theorem 2.2 based on results
and methods from [8, 10, 11].

The following procedure uses Godunov’s method for approximating hyperbolic
conservation laws by finite differences. This method chooses a piecewise constant
approximation of the data and solves a Riemann problem at each discontinuity.
The state variables are then averaged over each space interval. Defining a piecewise
constant approximation at the next time step requires that waves emerging from
one inter-cell boundary not interact with waves created at adjacent boundaries.
Controlling the Courant number as shown in (2.30) avoids such wave interactions.

The system (2.18) is a Temple system, for which shock and rarefaction curves
are straight lines that coincide in η − v space. For systems having two genuinely
nonlinear families, [11] shows the existence of an invariant region. However contact
discontinuity curves may not be straight lines and the existence of an invariant
region is then unclear. In fact, the shape of contact discontinuity curves in η − v
space depends on the function G(η, v).

In our problem, G is a function of v
η and the right eigenvectors corresponding to

the eigenvalues λ1 = 0 and λ2 = v
η2G

′( vη ) are

r1 =

(
1
v/η

)
and r2 =

(
1
0

)
.



EJDE-2022/78 LAGRANGIAN TRANSFORMS FOR SCALAR CONSERVATION LAWS 9

Thus the contact discontinuity curves are rays from the origin. From the given
initial data in (2.18), we see that there is a potential invariant region in the form
of a quadrilateral Q whose vertices are given by (m/M,m), (1,m), (M/m,M) and
(1,M), where 0 < m ≤ σ0 ≤ M . The sides of Q are either contact discontinuity
curves corresponding to λ1, i.e., rays from the origin, or are horizontal lines in the
η − v plane with λ2 decreasing as η increases. In Claim 1 below, we establish that
Q is indeed an invariant region for our problem. Hence, we can use the arguments
related to Godunov’s method from [11] that depend on having an invariant region
in the form of a quadrilateral, with alterations to accommodate the fact that one
family is linearly degenerate and the other is genuinely nonlinear.

For our problem, the 1-Riemann invariant is given by v/η and the 2-Riemann
invariant is v, with i-th Riemann invariant for i = 1, 2 defined as in [11]. We denote
these by p and q respectively. Let w = (η, v) and P =

(
−G(v/η), 0

)
. The Riemann

problem for (2.18) is

wt + P (w)x = 0

w(x, 0) = w0(x) =

{
wL, x ≤ 0

wR, x > 0
.

(2.29)

To apply Godunov’s method, take h to be the mesh length in x and k to be a time
step such that

k

h
sup
w∈Q
|λi(w)| < 1 . (2.30)

For j ∈ Z and n ∈ N, let xj = jh and tn = nk. Then the approximation to the
initial condition is defined at mesh points by

w0
j =

1

h

∫ xj

xj−1

w0(x)dx . (2.31)

Recall that Q is a quadrilateral containing wL and wR. In (p, q) coordinates, Q
is a rectangle. So if w0(x) ∈ Q for all x, w0

j lies in Q for all j. For every j, let
w̃(x, tn) = wnj denote the approximate solution at time tn for x ∈ (xj−1, xj ]. The
piecewise constant data at each tn give a Riemann problem at every xj . Solving the
local Riemann problem at each xj , let w̃(x, tn+1) denote the solution in [tn, tn+1).
By (2.30), the waves that emanate from (xj , tn) do not intersect before the next
time step tn+1. We average the local solution in each interval by defining

wn+1
j =

1

h

∫ xj

xj−1

w̃(x, tn+1)dx (2.32)

to approximate w at tn+1. As one family has characteristic speed 0, there is a
discontinuity at each xj . Define the extension wh of the grid function as

wh(x, t) = wn+1
j for (x, t) ∈ (xj−1, xj ]× [tn, tn+1) . (2.33)

Our aim is to show that given a sequence of mesh lengths {hl} converging to 0,
there is a subsequence of {hl} for which wh(x, t) converges boundedly a.e. to a
function w(x, t) and that w is a weak solution to the Riemann problem (2.29).
The following claims lead to our result.

Claim 1: Q is an invariant region for the Riemann problem (2.29). In other words,
if w0 ∈ Q then wnj ∈ Q for all j, n.



10 P. DUTTA EJDE-2022/78

Proof. We know that Q is a quadrilateral and wL, wR ∈ Q. The solution to the
Riemann problem always consists of a contact discontinuity with speed 0 and a
positive speed shock or rarefaction, with p constant on the contact discontinuity
and q constant on the other wave. In (p, q) coordinates, Q is a rectangle. Given
n, if wnj is in the rectangle Q for all j, Q contains w̃(x, tn+1) for x ∈ [xj−1, xj).

This implies wn+1
j ∈ Q as it is the average of all w̃(x, tn+1) over [xj−1, xj) at tn+1.

Thus, the claim follows by induction. �

We define distance between two points as

‖wL − wR‖ = |p(wL)− p(wR)|+ |q(wL)− q(wR)| . (2.34)

Claim 2: For all j, n, the following hold:

(1) ‖wnj−1 − w
n+1
j ‖+ ‖wn+1

j − wnj ‖ = ‖wnj − wnj−1‖;
(2)

∑
j ‖w

n+1
j − wn+1

j−1 ‖ ≤
∑
j ‖wnj − wnj−1‖.

Proof. (1) For any j and n, p(wnj )−p(wnj−1) = p(wnj )−p(wn+1
j )+p(wn+1

j )−p(wnj−1).

In (p, q) coordinates, wn+1
j is the intersection of a horizontal line from wnj−1 and a

vertical line from wnj . Note that wn+1
j is obtained by averaging the local solution

in [xj−1, xj) at tn+1. Therefore both its coordinates lie between the coordinates of

wnj−1 and wnj , so |p(wnj−1) − p(wn+1
j )| and |p(wn+1

j ) − p(wnj )| have the same sign.
Thus,

|p(wnj−1)− p(wn+1
j )|+ |p(wn+1

j )− p(wnj )| = |p(wnj )− p(wnj−1)|

and a similar equality holds for q. Hence, (1) follows using (2.34).
(2) By the triangle inequality and part (1),∑

j

‖wn+1
j − wn+1

j−1 ‖ ≤
∑
j

(
‖wn+1

j − wnj−1‖+ ‖wnj−1 − wn+1
j−1 ‖

)
=
∑
j

(
‖wn+1

j − wnj−1‖+ ‖wnj − wn+1
j ‖

)
=
∑
j

‖wnj − wnj−1‖ .

(2.35)

�

Integrating (2.29) over the rectangle (xj−1, xj ] × [tn, tn+1) and using the diver-
gence theorem, we obtain

wn+1
j = wnj +

k

h

[
P (wnj )− P (wnj−1)

]
. (2.36)

Claim 3: Given 0 < T1 < T2, there exists a constant C > 0 such that∫ ∞
−∞
|wh(x, T2)− wh(x, T1)| dx ≤ C|T2 − T1| · TV (w0) (2.37)

where TV (w0) is the total variation of initial data w0 measured in norm (2.34).



EJDE-2022/78 LAGRANGIAN TRANSFORMS FOR SCALAR CONSERVATION LAWS 11

Proof. Let 0 < T1 < T2 and r,R ∈ Z be such that

tr ≤ T1 < tr+1 ≤ tR ≤ T2 < tR+1 .

From this relation, using (2.33) we have∫ ∞
−∞
|wh(x, T2)− wh(x, T1)| dx = h

∞∑
j=−∞

|wR+1
j − wr+1

j |

≤ h
∞∑

j=−∞

R∑
n=r+1

|wn+1
j − wnj |

= k

∞∑
j=−∞

R∑
n=r+1

|P (wnj )− P (wnj−1)| .

(2.38)

As P is Lipschitz,

∞∑
j=−∞

|P (wnj )− P (wnj−1)| ≤ C · TV (wh) (2.39)

where C is the Lipschitz constant for P . (2.35) implies that {wh} has nonincreasing
total variation, so we have

∞∑
j=−∞

|P (wnj )− P (wnj−1)| ≤ C · TV (w0) . (2.40)

As k(R− r) ≤ |T2 − T1|, (2.38) and (2.39) together yield (2.37). �

Claims 2 and 3 show that wh is continuous in time and has bounded total
variation for every h. Moreover, {wh} is uniformly bounded on [0, T ] for T <
∞. By Helly’s theorem [12, Theorem 1.3], there exists a subsequence of {wh}
converging pointwise to a limit w(x, t). In addition, as {wh} is uniformly bounded,
this subsequence is said to converge boundedly a.e. to w(x, t). Hence, by Lax and
Wendroff’s theorem in [10, Section 1], w(x, t) is a weak solution to (2.29).

To establish that w(x, t) is an admissible weak solution, let us take a convex
extension (E ,Q) for (2.29) and denote

En+1
j = E(wn+1

j ) and Q̄(wn+1
j−1 , w

n+1
j ) = Q(w̃(xj−1, tn+2)) .

Recalling (2.32) and using Jensen’s inequality, we have

En+1
j = E

( 1

h

∫ xj

xj−1

w̃(x, tn+1) dx
)
≤ 1

h

∫ xj

xj−1

E(w̃(x, tn+1))dx . (2.41)

Given a convex extension (E ,Q), for a weak solution w of (2.29) to be admissible,
it must satisfy

E(w)t +Q(w)x ≤ 0 . (2.42)

In other words, for any smooth nonnegative test function ψ it must hold that

−
∫ ∞
0

∫ ∞
−∞

[E(w(x, t))ψt(x, t) +Q(w(x, t))ψx(x, t)] dx dt

−
∫ ∞
−∞
E(w(x, 0))ψ(x, 0) dx ≤ 0 .

(2.43)



12 P. DUTTA EJDE-2022/78

Since (2.42) holds for the solution to every local Riemann problem arising at each
xj , integrating (2.42) over (xj−1, xj ]× [tn, tn+1) we obtain

1

h

∫ xj

xj−1

E(w̃(x, tn+1))dx

≤ 1

h

∫ xj

xj−1

E(w̃(x, tn))dx− 1

h

∫ tn+1

tn

[Q(w̃(xj , t))−Q(w̃(xj−1, t))] dt,

which implies

En+1
j ≤ Enj −

k

h

[
Q̄(wnj , w

n
j+1)− Q̄(wnj−1, w

n
j )
]
. (2.44)

Then by [8, Theorem 1.1], a weak solution w to (2.29) obtained using Godunov’s
method satisfies the entropy condition (2.43). Thus w(x, t) is an admissible weak
solution to (2.29). Moreover, since w(x, t) has bounded total variation, by [4,
Theorem 9.4 ] it is the unique entropy admissible weak solution to the Cauchy
problem (2.29).

Acknowledgments. I thank Barbara Lee Keyfitz for her valuable suggestions on
this project. I would also like to thank the anonymous referees for their comments
which helped in improving the paper overall.

References

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applications à l’hydrodynamique des fluids parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966),
no. 1, 319–361.

[2] S. Bianchini, E. Marconi; On the structure of L∞-entropy solutions to scalar conservation

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fluid, Ann. of Math., (2), 92 (1970), no. 1, 102–163.
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Applied Mathematics, Wiley-Intersci., John Wiley & Sons, Inc., New York, 1999.

[8] A. Harten, P. D. Lax, B. V. Leer; On upstream differencing and Godunov-type schemes for
hyperbolic conservation laws, SIAM Rev., 25 (1983), no. 1, 35–61.

[9] J. Holmes, B. Keyfitz, F. Tiglay; Weak diffeomorphisms and solutions to conservation laws,
Matematica, 1 (2022), no. 1, 131–166.

[10] P. D. Lax, B. Wendroff; Systems of Conservation Laws, Comm. Pure Appl. Math., 13 (1960),

217–237.

[11] R. J. Leveque, B. Temple; Stability of Godunov’s method for a class of 2 x 2 systems of
conservation laws, Trans. Amer. Math. Soc., 288 (1985), no. 1, 115–123.

[12] J. Musielak, W. Orlicz; On generalized variations (I), Studia Math., 18 (1959), 11–41.
[13] W. Rudin; Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987.

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258, Springer-Verlag, New York, 1994.

Prerona Dutta
Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA

Email address: dutta.105@osu.edu


	1. Introduction
	2. Weak diffeomorphisms for scalar conservation law
	2.1. Transforming the problem
	2.2. Main results

	Appendix A
	Acknowledgments

	References