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A note on admissible solutions of 1d scalar conservation laws and 2d hamilton–jacobi equations

01 Dec 2004-Journal of Hyperbolic Differential Equations (World Scientific Publishing Company)-Vol. 01, Iss: 04, pp 813-826
TL;DR: In this paper, it was shown that entropy solutions of Dtu+Dxf(u) = 0 belong to SBVloc(Ω) for planar Hamilton-Jacobi PDEs with uniformly convex Hamiltonians.
Abstract: Let Ω⊂ℝ2 be an open set and f∈C2(ℝ) with f" > 0. In this note we prove that entropy solutions of Dtu+Dxf(u) = 0 belong to SBVloc(Ω). As a corollary we prove the same property for gradients of viscosity solutions of planar Hamilton–Jacobi PDEs with uniformly convex Hamiltonians.

Summary (1 min read)

Definition 2.4 (Characteristics). Let x /

  • The segment joining (t, x) with (0, y(t, x)) will be called (backward minimal) characteristic emanating from (t, x).
  • It turns out that the minimality of characteristics easily implies that two different characteristics starting even at different times are either one contained in the other or do not intersect .
  • As a consequence equality must hold and the two segments are parallel.
  • The authors first show how to conclude (3.7) from the lemma.

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ZurichOpenRepositoryand
Archive
UniversityofZurich
UniversityLibrary
Strickhofstrasse39
CH-8057Zurich
www.zora.uzh.ch
Year:2004
Anoteonadmissiblesolutionsof1Dscalarconservationlawsand2D
Hamilton-Jacobiequations
Ambrosio,L;DeLellis,C
Abstract:LetΩ฀฀2beanopensetandf฀C2(฀)withf>0. Inthisnoteweprovethatentropysolutions
ofDtu+Dxf(u)=0belongtoSBVloc(Ω).Asacorollaryweprovethesamepropertyforgradientsof
viscositysolutionsofplanarHamilton–JacobiPDEswithuniformlyconvexHamiltonians.
DOI:https://doi.org/10.1142/S0219891604000263
PostedattheZurichOpenRepositoryandArchive,UniversityofZurich
ZORAURL:https://doi.org/10.5167/uzh-21763
JournalArticle
Originallypublishedat:
Ambrosio,L;DeLellis,C(2004).Anoteonadmissiblesolutionsof1Dscalarconservationlawsand2D
Hamilton-Jacobiequations.JournalofHyperbolicDierentialEquations,1(4):813-826.
DOI:https://doi.org/10.1142/S0219891604000263

A NOTE ON ADMISSIBLE SOLUTIONS OF 1D SCALAR
CONSERVATION LAWS A ND 2D HAMILTON–JACOBI EQUATIONS
LUIGI AMBROSIO AND CAMILLO DE LELLIS
Abstract. Let R
2
be an open se t and f C
2
(R) w ith f
′′
> 0. In this note we prove
that entropy solutions of D
t
u + D
x
f(u) = 0 belong to SBV
loc
(Ω). As a corollary we prove
the s ame prop e rty for gradients of viscosity solutions of planar Hamilton–Jacobi PDEs with
uniformly convex hamiltonians.
1. Introduction
In this paper we consider entropy solutions of the scalar conservation law
D
t
u + D
x
[f(u) ] = 0 in (1.1)
and viscosity solutions of the planar Hamilton–Jacobi PDE
H(v) = 0 in Ω, (1.2)
where H a nd f are C
2
and locally uniformly convex. In these cases it is known that u and
v belong to BV (Ω
) for every open set
⊂⊂ Ω, i.e. that the distributions Du and Du
are vector (resp. matrix) valued Radon measures. The rough picture that one has in mind
when describing such solutions is the one of piecewise C
1
functions with discontinuities of
jump type. The space of BV functions enjoys good functional analytic properties, but the
behaviour of a generic BV function can be indeed very fa r from the picture above.
Following [3], given w BV (R
m
, R
k
) we decompose Dw into three mutually singular
measures: Dw = D
a
w + D
c
w + D
j
w. D
a
w is the part of the measure which is absolutely
continuous with resp ect to the Lebesgue measure L
m
. D
j
w is called jump part and it is
concentrated on the rectifiable m 1 dimensional set J where the function u has jump
discontinuities (in an appropriat e measure–theoretic sense: see Section 2). D
c
w is called the
Cantor part, it is singular with respect to L
m
and it satisfies D
c
w(E) = 0 fo r every Borel set
E with H
m1
(E) < . When m = 1, D
j
w consists of a countable sum of weighted Dirac
masses, whereas D
c
w is the non–atomic singular par t of the measure. A typical example of
D
c
w is the derivative of the Cantor–Vitali ternary function (see for instance Example 1.67
of [3]).
In [5] the authors introduced the space of special f unctions of bounded variations, denoted
by SBV , which consists of the functions w BV such that D
c
w = 0. This space played
an import ant role in the last years, in connection with problems coming from the theory
of image segmentation a nd with variationa l problems in fracture mechanics (see [3] and the
references quoted therein for a detailed present ation of this subject).
1

2 LUIGI AMBROSIO AND CAMILLO DE LELLIS
It is natural to ask whether entropy solutions of (1.1) and gradients of viscosity solutions
of (1.2) are locally SBV and, as far as we know, this question has never been addressed in
the literature. Our interest is in part motivated by some measure–theoretic questions arisen
in [2].
In the following remark we single out a canonical representative in the equivalence class
of u for which more precise informations, of pointwise tipe, are available.
Remark 1.1. Let u L
(Ω) be a weak solution of (1.1) and assume ]t
1
, t
2
[×J for some
open set J R. Using the equation one can prove that for every τ ]t
1
, t
2
[ the functions
f
ε
(x) =
R
τ+ε
τ
u(x, t)dt have a unique limit f in L
(J) weak
for ε 0 (see for instance
Theorem 4.1.1 of [4]). The refo re from now on we x the convention that u(τ, ·) = f(·).
The f ollowing is the main result of this note.
Theorem 1.2. Let u L
(Ω) be an entropy solution of (1.1) w i th f C
2
(R) locally
uniformly convex. Then there exists S R at most countable such that τ R \ S the
following holds:
u(τ, ·) SBV
loc
(Ω
τ
) with
τ
:= {x R : (τ, x) }. (1.3)
From this theorem, using the slicing theory of BV functions, we obtain:
Corollary 1.3. Let f C
2
(R) be locally uniform l y convex and let u L
(Ω) be an entropy
solution of (1.1). Then u SBV
loc
(Ω).
Event ually, via a local change of coordinates we a pply the previous result to the Hamilton–
Jacobi case:
Corollary 1.4. Let H C
2
(R
2
) be locally uniformly convex and let u W
1,
(Ω) be a
viscosity solution of H(u) = 0. Then u SBV
loc
(Ω).
As we show in Remark 3.3, Theorem 1.2 is optimal. Also t he regularity results obtained
in the two corollaries seem to be optimal, in view of the fact that shocks do occur and that
the gradients of viscosity solutions o f Hamilton-Jacobi PDEs can jump a long hypersurfaces.
Our result applies in particular to the distance function dist(x, K), which solves the eikonal
equation |∇u|
2
1 = 0 in the viscosity sense in = R
2
\ K. In this connection, we mention
the paper [8], where the authors establish among other things the SBV regularity in any
space dimension, but under some regularity assumptions on K.
It would be interesting to extend these results to
(a) BV admissible solutions of genuinely nonlinear systems of conservation laws in 1
space dimension;
(b) Viscosity solutions o f uniformly convex Hamilton–Jacobi PD Es in higher dimensions.
The proof of Theorem 1.2 uses at the very end a variational principle, due to Hopf and Lax.
However, it might be that combining part of this proof with the theory of characteristics for
systems of conservation laws (as developed in [4]) one could be able to extended Theorem 1.2
at least to the case (a).

ON ADMISSIBLE SOLUTIONS OF 1D SCALAR CONSERVATION LAWS 3
2. Preliminaries
2.1. BV and SBV spaces. In what follows L
d
and H
n
denote respectively the Lebesgue
measure on R
d
and the n–th dimensional Hausdorff measure on Euclidean spaces. A set
J R
d
is said countably H
n
–rec tifia b l e (or briefly rectifiable) if there exist countably many
n–dimensional Lipschitz graphs Γ
i
such that H
n
(J \
S
Γ
i
) = 0. Given a Borel measure µ
and a Borel set A we denote by µ
A the measure given by µ A(C) = µ(A C).
The approximate discontinuity set S
w
of a locally summable function w : R
d
R
m
and the approximate lim i t are defined as follows: x / S
w
if and only if there exists
z R
m
satisfying
lim
r0
r
d
Z
B
r
(x)
|w(y) z| dy = 0.
The vector z, if it exists, is unique and denoted by ˜w(x), the approximate limit of w at x.
It is easy to check that the set S
w
is Borel and that ˜w is a Borel function in its domain
(see §3.6 of [3] fo r the details). By Lebesgue differentiation theorem the set S
w
is Lebesgue
negligible and ˜w = w L
d
-a.e. in \ S
w
.
In a similar way one can define the approximate jump set J
w
S
w
, by requiring the
existence of a, b R
m
with a 6= b and of a unit vector ν such that
lim
r0
r
d
Z
B
+
r
(x,ν)
|w(y) a| dy = 0, lim
r0
r
d
Z
B
r
(x,ν)
|w(y) b| dy = 0,
where
(
B
+
r
(x, ν) := {y B
r
(x) : hy x, νi > 0} ,
B
r
(x, ν) := {y B
r
(x) : hy x, νi < 0} .
(2.1)
The triplet (a, b, ν), if it exists, is unique up to a permutation of a and b and a change of sign
of ν. We denote it by (w
+
(x), w
(x), ν(x)), where w
±
(x) are called approxim ate one-sided
limits of w at x. It is easy to check that the set J
w
is Bo r el and that w
±
and ν can be chosen
to be Borel functions in their domain (see again §3.6 of [3 ] for details).
The following structure theorem, essentially due to Federer and Vol’pert, holds (see for
instance Theorem 3.77 and Proposition 3.92 of [3]):
Theorem 2.1. Let w BV (Ω). Th en H
d1
(S
w
\ J
w
) = 0 and J
w
is a countably H
d1
rectifiable set. If we deno te by D
a
w the absolutely continuous part of Dw and by D
s
w the
singular part, then D
s
w can be written as D
j
w + D
c
w, where
D
j
w = (w
+
w
) ν
J
w
H
d1
J
w
, (2.2)
D
c
w(E) = 0 for any Borel set E with H
d1
(E) < . (2.3)
When R we have the following r efinement (see for instance Theorem 3.28 of [3]):
Proposition 2.2. Let w BV (Ω) and let R. Then S
w
= J
w
, ˜w is continuous on
\ J
w
and ˜w has classical left and righ t limits (which coinc i de with w
±
(x)) at any x J
w
.

4 LUIGI AMBROSIO AND CAMILLO DE LELLIS
Therefore
D
j
w =
X
xJ
w
(w
+
(x) w
(x))δ
x
.
2.2. Hopf–Lax formula and characteristics. Let f C
2
be locally uniformly convex,
u
0
L
1
(R) and let u L
(R
+
× R) be the entropy solution of the Cauchy problem
D
t
u + D
x
[f(u) ] = 0
u(0, ·) = u
0
.
(2.4)
Then u can be computed by using a variational principle, the so-called Hopf–Lax formula.
In particular we have the following well-known theorem.
Theorem 2.3 (Hopf–Lax formula). Let u
0
L
1
(R), let f : R R be C
2
and locally
uniformly convex and set
v
0
(y) :=
Z
y
−∞
u
0
(s) ds y R.
Let
v(t, x) := min
tf
x y
t
+ v
0
(y) : y R
. (2.5)
Then the fo llo wing statements hold:
(i) For any t > 0 there exists a countable set S
t
such that the minimum is attained at a
unique point y(t, x) for any x / S
t
.
(ii) The map x 7→ y(t, x) is nondecreas ing in its domain, its jump set is S
t
and v(t, ·) is
differentiable at any x / S
t
, with
f
(v
x
(t, x)) =
x y(t, x)
t
. (2.6)
In particular v
x
(t, ·) is continuous on R \ S
t
.
(iii) There exists a constant C such that
v
x
(t, x + y) v
x
(t, x) +
C
t
y whenever y 0 and x, x + y / S
t
. (2.7)
This is called Oleinik E–condition.
(iv) v is a Lipschitz map and u := v
x
is the unique entropy solution of (1.1) with the
initial condition u(0, ·) = u
0
.
(v) If t
n
t > 0, then v
x
(t
n
, ·) v
x
(t, ·) in L
1
loc
.
Proof. For a proof of point (i), of the fact that x 7→ y(t, x) is nondecreasing, and of the fact
that S
t
is the set of discontinuities of y(t, ·) we refer for instance to Theorem 1 of §3.4.2 of
[6]. For (iii) and (iv) we refer to Theorem 2 of §3.4.2 , to the first lemma o f §3.4.3 and to
Theorem 3 of the same section of [6].

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Book
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TL;DR: The Mumford-Shah functional minimiser of free continuity problems as mentioned in this paper is a special function of the Mumfordshah functional and has been shown to be a function of free discontinuity set.
Abstract: Measure Theory Basic Geometric Measure Theory Functions of bounded variation Special functions of bounded variation Semicontinuity in BV The Mumford-Shah functional Minimisers of free continuity problems Regularity of the free discontinuity set References Index

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Abstract: This is a masterly exposition and an encyclopedic presentation of the theory of hyperbolic conservation laws. It illustrates the essential role of continuum thermodynamics in providing motivation and direction for the development of the mathematical theory while also serving as the principal source of applications. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of analysis and the qualitative theory of partial differential equations, whereas prior exposure to continuum physics is not required. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. This new edition places increased emphasis on hyperbolic systems of balance laws with dissipative source, modeling relaxation phenomena. It also presents an account of recent developments on the Euler equations of compressible gas dynamics. Furthermore, the presentation of a number of topics in the previous edition has been revised, expanded and brought up to date, and has been enriched with new applications to elasticity and differential geometry. The bibliography, also expanded and updated, now comprises close to two thousand titles. From the reviews of the 3rd edition: "This is the third edition of the famous book by C.M. Dafermos. His masterly written book is, surely, the most complete exposition in the subject." Evgeniy Panov, Zentralblatt MATH "A monumental book encompassing all aspects of the mathematical theory of hyperbolic conservation laws, widely recognized as the "Bible" on the subject." Philippe G. LeFloch, Math. Reviews

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Frequently Asked Questions (2)
Q1. What have the authors contributed in "A note on admissible solutions of 1d scalar conservation laws and 2d hamilton-jacobi equations" ?

In this note the authors prove that entropy solutions of Dtu+Dxf ( u ) = 0 belong to SBVloc ( Ω ). As a corollary the authors prove the same property for gradients of viscosity solutions of planar Hamilton–Jacobi PDEs with uniformly convex Hamiltonians. In this note the authors prove that entropy solutions of Dtu + Dxf ( u ) = 0 belong to SBVloc ( Ω ). As a corollary the authors prove the same property for gradients of viscosity solutions of planar Hamilton–Jacobi PDEs with uniformly convex hamiltonians. 

The slicing theory of BV functions shows that the Cantor part of the 2-dimensional measure Dxu is the integral with respect to t of the Cantor parts of Du(t, ·) (see Theorem 3.108 of [3] for a precise statement).