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Journal ArticleDOI

Finer regularity of an entropy solution for 1-d scalar conservation laws with non uniform convex flux

04 Nov 2014-Rendiconti del Seminario Matematico della Università di Padova (European Mathematical Publishing House)-Vol. 132, pp 1-24
TL;DR: In this article, it was shown that for all most every t > 0, locally, the solution is in SBV (Special functions of bounded variations) in space variable, and that for almost everywhere in t ≥ 0, the entropy solution cannot be removed.
Abstract: Consider a scalar conservation law in one space dimension with initial data in LI : If the flux f is in C 2 and locally uniformly convex, then for all t > 0, the entropy solution is locally in BV (functions of bounded variation) in space variable. In this case it was shown in [5], that for all most every t > 0, locally, the solution is in SBV (Special functions of bounded variations). Furthermore it was shown with an example that for almost everywhere in t > 0 cannot be removed. This paper deals with the regularity of the entropy solutions of the strict convex C 1 flux f which need not be in C 2 and locally uniformly convex. In this case, the entropy solution need not be locally in BV in space variable, but the composition with the derivative of the flux function is locally in BV. Here we prove that, this composition is locally is in SBV on all most every t > 0. Furthermore we show that this is optimal.

Summary (1 min read)

This proves (2).

  • P 0; Where jI(t)j denotes the Lebesgue measure of I(t).
  • Then from the Besicovitch differentiation Theorem (see [13] ) it follows that there exists a set E t & [a(t); b(t)] with the property:.

PROOF. From (3.4), D(t) & [a(t); b(t)] and hence

  • For proving (3) of the theorem (1.3), the authors need the following backward construction which was proved in [1] in obtaining the solution of control problems for conservation laws.
  • Using the backward construction the authors can prove that it is also necessary.

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REND.SEM.MAT.UNIV.PADOVA, Vol. 132 (2014)
DOI 10.4171/RSMUP/132-1
Finer regularity of an entropy solution for 1-d scalar
conservation laws with non uniform convex flux
ADIMURTHI(*) - SHYAM SUNDAR GHOSHAL (**) - G.D. VEERAPPA GOWDA(***)
ABSTRACT - Consider a scalar conservation law in one space dimension with initial data
in L
1
: If the flux f is in C
2
and locally uniformly convex, then for all t > 0,the
entropy solution is locally in BV (functions of bounded variation) in space variable.
In this case itwas shownin [5], that for all most every t > 0, locally, the solution is in
SBV (Special functions of bounded variations). Furthermore it was shown with an
example that for almost everywhere in t > 0 cannot be removed. This paper deals
withthe regularity of theentropysolutions of the strict convexC
1
flux f which need
notbe in C
2
and locallyuniformly convex. In this case, the entropy solution neednot
be locallyin BV in space variable,but the composition with the derivative of the flux
functionis locally in BV. Here we prove that, thiscomposition is locally is in SBV on
all most every t > 0. Furthermore we show that this is optimal.
M
ATHEMATICS SUBJECT CLASSIFICATION (2010). 35B65, 35L65, 35L67.
K
EYWORDS. Hamilton-Jacobi equation, scalar conservation laws, characteristic
lines, sbv functions, uniformly convex.
1. Introduction
Let f : R ! R be a locally Lipschitz continuous function and u
0
2
L
1
(R): Consider the following scalar conservation law in one space di-
(*) Indirizzo dell'A.: Center for Applicable Mathematics, Tata Institute of
Fundamental Research, PO Bag No. 6503, Bangalore-560065, India.
E-mail: aditi@math.tifrbng.res.in
(**) Indirizzo dell'A.: Laboratoire de Mathe
Â
matiques, Universite
Â
de Franche-
Comte
Â
, 25030 BesancËon, France.
E-mail: ssghosha@univ-fcomte.fr
The second author would like to thank French ANR project ``CoToCoLa'' for
the support.
(***) Indirizzo dell'A.: Center for Applicable Mathematics, Tata Institute of
Fundamental Research, PO Bag No. 6503, Bangalore-560065, India.
E-mail: gowda@math.tifrbng.res.in

mension
@u
@t
@
@x
f (u) 0ifx 2 R; t > 0:(1:1)
u(x; 0) u
0
(x)if x 2 R:
A function u 2 L
1
(R R
) is said to be a solution of (1.1) if u is a weak
solution of (1.1) satisfying Kruzkov [14] entropy condition. By using van-
ishing viscosity, Kruzkov [14] proved that there exists a unique entropy
solution to (1.1).
Under the convexity assumption on the flux, Lax and Oleinik [12] have
obtained an explicit formula for the solution. In order to state their result,
let us recall some definitions and their properties of convex functions [4].
Let f : R ! R be a convex function. Then
(i) f is said to be a strictly convex function if for all x 6 y a 2 (0; 1),
f (ax (1 a)y) < af (x) (1 a) f (y):
(ii) f 2 C
2
(R) is said to be locally uniformly convex if for all compact set
K R, there exist a constant C(K) > 0 such that f
00
(x) C(K) for all
x 2 K. It is said to be uniformly convex in R if there exist C > 0 such that
f
00
(x) C for all x 2 R.
(iii) f is said to be of super linear growth if
lim
juj!1
f (u)
juj
1:
(iv)Associate to a convex function f ; let f
be its convex dual defined by
f
(p) sup
q
fpq f (q)g:
Then for any q 2 R;
lim
jpj!1
f
(p)
jpj
lim
jpj!1
q
p
jpj
f (q)
jpj

lim
jpj!1
q
p
jpj
since q is arbitrary and letting jqj!1to obtain
lim
jpj!1
f
(p)
jpj
1:
(v) Let f be in C
1
(R) and strictly convex. Then f
0
is a strictly increasing
function. Furthermore, it can be shown that if f is of super linear growth,
then f f

; ( f
)
0
( f
0
)
1
. Therefore f
is in C
1
(R) with super linear
2 Adimurthi - Shyam Sundar Ghoshal - G.D. Veerappa Gowda

growth and is strictly convex. With these preliminaries, we recall the fol-
lowing (see [12]):
T
HEOREM 1.1. (Lax and Olenik) Assume that f is in C
1
(R); strictly
convex and of super linear growth. Then there exist a function y(x; t) such
that x 7! y(x; t) is a non decreasing function and for all t > 0, a.e. x 2 R;
the solution u of (1.1) is explicitly given by
f
0
(u(x; t))
x y(x; t)
t
:(1:2)
As an immediate consequence of this theorem, we have
(1) For each t > 0; x 7! f
0
(u(x; t)) 2 BV
loc
(R):
(2) If f 2 C
2
and locally uniformly convex, then for all t > 0,x7! u(x; t)
is in BV
loc
(R):
In this context, Alberto Bressan raised the following question.
Problem 1. Assume that for all t > 0; x 7! u(x; t) 2 BV
loc
(R); then is it in
SBV
loc
(R)?
Under the suitable condition on f ; Ambrosio-De Lellis [5] proved the
following
T
HEOREM 1.2. (Ambrosio-De Lellis) Assume further that f 2 C
2
(R)
and locally uniformly convex. Then there exist a countable set S (0; 1)
such that for all t =2 S; x 7! u(x; t) 2 SBV
loc
(R): Furthermore, they gave an
example for which S 6 f:
Observe that from the explicit formula, if f
01
is a locally Lipschitz
continuous, then the solution is in BV
loc
(R) in space variable which is
guaranteed if f is C
2
and locally uniformly convex. This is what assumed in
[5]. In [7], the condition of locally uniformly convexity has been relaxed to
prove the same result as in [5] provided the zero set of f
00
is countable and u
is in BV
loc
: In [15] even the convexity condition is relaxed under the hy-
pothesis that f is C
2
and the zero set of f
00
is countable. For further gen-
eralization see [6], [8], [9], [10], [16].
Now the natural question is ``What happens if f 2 C
1
and satisfying the
hypothesis of Lax-Olenik Theorem?'' In this case, the problem of Bressan
can be reformulated as follows.
Finer regularity of an entropy solution for 1-d scalar etc. 3

Problem 2.
(i) Does x 7! u(x; t) 2 BV
loc
(R) for all t > 0.
(ii) Does x 7! f
0
(u(x; t)) 2 SBV
loc
(R) for a.e. t > 0:
(iii) If (ii) is true, does there exist a u
0
2 L
1
(R) and a countable set
S ft
n
g such that f
0
(u(; t
n
)) 62 SBV
loc
(R) for all n:
In view of (1.2), the main result of this paper is to show that the Problem
2 has an affirmative answer and we have the following
T
HEOREM 1.3. Let f satisfies the hypothesis of Theorem 1.1 and u be
the solution of (1.1). Then
(1) There exist a countable set S (0; 1) such that for all
t 62 S; x 7! f
0
(u(x; t)) 2 SBV
loc
(R):
(2) Assume that f (0) f
0
(0) 0 and there exist c > 0 and 0 < g < 1
such that for all a 0 b; f satisfies
f
0
1
(b) f
0
1
(a) c(b a)
g
:
Then there exist a u
0
2 L
1
(R) such that for all t > 0,x7! u(x; t) 62
BV
loc
(R):
(3) There exist a u
0
2 L
1
(R) and a decreasing sequence ft
n
g such that
x 7! f
0
(u(x; t
n
)) 62 SBV
loc
(R):
For (1), we follow the similar path as in [5]. The idea in the proof of (2)
lies in the construction of asymptotically single shock packets as in [1]. The
proof of (3) lies in the backward construction (see Appendix) as in [2], [3].
R
EMARK 1.4. There are many fluxes satisfying the hypothesis of (2).
For example, let p > 2; then f (u)
juj
p
p
is one of such function.
2. Preliminaries
Let f be as in Theorem 1.1. Let u
0
2 L
1
(R): Let A 2 R and define
v
0
(x)
Z
x
A
u
0
(u)du(2:1)
f
(p) max
q
fpq f (q)g(2:2)
4 Adimurthi - Shyam Sundar Ghoshal - G.D. Veerappa Gowda

the convex dual of f . Then from the hypothesis of f , it follows that
f
2 C
1
(R); strictly convex and having super linear growth.
Let t > 0; x 2 R; define
v(x; t) min
y
v
0
(y) tf
x y
t
no
:(2:3)
Then we have the following (see [12])
T
HEOREM 2.1. (Hopf): v 2 Lip(R R
) and is the unique viscosity
solution of the following Hamilton Jacobi equation.
@v
@t
f
@v
@x

0 if x 2 R; t > 0(2:4)
v(x; 0) v
0
(x) if x 2 R
furthermore it satisfies
(i) Dynamic programming principle: Let 0 s < t : then
v(x; t) min
y2R
v(y; s) (t s) f
x y
t s
no
(2:5)
(ii) There exist a M > 0 depending on u
0
and f
such that for all
minimizers of y of (2:5)
x y
t s
M:(2:6)
Next we define the different notions of characteristics and their
properties without proof.
D
EFINITION 2.2. Let 0 s < t; x 2 R: Then
(i) Characteristic sets ch(x; s; t); ch(x; t) are defined by
ch(x; s; t) fy 2 R; y is a minimizer in 2:5)g
ch(x; t) ch(x; 0; t):
(ii) y 2 ch(x; s; t) is called a characteristic point and the line joining
(x; t) and (y; s) is called a characteristic line segment.
(iii) Extreme characteristic points
y
(x; s; t) maxfy 2 ch(x; s; t)g
y
(x; s; t) minfy 2 ch(x; s; t)g
y
(x; t) y
(x; 0; t)
Finer regularity of an entropy solution for 1-d scalar etc. 5

Citations
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Journal ArticleDOI
TL;DR: In this article, the authors considered scalar conservation law with discontinuous flux in one space dimension and gave a complete picture of the bounded variation of the solution for all time for a uniform convex flux with only L ∞ data.

20 citations


Cites background from "Finer regularity of an entropy solu..."

  • ...We ask the question that can one expect the similar result when f 6= g? When f = g, it has been noticed in [3] that the assumption of uniformly convexity is optimal in order to prove the BV regularity without the the assumption that u0 / ∈ BV ....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the regularizing effect of the nonlinearity of the flux function has on the entropy solution of scalar conservation laws in one space dimension was studied. But the regularization effect was not considered in this paper.
Abstract: We deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function f has on the entropy solution. More precisely, if the set {w : f...

17 citations

Journal ArticleDOI
TL;DR: A new strategy is introduced for the optimal control problem for scalar conservation laws with convex flux by a new cost function and by the Lax–Oleinik explicit formula for entropy solutions, the nonlinear problem is converted to a linear problem.
Abstract: The optimal control problem for Burgers equation was first considered by Castro, Palacios and Zuazua. They proved the existence of a solution and proposed a numerical scheme to capture an optimal solution via the method of "alternate decent direction". In this paper, we introduce a new strategy for the optimal control problem for scalar conservation laws with convex flux. We propose a new cost function and by the Lax–Oleinik explicit formula for entropy solutions, the nonlinear problem is converted to a linear problem. Exploiting this property, we prove the existence of an optimal solution and, by a backward construction, we give an algorithm to capture an optimal solution.

14 citations

Book ChapterDOI
01 Aug 2016
TL;DR: For strictly smooth convex flux and the one-dimensional case, the proof of this conjecture in the framework of Sobolev fractional spaces and in fractional BV spaces was presented in this article.
Abstract: In 1994, Lions, Perthame and Tadmor conjectured the maximal smoothing effect for multidimensional scalar conservation laws in Sobolev spaces. For strictly smooth convex flux and the one-dimensional case, we detail the proof of this conjecture in the framework of Sobolev fractional spaces \(W^{s,1}\), and in fractional BV spaces: \(BV^{s}\). The \(BV^{s}\) smoothing effect is more precise and optimal. It implies the optimal Sobolev smoothing effect in \(W^{s,1}\) and also in \(W^{s,p}\) with the optimal \(p=1/s\). Moreover, the proof expounded does not use the Lax–Oleinik formula but a generalized one-sided Oleinik condition.

11 citations

Journal ArticleDOI
TL;DR: In this paper, the authors discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux and prove the smoothing effect of the equation forcing the $BV_{loc}$ solution near the interface for initial data without the assumption on the uniform convexity of the fluxes made as in [1,21].
Abstract: In this paper, we discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux. We prove the smoothing effect of the equation forcing the $BV_{loc}$ solution near the interface for $L^\infty$ initial data without the assumption on the uniform convexity of the fluxes made as in [1,21]. The proof relies on the method of characteristics and the explicit formulas.

11 citations

References
More filters
Book
01 Jan 1992
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
Abstract: GENERAL MEASURE THEORY Measures and Measurable Functions Lusin's and Egoroff's Theorems Integrals and Limit Theorems Product Measures, Fubini's Theorem, Lebesgue Measure Covering Theorems Differentiation of Radon Measures Lebesgue Points Approximate continuity Riesz Representation Theorem Weak Convergence and Compactness for Radon Measures HAUSDORFF MEASURE Definitions and Elementary Properties Hausdorff Dimension Isodiametric Inequality Densities Hausdorff Measure and Elementary Properties of Functions AREA AND COAREA FORMULAS Lipschitz Functions, Rademacher's Theorem Linear Maps and Jacobians The Area Formula The Coarea Formula SOBOLEV FUNCTIONS Definitions And Elementary Properties Approximation Traces Extensions Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Sobolev Functions Differentiability on Lines BV FUNCTIONS AND SETS OF FINITE PERIMETER Definitions and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions Isoperimetric Inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties of BV Functions Essential Variation on Lines A Criterion for Finite Perimeter DIFFERENTIABILITY AND APPROXIMATION BY C1 FUNCTIONS Lp Differentiability ae Approximate Differentiability Differentiability AE for W1,P (P > N) Convex Functions Second Derivatives ae for convex functions Whitney's Extension Theorem Approximation by C1 Functions NOTATION REFERENCES

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01 Jan 2000
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

4,299 citations

Journal ArticleDOI
TL;DR: By Luigi Ambrosio, Nicolo Fucso and Diego Pallara: 434 pp.
Abstract: By Luigi Ambrosio, Nicolo Fucso and Diego Pallara: 434 pp., £55.00, isbn 0-19-850254-1 (Clarendon Press, Oxford, 2000).

1,904 citations


"Finer regularity of an entropy solu..." refers background in this paper

  • ...Then from the hypothesis of f , it follows that f ∗ ∈ C(1)(IR), strictly convex and having super linear growth (see [4]) ....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a theory of generalized solutions in the large Cauchy's problem for the equations in the class of bounded measurable functions is constructed, and the existence, uniqueness and stability theorems for this solution are proved.
Abstract: In this paper we construct a theory of generalized solutions in the large of Cauchy's problem for the equations in the class of bounded measurable functions. We define the generalized solution and prove existence, uniqueness and stability theorems for this solution. To prove the existence theorem we apply the "vanishing viscosity method"; in this connection, we first study Cauchy's problem for the corresponding parabolic equation, and we derive a priori estimates of the modulus of continuity in of the solution of this problem which do not depend on small viscosity.Bibliography: 22 items.

1,799 citations

Journal ArticleDOI
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.

57 citations


"Finer regularity of an entropy solu..." refers background or methods in this paper

  • ...We follow the same the notations as in [5], for t > 0, x ∈ IR, θ ∈ [0, t] define m±(x, t) = x−y±(x,t) t γ±(θ) = x+m±(x, t)(θ − t) I(x, t) = (y−(x, t), y+(x, t)) C(x, t) = {(ξ, θ); 0 < θ < t, γ−(θ) < ξ < γ+(θ)} I(t) = ⋃ x∈IR I(x, t) c(t) = ⋃ x∈IR C(x, t) D(t) = {x ∈ IR; y+(x, t) has jump at x} = {set of discontinuities of y+(....

    [...]

  • ...Under the suitable condition on f, Ambrosio- Dellis [5] proved the following....

    [...]

  • ...Recently in [5], SBV regularity has been proved under the assumption that f is locally uniformly convex and for a....

    [...]

  • ...Assume that for all t > 0, x 7→ u(x, t) ∈ BVloc(IR), then is it in SBVloc(IR)? Under the suitable condition on f, Ambrosio- Dellis [5] proved the following....

    [...]

  • ...For (1), we follow the similar path as in [5]....

    [...]