Approximate Solutions of Nonlinear Conservation Laws

145 Pages·2007·2.12 MB·English

Approximate Solutions of
Nonlinear Conservation Laws
Eitan Tadmor
Department of Mathematics
UCLA, Los-Angeles CA 90095
and
School of Mathematical Sciences
Tel-Aviv University, Tel-Aviv 69978
Email: [email protected]
ABSTRACT
This is a summary of (cid:12)ve lectures delivered at the CIME course on "Advanced
Numerical Approximation of Nonlinear Hyperbolic Equations" held in Cetraro,
Italy, on June 1997.
Following the introductory lecture I | which provides a general overview of
approximate solutiontononlinear conservationlaws,theremaininglecturesdeal
with the speci(cid:12)cs of four complementing topics:
(cid:15) Lecture II. Finite-di(cid:11)erence methods { non-oscillatory central schemes;
(cid:15) Lecture III. Spectral approximations { the Spectral Viscosity method;
0
(cid:15) Lecture IV. Convergence rate estimates { a Lip convergence theory;
(cid:15) Lecture V. Kinetic approximations { regularity of kinetic formulations.
1
Acknowledgment. I thank Al(cid:12)o Quarteroni for the invitation,B. Cockburn, C.
Johnson & C.-WShu forthe teamdiscussions, the Italianparticipants fortheir
attention,andthehostingGrandHotelatSanMicheleforitsremarkablyunique
atmosphere. ResearchwassupportedinpartbyONRgrant#N00014-91-J-1076
and NSF grant #97-06827.
AMS Subject Classi(cid:12)cation. Primary 35L65, 35L60. Secondary 65M06,
65M12, 65M15,65M60, 65M70
Contents
1 A General Overview 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Hyperbolic Conservation Laws . . . . . . . . . . . . . . . . . . . 3
1.2.1 Averybriefoverview|mequationsindspatialdimensions 3
1.2.2 Scalar conservation laws (m=1;d(cid:21)1) . . . . . . . . . . 5
1.2.3 One dimensionalsystems (m(cid:21)1;d=1) . . . . . . . . . . 9
1.2.4 Multidimensionalsystems (m>1;d>1) . . . . . . . . . 12
1.3 Total VariationBounds . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 Finite Di(cid:11)erence Methods . . . . . . . . . . . . . . . . . . 13
1.3.2 TVD schemes (m=d=1) . . . . . . . . . . . . . . . . . 14
1.3.3 TVD (cid:12)lters . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.4 TVB approximations(m(cid:21)1;d=1) . . . . . . . . . . . . 21
1.4 Entropy Production Bounds . . . . . . . . . . . . . . . . . . . . . 25
1.4.1 Compensated compactness (m(cid:20)2;d=1) . . . . . . . . . 25
1.4.2 The streamline di(cid:11)usion (cid:12)nite-element method . . . . . . 26
1.4.3 The spectral viscosity method. . . . . . . . . . . . . . . . 26
1.5 Measure-valued solutions(m=1;d(cid:21)1) . . . . . . . . . . . . . . 27
1.5.1 Finite volumeschemes (d(cid:21)1) . . . . . . . . . . . . . . . 28
1.6 Kinetic Approximations . . . . . . . . . . . . . . . . . . . . . . . 28
1.6.1 Velocity averaging lemmas(m(cid:21)1;d(cid:21)1) . . . . . . . . . 29
1.6.2 Nonlinear conservation laws . . . . . . . . . . . . . . . . . 29
2 Non-oscillatory central schemes 42
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 A Short Guide to Godunov-Type schemes . . . . . . . . . . . . . 44
2.2.1 Upwind schemes . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2 Central schemes . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 Central schemes in one-space dimension . . . . . . . . . . . . . . 49
2.3.1 The second-order Nessyahu-Tadmorscheme . . . . . . . . 49
2.3.2 The third-order central scheme . . . . . . . . . . . . . . . 53
2.4 Central schemes in two space dimensions. . . . . . . . . . . . . . 57
2.4.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 63
2.5 Incompressible Euler equations . . . . . . . . . . . . . . . . . . . 68
2.5.1 The vorticity formulation . . . . . . . . . . . . . . . . . . 68
2
CONTENTS 1
2.5.2 The velocity formulation. . . . . . . . . . . . . . . . . . . 71
3 The Spectral Viscosity Method 77
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.2 The Fourier Spectral Viscosity (SV) method . . . . . . . . . . . . 78
3.2.1 The Fourier SV method { 2nd order viscosity . . . . . . . 80
3.2.2 Fourier SV method revisited { super viscosity . . . . . . . 81
3.3 Non-periodic boundaries . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.1 The Legendre SV approximation . . . . . . . . . . . . . . 89
3.3.2 Convergence of the Legendre SV method . . . . . . . . . 92
3.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.5 MultidimensionalFourier SV method . . . . . . . . . . . . . . . . 98
4 Convergence Rate Estimates 102
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2 Approximatesolutions . . . . . . . . . . . . . . . . . . . . . . . . 106
4.3 Convergence rate estimates . . . . . . . . . . . . . . . . . . . . . 108
4.3.1 Convex conservation laws . . . . . . . . . . . . . . . . . . 108
4.3.2 Convex Hamilton-Jacobiequations . . . . . . . . . . . . . 113
4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.4.1 Regularized Chapman-Enskogexpansion . . . . . . . . . . 114
4.4.2 Finite-Di(cid:11)erence approximations . . . . . . . . . . . . . . 115
4.4.3 Godunov type schemes . . . . . . . . . . . . . . . . . . . . 117
4.4.4 Glimmscheme . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.5 The Spectral Viscosity method . . . . . . . . . . . . . . . 123
5 Kinetic Formulations and Regularity 129
5.1 Regularizing e(cid:11)ect in one-space dimension . . . . . . . . . . . . . 129
5.2 Velocity averaging lemmas(m(cid:21)1;d(cid:21)1) . . . . . . . . . . . . . 131
5.3 Regularizing e(cid:11)ect revisited (m=1;d(cid:21)1) . . . . . . . . . . . . 132
5.3.1 Kinetic and other approximations . . . . . . . . . . . . . 134
5.4 Degenerate parabolic equations . . . . . . . . . . . . . . . . . . . 136
5.5 The 2(cid:2)2 isentropic equations. . . . . . . . . . . . . . . . . . . . 137
Chapter 1
A General Overview
Abstract. In this introductory lecture, we overview the development of mod-
ern, high-resolutionapproximationstohyperbolicconservationlawsandrelated
nonlinear equations. Since this overview also serves as an introduction for the
other lectures in this volume, it is less of a comprehensive overview, and more
of a bird's eye view of topics which play a pivotal role in the lectures ahead.
It consists of a dual discussion on the various mathematical concepts and the
related discrete algorithmswhichare the required ingredientsforthese lectures.
I start with a brief overview on the mathematicaltheory for nonlinear hy-
perbolic conservation laws. The theory of the continuum ( { and in this case,
the dis-continuum), is intimately related to the construction, analysis and im-
plementationofthe corresponding discrete approximations. Here, the basic the
notionsofviscosityregularization,entropy,monotonicity,totalvariationbounds
andRiemann'sproblemareintroduced. Thenfollowthethebasicingredientsof
the discrete theory: the Lax-Wendro(cid:11)theorem,andthe pivotal(cid:12)nite-di(cid:11)erence
schemes of Godunov, Lax-Friedrichs, and Glimm.
Toproceed, our dualpresentation of high-resolutionapproximationsis clas-
si(cid:12)edaccordingtotheanalyticaltoolswhichareusedinthedevelopmentoftheir
convergence theories. These include classical compactness arguments based on
Total Variation (TV) bounds, e.g., TVD (cid:12)nite-di(cid:11)erence approximations. The
(cid:0)1
use of compensated compactness arguments based on H -compact entropy
production is demonstrated inthe context ofstreamlinedi(cid:11)usion(cid:12)nite-element
method and spectral viscosity approximations. Measure valued solutions {
measured by their negative entropy production, are discussed in the context
of multidimensional (cid:12)nite-volume schemes. Finally, we discuss the recent use
of averaging lemmas which yield new compactness and regularity results for
approximate solutions of nonlinear conservation laws (as well as some related
equations), which admit an underlying kinetic formulation, e.g., (cid:12)nite-volume
and relaxation schemes.
1
CHAPTER 1. A GENERAL OVERVIEW 2
1.1 Introduction
The lectures in this volumedeal with modern algorithmsfor the accurate com-
putationofshock discontinuities,sliplines,andother similarphenomenawhich
could be characterized by spontaneous evolution of change in scales. Such phe-
nomenaposeaconsiderablecomputationalchallenge,whichisanswered,atleast
partially,by these newly constructed algorithms. New modernalgorithmswere
devised, that achieve one or more ofthe desirable properties of high-resolution,
e(cid:14)ciency, stability | in particular, lack of spurious oscillations, etc. The im-
pact of these new algorithms ranges from the original impetus in the (cid:12)eld of
ComputationalFluidDynamics(CFD),tothe(cid:12)eldsoilrecovery,movingfronts,
imageprocessing,... [75], [138], [132],[1].
In this introduction we survey a variety ofthese algorithmsfor the approxi-
mate solution of nonlinear conservation laws. The presentation is neither com-
prehensive nor complete | the scope is too wide for the present framework
Instead, we discuss the analytical tools which are used to study the stability
and convergence of these modern algorithms. We use these analytical issues
as our 'touring guide' to provide a readers' digest on the relevant approximate
methods, while studying there convergence properties. They include
(cid:15) Finite-di(cid:11)erence methods. These are the most widely used methods for
solving nonlinear conservation laws. Godunov-type di(cid:11)erence schemes
play a pivotal role. Two canonical examples include the upwind ENO
schemes(discussedinC.-W.Shu'slectures)andafamilyofhigh-resolution
non-oscillatory central schemes (discussed in Lecture II);
(cid:15) Finite element schemes. Here, the streamline di(cid:11)usion method and its
extensions are canonical example,discussed in C. Johnson's lectures;
(cid:15) Spectralapproximations. TheSpectralViscosity(SV)methodsisdiscussed
in Lecture III.
(cid:15) Finite-volume schemes. Finite-Volume (FV) schemes o(cid:11)er a particular
advantageforintegrationovermultidimensionalgeneraltriangulation,be-
yond the Cartesian grids. More can be found in B. Cockburn's lectures.
(cid:15) Kinetic formulations. Compactness and regularizing e(cid:11)ects of approxi-
mate solutions is quanti(cid:12)ed in terms of their underlying kinetic formula-
tions, Lecture V.
Somegeneral references are inorder. The theory ofhyperbolic conservation
laws is covered in [94], [178],[157], [149]. For the theory of their numerical ap-
proximationconsult [102],[58],[59],[159]. Weareconcerned withanalyticaltools
which are used in the convergence theories of such numerical approximations.
The monograph[50] could be consulted on recent development regarding weak
convergence. The reviews of [171], [123, 124] are recommended references for
the theory ofcompensatedcompactness, and[40,41],[17]dealwithapplications
CHAPTER 1. A GENERAL OVERVIEW 3
to conservationlawsand their numericalapproximations. Measure-valued solu-
tions inthe context ofnonlinearconservation lawswere introduced in[42]. The
articles [62], [53], [45] prove the averaging lemma, and [111],[112],[77] contain
applicationsinthecontextofkineticformulationfornonlinearconservationlaws
and related equations.
A (cid:12)nal word about notations. Di(cid:11)erent authors use di(cid:11)erent notations. In
this introduction, the conservative variable are denoted by the "density" (cid:26),
the spatial (cid:13)ux is A((cid:1)), ((cid:17);F) are entropy pairs, etc. In later lectures, these
are replaced by the more generic notations: conservative variables are u;v;:::,
(cid:13)uxes are denoted by f;g;:::, the entropy function is denoted U, etc.
1.2 Hyperbolic Conservation Laws
1.2.1 A very brief overview | m equations in d spatial
dimensions
The general set-up consists of m equations in d spatial dimensions
+ d
@t(cid:26)+rx(cid:1)A((cid:26))=0; (t;x)2Rt (cid:2)Rx: (1.2.1)
Here, A((cid:26)) := (A1((cid:26));:::;Ad((cid:26))) is the d-dimensional (cid:13)ux, and
(cid:26) := ((cid:26)1(t;x);:::;(cid:26)m(t;x)) is the unknown m-vector subject to initial condi-
tions (cid:26)(0;x)=(cid:26)0(x).
The basic facts concerning such nonlinear hyperbolic systems are, consult
[94],[113], [35],[157],[58],[149],
(cid:15) The evolution of spontaneous shock discontinuities which requires weak
(distributional) solutions of (1.2.1);
(cid:15) The existence of possibly in(cid:12)nitely many weak solutions of (1.2.1);
(cid:15) To single out a unique `physically relevant' weak solution of (1.2.1), we
seek a solution, (cid:26) = (cid:26)(t;x), which can be realized as a viscosity limit
"
solution, (cid:26)=lim(cid:26) ,
" " "
@t(cid:26) +rx(cid:1)A((cid:26) )="rx(cid:1)(Qrx(cid:26) ); "Q>0; (1.2.2)
(cid:15) The entropy condition. The notion of a viscosity limit solution is inti-
matelyrelated tothe notionofanentropy solution,(cid:26), whichrequires that
for allconvex entropy functions, (cid:17)((cid:26)), there holds, [93], [88, x5]
@t(cid:17)((cid:26))+rx(cid:1)F((cid:26))(cid:20)0: (1.2.3)
A scalar function, (cid:17)((cid:26)), is an entropy function associated with (1.2.1), if its
00 0
Hessian, (cid:17) ((cid:26)), symmetrizes the spatial Jacobians, Aj((cid:26)),
00 0 0 > 00
(cid:17) ((cid:26))Aj((cid:26))=Aj((cid:26)) (cid:17) ((cid:26)); j =1;:::;d:
CHAPTER 1. A GENERAL OVERVIEW 4
Itfollowsthatinthiscasethereexistsanentropy(cid:13)ux,F((cid:26)):=(F1((cid:26));:::;Fd((cid:26))),
which is determined by the compatibilityrelations,
0 > 0 0 >
(cid:17) ((cid:26)) Aj((cid:26)) =Fj((cid:26)) ; j =1;:::;d: (1.2.4)
Whatisthe relationbetween the entropy condition(1.2.3)and the viscosity
0 "
limitsolutionin(1.2.2)? multiplythe latter, on the left, by(cid:17) ((cid:26) ); the compat-
ibilityrelation (1.2.4)impliesthat the resulting two terms on the left of (1.2.2)
" "
amount to the sum of perfect derivatives, @t(cid:17)((cid:26)) +rx(cid:1)F((cid:26) ). Consider now
the right hand side of (1.2.2)(for simplicity,we assume the viscosity matrixon
the right to be the identity matrix,Q=I). Here we invoke the identity
0 " " " " > 00 " "
"(cid:17) ((cid:26) )(cid:1)x(cid:26) (cid:17)"(cid:1)x(cid:17)((cid:26) )(cid:0)"(rx(cid:26) ) (cid:17) ((cid:26) )rx(cid:26) :
The (cid:12)rst term tends to zero (in distribution sense); the second term is nonpos-
itive thanks to the convexity of (cid:17), and hence tend to a nonpositive measure.
Thus, a viscosity limitsolutionmustsatisfythe entropy inequality(1.2.3). The
" "
inverse implication: (1.2.3) =) (cid:26) = lim(cid:26) of viscosity solutions (cid:26) satisfying
(1.2.2), holds in the scalar case; the question requires a more intricate analysis
for systems, consult [93],[157] and the references therein.
Indeed, thebasicquestions regardingthe existence, uniqueness andstability
of entropy solutions for general systems are open. Instead, the present trend
seems to concentrate on special systems with additional properties which en-
able to answer the questions of existence, stability, large time behavior, etc.
One-dimensional 2(cid:2) 2 systems is a notable example for such systems: their
properties can be analyzed in view of the existence of Riemann invariants and
a familyofentropy functions,[56],[94,x6],[157],[40,41]. The systemofm(cid:21)2
chromatographic equations, [77], is another example for such systems.
The di(cid:14)culty of analyzing general systems of conservation laws is demon-
strated by the followingnegative result due to Temple,[174], which states that
already for systems with m (cid:21) 2 equations, there exists no metric, D((cid:1);(cid:1)), such
that the problem(1.2.1),(1.2.3) is contractive, i.e.,
1 2 1 2
69D : D((cid:26) (t;(cid:1));(cid:26) (t;(cid:1)))(cid:20)D((cid:26) (0;(cid:1));(cid:26) (0;(cid:1))); 0(cid:20)t(cid:20)T; (m(cid:21)2):
(1.2.5)
In this context we quote from[168] the following
Theorem 1.2.1 Assume the system (1.2.1) is endowed with a one-parameter
m
family of entropy pairs, ((cid:17)((cid:26);c); F((cid:26);c)); c 2 R , satisfying the symmetry
property
(cid:17)((cid:26);c)=(cid:17)(c;(cid:26)); F((cid:26);c)=F(c;(cid:26)): (1.2.6)
1 2
Let (cid:26) ;(cid:26) be two entropy solutions of (1.2.1). Then the following a priori esti-
mate holds Z Z
1 2 1 2
(cid:17)((cid:26) (t;x);(cid:26) (t;x))dx(cid:20) (cid:17)((cid:26)0(x);(cid:26)0(x))dx: (1.2.7)
x x
CHAPTER 1. A GENERAL OVERVIEW 5
Theorem 1.2.1 is based on the observation that the symmetry property
(1.2.6) is the key ingredient for Kru(cid:20)zkov's penetrating ideas in [88], which ex-
tends his scalar arguments into the case of general systems. This extension
seems to be part of the 'folklore' familiar to some, [36],[150]); a sketch of the
proof can be found in [168].
Remark. Theorem 1.2.1seems to circumvent the negative statement of (1.2.5).
This is done by replacing the metric D((cid:1);(cid:1)), with the weaker topology induced
by a family of convex entropies, (cid:17)((cid:1);(cid:1)). Many physically relevant systems are
endowed withatleastoneconvexentropy function({whichinturn, islinkedto
the hyperbolic character of these systems, [61],[52],[120]). Systems with \rich"
families of entropies like those required in Theorem 1.2.1 are rare, however,
consult [148]. The instructive (yet exceptional...) scalar case is dealt in x1.2.2.
If we relax the contractivity requirement, then we (cid:12)nd a uniqueness theory for
one-dimensional systems which was recently developed by Bressan and his co-
1
workers, [11]-[14]; Bressan's theory is based on the L -stability (rather than
contractivity) of the entropy solution operator of one-dimensionalsystems.
1.2.2 Scalar conservation laws (m = 1;d (cid:21) 1)
0
In the scalar case, the Jacobians Aj((cid:26)) are just scalars that can be always
symmetrized,sothatthecompatibilityrelation(1.2.4)infactde(cid:12)nestheentropy
(cid:13)uxes,Fj((cid:26)),forallconvex(cid:17)'s. Consequently,thefamilyofadmissibleentropies
inthescalarcaseconsistsofallconvexfunctions,andtheenvelopeofthisfamily
leads to Kru(cid:20)zkov's entropy pairs [88]
(cid:17)((cid:26);c)=j(cid:26)(cid:0)cj; F((cid:26);c)=sgn((cid:26)(cid:0)c)(A((cid:26))(cid:0)A(c)); c2R: (1.2.8)
Theorem 1.2.1applies in this case and (1.2.7) now reads
1 1 2
(cid:15) L -contraction. If (cid:26) ;(cid:26) are two entropy solutions of the scalar conserva-
tion law (1.2.1),then
2 1 2 1
k(cid:26) (t;(cid:1))(cid:0)(cid:26) (t;(cid:1))kL1(x) (cid:20)k(cid:26)0((cid:1))(cid:0)(cid:26)0((cid:1))kL1(x): (1.2.9)
Thus,the entropysolutionoperatorassociatedwithscalarconservationlaws
1
is L -contractive ({ or non-expansive to be exact), and hence, by the Crandall-
Tartar lemma(discussed below), it is also monotone
2 1 2 1
(cid:26)0((cid:1))(cid:21)(cid:26)0((cid:1))=)(cid:26) (t;(cid:1))(cid:21)(cid:26) (t;(cid:1)): (1.2.10)
1
Thenotionsofconservation,L contractionandmonotonicityplayanimpor-
tantroleinthetheoryofnonlinearconservationlaws,atleastinthescalarcase.
We discuss the necessary details of these notions, by proving the inverse impli-
cation: the monotonicityproperty (1.2.10)impliesthe all importantKru(cid:20)zkov's
entropy pairs (1.2.8) satisfying (1.2.3).
CHAPTER 1. A GENERAL OVERVIEW 6
Monotonicity and Kru(cid:20)zkov's entropy pairs
AnoperatorT iscalledmonotone(ororderpreserving)ifthefollowingimplication
1
holds for all(cid:26)'s (in some unspeci(cid:12)ed measure subspace of Lloc)
(cid:26)2 (cid:21)(cid:26)1 a:e:=)T((cid:26)2)(cid:21)T((cid:26)1) a:e: (1.2.11)
We use the terminology that if (cid:26)2 dominates (pointwise, a.e.) (cid:26)1, then T((cid:26)2)
dominates T((cid:26)1).
The followinglemmadue to Crandall & Tartar, [32], provides a useful charac-
terization for such monotone operators.
Lemma 1.2.1 (Crandall-Tartar [32]) ConsideranoperatorT,whichiscon-
R R
servative in the sense that T((cid:26))= (cid:26); 8(cid:26)'s. Then T is monotone i(cid:11) it is an
1
L -contraction, Z Z
jT((cid:26)2)(cid:0)T((cid:26)1)j(cid:20) j(cid:26)2(cid:0)(cid:26)1j: (1.2.12)
Proof. Thestandardnotations,(cid:26)1_(cid:26)2 :=max((cid:26)1;(cid:26)2)and(cid:26)1^(cid:26)2 :=min((cid:26)1;(cid:26)2)
will be used. Since j(cid:26)1(cid:0)(cid:26)2j(cid:17)(cid:26)1_(cid:26)2(cid:0)(cid:26)1^(cid:26)2, we (cid:12)nd by conservation that
Z Z Z Z Z
j(cid:26)1(cid:0)(cid:26)2j= (cid:26)1_(cid:26)2(cid:0) (cid:26)1^(cid:26)2 = T((cid:26)1_(cid:26)2)(cid:0) T((cid:26)1^(cid:26)2): (1.2.13)
Now, (cid:26)1 _(cid:26)2 dominates (pointwise a.e.) both (cid:26)1 and (cid:26)2; hence, if T is order
preserving,thenT((cid:26)1_(cid:26)2)dominatesbothT((cid:26)1)andT((cid:26)2),thatis,T((cid:26)1_(cid:26)2)(cid:21)
T((cid:26)1)_T((cid:26)2); similarly,(cid:0)T((cid:26)1^(cid:26)2)(cid:21)(cid:0)T((cid:26)1)^T((cid:26)2). We conclude that T is
1
an L -contraction, for
Z Z Z Z Z
j(cid:26)1(cid:0)(cid:26)2j(cid:21) T((cid:26)1)_T((cid:26)2)(cid:0) T((cid:26)1)^ T((cid:26)2)= jT((cid:26)1)(cid:0)T((cid:26)2)j:
(1.2.14)
The inverse implication (attributed to Stampacchia, I believe) starts with the
identity 2w+ (cid:17) jwj+ w, where w+ denotes, as usual, the 'positive part of',
w+ :=w_0. Setting w =T((cid:26)1)(cid:0)T((cid:26)2) the integrated version of this identity
reads
Z Z Z
2 (T((cid:26)1)(cid:0)T((cid:26)2)+ = jT((cid:26)1)(cid:0)T((cid:26)2)j+ T((cid:26)1)(cid:0)T((cid:26)2):
1
Given that T is L -contractive, then together with conservation it yields that
the two integrals on the right do not exceed
Z Z Z
2 (T((cid:26)1)(cid:0)T((cid:26)2)_0(cid:20) j(cid:26)1(cid:0)(cid:26)2j+ (cid:26)1(cid:0)(cid:26)2: (1.2.15)
Now, if (cid:26)2 dominates (cid:26)1, i.e., (cid:26)1 (cid:20) (cid:26)2 a.e., then the sum of the two integrals
on the RHSvanishes,andconsequently, the non-negativeintegrandonthe LHS
vanishes as well, i.e.,T((cid:26)1)(cid:0)T((cid:26)2)(cid:20)0.