Existence and uniqueness for a nonlinear parabolic/Hamilton-Jacobi coupled system describing the dynamics of dislocation densities

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Existence and uniqueness for a nonlinear

parabolic/Hamilton-Jacobi coupled system describing the dynamics of dislocation densities

Hassan Ibrahim

To cite this version:

Hassan Ibrahim. Existence and uniqueness for a nonlinear parabolic/Hamilton-Jacobi coupled system

describing the dynamics of dislocation densities. 2007. �hal-00138531�

hal-00138531, version 1 - 26 Mar 2007

paraboli/Hamilton-Jaobi oupled system desribing

the dynamis of disloation densities

Hassan IBRAHIM

∗

Marh27, 2007

Abstrat

We study a mathematial model desribing the dynamis of disloation densities in rystals. This

model is expressed as a one-dimensional system of a paraboliequation and a rst order Hamilton-

Jaobiequationthatareoupledtogether. Weshowtheexisteneanduniquenessofavisositysolution

amongthoseassumingalower-boundontheirgradientforalltimeinludingtheinitialdata. Moreover,

weshowtheexisteneofavisositysolutionwhenwehavenosuh restritionontheinitialdata. We

alsostatearesultofexisteneanduniquenessofanentropysolutionofthesystemobtainedbyspatial

derivation. Theuniquenessofthisentropysolutionholdsinthelassofboundedfrombelowsolutions.

Inordertoprovetheseresults,weusearelationbetweensalaronservationlawsandHamilton-Jaobi

equations, mainlyto getsomegradientestimates. This studywill takeplaein

R

, andon abounded domainwithsuitableboundaryonditions.

Resumé

Nous étudions unmodèle mathématique dérivant ladynamique de densités de disloations dans les

ristaux. Ce modèles'érit ommeunsystème 1Douplantune équationparaboliqueet uneéquation

deHamilton-Jaobidupremierordre.Onmontrel'existeneetl'uniitéd'unesolutiondevisositédans

lalassedesfontionsayantungradientminorépourtouttempsainsiqu'autempsinitial. Deplus,on

montre l'existened'unesolutionde visositésansette onditionsur ladonnéeinitiale. On présente

égalementun résultat d'existene et d'uniité pour une solutionentropiqued'un système obtenu par

dérivation spatiale. L'uniité deettesolutionentropiquealieu danslalassedessolutionsminorées.

Pourmontreresrésultats,onutiliseunerelationentrelesloisdeonservationsalaireetleséquations

deHamilton-Jaobi,prinipalementpourobtenirdesontrlesdugradient. Cetteétudealieu dans

R

etdansundomaineborné avedesonditionsauxbordsappropriées.

AMS Classiation: 70H20,35L65,49L25,54C70,74H20,74H25.

Key words: Hamilton-Jaobiequations, salaronservation laws,visositysolutions, entropy solu-

tions,dynamisofdisloationdensities.

∗

Cermis,EoledesPonts,ParisTeh,6et8avenueBlaisePasal,CitéDesartesChamps-sur-Marne,77455

Marne-la-ValléeCedex2,Frane

1.1 Physial motivation

A disloation is a defet, or irregularity within a rystal struture that an be observed by

eletron mirosopy. The theory was originally developed by Vito Volterra in 1905. Disloa-

tions area non-stationary phenomena and their motionis themain explanation of theplasti

deformation inmetallirystals (see[28 , 19℄for a reent and mathematial presentation).

Geometrially, eah disloation is haraterized by a physial quantity alled the Burgers

vetor, whih is responsible for its orientation and magnitude. Disloations are lassied as

being positive or negative due to the orientation of its Burgers vetor, and they an move in

ertainrystallographi diretions.

Startingfromthe motionofindividualdisloations, aontinuumdesriptionanbederived

by adopting a formulation of disloation dynamis in terms of appropriately dened disloa-

tion densities, namely the density of positive and negative disloations. In this paper we are

interested in the model desribed by Groma, Csikor and Zaiser [18℄, that sheads light on the

evolutionofthedynamisofthetwotypedensitiesofasystemofstraightparalleldisloations,

taking into onsideration the inuene of the short range disloation-disloation interations.

Themodelwasoriginally presentedin

R ² × (0, T )

^asfollows:



 



 



∂θ ⁺

∂t + b · ∂

∂ r

θ ⁺

(τ _sc + τ _{ef f} ) − AD b

(θ ⁺ + θ ⁻ ) · ∂

∂ r θ ⁺ − θ ⁻

= 0,

∂θ ⁻

∂t − b · ∂

∂ r

θ ⁻

(τ _sc + τ _{ef f} ) − AD b

(θ ⁺ + θ ⁻ ) · ∂

∂ r θ ⁺ − θ ⁻

= 0.

(1.1)

Where

T > 0

^,

r = (x, y)

^{represents the spatial variable,}

b

is the burger's vetor,

θ ⁺ ( r , t)

and

θ ⁻ ( r , t)

^{denote the densities of the positive and negative} disloations respetively. The quantity

A

^{isdened by the formula}

A = µ/[2π(1 − ν)]

^{, where}

µ

^{is theshear modulus and}

ν

is the Poisson ratio.

D

^{is a} non-dimensional onstant. Stress elds are represented through the self-onsistent stress

τ sc ( r , t)

^{, and the eetive stress}

τ _{ef f} ( r , t)

^.

_∂r ^∂

^{denotes the gradient}

withrespetto the oordinate vetor

r

. An earlier investigation of theontinuum desription of the dynamis of disloation densities has been done in [17℄. However, a major drawbak

of these investigations is that the short range disloation-disloation orrelations have been

negleted and disloation-disloation interations were desribed only bythe long-range term

whih isthe self-onsistent stress eld. Moreover, forthe modeldesribed in[17℄,we refer the

reader to [11 , 12 ℄ for a one-dimensional mathematial and numerial study, and to [4℄ for a

two-dimensional existeneresult.

In our work,we are interestedina partiular setting of(1.1) where we make thefollowing

assumptions:

(a1) thequantities inequations (1.1) areindependent of

y

^,

(a2)

b = (1, 0)

^{,and theonstants}

A

^and

D

^{aresetto be}

1

^,

(a3) theeetive stressis assumedto be zero.

Remark 1.1 (a1) gives that the self-onsistent stress

τ _sc

^{is null; this is a onsequene of the}

denitionof

τ _sc

^{(see [18℄).}

Assumptions(a1)-(a2)-(a3)permitrewritingtheoriginalmodelasa

1D

problemin

R × (0, T )

^:



 

 

 

 

θ ⁺ _t (x, t) −

θ ⁺ (x, t)

θ _x ⁺ (x, t) − θ _x ⁻ (x, t) θ ⁺ (x, t) + θ ⁻ (x, t)

x

= 0, θ ⁻ _t (x, t) +

θ ⁻ (x, t)

θ _x ⁺ (x, t) − θ _x ⁻ (x, t) θ ⁺ (x, t) + θ ⁻ (x, t)

x

= 0.

(1.2)

We onsideran integratedform of(1.2) and we let:

ρ ^± _x = θ ^± , θ = θ ⁺ + θ ⁻ , ρ = ρ ⁺ − ρ ⁻

^and

κ = ρ ⁺ + ρ ⁻ ,

^(1.3)

in order to obtain, for speial values of the onstants of integration, the following system of

PDEsinterms of

ρ

^and

κ

^:

( κ _t κ _x = ρ _t ρ _x

ⁱⁿ

Q _T = R × (0, T ), κ(x, 0) = κ ⁰ (x)

ⁱⁿ

R ,

(1.4)

and

(

ρ _t = ρ _xx

ⁱⁿ

Q _T , ρ(x, 0) = ρ ⁰ (x)

ⁱⁿ

R ,

(1.5)

where

T > 0

^{isa xedonstant. Enough regularityontheinitialdatawillbegiveninorderto}

impose thephysiallyrelevant ondition,

κ ⁰ _x ≥ | ρ ⁰ _x | .

^(1.6)

This ondition is natural: it indiates nothing but thepositivity of the disloation densities

θ ^± (x, 0)

^{atthe initial time(see(1.3)).}

1.2 Main results

In this paper, we show the existene and uniqueness of a visosity solution

κ

^{of (1.4) in the}

lassofallLipshitzontinuousvisositysolutions having speialbounded frombelow spatial

gradients. However, we show the existene of aLipshitz ontinuousvisosity solution of (1.4)

when this restrition is relaxed. A relation between salar onservation laws and Hamilton-

Jaobi equations will be exploited to get almost all our gradient ontrols of

κ

^{. This relation,}

that will be made preise later, will also lead to a result of existene and uniqueness of a

bounded entropy solutionof the following equation:







θ _t = ρ x ρ xx

θ

x

in

Q _T , θ(x, 0) = θ ⁰ (x)

ⁱⁿ

R ,

(1.7)

whih is dedued formally by taking a spatial derivation of (1.4). The uniqueness of this

entropy solution is always restrited to the lass of bounded entropy solutions witha speial

lower-bound.

Let

Lip( R )

^denotes:

Lip( R ) = { f : R 7→ R ; f

^{is aLipshitz ontinuousfuntion}

} .

We prove the following theorems:

Let

T > 0

^{. Take}

κ ⁰ ∈ Lip( R )

^and

ρ ⁰ ∈ C ₀ ^∞ ( R )

^{as initial data that satisfy:}

κ ⁰ _x ≥ p

(ρ ⁰ _x ) ² + ǫ ²

^{a.e. in}

R ,

^(1.8)

for some onstant

ǫ > 0

^{. Then, given the solution}

ρ

^{of (1.5), there exists a visosity solution}

κ ∈ Lip( ¯ Q _T )

^{of (1.4), unique among the visosity solutions} satisfying:

κ _x ≥ p

ρ ² _x + ǫ ²

^{a.e. in}

Q ¯ _T .

Theorem 1.3 (Existene and uniquenessof an entropy solution)

Let

T > 0

^{. Take}

θ ⁰ ∈ L ^∞ ( R )

^and

ρ ⁰ ∈ C ₀ ^∞ ( R )

^{suh that,}

θ ⁰ ≥ p

(ρ ⁰ _x ) ² + ǫ ²

^{a.e. in}

R ,

for some onstant

ǫ > 0

^{. Then, there exists an entropy solution}

θ ∈ L ^∞ ( ¯ Q T )

^{of (1.7), unique}

among the entropy solutions satisfying:

θ ≥ p

ρ ² _x + ǫ ²

^{a.e. in}

Q ¯ _T .

Moreover, we have

θ = κ _x

^{, where}

κ

^{isthe solution given by Theorem 1.2.}

The notion of visosity solutions and entropy solutions will be realled inSetion 2. We now

relate these results to our one-dimensional problem(1.2). Remarking that

ρ x = θ ⁺ − θ ⁻

^and

κ _x = θ ⁺ + θ ⁻

^{,wehave asa onsequene:}

Corollary 1.4 (Existene and uniqueness for problem (1.2))

Let

T > 0

^{. Let}

θ ₀ ⁺

^and

θ ₀ ⁻

^{be two given funtions} representingthe initial positive and negative disloation densities respetively. If the followingonditions are satised:

(1)

θ ⁺ ₀ − θ ₀ ⁻ ∈ C ₀ ^∞ ( R ),

(2)

θ ⁺ ₀

^,

θ ₀ ⁻ ∈ L ^∞ ( R )

^,

togetherwith,

θ ⁺ ₀ + θ ⁻ ₀ ≥ q

(θ ⁺ ₀ − θ ⁻ ₀ ) ² + ǫ ²

^{a.e. in}

R ,

thenthere existsasolution

(θ ⁺ , θ ⁻ ) ∈ (L ^∞ (Q _T )) ²

^{tothesystem(1.2),inthesenseofTheorems}

1.2and 1.3,unique among those satisfying:

θ ⁺ + θ ⁻ ≥ p

(θ ⁺ − θ ⁻ ) ² + ǫ ²

^{a.e. in}

Q ¯ _T .

Remark 1.5 Conditions (1)and (2)are suient requirements for the ompatibility withthe

regularity of

ρ ⁰

^and

κ ⁰

^{previously stated.}

Theorem 1.6 (Existene of a visosity solution, ase

ǫ = 0

⁾

Let

T > 0

^,

κ ⁰ ∈ Lip( R )

^and

ρ ⁰ ∈ C ₀ ^∞ ( R )

^{. If the ondition (1.6) is satised a.e. in}

R

, then there exists a visosity solution

κ ∈ Lip( ¯ Q T )

^{of (1.4)} satisfying:

κ _x ≥ | ρ _x |

^{a.e. in}

Q ¯ _T .

^(1.9)

Remark 1.7 Inthelimitasewhere

ǫ = 0

^{,weremarkthathaving(1.9)was}intuitivelyexpeted due tothe positivity of the disloation densities

θ ⁺

^and

θ ⁻

^{. Thisreets in somewaythe well-}

posedness of the model(1.2) of the dynamis ofdisloation densities. We also remark that our

result of existene of a solution of (1.4)under (1.9)still holds if we startwith

κ ⁰ _x = ρ ⁰ _x = 0

^on

someintervalof thereal line. In otherwords, we an imaginethat we startwiththeprobability

of the formationof no disloation zones.

Problem with boundary onditions.

We onsider one again problem(1.4), similar results to thatannouned above will be shown

ona bounded intervalof thereal linewithDirihlet boundaryonditions(see Setion 5). This

problemorresponds physiallyto thestudy of thedynamis of disloation densitiesina part

ofa materialwiththegeometry of aslab (see [18℄).

1.3 Organization of the paper

The paper isorganized asfollows. In Setion 2, we start bystating the denitionof visosity

and entropy solutions withsome of their properties. In Setion 3, we prove the existeneand

uniqueness of a visosity solution to an approximated problem of (1.4), namely Proposition

3.1, and we move on, giving additional properties of our approximated solution (Proposition

3.2) and onsequently proving Theorems 1.2 and 1.3. In Setion 4, we present the proof of

Theorem 1.6. Setion 5 is devoted to the study of problem (1.4) on a bounded domain with

suitableboundaryonditions. Finally,Setion6isanappendixontainingaskethoftheproof

to thelassial omparison priniple of salar onservationlaws adapted to our equation with

lowregularity.

2 Notations and Preliminaries

We rst x some notations. If

Ω

^{is an open subset of}

R ⁿ

,

k

^{is a positive integer, we denote}

by

C ^k (Ω)

^{the spae of all real valued}

k

^times ontinuously dierentiable funtions.

C ₀ ^k (Ω)

^is

the subspae of

C ^k (Ω)

^{onsisting offuntion of ompat support in}

Ω

^,and

C _b ^k (Ω) = C ^k (Ω) ∩ W ^k,∞ (Ω)

^where

W ^k,∞ (Ω)

^{is dened below. F}urthermore, let

U C(Ω)

^and

Lip(Ω)

^{denote the}

spaes of uniformlyontinuous funtionsand Lipshitzontinuousfuntions on

Ω

respetively.

Thesobolevspae

W ^m,p (Ω)

^with

m ≥ 1

^{an integer and}

p : 1 ≤ p ≤ ∞

^{a real,isdened by}

W ^n,p (Ω) =







u ∈ L ^p (Ω)

∀ α

^with

| α | ≤ n ∃ f _α ∈ L ^p (Ω)

^{suh that}

Z

Ω

uD ^α φ = ( − 1) ^|α|

Z

Ω

f _α φ ∀ φ ∈ C ₀ ^∞ (Ω)





 ,

where we denote

D ^α u = f _α

^{. Thisspae equippedwiththenorm}

|| u || W ^n,p = X

0≤|α|≤n

|| D ^α u || L ^p

isa Banah spae. Inwhat follows,

T > 0

^{. A map}

m : [0, ∞ ) 7→ [0, ∞ )

^{that satisfy}

• m

^{isontinuous and}non-dereasing;

• lim

x→0 ⁺ m(x) = 0

^;

• m(a + b) ≤ m(a) + m(b)

^for

a, b ≥ 0

^;

issaidto be amodulus, and

U C _x (Ω × [0, T ])

^{denotes thespae ofthose}

u ∈ C(Ω × [0, T ])

^for

whih thereis amodulus

m

^and

r > 0

^suhthat

| u(x, t) − u(y, t) | ≤ m( | x − y | )

^for

x, y ∈ Ω, | x − y | ≤ r

^and

t ∈ [0, T ].

We will dealwithtwo typesofequations:

1. Hamilton-Jaobi equation:

( u _t + F(x, t, u _x ) = 0

ⁱⁿ

Q _T , u(x, 0) = u ⁰ (x)

ⁱⁿ

R ,

(2.1)

2. Salar onservation laws:

( v _t + (F (x, t, v)) _x = 0

ⁱⁿ

Q _T , v(x, 0) = v ⁰ (x)

ⁱⁿ

R ,

(2.2)

where

F : R × [0, T ] × R → R (x, t, u) 7→ F (x, t, u)

isalledtheHamiltonian intheHamilton-Jaobiequations and theuxfuntion inthesalar

onservationlaws. Wewillagreeontheontinuityofthisfuntion,whileadditionalandspei

regularitywill be givenwhen itis needed.

Remark 2.1 We will use the funtion

F

^{as a notation for the} Hamiltonian/ux funtion.

Although

F

^{might dier from oneequation toanother, it will be laried in all whatfollows.}

Remark 2.2 The major part of this work onerns a Hamiltonian/ux funtion of a speial

form,namely:

F (x, t, u) = g(x, t)f (u),

^(2.3)

where suh forms often arise in problems of physial interest inluding tra ow [31℄ and

two-phase ow inporous media [16℄.

We start by dening the notion of visosity solution to Hamilton-Jaobi equations (2.1),

andentropysolutiontosalaronservationlaws(2.2)withauxfuntiongivenbyRemark2.2,

aswellassomeresultsaboutexistene,uniqueness,andregularitypropertiesofthesesolutions.

We will end bya lassial relation between these two problems. Theseresults will be needed

throughout this paper, preisereferenes for theproofswillbe mentioned later on.

2.1 Visosity solution: denition and properties

Denition 2.3 ([10℄, Visosity solution: non-stationary ase)

1) A funtion

u ∈ C(Q _T ; R )

^{is a visosity} sub-solution of

u _t + F (x, t, u _x ) = 0

ⁱⁿ

Q _T ,

^(2.4)

if for every

φ ∈ C ¹ (Q _T )

^{, whenever}

u − φ

^{attains a loal maximum at}

(x ₀ , t ₀ ) ∈ Q _T

^{, then}

φ _t (x ₀ , t ₀ ) + F (x ₀ , t ₀ , φ _x (x ₀ , t ₀ )) ≤ 0.

2) A funtion

u ∈ C(Q _T ; R )

^{is a visosity sup}er-solution of (2.4) if for every

φ ∈ C ¹ (Q _T )

^,

whenever

u − φ

^{attains a loal minimum at}

(x ₀ , t ₀ ) ∈ Q _T

^{, then}

φ _t (x ₀ , t ₀ ) + F (x ₀ , t ₀ , φ _x (x ₀ , t ₀ )) ≥ 0.

3) A funtion

u ∈ C(Q _T ; R )

^{is a visosity solution of (2.4) if it is both a visosity sub- and}

super-solutionof (2.4).

4) A funtion

u ∈ C( ¯ Q _T ; R )

^{is a visosity solution of the initial value problem (2.1) if}

u

^{is a}

visosity solutionof (2.4) and

u(x, 0) = u ⁰ (x)

ⁱⁿ

R

.

It is worth mentioning here that if a visosity solution of a Hamilton-Jaobi equation is dif-

ferentiable at a ertain point, then it solves the equation there (see [10 , Corollary I.6℄). An

equivalent denition depending on the sub- and super-dierential of a ontinuous funtion is

now presented. Thisdenition will be usedfor the demonstration of Proposition 2.10. Letus

reall thatthe sub-and the super-dierential of a ontinuous funtion

u ∈ C( R ⁿ × (0, T ))

^{, at}

apoint

(x, t) ∈ R ⁿ × (0, T )

^{, aredened asthelosed onvex sets:}

D ^1,− u(x, t) = n

(p, α) ∈ R ⁿ × R : lim inf

(y,s)→(x,t)

u(y, s) − u(x, t) − (p · (y − x) + α · (s − t))

| y − x | + | s − t | ≥ 0 o ,

and

D ^1,+ u(x, t) = n

(p, α) ∈ R ⁿ × R : lim sup

(y,s)→(x,t)

u(y, s) − u(x, t) − (p · (y − x) + α · (s − t))

| y − x | + | s − t | ≤ 0 o ,

respetively.

Denition 2.4 (Equivalent denition of visosity solution)

1) Afuntion

u ∈ C( R ⁿ × (0, T ))

^{is a visosity} super-solution of (2.1)if andonly if,for every

(x, t) ∈ R ⁿ × (0, T )

^:

∀ (p, α) ∈ D ^1,− u(x, t), α + F(x, t, p) ≥ 0.

^(2.5)

2) A funtion

u ∈ C( R ⁿ × (0, T ))

^{is a visosity} sub-solution of (2.1) if and only if, for every

(x, t) ∈ R ⁿ × (0, T )

^:

∀ (p, α) ∈ D ^1,+ u(x, t), α + F(x, t, p) ≤ 0.

^(2.6)

Thisdenitionismoreloal,foritpermitsveriationthatagivenexpliitfuntionisavisosity

solution in a more lassial way, i.e. using the derivative alulus. A similar denition, that

will be usedlater, ould be given inthe stationaryase. Let

Ω ⊂ R ⁿ

be anopen domain,and onsiderthe PDE

F(x, u(x), ∇ u(x)) = 0, ∀ x ∈ Ω,

^(2.7)

where

F : Ω × R × R ⁿ 7→ R

isa ontinuousmapping.

A ontinuous funtion

u : Ω 7→ R

is a visosity sub-solution of the PDE (2.7) if for any ontinuously dierentiable funtion

φ : Ω 7→ R

and any loal maximum

x ₀ ∈ Ω

^of

u − φ

^{, one}

has

F(x 0 , u(x 0 ), ∇ φ(x 0 )) ≤ 0.

Similarly,if at any loal minimum point

x ₀ ∈ Ω

^of

u − φ

^{, one has}

F(x ₀ , u(x ₀ ), ∇ φ(x ₀ )) ≥ 0,

then

u

^{isa visosity}super-solution. Finally, if

u

^{isboth a visosity} sub-solution anda visosity super-solution,then

u

^{isalled a visosity solution.}

Infat,this denitionisusedforinterpreting solutions of(1.4) inthevisositysense. Further-

more,we saythat

u

^{isa visositysolution of theDirihlet problem (2.7) with}

u = ζ ∈ C(∂Ω)

if:

(1)

u ∈ C( ¯ Ω)

^,

(2)

u

^{isa visositysolutionof (2.7) in}

Ω

^,

(3)

u = ζ

^on

∂Ω

^.

For abetter understandingof thevisosityinterpretationofboundaryonditionsof Hamilton-

Jaobi equations,we refer the readerto [2 , Setion 4.2℄.

Now, we will proeedbygiving themain results onerningvisositysolutions of (2.1). In

order to have existeneand uniqueness, the Hamiltonian

F

^{will be restritedbythefollowing}

onditions :

( F0 ) F ∈ C( R × [0, T ] × R )

^;

( F1 )

^{for eah}

R > 0

^{there is a onstant}

C _R

^{suh that for all}

(x, t, p)

^,

(y, t, q) ∈ R × [0, T ] × [ − R, R],

| F (x, t, p) − F (y, t, q) | ≤ C _R ( | p − q | + | x − y | );

( F2 )

^{there isa onstant}

C F

^{suh thatfor all}

(t, p) ∈ [0, T ] × R

and all

x, y ∈ R ,

| F (x, t, p) − F (y, t, p) | ≤ C _F | x − y | (1 + | p | ).

We usethese onditions towrite downsome results onvisositysolutions.

Theorem 2.6 (Comparison, [9, Theorem 1℄)

Let

F

^satisfy

( F0 )

^-

( F1 )

^-

( F2 )

^{. If}

u

^,

u ¯ ∈ U C _x ( ¯ Q _T )

^{are two visosity sub- and sup}er-solution of the Hamilton-Jaobiequation (2.1) respetively, with

u(x, 0) ≤ u(x, ¯ 0)

ⁱⁿ

R ,

then

u ≤ u ¯

ⁱⁿ

Q ¯ _T

^.

Theorem 2.7 (Existene, [9, Theorem 1℄)

Let

F

^satisfy

( F0 )

^-

( F1 )

^-

( F2 )

^{. If}

u ⁰ ∈ U C( R )

^{,then(2.1)hasavisositysolution}

u ∈ U C _x ( ¯ Q _T )

^.

Remark 2.8 The omparison theorem stated above gives the uniqueness of the visosity so-

lution.

F (x, t, u) = g(x, t)f (u),

the following onditions:

( V0 ) f ∈ C _b ¹ ( R ; R ), ( V1 ) g ∈ C _b ( ¯ Q _T ; R ), ( V2 ) g _x ∈ L ^∞ ( ¯ Q _T )

^,

imply

( F0 )

^-

( F1 )

^-

( F2 )

^{togetherwith the} boundedness of the Hamiltonian.

The next proposition reets the behavior of visosity solutions under additional regularity

assumptionson

u ⁰

^and

F

^.

Proposition 2.10 (Additional regularity of the visosity solution)

Let

F = gf

^satisfy

( V0 )

^-

( V1 )

^-

( V2 )

^{. If}

u ⁰ ∈ Lip( R )

^and

u ∈ U C x ( ¯ Q T )

^{isthe unique visosity}

solutionof (2.1), then

u ∈ Lip( ¯ Q _T )

^.

Proof. Consider the funtion

u ^ε

^{dened on}

R × [0, T ]

^by:

u ^ε (x, t) = sup

y∈R

u(y, t) − e ^kt | x − y | ² 2ε

.

By [20,Theorem 3℄, thefuntion

u

^satises,

| u(x, t) | ≤ c ^∗ ( | x | + 1)

^for

(x, t) ∈ R × [0, T ],

where

c

^and

c ^∗

^{are two positive onstants. Therefore,}

u

^{is a sublinear funtion for every}

time

t ∈ [0, T ]

^{. The funtion}

u ^ε

^{is dened via a supremum whih is attained beause of the}

sublinearityofthefuntion

u

^{(aquadratifuntionalwaysontrolalinearone);thesupremum}

an be ahievedat several points; let

x _ε

^{beone of them,sowe anwrite}

u ^ε (x, t) = u(x _ε , t) − e ^kt | x − x ε | ²

2ε .

We aregoingto prove thatfor

(p, α) ∈ R × R

,we have:

(p, α) ∈ D ^1,+ u ^ε (x, t) ⇒

p, α + ke ^kt | x − x _ε | ² 2ε

∈ D ^1,+ u(x _ε , t).

^(2.8)

Sine

(p, α) ∈ D ^1,+ u ^ε (x, t)

^{,thenwe an writefor}

(y, s) ∼ (x, t)

^that,

L = u ^ε (y, s) ≤ u ^ε (x, t) + α(s − t) + p(y − x) + o( | s − t | + | y − x | ) = R,

^(2.9)

where theleftside

L

^{of(2.9) satises,}

L ≥ u(z, s) − e ^ks | z − y | ²

2ε , z ∈ R ,

^(2.10)

and theright side

R

^{of(2.9) satises,}

R ≤ u(x _ε , t) − e ^kt | x − x _ε | ²

2ε + α(s − t) + p(y − x) + o( | s − t | + | y − x | ).

^(2.11)

Choose

z

^{suh that}

z − y = x _ε − x

^,then

z = x _ε + (y − x) ∼ x _ε ,

^sine

y ∼ x.

^(2.12)

Combining (2.9),(2.10), (2.11) and(2.12) together, we get

u(x _ε + (y − x), s) − e ^ks | x − x _ε | ²

2ε ≤

u(x ε , t) − e ^kt | x − x ε | ²

2ε + α(s − t) + p(z − x ε ) + o( | s − t | + | z − x ε | ),

and hene,

u(z, s) ≤ u(x ε , t) + (e ^ks − e ^kt ) | x − x ε | ² 2ε

+α(s − t) + p(z − x _ε ) + o( | s − t | + | z − x _ε | ).

^(2.13)

We have

(e ^ks − e ^kt ) | x − x _ε | ²

2ε = ke ^kt | x − x _ε | ²

2ε (s − t) + o( | s − t | ),

thenusing inequality(2.13), we get

u(z, s) ≤ u(x ε , t) +

α + ke ^kt | x − x ε | ² 2ε

(s − t)

+p(z − x _ε ) + o( | s − t | + | z − x _ε | ),

whih provesthat

α + ke ^kt | x − x _ε | ² 2ε , p

∈ D ^1,+ u(x ε , t),

and henestatement (2.8) is true. Sine

u

^isavisositysub-solution of(2.1), we have

α + ke ^kt | x − x _ε | ²

2ε + F(x _ε , t, p) ≤ 0.

We useondition

( F1 )

^with

p = q

^{,to get}

α + ke ^kt | x − x _ε | ²

2ε + F (x, t, p) ≤ F(x, t, p) − F (x _ε , t, p),

≤ C | x − x _ε | ,

therefore,

α + F(x, t, p) ≤ C | x − x _ε | − ke ^kt | x − x ε | ² 2ε ,

≤ Cr ε − k r _ε ² 2ε ,

≤ sup

r>0

Cr − kr ² 2ε

,

where

r _ε = | x − x _ε | .

^{At themaximum}

¯ r

^,wehave

C = ^k¯ _ε ^r

^{. Byhoosing}

k = ^C ₂ ²

^{,we get}

α + F (x, t, p) ≤ ε.

This inequality shows that

v ^ε = u ^ε − εt

^{is a visosity} sub-solution of (2.1) with

v ^ε (x, 0) = u ^ε (x, 0)

^{. By the omparison priniple,we have}

v ^ε (x, t) − u(x, t) ≤ sup

x∈R

(v ^ε (x, 0) − u ⁰ (x)),

≤ sup

x∈R

(u ^ε (x, 0) − u ⁰ (x)),

≤ sup

x∈R

sup

y∈R

u ⁰ (y) − | x − y | ² 2ε

− u ⁰ (x)

! ,

≤ sup

x,y∈R

γ | x − y | − | x − y | ² 2ε

,

≤ sup

r≥0

γr − r ² 2ε

= γ ² ε 2 ,

where

γ

^{is the Lipshitz onstant of the funtion}

u ⁰

^{, and}

r = | x − y |

^{. This altogether shows}

the following inequalityfor

x, y ∈ R

:

u(y, t) − e ^kt | x − y | ²

2ε ≤ u ^ε (x, t) ≤ u(x, t) + εt + γ ² ε

2 .

^(2.14)

Remarkherethat

k

^{isa xed; previously hosenonstant. Inequality(2.14) yields:}

u(y, t) − u(x, t) ≤ e ^kt | x − y | ²

2ε +

t + γ ²

2

ε = ζ/ε + βε,

^(2.15)

where

ζ = e ^{kt |x−y|} ₂ ²

^and

β =

t + ^γ ₂ ²

. We minimizeinequality(2.15) over

ε

^{to obtain,}

u(y, t) − u(x, t) ≤ 2 p

ζβ,

≤ e ^{kt 2} √ 2

r t + γ ²

2 | x − y | .

Sinethis inequality holds

∀ x, y ∈ R

,exhanging

x

^with

y

^yields,

| u(x, t) − u(y, t) | ≤ C(F, u 0 ) | x − y | ∀ x, y ∈ R

and

t ∈ [0, T ].

Thisshows thatthefuntion

u

^{isLipshitz ontinuousin}

x

^{,uniformlyintime}

t

^{. To prove the}

Lipshitz ontinuity in time, we mainly use the result of [20 , Theorem 3℄) with the fat that

u _t = − F (x, t, u _x )

^,andthe boundedness oftheHamiltonian.

2

Remark 2.11 It is worth mentioning that the spae Lipshitz onstant of the funtion

u

^de-

pendson

C

^,where

C

^appearsin

( F1 )

^for

p = q

^{,andonthe Lipshitzonstant}

γ

^ofthefuntion

u ₀

^{. While the time Lipshitzonstant depends on the bound of the} Hamiltonian.

Denition 2.12 (Entropy sub-/super-solution)

Let

F (x, t, v) = g(x, t)f (v)

^with

g, g _x ∈ L ^∞ _loc (Q _T ; R )

^and

f ∈ C ¹ ( R ; R )

^{. A funtion}

v ∈ L ^∞ (Q _T ; R )

^{is an entropy} sub-solution of (2.2) with bounded initial data

v ⁰ ∈ L ^∞ ( R )

^{if it}

satises:

Z

Q T

h η _i (v(x, t))φ _t (x, t) + Φ(v(x, t))g(x, t)φ _x (x, t)+

h(v(x, t))g _x (x, t)φ(x, t) i dxdt +

Z

R

η _i (v ⁰ (x))φ(x, 0)dx ≥ 0,

(2.16)

∀ φ ∈ C ₀ ¹ ( R × [0, T ); R ₊ )

^{, for any} non-dereasing onvex funtion

η i ∈ C ¹ ( R ; R )

^,

Φ ∈ C ¹ ( R ; R )

suh that:

Φ ^′ = f ^′ η _i ^′ ,

^and

h = Φ − f η ^′ _i .

^(2.17)

Anentropy super-solutionof (2.2)isdened by replaingin(2.16)

η _i

^with

η _d

^{; a}non-inreasing onvex funtion. An entropy solution isdened asbeing both entropy sub- and super-solution.

In other words, it veries (2.16) for any onvexfuntion

η ∈ C ¹ ( R ; R )

^.

A well knowharaterization of theentropysolution isthat:

Proposition 2.13 A funtion

v ∈ L ^∞ (Q T )

^{is an entropy} sub-solution of (2.2) if and onlyif

∀ k ∈ R

,

φ ∈ C ₀ ¹ ( R × [0, T ); R ₊ )

^{, onehas:}

Z

Q T

h (v(x, t) − k) ⁺ φ _t (x, t) +

^sgn

⁺ (v(x, t) − k)(f (v(x, t)) − f (k))g(x, t)φ _x (x, t) −

sgn

+ (v(x, t) − k)f (k)g x (x, t)φ(x, t) i dxdt +

Z

R

(v ⁰ (x) − k) ⁺ φ(x, 0)dx ≥ 0,

^(2.18)

Where

a ^± = ¹ ₂ ( | a | ± a)

^{and sgn}

^± (x) = ¹ ₂ (

^sgn

(x) ± 1)

^{. An entropy} super-solution of (2.2) is dened replaing in (2.18)

( · ) ⁺

^,sgn

⁺

^by

( · ) ⁻

^{, sgn}

⁻

^.

This haraterization an be dedued from (2.16), by using regularizations of the funtion

( · − k) ⁺

^{. Also(2.16)maybeobtainedfrom(2.18) by}approximatinganynon-dereasingonvex funtion

η _i ∈ C ¹ ( R ; R )

^{by a sequene of funtions of the form:}

η _i ⁽ⁿ⁾ ( · ) = P n

1 β _i ⁽ⁿ⁾ ( · − k _i ⁽ⁿ⁾ ) ⁺

^,

with

β _i ⁽ⁿ⁾ ≥ 0

^.

Entropy solution was rst introdued by Kru

z ˇ

^{kov [22 ℄ as the only physially admissible}

solution among all weak (distributional) solutions to salar onservation laws. These weak

solutions lak the fat of being unique for it is easy to onstrut multiple weak solutions to

Cauhyproblems (2.2),see [25 ℄.

Ournextdenitiononerns lassialsub-/super-solutionto salaronservationlaws. This

kindof solutions areshown tobe entropy solutions,for thedetails seelemma 3.3.

Denition 2.14 (Classial solutionto salar onservation laws)

Let

F (x, t, v) = g(x, t)f (v)

^with

g, g _x ∈ L ^∞ _loc (Q _T ; R )

^and

f ∈ C ¹ ( R ; R )

^{. A funtion}

v ∈ W ^1,∞ (Q _T )

^{is said tobe a lassial} sub-solution of (2.2)with

v ⁰ (x) = v(x, 0)

^{if it satises}

v _t (x, t) + (F (x, t, v(x, t))) _x ≤ 0

^{a.e. in}

Q _T .

^(2.19)

Classialsuper-solutionsare dened byreplaing

≤

^with

≥

^{in(2.19), andlassialsolutions}

are dened to be both lassial sub- andsuper-solutions.

Theorem 2.15 (Kru

z ˇ

^{kov's Existene Theorem)}

Let

F

^,

v ⁰

^{be given by Denition 2.12, andthe following onditions hold:}

( E0 ) f ∈ C _b ¹ ( R ), ( E1 ) g, g _x ∈ C _b ( ¯ Q _T ), ( E2 ) g _xx ∈ C( ¯ Q _T )

^,

then there exists an entropy solution

v ∈ L ^∞ (Q T )

^{of (2.2).}

In fat, Kru

ˇ z

^{kov'sonditions for existene were given for a general ux funtion [22 , Setion}

4℄. However, in Subsetion 5.4 of the same paper, a weak version of these onditions, that

an be easily heked in the ase

F (x, t, v) = g(x, t)f (v)

^and

( E0 )

^-

( E1 )

^-

( E2 )

^{, is presented.}

Furthermore, uniquenessfollows fromthe following omparison priniple.

Theorem 2.16 (Comparison Priniple)

Let

F

^{be given by Denition 2.12with}

f

^{satisfying(E0), and}

g

^satises,

( E3 ) g ∈ W ^1,∞ ( ¯ Q _T ).

Let

u(x, t)

^,

v(x, t) ∈ L ^∞ (Q _T )

^{be two entropy} sub-/super-solutions of (2.2) with initial data

u ⁰

^,

v ⁰ ∈ L ^∞ ( R )

^{. Suppose that,}

u ⁰ (x) ≤ v ⁰ (x)

^{a.e. in}

R ,

then

u(x, t) ≤ v(x, t)

^{a.e. in}

Q ¯ _T .

Proof. See Setion 6,Appendix.

2

It is worth notiing that in[22 ℄, theproof of the existene of entropy solutions of (2.2) is

made througha paraboli regularization of (2.2) and passingto the limit,with respet to the

L ¹

^{onvergene onompats, ina onvenient spae.}

At this stage,we are ready to present a relation thatsometimes hold between salar on-

servationlaws and Hamilton-Jaobiequations inone-dimensional spae.

2.3 Entropy-Visosity relation

Formally, by dierentiating (2.1) with respet to

x

^{and dening}

v = u _x

^{, we see that (2.1) is}

equivalent to thesalaronservation law(2.2)with

v ⁰ = u ⁰ _x

^andthesame

F

^{. Thisequivalene}

of the two problems has been exploited in order to translate some numerial methods for

hyperboli onservation laws to methods for Hamilton-Jaobi equations. Moreover, several

proofsweregivenintheonedimensionalase. Theusualproofofthisrelationdependsstrongly

on the known results about existene and uniqueness of the solutions of the two problems

together withthe onvergene of the visosity method (see [8, 23, 27℄). Another proof of this

proof ould alsobe found in[21℄ using the front traking method. The ase ofa Hamiltonian

of the form (2.3) is also treated even when

g(x, t)

^{is allowed to be} disontinuous inthe

(x, t)

planealong anite numberof (possiblyinterseted) urves, see[29 ℄.

In our work, the above stated relation will be suessfully used to get some gradient es-

timates of

κ

^{. Although several approahes were given to establish this onnetion, we will}

present for the reader's onveniene, a proof similar to that given in [8 , Theorem 2.2℄. For

every Hamiltionian/ux funtion

F = gf

^andevery

u ⁰ ∈ Lip( R )

^,let

EV = { ( V0 ), ( V1 ), ( V2 ), ( E0 ), ( E1 ), ( E2 ), ( E3 ) } ,

inother words,

EV =

The setof allonditions on

f

^and

g

^{ensuring the}

existene anduniquenessof aLipshitz ontinuousvisosity

solution

u ∈ Lip( ¯ Q _T )

^{of (2.1),and of anentropy}

solution

v ∈ L ^∞ (Q _T )

^{of (2.2),with}

v ⁰ = u ⁰ _x ∈ L ^∞ ( R )

^.

Theorem 2.17 (A link between visosity and entropy solutions)

Let

F = gf

^with

g ∈ C ² ( ¯ Q _T )

^,

u ⁰ ∈ Lip( R )

^and

EV

^{satised. Then,}

v = u _x

^{a.e. in}

Q _T .

Sketh of the proof. Let

ε > 0

^and

δ > 0

^{. We start the prove by making a paraboli}

regularization of equation (2.1) and a smooth regularization of

u ₀

^{and we solve the following}

paraboli equation:

( u ^ε,δ _t + F (x, t, u ^ε,δ _x ) = ǫu ^ε,δ _xx

ⁱⁿ

R × (0, T ), u ^ε,δ (x, 0) = u ^0,δ (x)

ⁱⁿ

R .

(2.20)

For thesakeofsimpliity,wewilldenote

u ^ε,δ

^by

w

^and

u ^0,δ

^by

w ⁰

^{. Notethattherstequation}

of(2.20) an beviewed asthe heatequationwitha soure term

F

^{. Thus, we have:}

( w _t − εw _xx = F[w](x, t)

ⁱⁿ

Q _T ,

w(x, 0) = w ⁰

ⁱⁿ

R ,

(2.21)

with

F [w](x, t) = F(x, t, w _x (x, t))

^{. From the lassial theory of heat equations, sine}

F[w] ∈ L ^p _loc (Q _T )

^and

w ⁰ ∈ W _loc ^1,p ( R )

^{, thereexistsa unique solution}

w

^{of(2.21) suh that}

w ∈ W _p ^2,1 (Ω) ∀ Ω ⊂⊂ Q _T

^and

1 < p < ∞ .

Here the spae

W p ^2,1 (Ω)

^,

p ≥ 1

^{is the Banah spae onsisting of all funtions}

w ∈ L ^p (Ω)

having generalized derivatives of the form

w _t

^and

w _xx

ⁱⁿ

L ^p (Ω)

^{. For more details, see [24,}

Theorem 9.1℄. We also notie that the spae

W _p ^2,1 (Ω)

^is ontinuously injeted in the Hölder spae

C ^α,α/2 (Ω)

^for

α = 2 − ³ _p

^and

p > ³ ₂

^{, see [24℄. We use now a bootstrap argument to}

inreasetheregularityof

w

^{,takingineahstage,the newregularityof}

F [w]

^{andtheregularity}

of

w ⁰

^{. Finally, we get that}

w ∈ C ^3,1 ( R × [0, T ))

^{(three times} ontinuously dierentiable in

L ^p

^{-estimatesoftheheatequation,see[24,3℄,itfollowstheuniformboundof}

u ^ε,δ

ⁱⁿ

W _loc ^1,p (Q T )

^,

for

p > 2

^{. Therefore,we getas}

δ → 0

^and

ε → 0

^that:

u ^ε,δ → u

ⁱⁿ

C( R × [0, T )),

with

u(x, 0) = u ⁰

^{. We now make use ofthe stabilitytheorem, [2 , Théorème 2.3℄, twieon the}

equation(2.20) togetthatthelimit

u

^{istheuniquevisositysolution of(2.1). Hene, wehave}

for any

φ ∈ C ₀ ^∞ (Q _T )

ε→0, δ→0 lim Z T

0

Z

R

u ^ε,δ _x φ dx dt = − lim

ε→0, δ→0

Z T 0

Z

R

u ^ε,δ φ _x dx dt

= −

Z _T

0

Z

R

uφ _x dx dt = Z _T

0

Z

R

u _x φ dx dt.

Theappearaneof

u _x

^followssine

u ∈ Lip( ¯ Q _T )

^{. Moreover,asa regularsolution, thefuntion}

v ^ε,δ = u ^ε,δ _x

^{solvesthe derived problem}

( v ^ε,δ _t + (F (x, t, v ^ε,δ )) _x = ǫv ^ε,δ _xx

ⁱⁿ

R × (0, T ), v ^ε,δ (x, 0) = u ^0,δ _x (x)

ⁱⁿ

R ,

(2.22)

and,aordingto[22 ,Theorem4℄,thesequene

v ^ε,δ

^onvergein

L _loc ¹ ( ¯ Q _T )

^,as

ε → 0

^and

δ → 0

^,

to theentropysolution

v

^{of(2.2). Then, for any}

φ ∈ C ₀ ^∞ (Q _T )

^,

ε→0 lim δ→0

Z T 0

Z

R

v ^ε,δ φ dx dt = Z T

0

Z

R

vφ dx dt.

Consequently,

Z T 0

Z

R

u _x φ dx dt = Z T

0

Z

R

vφ dx dt,

and

u _x = v

^{a.e. in}

Q _T

^.

2

Remark 2.18 The onverse ofthe previous theoremholdsunder ertain assumptions(see [21,

7℄).

Remark 2.19 In the multidimensional ase this one-to-one orrespondene no longer exists,

instead the gradient

v = ∇ u

^{satises formally a non-strit hyperboli system of} onservation laws (see [27,23 ℄).

Throughout Setions 3 and 4,

ρ

^{will always be the solution of the heat equation (1.5). The}

propertiesofthesolutionoftheheatequationwithsuharegularinitialdatawillbefrequently

used, we referthe reader to [3,13℄ fordetails.

3 The approximate problem

In this setion, we approximate (1.4) and we pose a more restritive ondition (see ondition

(1.8)) on the gradient of the initial data than of the physialy relevent one (1.6). We prove

thereaderwillnotie at theend ofthissetionthatthis restritiveonditionis satisedforall

time, and this what anels the approximation in the struture of (1.4) and returns it to its

original one. Finaly we present the proof ofTheorem 1.3.

For every

a > 0

^{, we build up an} approximation funtion

f a ∈ C _b ¹ ( R )

^{of the funtion}

¹ _x

dened by:

f a (x) =



 

  1

x

^if

x ≥ a,

2a − x

a ² + a ² (x − a) ²

^otherwise.

(3.1)

Proposition 3.1 For any

a > 0

^{, let}

f _a

^{be dened by (3.1) and}

H ∈ C ¹ ( R )

^{be a} salar-valued funtion. If

F _a (x, t, u) = − H(ρ _x (x, t))ρ _xx (x, t)f _a (u)

^(3.2)

and

κ ⁰ ∈ Lip( R )

^{, then the} Hamilton-Jaobi equation

( κ _t + F _a (x, t, κ _x ) = 0

ⁱⁿ

Q _T , κ(x, 0) = κ ⁰ (x)

ⁱⁿ

R ,

(3.3)

has a unique visosity solution

κ ∈ Lip( ¯ Q T )

^.

Proof. The proof is easily onluded from Theorems 2.6, 2.7 and Proposition 2.10, after

heking that the onditions

( V0 )

^-

( V1 )

^-

( V2 )

^{aresatised with}

g(x, t) = − H(ρ _x (x, t))ρ _xx (x, t).

^(3.4)

Theondition

( V0 )

^{istrivial,whilefor}

( V1 )

^{,wejustusethefatthat}

H

^{isboundedonompats}

andthefatthat

| ρ _x (x, t) | ≤ || ρ ⁰ _x || L ^∞ (R)

ⁱⁿ

Q ¯ _T

^{. Fortheondition}

( V2 )

^{,theregularityof}

ρ

^and

H

^{permits toompute thespatial derivative of}

g

ⁱⁿ

Q ¯ _T

^{,thus we have:}

g _x = − (H ^′ (ρ _x )ρ ² _xx + H(ρ _x )ρ _xxx ).

Theuniform bound ofthe spatialderivatives,up tothethirdorder, of thesolution oftheheat

equation, and the boundedness of

H ^′

^{onompats gives}immediately

( V2 )

^.

2

Inthefollowingproposition, weshowalower-boundestimateforthegradientof

κ

^obtained

in Proposition 3.1. It is worth mentioning that a result of lower-bound gradient estimates

for rst-order Hamilton-Jaobi equations ould be found in[26, Theorem 4.2℄. However, this

resultholdsforHamiltonians

F (x, t, u)

^{thatareonvexinthe}

u

^{-variable,usingonlythevisosity}

theorytehniques. Thisisnottheasehere, andinordertoobtainour lower-bound estimates,

we need to usethe visosity/entropy theorytehniques. Inpartiular,wehave thefollowing:

Proposition 3.2 Let

G ∈ C ³ ( R ; R )

^{satisfyingthe followingonditions:}

(G1)

G(x) ≥ G(0) > 0

^,

(G2)

G ^′′ ≥ 0.

H = GG ^′

^and

0 < a ≤ G(0).

If

κ ⁰

^satises:

κ ⁰ _x (x) ≥ G(ρ ⁰ _x (x)),

^{a.e. in}

R ,

then the solution

κ

^{obtained from}Proposition 3.1satises:

κ _x (x, t) ≥ G(ρ _x (x, t))

^{a.e. in}

Q ¯ _T .

^(3.5)

Inorder to proveProposition 3.2, we rstshow that

G(ρ _x )

^{is anentropy}sub-solution of

( ω _t + (F (x, t, ω)) _x = 0

ⁱⁿ

Q _T , ω(x, 0) = ω ⁰ (x)

ⁱⁿ

R ,

(3.6)

with

w ⁰ = G(ρ ⁰ _x )

^and

F

^{isthe sameasin(3.2). Before going further,we willpause to prove a}

lemmawhih makesiteasierto reahour goal.

Lemma 3.3 (Classial sub-solutions are entropy sub-solutions)

Let

v ∈ W ^1,∞ (Q _T )

^{bea lassial} sub-solutionof(2.2)with

v ⁰ (x) = v(x, 0)

^,then

v

^{isan entropy}

sub-solution.

Proof. Let

η _i

^,

Φ

^,

h

^and

φ

^{begiven byDenition2.12.} Multiplyinginequality(2.19) by

η _i ^′ (v)φ

doesnot hange its sign. Hene, afterdeveloping, we have:

η _i ^′ (v)v _t φ + η ^′ _i (v)g _x f (v)φ + η ^′ _i (v)gf ^′ (v)v _x φ ≤ 0,

^{a.e. in}

Q _T ,

^(3.7)

andsine

v

^{isLipshitzontinuous,weusethehain-ruleformulatogetherwith(2.17)torewrite}

(3.7) as:

(η _i (v)) _t φ + g _x f (v)η _i ^′ (v)φ + g(Φ(v)) _x φ ≤ 0,

^{a.e. in}

Q _T .

^(3.8)

Uponintegrating(3.8) over

Q _T

^and transferringderivativeswithrespetto

t

^and

x

^tothetest

funtion,we obtain:

Z

Q T

h

η i (v(x, t))φ t (x, t) + Φ(v(x, t))g(x, t)φ x (x, t)+

h(v(x, t))g _x (x, t)φ(x, t) i dxdt +

Z

R

η _i (v ⁰ (x))φ(x, 0)dx ≥ 0,

^(3.9)

whih endsthe proof.

2

Following same arguments, lassialsuper-solutions are shownto entropysuper-solutions. We

return now to the funtion

G(ρ _x )

^{and we areready to show thatit is indeed an entropysub-}

solutionof (3.6). In partiular,we have thefollowing:

Lemma 3.4 The funtion

G(ρ _x )

^{dened on}

Q _T

^{isa lassial} sub-solution of (3.6)with initial data

G(ρ ⁰ _x )

^{, hene an entropy} sub-solution.

Proof of Lemma 3.4. First, it is easily seen that

G(ρ _x ) ∈ W ^1,∞ (Q _T )

^{. Dene the salar}

valuedquantity

B

^on

Q T

^by:

B (x, t) = ∂ t (G(ρ x (x, t))) + ∂ x (F (x, t, G(ρ x (x, t)))).

Sine

0 < a ≤ G(0)

^{, we use(G1) to get}

f _a (G(ρ _x )) = 1/G(ρ _x )

^{and we observe that,}

B = G ^′ (ρ _x )ρ _xt − ∂ _x

H(ρ _x )ρ _xx G(ρ x )

= G ^′ (ρ _x )ρ _xxx − G(ρ _x )[H ^′ (ρ _x )ρ ² _xx + H(ρ _x )ρ _xxx ] − (G ^′ (ρ _x )ρ ² _xx H(ρ _x )) G ² (ρ _x )

!

= G(ρ _x )ρ _xxx (G(ρ _x )G ^′ (ρ _x ) − H(ρ _x )) − ρ ² _xx (H ^′ (ρ _x )G(ρ _x ) − H(ρ _x )G ^′ (ρ _x )) G ² (ρ _x )

= − ρ ² _xx G ^′′ (ρ _x ).

The ondition (G2) gives immediately that

B ≤ 0

^{. This proves that}

G(ρ _x )

^{is a lassial sub-}

solutionof equation(3.6) and henean entropysub-solution.

2

Proof of Proposition 3.2. From the denition of

H

^{and the properties of}

ρ

^{, it is easy to}

hek that

g ∈ C ² ( ¯ Q _T )

^{and that}

EV

^{is fully satised. Hene, we are in the framework of}

Theorem2.17with

u ⁰ = κ ⁰

^{. Thistheoremgivesthat}

κ _x

^{istheunique entropysolutionof(3.6)}

with

w ⁰ = κ ⁰ _x

^{. Moreover, by the previous lemma,}

G(ρ _x )

^{is an entropy} sub-solution of (3.6).

Sine

κ ⁰ _x ≥ G(ρ ⁰ _x ),

^{a.e. in}

R ,

we an applythe Comparison Theorem 2.16to getthedesiredresult.

2

It is worth notable here that we do not know how to obtain the lower-bound on the spatial

gradient

κ _x

^{using thevisosity framework diretly. However, for thease ofthe} upper-bound, we an doso(see Remark4.1). Atthis stage,xsome

ǫ > 0

^{,and let}

G ^ǫ (x) = p

x ² + ǫ ²

^and

a = G ^ǫ (0) = ǫ.

It islear that

G ^ǫ (x)

^{satisestheonditions (G1)-(G2) with}

H ^ǫ (x) = x,

and theHamiltonian

F

^{from (3.2) takesnowthefollowing shape:}

F _ǫ (x, t, u) = − ρ _x (x, t)ρ _xx (x, t)f _ǫ (u).

^(3.10)

Moreover,wehavethefollowingorollarywhihisisanimmediateonsequeneofPropositions

3.1and 3.2.

Corollary 3.5 There existsa unique visosity solution

κ ∈ Lip( ¯ Q _T )

^of

( κ _t + F _ǫ (x, t, κ _x ) = 0

ⁱⁿ

Q _T ,

κ(x, 0) = κ ⁰ ∈ Lip( R )

ⁱⁿ

R ,

(3.11)