Diagonal hyperbolic systems with large and monotone data Part I: Global continuous solutions

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Diagonal hyperbolic systems with large and monotone data Part I: Global continuous solutions

Ahmad El Hajj, Régis Monneau

To cite this version:

Ahmad El Hajj, Régis Monneau. Diagonal hyperbolic systems with large and monotone data Part I:

Global continuous solutions. Journal of Hyperbolic Differential Equations, World Scientific Publishing, 2010, 7 (1), pp.1-26. �10.1142/S0219891610002050�. �hal-00375045�

large and monotone data

Part I: Global ontinuous solutions

A. El Hajj

1

, R. Monneau

2

April 11, 2009

Abstrat

In this paper, we study diagonal hyperboli systems inone spae dimension. Based on anew

gradiententropyestimate,we provetheglobalexisteneofaontinuoussolution, forlargeand

non-dereasing initial data. We remark thatthese results overthe aseof systems whih are

hyperboli but not stritly hyperboli. Physially, this kind of diagonal hyperboli systems

appearsnaturally inthemodelling ofthedynamis of disloation densities.

AMS Classiation: 35L45, 35Q35, 35Q72, 74H25.

Key words: Global existene, system of Burgers equations, systemof non-linear transport

equations,non-linear hyperboli system,dynamis of disloation densities.

1 Introdution and main result

1.1 Setting of the problem

Inthispaperweareinterestedinontinuoussolutionstohyperbolisystemsindimension

one. Our workwillfous onsolutionsu(t, x) = (uⁱ(t, x))i=1,...,d^,where d^{is aninteger,of}

hyperboli systems whihare diagonal, i.e.

∂tuⁱ+λⁱ(u)∂xuⁱ = 0 ^on (0,+∞)×R, ^for i= 1, . . . , d, ^(1.1)

with the initialdata:

uⁱ(0, x) =uⁱ₀(x), x∈R^{, for} i= 1, . . . , d. ^(1.2)

1

Universitéd'Orléans,LaboratoireMAPMO,RoutedeChartres,45000Orléansedex2,Frane

2

Éole Nationale des Pontset Chaussées, CERMICS, 6et 8 avenue Blaise Pasal, Cité Desartes

Champs-sur-Marne,77455Marne-la-ValléeCedex2,Frane

Here ∂t= ∂

∂t ^and ∂x = ∂

∂x^{. Suh systems are} (sometimes) alled (d× d) ^hyperboli

systems. Our study of system (1.1) is motivated by onsideration of models desribing

the dynamis of disloationdensities (see the Appendix, Setion 5),whihis

∂tuⁱ+ X

j=1,...,d

Aiju^j

!

∂xuⁱ = 0 ^for i= 1, . . . , d,

where (Aij)i,j=1,...,d ^{is a} non-negative symmetri matrix. This model an be seen as a speial ase of system (1.1).

Forreal numbers αⁱ ≤β^{i,let usonsider the box}

U = Π^d_i=1[αⁱ, βⁱ]. ^(1.3)

We onsider a given funtion λ = (λⁱ)_i=1,...,d : U → R^{d, whih satises the following}

regularity assumption:

(H1)























the funtionλ∈C^∞(U)^,

there exists M0 >0 ^{suh that for} i= 1, ..., d,

|λⁱ(u)| ≤M₀ ^{for all} u∈U,

there exists M1 >0 ^{suh that for} i= 1, ..., d,

|λⁱ(v)−λⁱ(u)| ≤M₁|v−u| ^{for all} v, u∈U,

where|w|= X

i=1,...,d

|wⁱ|^,forw= (w¹, . . . , w^d)^{. Given anyBanahspae}(E,k · k^E)^,inthe

rest of the paper weonsider the norm onE^d: kwk^Ed = X

i=1,...,d

kwⁱk^E, ^for w= (w¹, . . . , w^d)∈E^d.

Weassume, for allu∈R^{d, that the matrix}

(λⁱ_,j(u))i,j=1,...,d^{, where} λⁱ_,j = ∂λⁱ

∂u^j,

is non-negative inthe positive one,namely

(H2)

for all u∈U, ^{we have} X

i,j=1,...,d

ξiξjλⁱ_,j(u)≥0 ^{for every} ξ = (ξ1, ..., ξd)∈[0,+∞)^d.

In (1.2), eah omponent uⁱ₀ ^{of the initialdata} u0 = (u¹₀, . . . , u^d₀) ^{is assumed satisfy the}

followingproperty:

(H3)















αⁱ ≤uⁱ₀ ≤β^i,

uⁱ₀ ^is non-dereasing,

∂xuⁱ₀ ∈LlogL(R)^,

for i= 1, . . . , d^,

where LlogL(R) ^{isthe following Zygmund spae:}

LlogL(R) =

f ∈ L¹(R) ^{suh that} Z

R

|f|ln (e+|f|)<+∞

.

This spae is equipped by the followingnorm:

kfk^LlogL(R) = inf

µ >0 : Z

R

|f| µ ln

e+ |f| µ

≤1

,

This norm isdue toLuxemburg (see Adams [1,(13), Page 234℄).

Ourpurposeistoshowtheexisteneofaontinuoussolutionu= (u¹, . . . , u^d)^suhthat,

for i= 1, . . . , d^{,the funtion}uⁱ(t,·)^satises (H3)^{for alltime.}

1.2 Main result

Itiswell-knownthatforthelassialsalarBurgersequation∂tu+∂x

u² 2

= 0^,theso-

lutionstaysontinuouswhentheinitialdataisLipshitz-ontinuousandnon-dereasing.

Wewantsomehowtogeneralizethisresulttotheaseofdiagonalhyperbolisystems. In

partiular,we say thata funtionu0 = (u¹₀, . . . , u^d₀) ^isnon-dereasing ifeahomponent

uⁱ₀ ^is non-dereasing fori= 1, . . . , d^.

Theorem 1.1 (Global existene of a non-dereasing solution)

Assume (H1)^, (H2) ^and (H3)^{. Then, there exists a funtion} u ^{whih satises for all} T >0^:

i) Existene of a weak solution:

The funtion u ^{is solution of} (1.1)-(1.2), where

u∈[L^∞((0,+∞)×R)]^d∩[C([0,+∞);LlogL(R))]^{d and} ∂xu∈[L^∞((0, T);LlogL(R))]^d,

suh that for a.e. t ∈[0, T) ^{the funtion} u(t,·) ^is non-dereasing in x ^{and satises the}

followingL^{∞ estimate:}

kuⁱ(t,·)k^L∞(R)≤ kuⁱ₀k^L∞(R), ^for i= 1, . . . , d, ^(1.4)

Z

R

X

i=1,...,d

f ∂xuⁱ(t, x) dx+

Z t 0

Z

R

X

i,j=1,...,d

λⁱ_,j(u)∂xuⁱ(s, x)∂xu^j(s, x)dx ds≤C1, (1.5)

where

0≤f(x) =

xln(x) + ¹_e ^if x≥1/e,

0 ^if 0≤x≤1/e, ^(1.6)

and C1 T, d, M1,ku0k[L^∞(R)]^d,k∂xu0k[LlogL(R)]^d

.

ii) Continuity of the solution:

Thesolutionu^{onstrutedin(i)belongsto}[C([0,+∞)×R)]^{dandthereexistsamodulus}

of ontinuity ω(δ, h)^{, suh that for all} δ, h ≥0 ^{and all} (t, x)∈(0, T −δ)×R^{, wehave:}

|u(t+δ, x+h)−u(t, x)| ≤C2 ω(δ, h) ^with ω(δ, h) = 1

ln(¹_δ + 1) + 1

ln(¹_h+ 1), ^(1.7)

where C2 T, d, M0, M1,ku0k[L^∞(R)]^d,k∂xu0k[LlogL(R)]^d

.

The key point toestablishTheorem 1.1 isthe gradient entropy estimate(1.5). We rst

onsider the paraboliregularization of the system (1.1) and we show that the smooth

solution admits the L^{∞ bound (1.4) and the} fundamental gradient entropy inequality (1.5). Then, these a priori estimates will allow us to pass to the limit when the regu-

larizationvanishes, whih will provide the existene of a solution. Letus mention that

a similar gradient entropy inequality was introduedin Cannone etal. [5℄ to prove the

existeneof a solutionof atwo-dimensionalsystem of two oupledtransportequations.

Remark 1.2 Remark that assumption (H2) ^{implies that the seond term on the left}

hand side of (1.5) isnon-negative. This will imply the LlogL ^{bound on the gradient of}

the solutions.

Up toour knowledge, the result stated inTheorem 1.1seems new. In relationwith our

result,weanite thepaperof Poupaud[24℄,wherearesult ofexisteneanduniqueness

of Lipshitz solutionsis proven for apartiular quasi-linearhyperbolisystem.

Hyperboli systems (1.1) in the ase d = 2 ^{are alled stritly hyperboli if and only if}

we have:

λ¹(u¹, u²)< λ²(u¹, u²). ^(1.8)

Inthis ase, aresultof Lax[19℄implies theexisteneof Lipshitz monotonesolutionsof

(1.1)-(1.2). This result wasalsoextended by Serre[25, VolII℄ inthe aseof (d×d)^rih

hyperboli systems (see also Subsetion 1.4 for more related referenes). Their results

are limited tothe ase of stritly hyperbolisystems. On the ontrary, inTheorem 1.1,

we do not assume that the hyperboli system is stritly hyperboli. See the following

remark for aquite detailedexample.

Condition(1.8)ontheeigenvaluesisnotrequiredinourframework(Theorem1.1). Here

isasimpleexampleofa(2×2)^{hyperbolibut notstritlyhyperbolisystem. We onsider}

solution u= (u¹, u²) ^of







∂tu¹+cos(u²)∂xu¹= 0,

∂tu²+u¹sin(u²)∂xu² = 0,

on (0,+∞)×R^{. (1.9)}

Assume:

i) u¹(−∞) = 1^, u¹(+∞) = 2 ^and ∂xu¹ ≥0^,

ii) u²(−∞) =−^π₂^, u²(+∞) = ^π₂ ^and ∂xu² ≥0^.

Here the eigenvaluesλ¹(u¹, u²) =cos(u²) ^andλ²(u¹, u²) =u¹sin(u²)^{ross eah other at}

the initial time (and indeed forany time). Nevertheless, we an ompute

(λⁱ_,j(u¹, u²))i,j=1,2 =

0 −sin(u²) sin(u²) u¹cos(u²)

,

whih satises (H2) ^(under assumptions (i) and (ii)). Therefore Theorem1.1 gives the existene of a solution to (1.9) with in partiular (i) and (ii).

Remark 1.4 (A generalization of Theorem 1.1)

In Theorem 1.1 we have onsidered a partiular system in order to simplify the presen-

tation. Our approah an be easily extended to the following generalized system:

∂tuⁱ+λⁱ(u, x, t)∂xuⁱ =hⁱ(u, x, t) ^on (0,+∞)×R, ^for i= 1, ..., d, ^(1.10)

with the following onditions:

- λⁱ ∈ W^1,∞(U × R×[0,+∞)) ^{and the matrix} (λⁱ_,j(u, x, t))i,j=1,...,d ^{is positive in the}

positive one for all (u, x, t)∈U ×R×[0,+∞) ^{(i.e. a ondition analogous to} (H2)^).

- hⁱ ∈W^1,∞(U ×R×[0,+∞))^, ∂xhⁱ ≥0 ^and hⁱ_,j ≥0 ^{for all}j 6=i^.

Let us remark that our system (1.1)-(1.2) does not reate shoks beause the solution

(given inTheorem 1.1) isontinuous. In this situation,it seems very naturalto expet

the uniqueness of the solution. Indeed the notionof entropy solution(in partiular de-

signed to deal with the disontinuities of weak solutions) does not seem so helpful in

this ontext. Even forsuh a simplesystem, we then ask the following:

Open question: Is there uniqueness of the ontinuous solution given in

Theorem 1.1 ?

this question.

1.3 Appliation to diagonalizable systems

Let us rst onsider a smooth funtion u = (u¹, . . . , u^d)^{, solution of the following non-}

onservative hyperbolisystem:







∂tu(t, x) +F(u)∂xu(t, x) = 0, u∈U, x∈R, t∈(0,+∞), u(x,0) = u0(x) x∈R,

(1.11)

wherethespaeofstatesU ^{isnowanopensubsetof}R^d,andforeahu^,F(u)^isa(d×d)^-

matrixand the mapF ^{is oflass} C¹(U)^{. Thesystem (1.11)issaid} (d×d)^hyperboli,if F(u) ^has d ^real eigenvalues and is diagonalizable for any given u ^{on the domain under}

onsideration. Bydenition, suhasystem is said tobediagonalizable,if thereexists a

smooth transformation w = (w¹(u), . . . , w^d(u)) ^with non-vanishing Jaobian suh that (1.11) an be equivalently rewritten (for smooth solutions)as the following system

∂twⁱ+λⁱ(w)∂xwⁱ = 0 ^for i= 1, . . . , d,

where λ^{i are smooth funtions of} w^{. Suh funtions} w^{i are alled strit} i^-Riemann

invariant.

Our approahan give ontinuous solutionstothe diagonalizedsystem, whihprovided

ontinuous solutionto the originalsystem (1.11).

1.4 A brief review of some related literature

For a salar onservation law, whih orresponds to system (1.11) in the ase d = 1

where F(u) = h^′(u) ^{is the derivative of some ux funtion} h^{, the global existene and}

uniqueness ofBV ^{solutionshas been} establishedbyOleinik[23℄ inonespaedimension.

The famous paper of Kruzhkov [18℄ overs the more general lass of L^{∞ solutions, in}

several spae dimensions. For an alternative approah based on the notion of entropy

proess solutions,seefor instane Eymardetal. [12℄. Fora dierentapproah basedon

a kineti formulation,see alsoLions etal. [22℄.

We now reall some well-known results for a lass of (2×2) ^{stritly hyperboli sys-}

tems in one spae dimension. This means that F(u) ^{has two real, distint} eigenvalues satisfying (1.8). As mentioned above, Lax [19℄ proved the existene and uniqueness of

non-dereasingandsmoothsolutionsfordiagonalized(2×2)^{stritlyhyperbolisystems.}

Inthe ase ofsome (2×2)^{stritlyhyperbolisystems,DiPerna [6,7℄showed the global}

existene of aL^{∞ solution. The proof of DiPerna relies ona ompensated ompatness}

argument, based on the representation of the weak limit in terms of Young measures,

whih must redue to a Dira mass due to the presene of a large family of entropies.

For general (d ×d) ^{stritly hyperboli systems; i.e. where} F(u) ^has d ^{real, distint}

eigenvalues

λ¹(u)<· · ·< λ^d(u), ^(1.12)

Bianhini and Bressan proved in a very omplete paper [3℄, a striking global existene

and uniqueness result of solutions to system (1.11), assuming that the initialdata has

smalltotal variation. This approahismainlybasedona arefulanalysis ofthe vanish-

ingvisosityapproximation. An existeneresult has rst been proved by Glimm[15℄ in

thespeialaseofonservativeequations,i.e. F(u) =Dh(u)^{istheJaobianofsomeux}

funtion h^{. Letus mention that an existene result has been also obtained by LeFloh}

and Liu[20, 21℄ in the non-onservative ase.

Weanalsomentionthat,oursystem(1.1)isrelatedtoothersimilarmodelsindimension

N ≥1^{,suhassalartransportequationsbasedonvetoreldswithlow}regularity. Suh equationswereforinstanestudiedbyDipernaandLionsin[8℄. Theyhaveprovedtheex-

istene(anduniqueness) ofasolution(inthe renormalizedsense),forgivenvetorelds

inL¹((0,+∞);W_loc^1,1(R^N))^{whosedivergene isin}L¹((0,+∞);L^∞(R^N))^{. This studywas}

generalized by Ambrosio [2℄, who onsidered vetor elds in L¹((0,+∞);BVloc(R^N))

with bounded divergene. Inthe present paper, wework indimensionN = 1^{and prove}

the existene (and some uniqueness results) of solutions of the system (1.1)-(1.2) with

a veloity vetor eld λⁱ(u)^, i = 1, . . . , d^{. Here, in Theorem 1.1, the divergene of our}

vetor eld is only in L^∞((0,+∞), LlogL(R))^{. In this ase we proved the existene}

result thanks to the gradient entropy estimate (1.5), whih gives a better estimate on

the solution.

Let us also mention that for hyperboli and symmetri systems in dimension N ≥ 1^,

Ga

◦

rdinghasprovedin[13℄aloalexisteneanduniquenessresultinC([0, T);H^s(R^N))∩ C¹([0, T);H^s−1(R^N))^{, with} s > ^N₂ + 1 ^{(see also Serre [25, Vol I, Th 3.6.1℄), this result}

being only loalin time, even indimension N = 1^.

1.5 Organization of the paper

This paper is organized as follows: in Setion 2, we approximate the system (1.1), by

adding the visosity term (ε∂xxu^{i). Then we show a global in time existene for this}

approximated system. Moreover, we show that these solutions are regular and non-

dereasing in x ^{for all} t > 0^{. In Setion 3, we prove the gradient entropy inequality}

and some other ε^{-uniform a priori estimates. In Setion 4, we prove the main result}

(Theorem 1.1) passing to the limitas ε ^{goes to}0^{. Finally,in the appendix (Setion 5),}

we derive amodelfor the dynamis of disloationdensities.

The system (1.1) an bewritten as:

∂tu+λ(u)⋄∂xu= 0, ^(2.13)

where u := (uⁱ)1,...,d^, λ(u) = (λⁱ(u))1,...,d ^and u⋄ v ^{is the omponent by omponent}

produt of the two vetors u = (u¹, . . . , u^d), v = (v¹, . . . , v^d) ∈ R^{d. This is the vetor}

in R^{d whose oordinates are given by} (u⋄v)ⁱ := uⁱv^{i. We now onsider the following}

paraboliregularization of system (2.13),for all 0< ε≤1^:







∂tu^ε+λ(u^ε)⋄∂xu^ε =ε∂xxu^ε

u^ε(x,0) =u^ε₀(x), ^with u^ε₀(x) :=u0∗ηε(x)^,

(2.14)

where ∂xx = ∂²

∂x^{2 and} ηε ^{is a mollier verify,} ηε(·) = ¹_εη(_ε^·)^{, suh that} η ∈ C_c^∞(R) ^{is a}

non-negativefuntion satisfying

R

Rη= 1^.

Remark 2.1 By lassial properties of the mollier (ηε)ε ^{and the fat that} u^ε₀ ∈ [L^∞(R)]^{d, then} u0 ∈ [C^∞(R)]^d∩[W^2,∞(R)]^{d. Moreover using the} non-negativity of ηε^,

the seond equation of (2.14) we get that

ku^ε,i₀ k^L∞(R)≤ kuⁱ₀k^L∞(R), ^for i= 1, . . . , d^,

and (H3) ^{alsoimplies that}u^ε₀ ^is non-dereasing.

The followingtheorem is a globalexistene result for the regularized system (2.14).

Theorem 2.2 (Global existene of non-dereasing smooth solutions)

Assume(H1)^{andthattheinitial data}u^ε₀ ^isnon-dereasingandsatisesu^ε₀ ∈[C^∞(R)]^d∩ [W^2,∞(R)]^{d. Then the system (2.14), admits a solution} u^ε ∈ [C^∞([0,+∞) ×R)]^d∩ [W^2,∞((0,+∞)× R)]^{d suh that the funtion} u^ε(t,·) ^is non-dereasing for all t > 0^.

Moreover, for allt > 0^{, we have the a prioribounds:}

ku^ε,i(t,·)k^L∞(R)≤ ku^ε,i₀ k^L∞(R), ^for i= 1, . . . , d, ^(2.15) ∂xu^ε,i

L^∞([0,+∞);L¹(R)) ≤2ku^ε,i₀ k^L∞(R), ^for i= 1, . . . , d. ^(2.16)

The lines of the proof of this theorem are very standard (see for instane Cannone et

al. [5℄for a similar problem). Forthis reason,we skip the details of the proof. First of

allweremark that the estimate(2.15)isa diretappliationof the Maximum Priniple

Theorem for paraboli equations (see Gilbarg-Trudinger [14, Th.3.1℄). The regularity

of the solution follows from a bootstrap argument. The monotoniity of the solution

is a onsequene of the maximum priniple for salar paraboli equations applied to

w^ε =∂xu^{ε satisfying}

∂tw^ε+λ(u^ε)⋄∂xw^ε+∂x(λ(u^ε))⋄w^ε=ε∂xxw^ε.

Sine ∂xu^ε ≥0 ^{this implies easilythe seond estimate (2.16).}

3

ε

In this setion, weshow some ε^{-uniform estimates onthe solutionsof system (2.14).}

Before going into the proof of the gradient entropy inequality dened in (1.5), we an-

noune the main idea to establishthis estimate. Now, let us set for w≥ 0 ^{the entropy}

funtion

f¯(w) =wlnw.

For any non-negative test funtion ϕ ∈ C_c¹([0,+∞)× R)^{, let us dene the following}

gradient entropy with wⁱ :=∂xu^i: S(t) =¯

Z

R

ϕ(t,·) X

i=1,...,d

f¯(wⁱ(t,·))

! dx.

It is very natural to introdue suh quantity S(t)¯ ^{whih in the ase} ϕ ≡ 1^{, appears}

to be nothing else than the total entropy of the system of d ^{type of partiles of non-}

negative densities wⁱ ≥ 0^{. Then after two} integration by parts, it is formally possible todedue from(1.1)theequalityinthefollowinggradient entropy inequalityforallt ≥0

dS(t)¯ dt +

Z

R

ϕ X

i,j=1,...,d

λⁱ_,jwⁱw^j

!

dx≤R(t), ^for t≥0, ^(3.17)

with the rest

R(t) = Z

R

(

(∂tϕ) X

i=1,...,d

f(w¯ ⁱ)

!

+ (∂xϕ) X

i=1,...,d

λⁱf(w¯ ⁱ)

!) dx,

where we do not show the dependene on t ^{in the integrals. We remark in partiular}

that this rest isformallyequal tozero if ϕ ≡1^.

To guarantee the existene of ontinuous solutions when ε = 0^{, we will assume later} (H2) ^{whih guarantees the} non-negativity on the seond term of the left hand side of inequality(3.17).

Comingbak to arigorous statement, we willprovethe following result.

Proposition 3.1 (Gradient entropy inequality)

Assume (H1) ^{and onsider a funtion} u0 ∈ [L^∞(R)]^{d satisfying} (H3)^{. For any}

0 < ε ≤ 1^{, we onsider the solution} u^{ε of the system (2.14) given in Theorem}

2.2 with initial data u^ε₀ = u0 ∗ ηε^{. Then for any} T > 0^{, there exists a onstant} C T, d, M1,ku0k[L^∞(R)]^d,k∂xu0k[LlogL(R)]^d

suh that

S(t) + Z t

0

Z

R

X

i,j=1,...,d

λⁱ_,j(u^ε)w^ε,iw^ε,j ≤C, ^with S(t) = Z

R

X

i=1,...,d

f(w^ε,i(t,·))dx. ^(3.18)

where f ^{is dened in (1.6) and} w^ε = (w^ε,i)i=1,...,d =∂xu^ε.

Forthe proofof Proposition 3.1, we need the followingtehnial lemma:

Lemma 3.2 (LlogL ^estimate)

Let (ηε)ε∈(0,1] ^{be a} non-negative mollier satisfying

R

Rηε = 1^{, let} f ^{be the funtion de-}

ned in (1.6) and h∈L¹(R)^{be a} non-negative funtion. Then i)

Z

R

f(h)<+∞^ifandonlyifh∈LlogL(R)^{. Moreoverwehavethefollowingestimates:}

Z

R

f(h)≤1 +khk^LlogL(R)+khk^L1(R)ln 1 +khk^LlogL(R)

, ^(3.19)

khk^LlogL(R)≤1 + Z

R

f(h) + ln(1 +e²)khk^L1(R). ^(3.20)

ii) If h∈LlogL(R)^{, then for every} ε∈ (0,1] ^{the funtion} hε =h∗ηε∈LlogL(R) ^and

satises

khεk^LlogL(R) ≤Ckhk^{LlogL(R) and} kh−hεk^LlogL(R) →0 ^as ε →0,

where C ^{is a universal onstant.}

Proof of Lemma 3.2:

The proof of (i) is trivial. To prove estimate (3.19), we rst remark that, for allh ≥ 0

and µ∈(0,1]^{,we have}

hln(h) + 1 e

11_{h≥¹

e} ≤hln(h+e)≤hln(e+µh) +|ln(µ)|h.

Weapply this inequality with µ= 1

max(1,khk^LlogL(R)) ^{and integrate,weget} Z

R

f(h) ≤ 1 µ

Z

R

µhln(e+µh) +|ln(µ)|khk^L1(R)

≤ 1

µ+|ln(µ)|khk^L1(R),