On decay of periodic entropy solutions to a scalar conservation law

(1)

Ann. I. H. Poincaré – AN 30 (2013) 997–1007

www.elsevier.com/locate/anihpc

On decay of periodic entropy solutions to a scalar conservation law

^✩

E.Yu. Panov

Novgorod State University, 41, B. St-Peterburgskaya, 173003 Veliky Novgorod, Russia Received 6 March 2012; accepted 21 December 2012

Available online 11 January 2013

Abstract

We establish a necessary and sufficient condition for decay of periodic entropy solutions to a multidimensional conservation law with merely continuous flux vector.

Résumé

Nous considérons les lois de conservation [hyperboliques] en plusieurs dimensions d’espace avec la fonction de flux seulement continue. Nous établissons une condition nécessaire et suffisante pour la décroissance des solutions entropiques périodiques de ce problème.

MSC:35L65; 35B10; 35B35

Keywords:Conservation laws; Periodic entropy solutions; Decay property;H-measures

1. Introduction

In the half-spaceΠ=R+×Rⁿ,R+=(0,+∞), we consider a first order multidimensional conservation law

u_t+divxϕ(u)=0, (1.1)

where the flux vector ϕ(u)is supposed to be only continuous: ϕ(u)=(ϕ₁(u), . . . , ϕ_n(u))∈C(R,Rⁿ). Recall the notion of entropy solution of (1.1) in the sense of S.N. Kruzhkov[4].

Definition 1. A bounded measurable function u=u(t, x)∈L^∞(Π ) is called an entropy solution (e.s. for short) of (1.1) if for allk∈R

|u−k|t+divx

sign(u−k)

ϕ(u)−ϕ(k)

0 (1.2)

in the sense of distributions onΠ(inD(Π )).

✩ This research was carried out under financial support of the Russian Foundation for Basic Research (grant No. 12-01-00230-a).

E-mail address:Eugeny.Panov@novsu.ru.

http://dx.doi.org/10.1016/j.anihpc.2012.12.009

(2)

Condition (1.2) means that for all non-negative test functionsf =f (t, x)∈C₀¹(Π )

Π

|u−k|f_t+sign(u−k)

ϕ(u)−ϕ(k)

· ∇xf

dt dx0 (here “·” denotes the inner product inRⁿ).

As was shown in [13](see also[14]), an e.s.u(t, x)always admits a strong trace u0=u0(x)∈L^∞(Rⁿ)on the initial hyperspacet=0 in the sense of relation

ess lim

t→0 u(t,·)=u0 inL¹_loc Rⁿ

, (1.3)

that is,u(t, x)is an e.s. to the Cauchy problem for Eq. (1.1) with initial data

u(0, x)=u₀(x). (1.4)

Remark 1.1.It was also established in[13, Corollary 7.1]that, after possible correction on a set of null measure, an e.s.u(t, x)is continuous onR+as a mapt→u(t,·)ofR+intoL¹_loc(Rⁿ).

When the flux vector is Lipschitz continuous, the existence and uniqueness of e.s. to the problem (1.1), (1.4) are well-known (see[4]). In the case under consideration when the flux functions are merely continuous, the effect of infinite speed of propagation for initial perturbations appears, which leads even to the nonuniqueness of e.s. to problem (1.1), (1.4) ifn >1 (see examples in[5,6]).

But, if initial function is periodic (at least inn−1 independent directions), the uniqueness holds: an e.s. of (1.1), (1.4) is unique and space-periodic, see the proof in[11,12]. In the present paper we assume that the requirement of space-periodicity holds:u(t, x+e_i)=u(t, x)for almost all(t, x)∈Π and alli=1, . . . , n, where{e_i}ⁿ_i₌₁is a basis of periods inRⁿ. We assume that this basis is fixed. Then, without loss of generality, we may suppose that{e_i}ⁿ_i₌₁ is the canonical basis. We denote byP= [0,1)ⁿthe corresponding fundamental parallelepiped (cube) (which can be identified with a torus).

As was established by G.-Q. Chen and H. Frid[1], under the conditionsϕ(u)∈C²(R,Rⁿ)and

∀(τ, ξ )∈Rⁿ⁺¹, (τ, ξ ) =0, meas

u∈Rτ +ϕ(u)·ξ =0 =0, (1.5)

the following decay result holds for space-periodic e.s.u(t, x)of (1.1), (1.4):

ess lim

t→∞ u(t,·)=const= 1

|P|

P

u₀(x) dx inL¹(P ). (1.6)

Here |P| denotes the Lebesgue measure ofP (in the case under consideration P is a unite cube and, therefore,

|P| =1).

Definition 2.We will say that Eq. (1.1) satisfiesthe decay propertyif (1.6) holds for every periodic e.s.

In the present paper we propose the following necessary and sufficient condition for the decay property (byZwe denote the set of integers)

∀(τ, ξ )∈R×Zⁿ, (τ, ξ ) =0, the functionu→τ u+ϕ(u)·ξ is not constant on non-empty intervals, (1.7) in the general case of only continuous flux vectorϕ(u). Obviously, this condition is equivalent to the requirement that

∀ξ∈Zⁿ,ξ =0,the functionsu→ϕ(u)·ξ are not affine on non-empty intervals.

Thus, our main result is the following theorem.

Theorem 1.1.Eq.(1.1)satisfies the decay property if and only if condition(1.7)holds.

Remark 1.2.In the case of arbitrary basis of periods{e_i}ⁿ_i₌₁one can make the linear change of variablesx=x(y)= _n

i=1y_ie_i. Then, as is easily verified, ifu(t, x) is a periodic entropy solution of (1.1) then the functionv(t, y)= u(t, x(y))is an entropy solution of the equation

(3)

v_t+divyϕ(v)˜ =0,

whereϕ˜_j(v)=ϕ(v)·e_j,{e_j}ⁿ_j₌₁being the dual basis:

e_j·e_i=δ_ij=

1, i=j, 0, i =j.

Clearly,v(t, y)is ay-periodic function with the canonical basis of periods. By Theorem1.1the necessary and sufficient condition forv(t, y)to satisfy the decay property is condition (1.7):

∀η∈Zⁿ, η =0, the functionv→ ˜ϕ(v)·ηis not affine on non-empty intervals.

Observe thatϕ(v)˜ ·η=ϕ(v)·ξ, whereξ=n

j=1ηje_j ∈LandL= {n

j=1ηje_j |ηj∈Z}is the dual lattice to the lattice of periodsL= {_n

i=1ξ_ie_i|ξ_i ∈Z}. It is obvious that the decay property forv(t, y)holds if and only if this property holds for original solutionu(t, x).

Therefore, in the case of arbitrary basis of periods, Theorem1.1remains valid after replacement of the latticeZⁿ in (1.7) byL:

∀ξ∈L, ξ =0, the functionu→ϕ(u)·ξ is not affine on non-empty intervals.

By Remark1.2in the case when the basis of periods is not fixed and may depend on a solution, the statement of Theorem1.1remains valid after replacement of condition (1.7) by the following stronger one:

∀(τ, ξ )∈R×Rⁿ, (τ, ξ ) =0, the functionu→τ u+ϕ(u)·ξ is not constant on non-empty intervals. (1.8) Obviously, condition (1.8) is strictly weaker than (1.5) even in the case of smooth fluxϕ(u).

2. Preliminaries

To prove Theorem1.1, we use, as in[1], the strong pre-compactness property for the self-similar scaling sequence u(kt, kx),k∈N. This pre-compactness property will be obtained under condition (1.7) with the help of localization principles forH-measures with “continuous indexes”, introduced in[8]. The strong pre-compactness property for arbitrary sequences of e.s. of (1.1) under exact non-degeneracy condition (1.8) was derived in[9](see also[10,15]

for the case of general flux vectorϕ=ϕ(t, x, u)). In the present paper we take into account the periodicity condition, which allows to refine the localization principle.

First, we recall the original concept ofH-measure introduced by L. Tartar[17]and P. Gerárd[3]. LetF (u)(ξ ), ξ ∈R^N, be the Fourier transform of a functionu(x)∈L²(R^N),S=S^N⁻¹= {ξ∈R^N| |ξ| =1}be the unit sphere inR^N. Denote byu→u,u∈Cthe complex conjugation.

LetΩ be an open domain inR^N, l∈N, and let U_k(x)=(U_k¹(x), . . . , U_k^l(x))∈L²_loc(Ω,R^l)be a sequence of vector-functions weakly convergent to the zero vector.

Proposition 2.1.(See Theorem 1.1 in[17].) There exist a family of complex Borel measuresμ= {μ^ij}^l_i,j₌₁inΩ×S and a subsequenceU_r(x)=U_k(x),k=k_r, such that

μ^ij, Φ1(x)Φ2(x)ψ (ξ )

= lim

r→∞

R^N

F Φ1U_rⁱ

(ξ )F Φ2U_r^j

(ξ )ψ ξ

|ξ|

dξ (2.1)

for allΦ₁(x), Φ₂(x)∈C₀(Ω)andψ (ξ )∈C(S).

The familyμ= {μ^ij}^l_i,j₌₁is called theH-measure corresponding toU_r(x).

Remark 2.1.In the case when the sequenceU_k(x)is bounded inL^∞(Ω)it follows from (2.1) and the Plancherel identity that pr_x|μ^pq|Cmeas, and that (2.1) remains valid for allΦ₁(x), Φ₂(x)∈L²(Ω), cf.[15, Remark 2(a)].

Here we denote by|μ|the variation of measureμ(it is a non-negative measure), and by meas the Lebesgue measure onΩ.

(4)

We need also the concept of measure-valued functions (Young measures). LetΩ⊂R^Nbe an open domain. Recall (see[2,16]) that a measure-valued function onΩis a weakly measurable mapx→ν_xofΩ into the space Prob0(R) of probability Borel measures with compact support inR.

The weak measurability of ν_x means that for each continuous function g(λ) the function x → ν_x, g(λ) = g(λ) dν_x(λ)is measurable onΩ.

Measure-valued functions of the kindν_x(λ)=δ(λ−u(x)), whereu(x)∈L^∞(Ω)andδ(λ−u^∗)is the Dirac measure atu^∗∈R, are calledregular. We identify these measure-valued functions and the corresponding functionsu(x), so that there is a natural embedding ofL^∞(Ω)into the set MV(Ω)of measure-valued functions onΩ.

Measure-valued functions naturally arise as weak limits of bounded sequences in L^∞(Π )in the sense of the following theorem by L. Tartar[16].

Theorem 2.1.Letu_m(x)∈L^∞(Ω),m∈N, be a bounded sequence. Then there exist a subsequence (we keep the notationu_m(x)for this subsequence)and a measure-valued functionν_x∈MV(Ω)such that

∀g(λ)∈C(R) g(u_m) →

m→∞

ν_x, g(λ)

weakly-∗ inL^∞(Ω). (2.2)

Besides,ν_xis regular, i.e.,ν_x(λ)=δ(λ−u(x))if and only ifu_m(x)→m→∞u(x)inL¹_loc(Ω)(strongly).

In[8]the new concept ofH-measures with “continuous indexes” was introduced, corresponding to sequences of measure-valued functions. We describe this concept in the particular case of “usual” sequences inL^∞(Ω). Letu_m(x) be a bounded sequence inL^∞(Ω). Passing to a subsequence if necessary, we can suppose that this sequence converges to a measure-valued functionν_x∈MV(Ω)in the sense of relation (2.2). We introduce the measuresγ_x^m(λ)=δ(λ− u_m(x))−ν_x(λ)and the corresponding distribution functionsU_m(x, p)=γ_x^m((p,+∞)),u₀(x, p)=ν_x((p,+∞))on Ω×R. Observe thatU_m(x, p), u₀(x, p)∈L^∞(Ω)for allp∈R, see[8, Lemma 2]. We define the set

E=E(ν_x)=

p₀∈Ru₀(x, p) →

p→p0

u₀(x, p₀)inL¹_loc(Ω)

.

As was shown in[8, Lemma 4], the complementR\Eis at most countable and ifp∈EthenU_m(x, p) _m_→∞ 0 weakly-∗inL^∞(Ω).

The next result, similar to Proposition2.1, has been established in[8, Theorem 3],[10, Proposition 2, Lemma 2].

Proposition 2.2.

(1) There exist a family of locally finite complex Borel measures{μ^pq}p,q∈EinΩ×Sand a subsequenceU_r(x, p)= Um_r(x, p)such that for allΦ1(x), Φ2(x)∈C0(Ω)andψ (ξ )∈C(S)

μ^pq, Φ₁(x)Φ₂(x)ψ (ξ )

= lim

r→∞

R^N

F

Φ₁U_r(·, p) (ξ )F

Φ₂U_r(·, q) (ξ )ψ

ξ

|ξ|

dξ; (2.3)

(2) The correspondence(p, q)→μ^pq is a continuous map fromE×Einto the spaceMloc(Ω×S)of locally finite Borel measures onΩ×S(with the standard locally convex topology);

(3) For anyp1, . . . , pl∈Ethe matrix{μ^pⁱ^p^j}^l_i,j₌₁is Hermitian and positive semidefinite, that is, for allζ1, . . . , ζl∈ Cthe measure

l

i,j=1

μ^pⁱ^p^jζ_iζ_j0.

Notice that assertion (3) readily follows from relation (2.3).

We call the family of measures{μ^pq}p,q∈EtheH-measure corresponding to the subsequenceu_r(x)=u_m_r(x).

As was demonstrated in[8], theH-measureμ^pq=0 for allp, q∈Eif and only if the subsequenceu_r(x)converges asr→ ∞strongly (inL¹_loc(Ω)). Observe also that assertion (3) in Proposition2.2implies that measuresμ^pp0 for allp∈E, and that

(5)

μ^pq(A)

μ^pp(A)μ^qq(A) (2.4)

for any Borel set A ⊂Ω ×S and all p, q ∈ E. Indeed, this directly follows from the fact that the matrix μ^pp(A) μ^pq(A)

μ^qp(A) μ^qq(A)

is Hermitian and positive semidefinite.

3. Main results

We fix a periodic e.s.u=u(t, x)of (1.1). Without loss of generality, we may assume thatu(t,·)∈C(R₊, L¹(P )) (see Remark1.1above).

Lemma 3.1.Lets(u)be a Lipschitz function,v(t, x)=s(u(t, x)), and

v(t, x)=

κ∈Zⁿ

a_κ(t)e^{2π iκ}^·^x

be the Fourier series ofv(t,·)inL²(P ), so thataκ(t)=

Pe⁻^{2π iκ}^·^xv(t, x) dx. Then this series converges tov(t,·)in L²(P )uniformly with respect tot, that is, for eachε >0there exists a valueN∈Nsuch that

|κ|>N

a_κ(t)²< ε² ∀t >0. (3.1)

Proof. Let u₀(x)∈L^∞(Rⁿ)be a strong trace of u(t, x) on the initial hyper-plane t=0 (recall that its existence follows from the results of[13,14]). Obviously,u₀(x)is a periodic function. Since for eachh∈Rⁿ u(t, x+h)is a periodic e.s. of the Cauchy problem for (1.1) with initial datau₀(x+h)then for allt >0

P

v(t, x+h)−v(t, x)²dx2L²u∞

P

u(t, x+h)−u(t, x)dx 2L²u_∞

P

u₀(x+h)−u₀(x)dx (3.2)

by the L¹-contraction property, see for example [7, Corollary 3.3]. Here L is a Lipschitz constant of s(u), i.e.,

|s(u2)−s(u1)|L|u2−u1|for everyu1, u2∈R.

In view of (3.2), the set of functionsF = {v(t,·)|t >0}is precompact inL²(P ). By Hausdorff’s compactness criterion there exists a finiteε/2-net{gk(x)}^m_k₌₁forF inL²(P ). Letbκ,k=

Pe⁻^{2π iκ}^·^xgk(x) dx,κ∈Zⁿ, be Fourier coefficients ofgk(x). Observe that

κ∈Zⁿ

|b_κ,k|²= g_k²_L2(P )<+∞. Therefore, there exists an integerNsuch that

|κ|>N

|bκ,k|²< ε²/4 (3.3)

for allk=1, . . . , m. Since{gk(x)}^m_k₌₁is anε/2-net forF then for eacht >0 one can find suchk∈ {1, . . . , m}that

κ∈Zⁿ

aκ(t)−bκ,k²=v(t,·)−gk²

L²(P )< ε²/4. (3.4)

In view of (3.3), (3.4) and Minkowski inequality we find

|κ|>N

a_κ(t)²1/2

|κ|>N

a_κ(t)−b_κ,k²1/2

+

|κ|>N

|b_κ,k|² 1/2

< ε,

and (3.1) follows. 2

(6)

Let v=v(t, x)∈L^∞(Π )be the function introduced in Lemma3.1. We consider the sequencev_k =v(kt, kx), k∈N, and the Tartar’sH-measureμˆ corresponding to the scalar sequencev_r −v^∗, wherev_r =v_k_r(t, x)is a subsequence ofv_k, andv^∗=v^∗(t, x)is a weak-∗limit ofv_r asr→ ∞inL^∞(Π ).

Lemma 3.2.

(i) The functionv^∗(t, x)=v^∗(t)does not depend onx; (ii) suppμˆ⊂Π×S0, where

S₀=ξ /ˆ |ˆξ| ∈Sξˆ=(τ, ξ ) =0, τ∈R, ξ∈Zⁿ . Proof. Form∈Nwe introduce the sets

Sm=ξ /ˆ |ˆξ| ∈Sξˆ=(τ, ξ ) =0, τ∈R, ξ∈Zⁿ, |ξ|m .

It is clear thatS_mis a closed subset of the sphereS(it is the union of the finite set of circles{(p, qξ /|ξ|)|p²+q²=1}, whereξ∈Zⁿ, 0<|ξ|m), andS₀=_∞

m=1S_m. Let v(t, x)=s

u(t, x)

=

κ∈Zⁿ

a_κ(t)e^{2π iκ}^·^x

be the Fourier series forv(t,·)inL²(P ). Then v_r(t, x)=v(k_rt, k_rx)=

κ∈Zⁿ

a_κ(k_rt )e^{2π ik}^r^κ^·^x. (3.5)

It follows from (3.5) that the functionv^∗(t, x)does not actually depend onx:v^∗(t, x)=v^∗(t), andv^∗(t)is the weak-∗

limit of the sequencea0(k_rt ),r∈N, inL^∞(R₊). Thus, statement (i) is proved.

We denoteb0,r=a0(k_rt )−v^∗(t);b_κ,r=a_κ(k_rt ), whereκ∈Zⁿ,κ =0. Letα(t )∈C0(R₊), andβ(x)∈L²(Rⁿ)∩ C^∞(Rⁿ)be such that its Fourier transform is a continuous compactly supported function:

β(ξ )˜ =

Rⁿ

e⁻^{2π iξ}^·^xβ(x) dx∈C₀ Rⁿ

. (3.6)

We takeR=max_ξ_∈_supp_β_˜|ξ|. LetΦ(t, x)=α(t )β(x). By (3.5) we find that v_r(t, x)−v^∗(t)

Φ(t, x)=

κ∈Zⁿ

b_κ,r(t)α(t)e^{2π ik}^r^κ^·^xβ(x). (3.7)

Observe that the Fourier transform ofe^{2π ik}^r^κ^·^xβ(x)inRⁿ coincides withβ(ξ˜ −k_rκ). Since fork_r>2R supports of these functions do not intersect, then for suchrthe series

κ∈Zⁿ

b_κ,r(t)α(t)β(ξ˜ −k_rκ) (3.8)

is orthogonal inL²(Rⁿ)for eacht >0. Besides, by the Plancherel equality ˜β(ξ−k_rκ)_L²₍_Rⁿ₎= ˜β2= β2, and

κ∈Zⁿ

b_κ,r(t)α(t)²β(ξ˜ −k_rκ)²

L²(Rⁿ)=α(t )²β²2

κ∈Zⁿ

b_κ,r(t)²

=α(t )²β²2·v(krt,·)−v^∗(t)²

L²(P )<+∞. Therefore, orthogonal series (3.8) converges inL²(Rⁿ)for eacht >0. Moreover, by Lemma3.1

κ∈Zⁿ,|κ|>N

b_κ,r(t)²=

κ∈Zⁿ,|κ|>N

a_κ(k_rt )² →

N→∞0

(7)

uniformly with respect to t >0. Hence, series (3.8) converges in L²(Rⁿ)uniformly with respect to t. Since the Fourier transformation is an isomorphism onL²(Rⁿ), we conclude that series (3.7) also converges in L²(Rⁿ)(not only inL²(P )) uniformly with respect tot. Sinceα(t )∈C₀(R), this implies that (3.7) converges inL²(Π ), and

F

(v_r−v)Φ(ˆξ )=

κ∈Zⁿ

F^t(αb_κ,r)(τ )β(ξ˜ −k_rκ), ξˆ=(τ, ξ ), (3.9) whereF^t(h)(τ )=

Re⁻^{2π iτ t}h(t) dtdenotes the Fourier transform over the time variable (we extend functionsh(t)∈ L²(R₊)on the whole lineR, settingh(t)=0 fort <0). It follows from (3.9) that fork_r>2R

Rⁿ⁺¹

F Φ

v_r−v^∗(ˆξ )²ψξ /ˆ |ˆξ| dξˆ

=

κ∈Zⁿ

Rⁿ⁺¹

F^t(αb_κ,r)(τ )²β(ξ˜ −k_rκ)²ψξ /ˆ |ˆξ|

dξ ,ˆ (3.10)

whereψ (ξ )ˆ ∈C(S)is arbitrary. Now we fixε >0. Recall thatb_κ,r=a_κ(k_rt )for κ =0, and by Lemma3.1there existsm∈Nsuch that

κ∈Zⁿ,|κ|>m

Rⁿ⁺¹

F^t(αb_κ,r)(τ )²β(ξ˜ −k_rκ)²dξˆ

=

κ∈Zⁿ,|κ|>m

Π

α(t )a_κ(k_rt )²β(x)²dt dx Φ²2·sup

t >0

κ∈Zⁿ,|κ|>m

aκ(t)²< ε. (3.11)

Now we suppose thatψ_∞1 andψ (ˆξ )=0 on the setS_m. By (3.11)

κ∈Zⁿ,|κ|>m

Rⁿ⁺¹

F^t(αb_κ,r)(τ )²β(ξ˜ −k_rκ)²ψξ /ˆ |ˆξ|dξˆε. (3.12) Since continuous function ψ (ξ )ˆ is uniformly continuous on the compact S then we can find such δ >0 that

ξˆ

|ˆξ|− ηˆ

| ˆη|

2|ˆξ− ˆη|

| ˆη| =2|ξ−k_rκ|

| ˆη| 2R/| ˆη|. (3.13)

Observe that for each nonzeroκ∈Zⁿ,| ˆη|k_r. Then, by (3.13) we see that for allr∈Nsuch thatk_r >2R/δand all κ∈Zⁿ, 0<|κ|m,

ψξ /ˆ |ˆξ|=ψξ /ˆ |ˆξ|

−ψ ˆ

η/| ˆη|< ε. (3.14)

We use here thatη/ˆ | ˆη| ∈S_mand, therefore,ψ (η/ˆ | ˆη|)=0. In view of (3.14), for allk_r>2R/δ

κ∈Zⁿ,0<|κ|m

Rⁿ⁺¹

F^t(αbκ,r)(τ )²β(ξ˜ −krκ)²ψξ /ˆ |ˆξ|dξˆ

ε

κ∈Zⁿ,0<|κ|m

Rⁿ⁺¹

F^t(αbκ,r)(τ )²β(ξ˜ −krκ)²dξˆ εβ²2

κ∈Zⁿ

R

α(t )b_κ,r(t)²dtεΦ²2sup

t >0

κ∈Zⁿ

b_κ,r(t)²

=εΦ²2sup

t >0

v(k_rt,·)−v^∗(t)²

L²(P )CεΦ²2, (3.15)

(8)

whereC=4v²_∞. Further, it follows from (3.13) withηˆ=(τ,0)that for|ξ|Rand|τ|> R₁=2R/δ ψξ /ˆ |ˆξ|=ψξ /ˆ |ˆξ|

−ψ

τ/|τ|,0< ε.

Therefore,

Rⁿ⁺¹

θ

|τ| −R1F^t(αb0,r)(τ )²β(ξ )˜ ²ψξ /ˆ |ˆξ|dξˆCεΦ²2. (3.16) Hereθ (r)=_{1, r>0,}

0, r0 is the Heaviside function.

For|τ|R₁we are reasoning in the following way. Sinceα(t )b_0,r(t)=α(t)(a_0,r(t)−v^∗(t)) 0 asr→ ∞, and αb_0,r1C₁=2v∞α1, the Fourier transformF^t(αb_0,r)(τ )→r→∞0 for allτ∈Rand uniformly bounded:

|F^t(αb_0,r)(τ )|C₁. By Lebesgue dominated convergence theorem

R

θ

R₁− |τ|F^t(αb_0,r)(τ )²dτ →

r→∞0.

Therefore (recall thatψ_∞1),

Rⁿ⁺¹

θ

R₁− |τ|F^t(αb_0,r)(τ )²β(ξ )˜ ²ψξ /ˆ |ˆξ|dξˆ β2

R

θ

R₁− |τ|F^t(αb_0,r)(τ )²dτ →

r→∞0. (3.17)

In view of (3.16), (3.17) we find

rlim→∞

Rⁿ⁺¹

F^t(αb_0,r)(τ )²β(ξ )˜ ²ψξ /ˆ |ˆξ|dξˆCεΦ²2. (3.18)

Using (3.10), (3.12), (3.15) and (3.18), we arrive at the relation

rlim→∞

Rⁿ⁺¹

F Φ

v_r−v^∗(ˆξ )²ψξ /ˆ |ˆξ|dξˆC₂ε, (3.19)

whereC₂is a constant independent onψandm. By the definition ofH-measure and Remark2.1

rlim→∞

Rⁿ⁺¹

F Φ

vr−v^∗(ˆξ )²ψξ /ˆ |ˆξ|dξˆ

= ˆ

μ,Φ(t, x)²ψ (ξ )ˆ =

Π×(S\Sm)

Φ(t, x)²ψ (ξ )ˆ dμ(t, x,ˆ ξ ),ˆ and (3.19) implies that

Π×(S\S_m)

Φ(t, x)²ψ (ξ ) dˆ μ(t, x,ˆ ξ )ˆ C2ε

for allψ (ξ )ˆ ∈C0((S\Sm))such that 0ψ (ξ )ˆ 1. Therefore, we can claim that

Π×(S\Sm)

Φ(t, x)²dμ(t, x,ˆ ξ )ˆ C₂ε, and sinceS\S₀⊂S\S_m, we obtain the relation

(9)

Π×(S\S₀)

Φ(t, x)²dμ(t, x,ˆ ξ )ˆ C2ε, which holds for arbitrary positiveε. Therefore,

Π×(S\S0)

Φ(t, x)²dμ(t, x,ˆ ξ )ˆ =0. (3.20)

Since for every point(t₀, x₀)∈Π one can find functionsα(t ),β(x) with the prescribed above properties in such a way thatΦ(t, x)=α(t )β(x) =0 in a neighborhood of(t₀, x₀), we derive from (3.20) the desired inclusion suppμˆ ⊂ Π×S₀. 2

We consider theH-measure{μ^pq}p,q∈Ecorresponding to a subsequenceu_r=u_k_r(t, x)of the sequenceu_k(t, x)= u(kt, kx),k∈N, defined in accordance with Proposition2.2.

Theorem 3.1.For everyp, q∈E,suppμ^pq⊂Π×S₀.

Proof. Letνt,xbe a weak measure valued limit of the sequenceur. We introduce measures γ_t,x^r (λ)=δ

λ−ur(t, x)

−νt,x(λ),

and setUr(t, x, p)=γ_t,x((p,+∞)). Lets(u)∈C¹(R)be such that its derivatives(u)is compactly supported, and v_r(t, x)=s(u_r(t, x)),r∈N. Thenv_r v^∗(t)=

s(λ) dν_t,x(λ)asr→ ∞weakly-∗inL^∞(Π )(by Lemma3.2(i), the limit functionv^∗(t)does not depend onx). Integrating by parts, we find that

v_r(t, x)−v^∗(t)=

s(λ) dγ_t,x^r (λ)=

s(λ)U_r(t, x, λ) dλ. (3.21)

LetΦ(t, x)∈C₀(Π ),ψ (ξ )ˆ ∈C(S). Then, in view of (3.21), we find

Rⁿ⁺¹

F Φ

v_r−v^∗(ˆξ )²ψξ /ˆ |ˆξ| dξˆ

= s(p)s(q)

Rⁿ⁺¹

F

ΦU_r(·, p)(ˆξ )F

ΦU_r(·, q)

(ξ )ψˆ ξ /ˆ |ˆξ| dξˆ

dp dq. (3.22)

By the definition ofH-measure, for eachp, q∈E

rlim→∞

Rⁿ⁺¹

F

ΦU_r(·, p)(ˆξ )F

ΦU_r(·, q)(ˆξ )ψξ /ˆ |ˆξ| dξˆ=

μ^pq,Φ(t, x)²ψ (ξ )ˆ .

Using Lebesgue dominated convergence theorem, we can pass to the limit asr→ ∞in equality (3.22) and arrive at μ,ˆ Φ(t, x)²ψ (ξ )ˆ

= lim

r→∞

Rⁿ⁺¹

F

Φ(vr−v)(ˆξ )²ψξ /ˆ |ˆξ| dξˆ

= s(p)s(q)

μ^pq,Φ(t, x)²ψ (ξ )ˆ

dp dq, (3.23)

whereμˆ = ˆμ(t, x,ξ )ˆ is the Tartar’sH-measure, corresponding to the scalar sequencev_r−v^∗. Clearly, the equality μ,ˆ Φ(t, x)²ψ (ξ )ˆ

= s(p)s(q)

μ^pq,Φ(t, x)²ψ (ξ )ˆ dp dq

remains valid for every Borel functionψ (ξ ). Takingˆ ψ (ξ )ˆ being the indicator function of the setS\S₀ and using Lemma3.2, we obtain the relation

(10)

s(p)s(q)

μ^pq,Φ(t, x)²ψ (ξ )ˆ

dp dq=0. (3.24)

Now we take in (3.24)s(p)=lω(l(p−p0)), wherep0∈E,l∈N, andω(y)∈C0((0,1))be a non-negative function such that

ω(y)dy=1. Since theH-measureμ^pqis strongly continuous with respect to(p, q)at point(p0, p0), we derive from (3.24) in the limit asl→ ∞that

μ^p⁰^p⁰,Φ(t, x)²ψ (ξ )ˆ

= lim

l→∞l² ω

l(p−p₀) ω

l(q−p₀)

μ^pq,Φ(t, x)²ψ (ξ )ˆ

dp dq=0.

SinceΦ(t, x)∈C₀(Π )is arbitrary, we conclude thatμ^p⁰^p⁰(Π×(S\S₀))=0 (remark thatμ^p⁰^p⁰0). Hence, for everyp=p₀∈E, suppμ^pp⊂Π×S₀. Finally, as directly follows from (2.4), forp, q∈Esuppμ^pq⊂suppμ^pp⊂ Π×S₀. The proof is complete. 2

Let us define the minimal linear subspace L=L(p)⊂Rⁿ⁺¹ such that suppμ^pp ⊂Π ×L. Since u_r(t, x) is a bounded sequence of e.s. of Eq. (1.1) from the results of[9] (see Lemma 2 withq =p₀ and the proof of Theo- rem 4) it follows the localization principle:

Theorem 3.2.There existsδ >0such that the functionu→τ u+ξ·ϕ(u)is constant on the interval(p−δ, p+δ) for allξˆ=(τ, ξ )∈L.

Now we are ready to prove our main Theorem1.1.

Proof of Theorem 1.1. We fixp∈Eand assume thatμ^pp =0. Then the spaceL=L(p)is not trivial: dimL >0.

By Theorem3.1there exists a nonzero vectorξˆ=(τ, ξ )∈(R×Zⁿ)∩L. Then, by Theorem3.2the functionu→ τ u+ξ ·ϕ(u)is constant on some interval(p−δ, p+δ), which contradicts to condition (1.7). Henceμ^pp=0 for allp∈E. In view of (2.4) this implies that theH-measureμ^pq ≡0. Therefore the sequenceu_r(t, x)converges as r→ ∞to a functionu^∗(t, x)∈L^∞(Π )strongly, inL¹_loc(Π ). By Lemma3.2(i) the limit function does not depend on x:u^∗(t, x)=u^∗(t). Passing to the limit asr→ ∞in equalities(u_r)_t+divxϕ(u_r)=0 inD(Π )we derive, in view of the strong convergenceu_r→u^∗,ϕ(u_r)→ϕ(u^∗)asr→ ∞, the relation(u^∗)_t+divxϕ(u^∗)=0 inD(Π ). Since u^∗=u^∗(t)we find(u^∗)=0 inD(R₊), which yieldsu^∗=const. The relationu_r(t, x)→r→∞u^∗inL¹_loc(Π )implies that (after possible extraction of a subsequence) for a.e.t >0,ur(t, x)→r→∞u^∗inL¹_loc(Rⁿ). By the periodicity, this reads

P

u(k_rt, k_rx)−u^∗dx →

r→∞0.

Making the change of variablesy=k_rxand using the space periodicity ofu, we find that for a.e.t >0

P

u(k_rt, y)−u^∗dy=

P

u(k_rt, k_rx)−u^∗dx →

r→∞0. (3.25)

We fix sucht=t0>0. Then for a.e.t > krt0

P

u(t, y)−u^∗dy

P

u(krt0, y)−u^∗dy, (3.26)

by theL¹(P )-contraction property. In view of (3.25) it follows from (3.26) that ess limt→∞u(t, x)=u^∗ inL¹(P ).

Finally, by the conservation of “mass” (see[11]), for a.e.t >0

P

u(t, x) dx=

P

u₀(x) dx,

whereu₀(x)is a strong trace ofu(t, x)on the initial hyper-spacet=0. Passing in this relation to the limit ast→ ∞, we obtain that

(11)

u^∗= 1

|P|

P

u₀(x) dx=

P

u₀(x) dx.

Hence, ess lim

t→∞ u(t, x)=

P

u₀(x) dx inL¹(P ), and decay property (1.6) holds.

Conversely, assume that Eq. (1.1) satisfies the decay property. Let us demonstrate that it satisfies condition (1.7).

Assuming the contrary, we can find the segment [a, b],a < b, and a nonzero point(τ, ξ )∈R×Zⁿ such that the functionu→τ u+ξ·ϕ(u)is constant on the segment[a, b]. Then, as is easy to verify, the function

u(t, x)=a+b

2 +b−a 2 sin

2π(τ t+ξ·x)

is a periodic e.s. of (1.1), which does not satisfy the decay property. The obtained contradiction shows that condition (1.7) holds. We conclude that this condition is necessary and sufficient for the decay property. 2

References

[1] G.-Q. Chen, H. Frid, Decay of entropy solutions of nonlinear conservation laws, Arch. Ration. Mech. Anal. 146 (2) (1999) 95–127.

[2] R.J. DiPerna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal. 88 (1985) 223–270.

[3] P. Gerárd, Microlocal defect measures, Comm. Partial Differential Equations 16 (1991) 1761–1794.

[4] S.N. Kruzhkov, First order quasilinear equations in several independent variables, Mat. Sb. 81 (1970) 228–255; English transl. in: Math.

USSR-Sb. 10 (1970) 217–243.

[5] S.N. Kruzhkov, E.Yu. Panov, First-order conservative quasilinear laws with an infinite domain of dependence on the initial data, Dokl. Akad.

Nauk SSSR 314 (1990) 79–84; English transl. in: Soviet Math. Dokl. 42 (1991) 316–321.

[6] S.N. Kruzhkov, E.Yu. Panov, Osgood’s type conditions for uniqueness of entropy solutions to Cauchy problem for quasilinear conservation laws of the first order, Ann. Univ. Ferrara Sez. VII (N.S.) 40 (1994) 31–54.

[7] P.V. Lysuho, E.Yu. Panov, Renormalized entropy solutions to the Cauchy problem for first order quasilinear conservation laws in the class of periodic functions, J. Math. Sci. 177 (1) (2011) 27–49.

[8] E.Yu. Panov, On sequences of measure-valued solutions of first-order quasilinear equations, Mat. Sb. 185 (2) (1994) 87–106; English transl.

in: Russian Acad. Sci. Sb. Math. 81 (1) (1995) 211–227.

[9] E.Yu. Panov, On strong precompactness of bounded sets of measure valued solutions for a first order quasilinear equation, Mat. Sb. 186 (5) (1995) 103–114; English transl. in: Russian Acad. Sci. Sb. Math. 186 (5) (1995) 729–740.

[10] E.Yu. Panov, Property of strong precompactness for bounded sets of measure valued solutions of a first-order quasilinear equation, Mat. Sb.

190 (3) (1999) 109–128; English transl. in: Russian Acad. Sci. Sb. Math. 190 (3) (1999) 427–446.

[11] E.Yu. Panov, A remark on the theory of generalized entropy sub- and supersolutions of the Cauchy problem for a first-order quasilinear equation, Differ. Uravn. 37 (2) (2001) 252–259; English transl. in: Differ. Equ. 37 (2) (2001) 272–280.

[12] E.Yu. Panov, Maximum and minimum generalized entropy solutions to the Cauchy problem for a first-order quasilinear equation, Mat. Sb.

193 (5) (2002) 95–112; English transl. in: Russian Acad. Sci. Sb. Math. 193 (5) (2002) 727–743.

[13] E.Yu. Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ.

2 (4) (2005) 885–908.

[14] E.Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ. 4 (4) (2007) 729–770.

[15] E.Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal. 195 (2) (2010) 643–673.

[16] L. Tartar, Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics: Heriot. Watt Symposium, vol. 4, Edinburgh, 1979, in: Res. Notes Math., vol. 39, 1979, pp. 136–212.

[17] L. Tartar,H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115 (3–4) (1990) 193–230.