On compactness of bounded solutions to multidimensional conservation laws with memory

NoDEANonlinear differ. equ. appl. 5 (1998) 193 – 2041021-9722/98/020193-12 $ 1.50+0.20/0

c© Birkhauser Verlag, Basel, 1998

Nonlinear Differential Equationsand Applications NoDEA

On compactness of bounded solutions tomultidimensional conservation laws with memory ∗

Eduard FEIREISLInstitute of Mathematics AVCR, Zitna 25

CZ-115 67 Praha 1, Czech Republic

Hana PETZELTOVAInstitute of Mathematics AVCR, Zitna 25

Cz-115 67 Praha 1, Czech Republic

AbstractWe prove compactness of bounded sets of entropy solutions to conservationlaws with memory of the form

(u+K ∗ u)t + Σai(u)uxi = 0.

The kernel K is assumed to be a nonincreasing and integrable function of t.

1 Introduction

In this paper, we prove compactness of bounded solutions of conservation lawswith memory, generalizing thus the celebrated result of TARTAR [7] and LIONS,PERHAME and TADMOR [5]. Specifically, we consider the multidimensional con-servation law of the form:

∂

∂t

(u+

∫ t

−∞K(t− s)u(s) ds

)+

N∑i=1

ai(u)∂u

∂xi= 0 on (0, T )×RN , (1.1)

where, for simplicity, the solution u,

u = u(t, x), t ∈ (−∞, T ), x ∈ RN

is spatially periodic with the period 1, i.e.

u ∈ C((−∞, T );L1(T N )

)∩ L∞

((−∞, T )× T N

)(1.2)

where the symbol T N ≡ ([0, 1]|0,1)N stands for the N-dimensional torus.

∗Supported by grants A1019703 GA AV CR and 201/96/0432 GA CR.

194 Eduard Feireisl and Hana Petzeltova NoDEA

The kernel K will satisfy

K ∈ L1(R1), K(s) ≡ 0 for s < 0, K nonincreasing on (0,∞). (1.3)

The main result may be formulated as follows:

Theorem 1.1 Let K satisfy the hypothesis (1.3). Let u be an entropy solution ofthe equation (1.1) on the time interval (0, T ),

−M ≤ u(t, x) ≤M for a.e. t ∈ (−∞, T ], x ∈ T N (1.4)

and‖K‖L1(0,∞) ≤ κ. (1.5)

Moreover, let ai ∈ C2[−M,M ], i = 1, . . . , N and

measv ∈ [−M,M ] ; ξ0 + ΣNi=1ai(v)ξi = 0 = 0 (1.6)

for any fixed ξ 6= 0, ξ ∈ RN+1.Then u|[0,T ] belongs to a compact set of L1

((0, T )×T N

)depending on M, κ

only.

Although nonlinearity generally excludes good uniform estimates on solutionsto conservation laws, in certain settings the very fact that the equation is nonlinearprovides enough extra information sufficient for the solution set to be compact.

Using the theory of compensated compactness TARTAR [7] exploited thisidea to show that bounded solutions to nonlinear conservation laws (without mem-ory) in one spatial variable are in fact compact in L1. Even though the scalarproblems are often considered far better understood than systems, it took quite along time to extend Tartar’s result to the scalar conservation law in several spacedimensions. The two-dimensional case was treated by ENGQUIST and WEINAN[2] using careful analysis of the behaviour of shocks in entropy solutions. Finally,employing the kinetic method LIONS, PERTHAME and TADMOR [5] solved theproblem in general N-dimensional setting under the nondegeneracy assumption(1.6), which means, roughly speaking, that the curve v 7→ (ai(v), . . . , aN (v)) can-not remain in a given hyperplane.

When the memory effects come into play, however, considerably less seemsto be known about the behaviour of large norm solutions. The very notion ofthe entropy solution and related questions of existence and uniqueness have beenstudied quite recently by COCKBURN, GRIPENBERG and LONDEN [1] (see thenext section). Such a concept of solutions makes it possible to derive the kineticformulation also for (1.1). This, in turn, together with deep results on Fouriermultipliers in Lp spaces enables to prove the desired compactness result. Morespecifically, we use the Lizorkin theorem to minimize the hypotheses concerningthe kernels K. Moreover, the reader will observe that Theorem 1.1 still holds evenif weaker conditions on the solution are imposed.

Vol. 5, 1998 Multidimensional conservation laws with memory 195

The physical intuition suggests that various types of dissipation when actingtogether should strengthen their effects on the solutions and thus lead to betterregularity and smoothness properties. However, the cooperation of those mecha-nisms is very subtle and sometimes the fact that the material may “remember” itsinitial shape can slow down the regularization process rather than to speed it up.For instance, the well-known L1 − L∞ regularizing effect for genuinely nonlinearconservation laws in 1-D does not seem to have its counterpart when memory isinvolved (cf. e.g. GREENBERG [4]).

2 Entropy solutions and kinetic formulation

Following COCKBURN, GRIPENBERG and LONDEN [1] we introduce the con-cept of the entropy solution:

Definition: We shall say that a function u,

u ∈ L∞((−∞, T )× T N

)∩ C

((−∞, T ];L1(T N )

),

is an entropy solution of the equation (1.1) on the time interval [0, T ] if

∂

∂t

(S(u(t)) +

∫ t

−∞Ks(t− s)S(u(s)) ds

)+ (2.1)

(Kr(0+)u(t) +

∫ t

−∞u(s) dKr(t− s)

)S′(u(t)) +

N∑i=1

∂

∂xiηi(u(t)) ≤ 0

in the sense of distributions for any convex entropy S and the corresponding flux η,

ηi(v) =∫ v

0S′(z)ai(z) dz,

and for any decomposition of the type

K = Kr +Ks, Kr,Ks ≥ 0, Kr,Ks nonincreasing on (0,∞), Kr(0+) <∞.

Remark: Since any convex function can be approximated by a linear combinationof affine functions and functions of the form |.−c|, the above definition is equivalentto that of COCKBURN, GRIPENBERG and LONDEN [1, Definition 6].

Remark: It was proved in [1] that given the function u|(−∞,0], there exists exactlyone entropy solution of (1.1). In fact, the conclusion of Theorem 1.1 remains trueif we require (2.1) to hold only for Kr ≡ 0, Ks ≡ K. In this case, however,uniqueness of solutions is not guaranteed.

Next, following the ideas of LIONS, PERTHAME and TADMOR [5] we intro-duce the kinetic formulation of the problem. To a solution u, we assign a function f :

f(t, x, v) =

1 if 0 < v ≤ u(t, x)−1 if 0 > v ≥ u(t, x)

0 otherwise.(2.2)


Observe that ∫ ∞−∞

f(t, x, v)h′(v) dv = h(u(t, x)) for all t, x. (2.3)

and for any h ∈ C1, h(0) = 0.The future analysis is based on the following result:

Proposition 2.1 Under the hypotheses of Theorem 1.1, the function f defined by(2.2) satisfies

supp(f) ⊂ (−∞, T ]× T N × [−M,M ], (2.4)

and

∂

∂t

(f(t, x, v) +

∫ t

−∞K(t− s)f(s, x, v) ds

)+

N∑i=1

ai(v)∂f(t, x, v)

∂xi=∂m

∂v(2.5)

on (0, T )×T N×R1 in the sense of distributions where m is a nonnegative boundedmeasure supported on the set [0, T ]× T N × [−M,M ],

‖m‖M(RN+2) ≤ C(κ,M) (2.6)

Remark: In contrast with the case of a conservation law without memory, it isnot clear if the formulation (2.5) is really equivalent to the original concept of theentropy solution.

Proof: We define a distribution m,

< m, g(t, x, v) >≡∫ T

0

∫T N

∫ ∞−∞

(f +K ∗ f)Gt(t, x, v) +∑

ai(v)fGxi(t, x, v)

where

G(t, x, v) =∫ v

−Mg(t, x, z) dz and K ∗ f(t) ≡

∫ t

−∞K(t− s)f(s) ds

for any g periodic in x and compactly supported in t ∈ (0, T ), v ∈ R1. Observethat supp(m) ⊂ [0, T ]× T N × [−M,M ].

Moreover, we have

<∂m

∂v, g >= − < m, gv >=

∫ T

0

∫T N

∫ ∞−∞

(f +K ∗ f)gt(t, x, v) +∑

ai(v)fgxi(t, x, v) dv dx dt−∫ T

0

∫T N

∫ ∞−∞

(f +K ∗ f)gt(t, x,−M) +∑

ai(v)fgxi(t, x,−M) dv dx dt


where, by virtue of (2.3),∫ T

0

∫T N

∫ ∞−∞

(f +K ∗ f)gt(t, x,−M) +∑

ai(v)fgxi(t, x,−M) dv dx dt =

∫ T

0

∫T N

(u+K ∗ u)gt(t, x,−M) +∑

Ai(u)gxi(t, x,−M) dx dt = 0,

Ai(v) ≡∫ v

0ai(z) dz,

which yields the formula (2.5).Consequently, it remains to show that m is a nonnegative bounded measure.

To this end, consider the functions g = g(t, x) ≥ 0, h = h(v) ≥ 0, supp (h)compact and set

S(v) =∫ v

0(∫ z

−Mh(s) ds) dz. (2.7)

We compute< m, g ⊗ h >=∫ T

0

∫T N

∫ M

−M(f +K ∗ f)S′(v)gt +

∑ai(v)fS′(v)gxi dv dx dt =

∫ T

0

∫T N

(S(u) +K ∗ S(u)) gt +∑

ηi(u)gxi dx dt ≥ 0

where the last inequality follows from (2.1) with Ks = K, Kr = 0. Consequently,m is a nonnegative distribution on [0, T ] × T N × R1 with support contained in[0, T ]× T N × [−M,M ].

Now, taking

gε(t) 1 ∈ C1(K) for any compact K ⊂ (0, T ), 0 ≤ h(v) ≤ 1

and using (2.3) we obtain

< m, gε ⊗ h >=∫ T

0gεt(t)

∫T N

S(u) +K ∗ S(u) dx dt

where S is defined as in (2.7). Passing to the limit for ε→ 0 we deduce

< m, 1⊗h >=∫T N

S(u)(0)+K∗S(u)(0) dx−∫T N

S(u)(T )+K∗S(u)(T ) dx. (2.8)

We conclude that m is a nonnegative bounded distribution with supp(m) ⊂[0, T ] × T N × [−M,M ], in particular, m is extendible to the set of compactlysupported continuous functions, i.e by virtue of the Riesz representation theorem,m is a measure. Finally, the relation (2.6) follows from (2.8).


3 The proof of Theorem 1.1

Multiplying (2.5) by a function ψ = ψ(t, x) ∈ D((0, T )×(0, 1)N ), |ψ| ≤ 1 we obtain

(F +K ∗ F )t +∑

ai(v)Fxi = (3.1)

∂ψm

∂v+ fψt +

∑ai(v)fψxi + [K ∗ (fψ)]t − (K ∗ f)tψ

in the sense of distributions on RN+2 where we have denoted

F (t, x, v) ≡ f(t, x, v)ψ(t, x).

Moreover, we compute

G(t, x, v) ≡ [K ∗ (fψ)]t − (K ∗ f)tψ =

−∫ t

−∞

ψ(t, x)− ψ(s, x)t− s f(s, x, v)(t − s) dK(t− s).

On the other hand, by virtue of (1.3), we have

0 ≤ −∫ ∞

0z dK(z) = − lim

ε→0+

∫ 1ε

ε

z dK(z) = (3.2)

limε→0+

[εK(ε)− 1εK(

1ε

)] +∫ ∞

0K(z) dz ≤ κ

aslimz→0+

zK(z) = limz→∞

zK(z) = 0

for any K satisfying (1.3).Consequently, we have obtained the estimate

‖G‖L∞ ≤ C(ψ)κ. (3.3)

Moreover, G = 0 for t ≤ 0 and

G(t, x, v) =∫ T

−∞(ψ(t, x) − ψ(s, x))f(s, x, v) dK(t− s)

for all t ≥ T . Thus we have

|G(t, x, v)| ≤ −C(ψ)∫ T

−∞dK(t− s) = CK(t− T ) ≤ Cκ

(t− T ), t > T (3.4)

where the last inequality follows from (1.3).


We conclude that

(F +K ∗ F )t +∑

ai(v)Fxi =∂m1

∂v+m2 (3.5)

where, by virtue of Proposition 2.1,

m1 ∈ K1 (3.6)

where K1 is a compact subset of the space W−1,s(RN+2), 1 < s < N+2N+1 .

Similarly, using (3.3),(3.4) we have

m2 ∈ K2 (3.7)

where K2 is a compact subset of W−1,s(RN+2). Given s, both K1, K2 depend onlyon ψ, T, κ,M .

At this stage, analogously as in LIONS, PERHAME and TADMOR [5], wetake advantage of the fact that (3.5) is linear in (x, t) for any fixed v and, con-sequently, we can use the Fourier transform technique to obtain the desirableestimates.

To this end, we introduce the complex valued quantity Λ,

Λ(ξ, v) ≡ξ0(1 + K(ξ0)

)+ ΣNi=1ai(v)ξi|ξ| , (3.8)

ξ = (ξ0, . . . , ξN ) ∈ RN+1, ξ 6= 0, v ∈ R1

where K denotes the Fourier transform of K, and an auxiliary function φ(z) suchthat

φ ∈ D(R1) , supp(φ) ⊂ (−1, 1), φ(z) = 1 for all z ∈ [−12,

12

]

andφ(z) ∈ [0, 1] for all z ∈ R1.

Now, we write

ψ(t, x)u(t, x) =∫ ∞−∞

ψf dv =∫ ∞−∞

F (t, x, v) dv =

∫ ∞−∞F−1[φ(

Λδ

) + φ(ξ

δ)(1− φ(

Λδ

))+

(1− φ(

ξ

δ))(

1− φ(Λδ

))]FF dv

where FF denotes the Fourier transform of F in the variables (x0, x1, . . . , xN )with x0 ≡ t .

Since F is a compactly supported function and |F | ≤ 1, the conclusion ofTheorem 1.1 will follow from the following three assertions:


Lemma 3.1 Under the hypotheses (1.3),(1.6), let

χ(δ) = sup‖K‖L1≤κ, ξ 6=0

meas Ω(K, ξ, δ)

where we have denoted

Ω(K, ξ, δ) ≡ v ∈ [−M,M ] ; |Λ(ξ, v)| ≤ δ.

Thenlimδ→0+

χ(δ) = 0 (3.9)

Proof: First of all, observe that

limδ→0+

measv ∈ [−M,M ] ; |ξ0 + ΣNi=1ai(v)ξi| ≤ δ = (3.10)

meas ∩δ>0 v ∈ [−M,M ] ; |ξ0 + ΣNi=1ai(v)ξi| ≤ δ =

measv ∈ [−M,M ] ; ξ0 + ΣNi=1ai(v)ξi = 0 = 0

for any fixed ξ 6= 0 where the last equality follows from (1.6).Now, arguing by contradiction we construct a sequence

ξj

|ξj | = zj → z, |z| = 1, Kj(ξj0) ≡ kj → k

such that

measv ∈ [−M,M ] ; |zj0(1 + kj) + ΣNi=1ai(v)zji | ≤1j ≥ ε > 0 for all j. (3.11)

Moreover, it follows from GRIPENBERG [3, Theorem 2] that

1 + k 6= 0. (3.12)

On the other hand, we deduce

v ∈ [−M,M ] ; |zj0(1 + kj) + ΣNi=1ai(v)zji | ≤1j ⊂

v ∈ [−M,M ] ; |z0(1 + Re k) + Σai(v)zi| ≤

|z0(1 + k) + Σai(v)zi| ≤1j

+ |zj0(1 + kj)− z0(1 + k) + ΣNi=1ai(v)(zji − zi)|.

By virtue of (3.10), the measure of the set on the right-hand side of theinclusion goes to zero in contrast with (3.11). Indeed, if z0 = 0, such a conclusionis straightforward as (z1, . . . , zN ) 6= 0 while, if z0 6= 0, the relation

|zj0(1 + kj) + ΣNi=1ai(v)zji | ≤1j

implies Im kj → 0.

Therefore, by virtue (3.12), Re(1 + k) 6= 0, and, consequently, the relation (3.10)may be used with ξ0 = z0(1 + Re k)) 6= 0.


Proposition 3.1 Under the hypotheses (1.3)-(1.6), we have

‖∫ ∞−∞F−1[φ(

|Λ|δ

)FF ] dv‖L2(t,x)→ 0 as δ → 0+, (3.13)

and

‖∫ ∞−∞F−1[φ(

ξ

δ)(1− φ(

|Λ|δ

))FF

]dv‖L2

(t,x)→ 0 as δ → 0+ (3.14)

uniformly for K satisfying (1.5).

Proof: We have

‖∫ ∞−∞F−1[φ(

|Λ|δ

)FF ] dv‖L2(t,x)

= ‖F−1[∫ ∞−∞

φ(|Λ|δ

)FF dv]‖L2(t,x)

=

‖∫ ∞−∞

φ(|Λ|δ

)FF dv‖L2ξ

by Plancherel equality.Furthermore, ∫ ∞

−∞· · ·∫ ∞−∞|∫ ∞−∞

φ(|Λ|δ

)FF dv|2 dξ =

∫ ∞−∞· · ·∫ ∞−∞|∫

Ω(ξ,δ)φ(|Λ|δ

)FF meas(Ω)dv

meas(Ω)|2 dξ.

By virtue of the Jensen inequality, we obtain∫ ∞−∞· · ·∫ ∞−∞|∫

Ω(ξ,δ)φ(|Λ|δ

)FF meas(Ω)dv

meas(Ω)|2 dξ ≤

∫ ∞−∞· · ·∫ ∞−∞

( ∫ ∞−∞|φ(|Λ|δ

)FF |2meas(Ω) dv)dξ ≤ χ(δ)‖F‖2L2

where the right-hand side goes to zero as δ → 0+ by Lemma 3.1.The proof of (3.14) is straightforward since

φ(ξ

δ) 6= 0 only for |ξ| ≤ δ

and F has compact support in t, x, v, |F | ≤ 1.


Proposition 3.2 Suppose that

F = F (t, x, v) ∈ L∞(Q), supp(F ) ⊂ int (Q)

satisfies the identity

(F +K ∗ F )t +N∑i=1

ai(v)Fxi =∑

α=0,1,2,β=0,1,i=0,...,N

∂αv ∂βxigα,β,i (3.15)

on RN+1 ×R1 in the sense of distributions, where we have denoted

Q ≡ [0, T ]× [0, 1]N × [−M,M ].

Moreover, let s ∈ (1,∞), gα,β,i ∈ Ls(RN+2) and ai ∈ C2[−M,M ]. Then

‖∫ ∞−∞F−1[(1− φ(

ξ

δ))(

1− φ(Λδ

))]FF dv‖Lst,x (3.16)

≤ C(δ, κ) maxα,β,i‖gα,β,i‖Ls .

Proof: As (3.15) is linear, it suffices to show (3.16) if the right-hand side of (3.15)takes the form ∂2

v∂βxjg.

Fixing v and taking the Fourier transform of (3.15) with respect to (t, x) weobtain

Λ(ξ, v)|ξ|F (ξ, v) = ξβj ∂2v g(ξ, v). (3.17)

Multiplying (3.17) by the expression(1− φ( ξδ )

)(1− φ(Λ

δ ))

Λ|ξ| Ψ(v)

and integrating by parts we deduce∫ ∞−∞

(1− φ(

ξ

δ))(

1− φ(Λδ

))F (ξ, v) dv =

∫ ∞−∞

(1− φ(

ξ

δ))(

1− φ(Λδ

))F (ξ, v)Ψ(v) dv = (3.18)

∫ ∞−∞

∂2v

[ξβj|ξ|

(1− φ( ξδ )

)(1− φ(Λ

δ ))

ΛΨ(v)

]g(ξ, v) dv

where Ψ ∈ D(R1) is a fixed function such that Ψ(v) ≡ 1 on [−M,M ].


Next, applying the inverse Fourier transform to both sides of (3.18) we con-clude

‖∫ ∞−∞F−1(1− φ(

ξ

δ))(

1− φ(Λδ

))FF (v) dv‖Lst,x ≤ (3.19)

∫ ∞−∞‖F−1∂2

v

[ξβj|ξ|

(1− φ( ξδ )

)(1− φ(Λ

δ ))

ΛΨ(v)

]Fg(v) ‖Lst,x dv.

Thus it suffices to show that the quantity

N (ξ, v, n) ≡ ∂2v

[ξβj|ξ|

(1− φ( ξδ )

)(1− φ(Λ

δ ))

ΛΨ(v)

](3.20)

is an Ls−Fourier multiplier with the norm C(δ, κ) independent of v. The right-hand side of (3.19) may be then estimated from above by the quantity

C

∫supp (Ψ)

‖g(v)‖Lst,x dv

which yields (3.16).The rest of the proof will consist of showing that N is a Fourier multiplier.We shall say that a function h(ξ, v) has the property (L) if

h(., v) ∈ C1(RN+1 \ 0) and (L)

sup|ξγ∂γvh(ξ, v)| ; ξ 6= 0, γ = (γ0, . . . , γN ), 0 ≤ γi ≤ 1, i = 0, . . . , N ≤ C

independently of v ∈ supp(Ψ).

According to the Lizorkin theorem (see LIZORKIN [6]), the function h en-joying the property (L) is an Ls−multiplier with the norm depending only on Cand s, 1 < s <∞.

Consequently, a straightforward computation yields that

ξβj (1− φ( ξδ ))|ξ| ∂αv Ψ(v), α = 0, 1, 2

are the Ls−multipliers, 1 < s <∞, therefore it is enough to show that

∂αvG(Λ) where G(Λ) ≡1− φ(Λ

δ )Λ

, α = 0, 1, 2

are also multipliers.It is easy to check, using the chain rule, that a superposition of the type

H(h), where h satisfies (L) and H is smooth on a neighborhood of the range of h,has again the property (L).


Consequently, it is enough to show that Λ and its derivatives with respect tov satisfy (L). As ai ∈ C2, the only nonstandard part is to verify (L) for

ξ0K(ξ0)|ξ| .

After a direct computation, this reduces to showing that

|∂ξ0(ξ0K(ξ0))| is bounded (3.21)

for all ξ0 6= 0.To this end, we compute

∂ξ0 [ξ0K(ξ0)] = −∫ ∞

0exp(−iξ0z)z dK(z), ξ0 6= 0. (3.22)

Consequently, (3.21) follows from (3.22), (3.2) where the bound depends only on κ.

References

[1] B. COCKBURN, G. GRIPENBERG, and S.-O. LONDEN, On convergenceto entropy solutions of a single conservation law. J. Differential Equations,128(1):206–251, 1996.

[2] B. ENGQUIST and E. WEINAN, Large time behavior and homogenization ofsolutions of two-dimensional conservation laws. Commun. Pure Appl. Math.,46:1–26, 1993.

[3] G. GRIPENBERG, On the resolvents of Volterra equations with nonincreasingkernels. J. Math. Anal. Appl., 76:134–145, 1980.

[4] J.M. GREENBERG. The existence and qualitative properties of solutions of∂u∂t + 1

2∂∂x [u2 +

∫ t0 c(s)u

2(x, t − s) ds] = 0. J. Math. Anal. Appl., 42:205–220,1973.

[5] P.-L. LIONS, B. PERTHAME, and E. TADMOR. A kinetic formulation ofmultidimensional scalar conservation laws and related equations. J. Amer.Math. Soc., 7(1):169–191, 1994.

[6] P.I. LIZORKIN, Generalized Liouville differentiation and function spacesLrp(En). Embeddings theorems (in Russian). Matem. Sbornik, 60(3):325–353,1963.

[7] L. TARTAR, Compensated compactness and applications to partial differentialequations. Nonlinear Anal. and Mech., Heriot-Watt Sympos., L.J. Knoppseditor, Research Notes in Math 39, Pitman, Boston, pages 136–211, 1975.

Received January 25, 1997

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On compactness of bounded solutions to multidimensional conservation laws with memory