Nonlinear Analysis 63 (2005) e2483–e2489www.elsevier.com/locate/na

On renormalized dissipative solutions forconservation laws�

Satoru Takagi∗,1

Department of Mathematics, School of Education, Waseda University, 1–6–1 Nishi-waseda, Shinjuku-ku,Tokyo 169–8050, Japan

Abstract

We introduce a new notion of renormalized dissipative solutions for a scalar conservation lawut + div F(u) = f with locally Lipschitz F and L1 data, and prove the equivalence of such solutionsand renormalized entropy solutions in the sense of Bénilan et al. The structure of renormalizeddissipative solutions is useful to deal with relaxation systems than the renormalized entropy scheme.As an application of our result, we prove the existence of renormalized dissipative solutions viarelaxation.� 2005 Elsevier Ltd. All rights reserved.

MSC: 35L65; 47H06

Keywords: Renormalized dissipative solutions; Renormalized entropy solutions; Conservation laws; LocallyLipschitz continuous

� This is a joint work with Professor Kazuo Kobayasi.∗ Tel.: +81 3 3208 8443; fax: +81 3 5286 1308.

E-mail address: satoru@edu.waseda.ac.jp.1 The author was supported by Waseda University Grant for Special Research Projects #2003A-856.

0362-546X/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2005.03.002

e2484 S. Takagi / Nonlinear Analysis 63 (2005) e2483–e2489

1. Introduction

We consider the following Cauchy problem:

(CP)

{ �u

�t+ div F(u) = f in Q := (0, T ) × RN,

u(0, ·) = u0 in RN,

where T > 0 and N �1. Here f ∈ L1(Q) and u0 ∈ L1(RN) are given functions and theflux F : R → RN is a locally Lipschitz continuous function.

Kružkov [8] proved that if u0 ∈ L1(RN) ∩ L∞(RN), then (CP) has a unique weaksolution u ∈ C([0, T ); L1(RN)) ∩ L∞(Q) satisfying the entropy inequalities, which isthe so-called entropy solution. In the case that the flux F is globally Lipschitz, Portilheirointroduced a notion of dissipative solutions for (CP) and proved the equivalence of suchsolutions and entropy solutions. The relationship between the notions of various solutionsfor degenerate parabolic equations is also investigated in [6,9]. The dissipative solutions aremore suitable to obtain relaxation limits for some hyperbolic systems than entropy solutions.Indeed, Portilheiro [11] used this notion to obtain certain relaxation limits for hyperbolicsystems describing discrete velocity models and chemical reaction models. His idea is alsobased on the perturbed test function method introduced by Evans [3] for conservation laws.These systems have already been studied by Katsoulakis and Tzavaras [4,5], who obtainedseveral important results including comparison results. On the other hand, it is known thatif f ∈ L1(Q) and u0 ∈ L1(RN), then the mild solution u of (CP) constructed by nonlinearsemigroup theory is a unique entropy solution which is unbounded in general. In the casethat F is only locally Lipschitz, the flux function F(u) may fail to be locally integrable sinceno growth condition is assumed on the flux F, and hence (CP) does not possess a solutioneven in the sense of distributions. To overcome this, the notion of renormalized entropysolutions has been introduced in [1], where the existence and uniqueness of a renormalizedentropy solution of (CP) has been established and the semigroup solutions of (CP) in L1

spaces are characterized. Renormalized solutions have been introduced first by DiPernaand Lions [2] for the Boltzmann equation and utilized for degenerate elliptic and parabolicproblems in the L1-setting in the last decade. However, the argument in [10] does not workwell in the case that F is only locally Lipschitz and the solution u is unbounded.

Our purpose in this paper is to extend some results in [10] to the case of locally Lipschitzcontinuous flux. In this paper, we introduce a new notion of renormalized dissipative solu-tions which is a generalization of dissipative solutions in [10], and show the equivalence ofrenormalized dissipative solutions and renormalized entropy solutions. In the final section,as an application, we apply our result to contractive relaxation systems in merely an L1-setting and construct a renormalized dissipative solution via relaxation. Some relaxationsystems can be found in [7,12].

2. The main result

We begin with some notations and definitions. For r, s ∈ R, we set r ∧ s := min(r, s),r ∨ s := max(r, s), r+ := r ∨ 0 and r− := (−r) ∨ 0. For r ∈ R and j = 0, 1, we define

S. Takagi / Nonlinear Analysis 63 (2005) e2483–e2489 e2485

a sign function Sj by Sj (r) = 1 if r > 0, Sj (r) = −1 if r < 0, Sj (0) = j . Then we denoteS+

j (r) := Sj (r) ∨ 0 and S−j (r) := Sj (r) ∧ 0.

Let u ∈ L1(Q). For (t, x) ∈ Q and r > 0, we set

Br(t, x) := {(s, y) ∈ Q; (s − t)2 + |y − x|2 �r2},and define the upper and lower semicontinuous envelopes of u as

u∗(t, x) := limr↓0

sup {u(s, y); (s, y) ∈ Br(t, x)}

and

u∗(t, x) := limr↓0

inf {u(s, y); (s, y) ∈ Br(t, x)},

respectively. Then we see that u∗ �u�u∗, u∗ is upper semicontinuous and u∗ is lowersemicontinuous.

We now recall from [1] the definition of renormalized entropy solutions.

Definition 2.1. (i) We say u ∈ L1(Q) is a renormalized entropy subsolution of (CP) if forany k, � ∈ R,

�k,� := (u ∧ � − k)+t + div{S+0 (u ∧ � − k)(F(u ∧ �) − F(k))}

− S+0 (u ∧ � − k)f (2.1)

is a Radon measure on Q such that for each k ∈ R, lim�→∞ �+k,�(Q) = 0, and for each

� ∈ R, (u(t, ·) ∧ � − u0 ∧ �)+ → 0 in L1loc(R

N) as t → 0 essentially.(ii) We say u ∈ L1(Q) is a renormalized entropy supersolution of (CP) if for any k, � ∈ R,

�k,� := (u ∨ � − k)−t + div{S−0 (u ∨ � − k)(F(u ∨ �) − F(k))}

− S−0 (u ∨ � − k)f (2.2)

is a Radon measure on Q such that for each k ∈ R, lim�→−∞ �+k,�(Q) = 0, and for each

� ∈ R, (u(t, ·) ∨ � − u0 ∨ �)− → 0 in L1loc(R

N) as t → 0 essentially.(iii) We say u ∈ L1(Q) is a renormalized entropy solution of (CP) if u is a renormalized

entropy subsolution of (CP) and also a renormalized entropy supersolution of (CP).

Next, we introduce a new notion of renormalized dissipative solutions of (CP).

Definition 2.2. (i) We say u ∈ L1(Q) is a renormalized dissipative subsolution of (CP) ifthere is a sequence {��} ⊂ Mb(Q)+ with ��(Q) → 0 as � → ∞ such that for each ��1and � ∈ T�,∫

Q

S+0 (u ∧ � − �)(f − �t − div F(�)) dx dt +

∫Q

S+0 (u∗ ∧ � − �) d�� �0 (2.3)

and (u(t, ·) ∧ � − u0 ∧ �)+ → 0 in L1loc(R

N) as t → 0 essentially, where T� := C10(Q) ∩

{�; �(t, x) ≡ k for (t, x) ∈ Q if |x| > R for some k ∈ (−�, �) and R > 0} and Mb(Q)+denotes the space of all nonnegative bounded measures on Q.

e2486 S. Takagi / Nonlinear Analysis 63 (2005) e2483–e2489

(ii) We say u ∈ L1(Q) is a renormalized dissipative supersolution of (CP) if there isa sequence {��} ⊂ Mb(Q)+ with ��(Q) → 0 as � → ∞ such that for each ��1 and� ∈ T�,∫

Q

S−0 (u∨(−�)−�)(f −�t−div F(�)) dx dt+

∫Q

S−0 (u∗∨(−�)−�) d�� �0 (2.4)

and (u(t, ·) ∨ � − u0 ∨ �)− → 0 in L1loc(R

N) as t → 0 essentially.(iii) We say u ∈ L1(Q) is a renormalized dissipative solution of (CP) if u is a renormalized

dissipative subsolution of (CP) and also a renormalized dissipative supersolution of (CP).

Then we obtain the following main result.

Theorem 2.3. Suppose that u ∈ L1(Q) and u∗(t, x) < ∞ and u∗(t, x) > − ∞ for a.e.(t, x) ∈ Q. Then u is a renormalized entropy subsolution (resp. supersolution) of (CP) ifand only if u is a renormalized dissipative subsolution (resp. supersolution) of (CP).

3. Proof of Theorem 2.3

Lemma 3.1. If u ∈ L1(Q) and u∗(t, x) < ∞ for a.e. (t, x) ∈ Q, then a renormalizedentropy subsolution u of (CP) implies a renormalized dissipative subsolution.

Proof (Sketch). It follows from [1, Proposition 2.7] that there exists a sequence {��} ⊂Mb(Q)+ such that ��(Q) → 0 as � → ∞ and �k,� = �� − �k − �{u>�}f for k < �. Thenwe have∫

u∗∧�<k

� d�� =∫

u∗∧�<k

� d�k

for each � ∈ C∞0 (Q). Then, from (2.1) again we obtain for any � ∈ C∞

0 (Q)+ and k < �,

∫Q

S+0 (u ∧ � − k){(u ∧ � − k)�t + f � + (F(u ∧ �) − F(k)) · ∇�} dx dt

+∫

Q

S+0 (u∗ ∧ � − k){�d�� − �{u>�}f � dx dt}�0 (3.1)

for any � ∈ C∞0 (Q)+ and k, � ∈ R.

Let �� and � be standard mollifiers on RN and R, respectively, and let n be a nonnegativesmooth function satisfying

n(t, x) :={

1 if |x|�n,

0 if |x|�2n,

S. Takagi / Nonlinear Analysis 63 (2005) e2483–e2489 e2487

and |∇n|�C/n with positive constant C. Take � ∈ T�. Then we put � = ��(x − y)�(t − s)n(t, x) and k = �(s, y) in (3.1), and integrate in s and y over Q to get∫

Q2S+

0 (u ∧ � − �){(u ∧ � − �)(���n)t + f ���n

+ (F(u ∧ �) − F(�)) · ∇x(���n)} dy ds dx dt

+∫

Q2S+

0 (u∗ ∧ � − �){���n d�� − �{u>�}f ���n dx dt} dy ds� 0.

Passing to the limit as first � ↓ 0, ↓ 0 and then n → ∞, we obtain∫Q

S+0 (u ∧ � − �)(f − �t − div F(�)) dx dt

+∫

Q

S+0 (u∗ ∧ � − �){d�� + �{u>�}f dx dt}�0

which means that (2.3) holds with d�� replaced by d�� + �{u>�}f dx dt . �

Lemma 3.2. If u ∈ L1(Q) and u∗(t, x) < ∞ for a.e. (t, x) ∈ Q, then a renormalizeddissipative subsolution u of (CP) implies a renormalized entropy subsolution.

Proof (Sketch). Let k ∈ (−�, �) and � ∈ C∞0 (Q)+. For each �, � > 0, choose a function

��,� ∈ C∞0 (Q) such that

��,�(t, x) :={0 if (t, x) ∈ B�(0, 0),

1/� if (t, x) ∈ Q\B�+�(0, 0),

where B�(0, 0) = {(t, x) ∈ Q; t2 + |x|2 ��2}. Putting ��,� := k + ��,�(t − s, x − y)

in (2.3) for each (s, y) ∈ Q, multiplying (2.3) by an arbitrary function �(s, y) ∈ C∞0 (Q)+

and integrating in s and y over Q, we have∫Q2

S+0 (u ∧ � − ��,�){f − (��,�)t − divx F(��,�)}� dy ds dx dt

+∫

Q2S+

0 (u∗ ∧ � − ��,�)� dy ds d�� � 0. (3.2)

Passing to the limit as � ↓ 0 and using the Lebesgue differentiation theorem, we obtain∫Q

S+0 (u ∧ � − k){f � + (u ∧ � − k)�t + (F(u ∧ �) − F(k)) · ∇�} dx dt

+∫

Q

S+0 (u∗∧ � − k)� d�� �0

for each � ∈ C∞0 (Q)+. This means that

(u ∧ � − k)+t + div{S+0 (u ∧ � − k)(F(u ∧ �) − F(k))}

− S+0 (u ∧ � − k)f − S+

0 (u∗ ∧ � − k)��

e2488 S. Takagi / Nonlinear Analysis 63 (2005) e2483–e2489

is a Radon measure on Q and hence

�k,� = (u ∧ � − k)+t + div{S+0 (u ∧ � − k)(F(u ∧ �) − F(k))} − S+

0 (u ∧ � − k)f

is also a Radon measure on Q. Moreover, �+k,�(Q)���(Q) → 0 as � → ∞. �

In a similar way, we can also show the equivalence between renormalized entropy super-solutions and renormalized dissipative solutions of (CP). Thus, we complete the proof ofthe theorem.

4. Applications

We prove the existence of renormalized dissipative solutions of (CP) via relaxation meth-ods. Let i > 0 and suppose that for k = 1, 2, . . . and i = 1, 2, . . . , N , Vk,i satisfy theconditions

1 +N∑

i=1

1

Vk,i

inf|u|�kF ′

i (u) > 0,

1 + �

1 + ∑Nj=1 V −1

k,j inf |u|�k F ′j (u)

1

Vk,i

sup|u|�k

F ′i (u) < i ,

where � = ∑Ni=1 i . It is proved in [4, Lemma 4.1] that there are a strictly increasing

function rk : [−k, k] → R defined by w = rk(u) := (u + ∑Ni=1 V −1

k,i Fi(u))/(1 + �) andfunctions hk,i : [rk(−k), rk(k)] → R, satisfying the conditions dhk,i/dw < 0, hk,i(0) = 0such that

w −N∑

i=1

hk,i(w) = u and iw + hk,i(w) = Fi(u)

Vk,i

, u ∈ [−k, k].

We consider the following family of relaxation systems for w� and z� = (z�1, . . . , z

�N):

(CRS)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

�w�

�t+

N∑i=1

iVk,i

�w�

�xi

= 1

�

N∑i=1

(hk,i(w�) − z�

i ),

�z�i

�t− Vk,i

�z�i

�xi

= 1

�(hk,i(w

�) − z�i ), i = 1, . . . , N, � > 0,

with the initial conditions w�(0, x) = w0(x), z�(0, x) = z0(x) for x ∈ RN and a�w0 �b,hk,i(b)�zi0 �hk,i(a), where a < 0 and b > 0 are constants such that

−k�a +N∑

i=1

hk,i(b)�b +N∑

i=1

hk,i(a)�k.

The following proposition is obtained in [4, Theorem 4.2], and we can prove Theorem 4.2.

S. Takagi / Nonlinear Analysis 63 (2005) e2483–e2489 e2489

Proposition 4.1. Let k�1, u�=w�−∑Ni=1 z�

i and let u0 =w0 −∑Ni=1 zi0 ∈ L1(RN). Then

uk = lim�↓0u� exists in L1(Q) and uk is an entropy solution of (CP) with f = 0 satisfying

−k�uk �k.

Theorem 4.2. The limit function u above is a unique renormalized dissipative solution of(CP) with f = 0.

References

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[10] M. Portilheiro, Proc. R. Soc. Edinburgh Sect. A 133 (2003) 1193–1207.[11] M. Portilheiro, J. Differential Equations 195 (2003) 66–81.[12] S. Takagi, GAKUTO Internat. Ser. Math. Sci. Appl., to appear.