Stability properties for quasilinear parabolic equations with measure data

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Stability properties for quasilinear parabolic equations with measure data

Marie-Françoise Bidaut-Véron, Quoc-Hung Nguyen

To cite this version:

Marie-Françoise Bidaut-Véron, Quoc-Hung Nguyen. Stability properties for quasilinear parabolic

equations with measure data. 2014. �hal-01060682�

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Stability properties for quasilinear parabolic equations with measure data

Marie-Fran¸coise BIDAUT-VERON

^∗

Quoc-Hung NGUYEN

^†

.

Abstract

Let Ω be a bounded domain ofR^N, andQ= Ω×(0, T).We study problems of the model type







ut−∆pu=µ inQ, u= 0 on∂Ω×(0, T), u(0) =u⁰ in Ω,

where p > 1, µ ∈ M_b(Q) and u⁰ ∈ L¹(Ω). Our main result is a stability theorem extending the results of Dal Maso, Murat, Orsina, Prignet, for the elliptic case, valid for quasilinear operators u 7−→

A(u) =div(A(x, t,∇u)).

1 Introduction

Let Ω be a bounded domain ofR^N, and Q= Ω×(0, T), T >0.We denote byMb(Ω) andMb(Q) the sets of bounded Radon measures on Ω andQrespectively. We are concerned with the problem





ut−div(A(x, t,∇u)) =µ inQ,

u= 0 on∂Ω×(0, T),

u(0) =u0 in Ω,

(1.1)

whereµ∈ Mb(Q),u0 ∈L¹(Ω) andAis a Caratheodory function on Q×R^N, such that fora.e.(x, t)∈Q, and anyξ, ζ ∈R^N,

A(x, t, ξ).ξ≥Λ1|ξ|^p, |A(x, t, ξ)| ≤a(x, t) + Λ2|ξ|^p−1, Λ1,Λ2>0, a∈L^p^′(Q), (1.2) (A(x, t, ξ)−A(x, t, ζ)).(ξ−ζ)>0 ifξ6=ζ, (1.3) forp >1.This includes the model problem where div(A(x, t,∇u)) = ∆pu,where ∆p is thep-Laplacian.

The corresponding elliptic problem:

−∆pu=µ in Ω, u= 0 on∂Ω,

∗Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350, Faculté des Sciences, 37200 Tours France. E-mail:

veronmf@univ-tours.fr

†Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350, Faculté des Sciences, 37200 Tours France. E-mail:

Hung.Nguyen-Quoc@lmpt.univ-tours.fr

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withµ∈ Mb(Ω),was studied in [9, 10] forp >2−1/N,leading to the existence of solutions in the sense of distributions. For anyp >1,andµ∈L¹(Ω),existence and uniqueness are proved in [4] in the class ofentropy solutions. For anyµ∈ Mb(Ω) the main work is done in [14, Theorems 3.1, 3.2], where not only existence is proved in the class ofrenormalized solutions, but also a stability result, fundamental for applications.

Concerning problem (1.1), the first studies concern the caseµ∈L^p^′(Q) andu0∈L²(Ω), where existence and uniqueness are obtained by variational methods, see [19]. In the general case µ ∈ Mb(Q) and u0 ∈ Mb(Ω),the pionner results come from [9], proving the existence of solutions in the sense of distributions for

p > p1= 2− 1

N+ 1, (1.4)

see also [11]. The approximated solutions of (1.1) lie in Marcinkiewicz spacesu ∈ L^p^c^,∞(Q) and |∇u| ∈ L^m^c^,∞(Q),where

pc=p−1 + p

N, mc=p− N

N+ 1. (1.5)

This condition (1.4) ensures thatuand|∇u|belong toL¹(Q), sincemc>1 meansp > p1andpc>1 means p >2N/(N+ 1).Uniqueness follows in the case p= 2, A(x, t,∇u) =∇u,by duality methods, see [21].

For µ∈L¹(Q), uniqueness is obtained in new classes of entropy solutions, and renormalized solutions, see [5, 26, 27].

A larger set of measures is studied in [15]. They introduce a notion of parabolic capacity initiated and inspired by [24], used after in [22, 23], defined by

c^Q_p(E) = inf( inf

E⊂Uopen⊂Q{||u||W :u∈W, u≥χU a.e.in Q}), for any Borel setE⊂Q,where setting X=L^p((0, T);W₀^1,p(Ω)∩L²(Ω)),

W =

z:z∈X, zt∈X^′ , embedded with the norm||u||W =||u||X+||ut||X^′. LetM0(Q) be the set of Radon measuresµonQthat do not charge the sets of zeroc^Q_p-capacity:

∀E Borel set ⊂Q, c^Q_p(E) = 0 =⇒ |µ|(E) = 0.

Then existence and uniqueness of renormalized solutions of (1.1) hold for any measureµ∈ Mb(Q)∩ M0(Q), calledsoft (or diffuse, or regular) measure, andu0∈L¹(Ω), andp >1. The equivalence with the notion of entropy solutions is shown in [16]. For such a soft measure, an extension to equations of type (b(u))t−∆pu=µ is given in [6]; another formulation is used in [23] for solving a perturbed problem from (1.1) by an absorption term.

Next consider an arbitrary measure µ∈ Mb(Q).LetMs(Q) be the set of all bounded Radon measures onQwith support on a set of zero c^Q_p-capacity, also calledsingular. LetM⁺_b(Q),M⁺₀(Q),M⁺_s(Q) be the positive cones ofMb(Q),M0(Q),Ms(Q). From [15],µcan be written (in a unique way) under the form

µ=µ0+µs, µ0∈ M0(Q), µs=µ⁺_s −µ⁻_s, µ⁺_s, µ⁻_s ∈ M⁺_s(Q), (1.6) andµ0∈ M0(Q) admits (at least) a decomposition under the form

µ0=f−divg+ht, f ∈L¹(Q), g∈(L^p^′(Q))^N, h∈X, (1.7) and we writeµ0 = (f, g, h). Conversely, any measure of this form, such that h∈ L^∞(Q), lies in M0(Q), see [23, Proposition 3.1]. The solutions of (1.1) are searched in a renormalized sense linked to this decomposition, introduced in [15, 22]. In the range (1.4) the existence of a renormalized solution relative to the

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decomposition (1.7) is proved in [22], using suitable approximations ofµ0and µs. Uniqueness is still open, as well as in the elliptic case.

Inall the sequel we suppose that psatisfies (1.4). Then the embedding W₀^1,p(Ω)⊂L²(Ω) is valid, that means

X =L^p((0, T);W₀^1,p(Ω)), X^′=L^p^′((0, T);W^−1,p^′(Ω)).

In Section 2 we recall the definition of renormalized solutions, given in [22], that we call R-solutions of (1.1), relative to the decomposition (1.7) of µ0, and study some of their properties. Our main result is a stability theorem for problem (1.1), proved in Section 3, extending to the parabolic case the stability result of [14, Theorem 3.4]. In order to state it, we recall that a sequence of measuresµn∈ Mb(Q) converges to a measureµ∈ Mb(Q) in thenarrow topology of measures if

n→∞lim Z

Q

ϕdµn= Z

Q

ϕdµ ∀ϕ∈C(Q)∩L^∞(Q).

Theorem 1.1 Let A:Q×R^N →R^N satisfy (1.2),(1.3). Let u0∈L¹(Ω), and µ=f−divg+ht+µ⁺_s −µ⁻_s ∈ Mb(Q), withf ∈L¹(Q), g∈(L^p^′(Q))^N, h∈X andµ⁺_s, µ⁻_s ∈ M⁺_s(Q). Letu0,n∈L¹(Ω),

µn=fn−divgn+ (hn)t+ρn−ηn∈ Mb(Q), with fn∈L¹(Q), gn∈(L^p^′(Q))^N, hn∈X,andρn, ηn∈ M⁺_b(Q),such that

ρn =ρ¹_n−divρ²_n+ρn,s, ηn =η¹_n−divη_n²+ηn,s, withρ¹_n, η¹_n∈L¹(Q), ρ²_n, η_n² ∈(L^p^′(Q))^N andρn,s, ηn,s∈ M⁺_s(Q).Assume that

sup

n

|µn|(Q)<∞,

and {u0,n} converges to u0 strongly in L¹(Ω), {fn} converges to f weakly in L¹(Q), {gn} converges to g strongly in(L^p^′(Q))^N,{hn} converges to hstrongly in X,{ρn} converges to µ⁺_s and {ηn} converges to µ⁻_s in the narrow topology; and

ρ¹_n ,

η_n¹ are bounded in L¹(Q), and ρ²_n ,

η²_n bounded in(L^p^′(Q))^N. Let {un} be a sequence of R-solutions of





un,t−div(A(x, t,∇un)) =µn in Q, un= 0 on∂Ω×(0, T),

un(0) =u0,n in Ω.

(1.8)

relative to the decomposition (fn+ρ¹_n−η¹_n, gn+ρ²_n−η²_n, hn)of µn,0.Let Un=un−hn.

Then up to a subsequence,{un} converges a.e.in Qto a R-solution uof (1.1), and{Un} convergesa.e.

in Q to U = u−h. Moreover, {∇un},{∇Un} converge respectively to ∇u,∇U a.e. in Q, and {Tk(Un)}

converge toTk(U)strongly inX for any k >0.

In Section 4 we check that any measureµ ∈ Mb(Q) can be approximated in the sense of the stability Theorem, hence we find again the existence result of [22]:

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Corollary 1.2 Let u0∈L¹(Ω) andµ∈ Mb(Q). Then there exists a R-solution uto the problem (1.1) with data(µ, u0).

Moreover we give more precise properties of approximations ofµ∈ Mb(Q),fundamental for applications, see Propositions 4.1 and 4.2. As in the elliptic case, Theorem 1.1 is a key point for obtaining existence results for more general problems, and we give some of them in [2, 3, 20], for measuresµsatisfying suitable capacitary conditions. In [2] we study perturbed problems of order 0, of type

ut−∆pu+G(u) =µ in Q, (1.9) whereG(u) is an absorption or a source term with a growth of power or exponential type, and µis a good in time measure. In [3] we use potential estimates to give other existence results in case of absorption with p >2. In [20], one considers equations of the form

ut−div(A(x, t,∇u)) +G(u,∇u) =µ

under (1.2),(1.3) withp= 2,and extend in particular the results of [1] to nonlinear operators.

2 Renormalized solutions of problem (1.1)

2.1 Notations and Definition

For any functionf ∈ L¹(Q), we write R

Qf instead of R

Qf dxdt, and for any measurable set E ⊂Q, R

Ef instead ofR

Ef dxdt. For any open set̟ of R^m andF ∈(L^k(̟))^ν, k∈[1,∞], m, ν∈N^∗,we setkFk_k,̟ = kFk_(Lk(̟))^ν

We setTk(r) = max{min{r, k},−k},for any k >0 andr∈R. We recall that ifuis a measurable function defined and finitea.e. inQ, such thatTk(u)∈X for any k >0, there exists a measurable functionwfrom Qinto R^N such that∇Tk(u) = χ|u|≤kw, a.e.in Q,and for any k >0. We define the gradient∇uof uby w=∇u.

Letµ=µ0+µs∈ Mb(Q),and (f, g, h) be a decomposition ofµ0given by (1.7), andcµ0=µ0−ht=f−divg.

In the general caseµc0∈ M(Q),/ but we write, for convenience, Z

Q

wdcµ0:=

Z

Q

(f w+g.∇w), ∀w∈X∩L^∞(Q).

Definition 2.1 Let u0 ∈ L¹(Ω), µ = µ0+µs ∈ Mb(Q). A measurable function u is a renormalized solution, called R-solution of (1.1) if there exists a decompostion(f, g, h)of µ0 such that

U =u−h∈L^σ((0, T);W₀^1,σ(Ω))∩L^∞((0, T);L¹(Ω)), ∀σ∈[1, mc) ; Tk(U)∈X, ∀k >0, (2.1) and:

(i) for any S∈W^2,∞(R)such that S^′ has compact support on R, andS(0) = 0,

− Z

Ω

S(u0)ϕ(0)dx− Z

Q

ϕtS(U) + Z

Q

S^′(U)A(x, t,∇u).∇ϕ+ Z

Q

S^′′(U)ϕA(x, t,∇u).∇U = Z

Q

S^′(U)ϕdcµ0, (2.2) for anyϕ∈X∩L^∞(Q)such thatϕt∈X^′+L¹(Q)andϕ(., T) = 0;

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(ii) for anyφ∈C(Q),

m→∞lim 1 m

Z

{m≤U<2m}

φA(x, t,∇u).∇U = Z

Q

φdµ⁺_s (2.3)

m→∞lim 1 m

Z

{−m≥U>−2m}

φA(x, t,∇u).∇U = Z

Q

φdµ⁻_s. (2.4)

Remark 2.2 As a consequence, S(U)∈ C([0, T];L¹(Ω)) and S(U)(.,0) =S(u₀) in Ω; and u satisfies the equation

(S(U))_t−div(S^′(U)A(x, t,∇u)) +S^′′(U)A(x, t,∇u).∇U=f S^′(U)−div(gS^′(U)) +S^′′(U)g.∇U, (2.5) in the sense of distributions inQ, see [22, Remark 3]. Moreover assume that [−k, k]⊃ suppS^′. then from (1.2) and the H¨older inequality, we find easily that

kS(U)_tk_X^′_+L1(Q)≤CkSk_W2,∞(R)(|∇u|^pχ|U|≤k

^1/p_1,Q^′+|∇u|^pχ|U|≤k

_1,Q+k|∇Tk(U)|k^p_p,Q +kak_p′,Q+kak^p_p^′′,Q+kfk_1,Q+kgk_p′,Q

|∇u|^pχ|U|≤k

^1/p_1,Q+kgk_p′,Q), (2.6) whereC=C(p,Λ2). We also deduce that, for anyϕ∈X∩L^∞(Q), such thatϕt∈X^′+L¹(Q),

Z

Ω

S(U(T))ϕ(T)dx− Z

Ω

S(u0)ϕ(0)dx− Z

Q

ϕtS(U) + Z

Q

S^′(U)A(x, t,∇u).∇ϕ +

Z

Q

S^′′(U)A(x, t,∇u).∇U ϕ= Z

Q

S^′(U)ϕdcµ0. (2.7) Remark 2.3 Let u, U satisfy (2.1). It is easy to see that the condition (2.3) ( resp. (2.4) ) is equivalent to

m→∞lim 1 m

Z

{m≤U<2m}

φA(x, t,∇u).∇u= Z

Q

φdµ⁺_s (2.8)

resp.

m→∞lim 1 m

Z

{m≥U>−2m}

φA(x, t,∇u).∇u= Z

Q

φdµ⁻_s. (2.9)

In particular, for anyϕ∈L^p^′(Q)there holds

m→∞lim 1 m

Z

m≤|U|<2m

|∇u|ϕ= 0, lim

m→∞

1 m

Z

m≤|U|<2m

|∇U|ϕ= 0. (2.10)

Remark 2.4 (i) Any function U ∈ X such that Ut ∈ X^′+L¹(Q) admits a unique c^Q_p-quasi continuous representative, definedc^Q_p-quasia.e. in Q, still denoted U. Furthermore, ifU ∈L^∞(Q), then for any µ0 ∈ M0(Q),there holds U ∈L^∞(Q, dµ0), see [22, Theorem 3 and Corollary 1].

(ii) Letube any R- solution of problem (1.1). Then,U =u−hadmits ac^Q_p-quasi continuous functions representative which is finitec^Q_p-quasia.e.inQ,andusatisfies definition 2.1 for every decomposition( ˜f ,˜g,˜h) such thath−˜h∈L^∞(Q), see [22, Proposition 3 and Theorem 4 ].

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2.2 Steklov and Landes approximations

A main difficulty for proving Theorem 1.1 is the choice of admissible test functions (S, ϕ)in (2.2), valid for any R-solution. Because of a lack of regularity of these solutions, we use two ways of approximation adapted to parabolic equations:

Definition 2.5 Let ε∈(0, T) andz ∈L¹_loc(Q). For any l ∈ (0, ε)we define the Steklov time-averages [z]l,[z]−l of zby

[z]l(x, t) =1 l

Zt+l t

z(x, s)ds for a.e.(x, t)∈Ω×(0, T−ε),

[z]−l(x, t) = 1 l

Zt t−l

z(x, s)ds for a.e.(x, t)∈Ω×(ε, T).

The idea to use this approximation for R-solutions can be found in [7]. Recall some properties, given in [23].

Letε ∈(0, T),and ϕ1 ∈ C_c^∞(Ω×[0, T)), ϕ2 ∈C_c^∞(Ω×(0, T]) with Suppϕ1 ⊂Ω×[0, T −ε], Suppϕ2 ⊂ Ω×[ε, T]. There holds:

(i) Ifz∈X, thenϕ1[z]l andϕ2[z]−l∈W.

(ii) Ifz ∈X andzt∈X^′+L¹(Q), then, asl →0, (ϕ1[z]l) and (ϕ2[z]−l) converge respectively toϕ1zand ϕ2z inX,anda.e.in Q; and (ϕ1[z]l)_t,(ϕ2[z]−l)_tconverge to (ϕ1z)t,(ϕ2z)tin X^′+L¹(Q).

(iii) If moreoverz ∈ L^∞(Q), then from any sequence{ln} → 0, there exists a subsequence{lν} such that {[z]lν},{[z]−lν} converge toz, c^Q_p-quasi everywhere inQ.

Next we recall the approximation used in several articles [8, 12, 11], first introduced in [17].

Definition 2.6 Let k >0, and y ∈L^∞(Ω) and Y ∈X such that||y||_L^∞_(Ω) ≤k and||Y||_L^∞_(Q) ≤k. For anyν ∈N,aLandes-time approximationhYiν of the functionY is defined as follows:

hYiν(x, t) =ν Z t

0

Y(x, s)e^ν(s−t)ds+e^−νtzν(x), ∀(x, t)∈Q.

where{zν} is a sequence of functions in W₀^1,p(Ω)∩L^∞(Ω), such that ||zν||_L^∞_(Ω) ≤k,{zν} converges to y a.e.inΩ, andν⁻¹||zν||^p

W₀^1,p(Ω) converges to 0.

Therefore, we can verify that (hYiν)t∈X, hYiν ∈X∩L^∞(Q), ||hYiν||∞,Q ≤kand {hYiν} converges toY strongly inX anda.e.inQ. Moreover,hYiν satisfies the equation (hYiν)t=ν(Y − hYiν) in the sense of distributions inQ,andhYiν(0) =zν in Ω. In this paper, we only use theLandes-time approximation of the functionY =Tk(U),where y=Tk(u0).

2.3 First properties

In the sequel we use the following notations: for any functionJ ∈W^1,∞(R), nondecreasing withJ(0) = 0, we set

J(r) = Z r

0

J(τ)dτ, J(r) = Z r

0

J^′(τ)τ dτ. (2.11)

It is easy to verify thatJ(r)≥0,

J(r) +J(r) =J(r)r, and J(r)− J(s)≥s(J(r)−J(s)) ∀r, s∈R. (2.12)

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In particular we define, for anyk >0,and anyr∈R, Tk(r) =

Z r 0

Tk(τ)dτ, Tk(r) = Z r

0

T_k^′(τ)τ dτ, (2.13)

and we use several times a truncature used in [14]:

Hm(r) =χ[−m,m](r) +2m− |s|

m χm<|s|≤2m(r), Hm(r) = Z r

0

Hm(τ)dτ. (2.14) The next Lemma allows to extend the range of the test functions in (2.2).

Lemma 2.7 Letube a R-solution of problem (1.1). LetJ ∈W^1,∞(R)be nondecreasing withJ(0) = 0, and J defined by (2.11). Then,

Z

Q

S^′(U)A(x, t,∇u).∇(ξJ(S(U))) + Z

Q

S^′′(U)A(x, t,∇u).∇U ξJ(S(U))

− Z

Ω

ξ(0)J(S(u0))S(u0)dx−

Z

Q

ξtJ(S(U))≤ Z

Q

S^′(U)ξJ(S(U))dcµ0, (2.15) for any S ∈ W^2,∞(R) such that S^′ has compact support on R and S(0) = 0, and for any ξ ∈ C¹(Q)∩ W^1,∞(Q), ξ≥0.

Proof. Let J be defined by (2.11). Let ζ ∈ C_c¹([0, T)) with values in [0,1], such that ζt ≤ 0, and ϕ=ζξ[j(S(U))]l. Clearly,ϕ∈X∩L^∞(Q); we choose the pair of functions (ϕ, S) as test function in (2.2).

From the convergence properties of Steklov time-averages, we easily will obtain (2.15) if we prove that lim

l→0,ζ→1

(−

Z

Q

(ζξ[j(S(U))]_l)_tS(U))≥ − Z

Q

ξtJ(S(U)).

We can write−R

Q(ζξ[j(S(U))]_l)_tS(U) =F+G,with F =−

Z

Q

(ζξ)t[j(S(U))]_lS(U), G=− Z

Q

ζξS(U)1

l (j(S(U))(x, t+l)−j(S(U))(x, t)). Using (2.12) and integrating by parts we have

G≥ − Z

Q

ζξ1

l (J(S(U))(x, t+l)−J(S(U))(x, t)) =− Z

Q

ζξ∂

∂t([J(S(U))]_l)

= Z

Q

(ζξ)t[J(S(U))]_l+ Z

Ω

ζ(0)ξ(0)[J(S(U))]_l(0)dx≥ Z

Q

(ζξ)t[J(S(U))]_l, sinceJ(S(U))≥0.Hence,

− Z

Q

(ζξ[j(S(U))]_l)_tS(U)≥ Z

Q

(ζξ)t[J(S(U))]_l+F = Z

Q

(ζξ)t([J(S(U))]_l−[J(S(U))]_lS(U)). Otherwise,J(S(U)) and J(S(U))∈C([0, T] ;L¹(Ω)), thus {(ζξ)t([J(S(u))]_l−[J(S(u))]_lS(u))} converges to−(ζξ)tJ(S(u)) inL¹(Q) asl→0. Therefore,

lim

l→0,ζ→1(−

Z

Q

(ζξ[J(S(U))]_l)_tS(U))≥ lim

ζ→1

− Z

Q

(ζξ)_tJ(S(U))

≥ − Z

Q

ξtJ(S(U)),

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which achieves the proof.

Next we give estimates of the function and its gradient, following the first ones of [11], inspired by the estimates of the elliptic case of [4]. In particular we extend and make more precise the a priori estimates of [22, Proposition 4] given for solutions with smooth data; see also [15, 18].

Proposition 2.8 If uis a R-solution of problem (1.1), then there exists C1 =C1(p,Λ1,Λ2)such that, for

anyk≥1 andℓ ≥0, Z

ℓ≤|U|≤ℓ+k

|∇u|^p+ Z

ℓ≤|U|≤ℓ+k

|∇U|^p ≤C1kM, (2.16)

kUk_L∞(((0,T));L¹(Ω)) ≤C1(M +|Ω|), (2.17) whereM =ku0k_1,Ω+|µs|(Q)+kfk_1,Q+kgk^p_p^′′,Q+khk^p_X+||a||^p_p^′′,Q.

As a consequence, for anyk ≥1,

meas{|U|> k} ≤C2M1k^−p^c, meas{|∇U|> k} ≤C2M2k^−m^c, (2.18) meas{|u|> k} ≤C2M2k^−p^c, meas{|∇u|> k} ≤C2M2k^−m^c, (2.19) whereC2=C2(N, p,Λ1,Λ2), andM1= (M+|Ω|)^N^pM andM2=M1+M.

Proof. Set for anyr∈R, andm, k, ℓ >0,

Tk,ℓ(r) = max{min{r−ℓ, k},0}+ min{max{r+ℓ,−k},0}.

Form > k+ℓ, we can choose (J, S, ξ) = (Tk,ℓ, Hm, ξ) as test functions in (2.15), where Hm is defined at (2.14) andξ∈C¹([0, T]) with values in [0,1], independent onx. SinceTk,ℓ(Hm(r)) =Tk,ℓ(r) for allr∈R, we obtain

−R

Ωξ(0)Tk,ℓ(u0)Hm(u0)dx−R

QξtTk,ℓ(Hm(U))

+ R

{ℓ≤|U|<ℓ+k}

ξA(x, t,∇u).∇U−_m^k R

{m≤|U|<2m}

ξA(x, t,∇u).∇U ≤R

QHm(U)ξTk,ℓ(U)dcµ0.

And Z

Q

Hm(U)ξTk,ℓ(U)dµc0= Z

Q

Hm(U)ξTk,ℓ(U)f+ Z

{ℓ≤|U|<ℓ+k}

ξ∇U.g−k m

Z

{m≤|U|<2m}

ξ∇U.g.

Letm→ ∞; then, for anyk≥1,sinceU ∈L¹(Q) and from (2.3), (2.4), and (2.10), we find

− Z

Q

ξtTk,ℓ(U) + Z

{ℓ≤|U|<ℓ+k}

ξA(x, t,∇u).∇U ≤ Z

{ℓ≤|U|<ℓ+k}

ξ∇U.g+k(ku0k_1,Ω+|µs|(Q)+kfk_1,Q). (2.20)

Next, we takeξ≡1. We verify that

A(x, t,∇u).∇U− ∇U.g≥Λ1

4 (|∇u|^p+|∇U|^p)−c1(|g|^p^′+|∇h|^p+|a|^p^′)

for somec1=c1(p,Λ1,Λ2)>0. Hence (2.16) follows. Thus, from (2.20) and the H¨older inequality, we get, for anyξ∈C¹([0, T]) with values in [0,1],

− Z

Q

ξtTk,ℓ(U)≤c2kM

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for somec2=c2(p,Λ1,Λ2)>0.ThusR

ΩTk,ℓ(U)(t)dx≤c2kM,fora.e. t∈(0, T).We deduce (2.17) by taking k= 1, ℓ= 0, sinceT1,0(r) =T1(r)≥ |r| −1,for anyr∈R.

Next, from the Gagliardo-Nirenberg embedding Theorem, see [13, Proposition 3.1], we have Z

Q

|Tk(U)|^p(N+1)^N ≤c3kUk_L^N^p∞(((0,T));L¹(Ω))

Z

Q

|∇Tk(U)|^p, wherec3=c3(N, p).Then, from (2.16) and (2.17), we get, for anyk≥1,

meas{|U|> k} ≤k⁻^p(N+1)^N Z

Q

|Tk(U)|^p(N+1)^N ≤c3kUk

p N

L^∞((0,T);L¹(Ω))k⁻^p(N+1)^N Z

Q

|∇Tk(U)|^p≤c4M1k^−p^c, withc4=c4(N, p,Λ1,Λ2). We obtain

meas{|∇U|> k} ≤ 1 k^p

Z k^p 0

meas ({|∇U|^p > s})ds

≤measn

|U|> k^N+1^N o + 1

k^p Z k^p

0

measn

|∇U|^p > s,|U| ≤k^N+1^N o ds

≤c4M1k^−m^c+ 1 k^p

Z

|U|≤k^N+1^N

|∇U|^p≤c5M2k^−m^c,

withc5=c5(N, p,Λ1,Λ2).Furthermore, for anyk≥1,

meas{|h|> k}+ meas{|∇h|> k} ≤c6k^−pkhk^p_X, wherec6=c6(N, p). Therefore, we easily get (2.19).

Remark 2.9 If µ∈L¹(Q)anda≡0 in (1.2), then (2.16) holds for allk >0 and the term|Ω|in inequality (2.17) can be removed, whereM =||u0||1,Ω+|µ|(Q). Furthermore, (2.19) is stated as follows:

meas{|u|> k} ≤C2M^p+N^N k^−p^c, meas{|∇u|> k} ≤C2M^N+2^N+1k^−m^c,∀k >0. (2.21) withC2=C2(N, p,Λ1,Λ2).To see last inequality, we do in the following way:

meas{|∇U|> k} ≤measn

|U|> M^N+1¹ k^N+1^N o + 1

k^p Z k^p

0

measn

|∇U|^p> s,|U| ≤M^N+1¹ k^N+1^N o ds

≤C2M^N+2^N+1k^−m^c.

Proposition 2.10 Let {µn} ⊂ Mb(Q), and{u0,n} ⊂L¹(Ω), such that sup

n

|µn|(Q)<∞, and sup

n

||u0,n||1,Ω<∞.

Letun be a R-solution of (1.1) with data µn=µn,0+µn,sandu0,n,relative to a decomposition (fn, gn, hn) of µn,0, and Un =un−hn. Assume that {fn} is bounded in L¹(Q), {gn} bounded in (L^p^′(Q))^N and {hn} bounded inX.

Then, up to a subsequence,{Un} converges a.e. to a functionU ∈L^∞((0, T);L¹(Ω)), such that Tk(U)∈X for anyk >0andU ∈L^σ((0, T);W₀^1,σ(Ω))for any σ∈[1, mc). And

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(i){Un} converges to U strongly inL^σ(Q)for any σ∈[1, mc),andsupkUnk_L^∞_((0,T);L1(Ω)) <∞, (ii)sup_k>0sup_n _k+1¹ R

Q|∇Tk(Un)|^p <∞,

(iii){Tk(Un)} converges to Tk(U)weakly inX,for any k >0,

(iv){A(x, t,∇(Tk(Un) +hn))} converges to some Fk weakly in (L^p^′(Q))^N.

Proof. TakeS∈W^2,∞(R) such thatS^′has compact support onRandS(0) = 0. We combine (2.6) with (2.16), and deduce that{S(Un)t} is bounded inX^′+L¹(Q) and {S(Un)} bounded inX. Hence,{S(Un)}

is relatively compact inL¹(Q). On the other hand, we chooseS =Sk such thatSk(z) = z,if|z|< kand S(z) = 2ksignz,if|z|>2k. From (2.17), we obtain

meas{|Un−Um|> σ} ≤meas{|Un|> k}+ meas{|Um|> k}+ meas{|Sk(Un)−Sk(Um)|> σ}

≤ c

k+ meas{|Sk(Un)−Sk(Um)|> σ},

where c does not depend of n, m. Thus, up to a subsequence{un} is a Cauchy sequence in measure, and convergesa.e.inQto a functionu. Thus,{Tk(Un)}converges toTk(U) weakly inX, since sup_nkTk(Un)k_X <

∞for anyk >0. And

|∇(Tk(Un) +hn)|^p−2∇(Tk(Un) +hn) converges to someFk weakly in (L^p^′(Q))^N. Furthermore, from (2.18),{Un}strongly converges toU in L^σ(Q),for anyσ < pc.

3 The convergence theorem

We first recall some properties of the measures, see [22, Lemma 5], [14].

Proposition 3.1 Let µs = µ⁺_s −µ⁻_s ∈ Mb(Q), where µ⁺_s and µ⁻_s are concentrated, respectively, on two disjoint setsE⁺ andE⁻ of zeroc^Q_p-capacity. Then, for any δ >0, there exist two compact sets K_δ⁺⊆E⁺ andK_δ⁻ ⊆E⁻ such that

µ⁺_s(E⁺\K_δ⁺)≤δ, µ⁻_s(E⁻\K_δ⁻)≤δ,

and there exist ψ_δ⁺, ψ_δ⁻ ∈ C_c¹(Q) with values in [0,1], such that ψ⁺_δ, ψ_δ⁻ = 1 respectively on K_δ⁺, K_δ⁻, and supp(ψ_δ⁺)∩supp(ψ_δ⁻) =∅, and

||ψ⁺_δ ||X+||(ψ_δ⁺)t||_X^′_+L¹_(Q)≤δ, ||ψ_δ⁻||X+||(ψ_δ⁻)t||_X^′_+L¹_(Q)≤δ.

There exist decompositions(ψ⁺_δ)t= ψ⁺_δ1

t+ ψ⁺_δ2

t and(ψ⁻_δ)t= ψ⁻_δ1

t+ ψ⁻_δ2

t in X^′+L¹(Q), such that

ψ_δ⁺1 t

X^′ ≤ δ

3, ψ⁺_δ2 t

1,Q≤ δ

3, ψ⁻_δ1 t

X^′ ≤ δ

3, ψ⁻_δ2 t

1,Q≤ δ

3. (3.1)

Both

ψ_δ⁺ and

ψ_δ⁻ converge to0, weak-^∗ inL^∞(Q), and strongly inL¹(Q)and up to subsequences,a.e.

inQ,asδ tends to0.

Moreover ifρn andηn are as in Theorem 1.1, we have, for anyδ, δ1, δ2>0, Z

Q

ψ⁻_δdρn+ Z

Q

ψ_δ⁺dηn=ω(n, δ), Z

Q

ψ⁻_δ dµ⁺_s ≤δ, Z

Q

ψ_δ⁺dµ⁻_s ≤δ, (3.2) Z

Q

(1−ψ⁺_δ₁ψ_δ⁺₂)dρn=ω(n, δ1, δ2), Z

Q

(1−ψ⁺_δ₁ψ_δ⁺₂)dµ⁺_s ≤δ1+δ2, (3.3) Z

Q

(1−ψ⁻_δ₁ψ_δ⁻₂)dηn =ω(n, δ1, δ2), Z

Q

(1−ψ_δ⁻₁ψ_δ⁻₂)dµ⁻_s ≤δ1+δ2. (3.4)

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Hereafter, if n, ε, ..., ν are real numbers, and a function φdepends onn, ε, ..., ν and eventual other pa- rametersα, β, .., γ, and n → n0, ε → ε0, .., ν → ν0, we write φ = ω(n, ε, .., ν), then this means that, for fixed α, β, .., γ, there holds limν→ν0..limε→ε0limn→n0|φ| = 0. In the same way, φ≤ ω(n, ε, δ, ..., ν) means limν→ν0..limε→ε0limn→n0φ≤0, andφ≥ω(n, ε, .., ν) means−φ≤ω(n, ε, .., ν).

Remark 3.2 In the sequel we recall a convergence property still used in [14]: If {b1,n} is a sequence in L¹(Q)converging to b1 weakly inL¹(Q) and{b2,n} a bounded sequence in L^∞(Q) converging tob2, a.e. in Q,thenlimn→∞R

Qb1,nb2,n=R

Qb1b2. Next we prove Thorem 1.1.

Scheme of the proof. Let {µn},{u0,n} and {un} satisfy the assumptions of Theorem 1.1. Then we can apply Proposition 2.10. Setting Un = un −hn, up to subsequences, {un} converges a.e. in Q to some function u, and {Un} converges a.e. to U = u−h, such that Tk(U) ∈ X for any k > 0, and U ∈L^σ((0, T);W₀^1,σ(Ω))∩L^∞((0, T);L¹(Ω)) for everyσ∈[1, mc). And{Un} satisfies the conclusions (i) to (iv) of Proposition 2.10. We have

µn= (fn−divgn+ (hn)t) + (ρ¹_n−divρ²_n)−(η_n¹−divη_n²) +ρn,s−ηn,s

=µn,0+ (ρn,s−ηn,s)⁺−(ρn,s−ηn,s)⁻, where

µn,0=λn,0+ρn,0−ηn,0, withλn,0=fn−divgn+ (hn)t, ρn,0=ρ¹_n−divρ²_n, ηn,0=η_n¹−divη_n². (3.5) Hence

ρn,0, ηn,0∈ M⁺_b(Q)∩ M0(Q), and ρn≥ρn,0, ηn≥ηn,0. (3.6) LetE⁺, E⁻ be the sets where, respectively,µ⁺_s andµ⁻_s are concentrated. For anyδ1, δ2>0, letψ⁺_δ₁, ψ_δ⁺₂and ψ_δ⁻₁, ψ⁻_δ₂ as in Proposition 3.1 and set

Φδ1,δ2 =ψ_δ⁺₁ψ_δ⁺₂+ψ_δ⁻₁ψ⁻_δ₂. Suppose that we can prove the two estimates, near E

I1:=

Z

{|Un|≤k}

Φδ1,δ2A(x, t,∇un).∇(Un−hTk(U)iν)≤ω(n, ν, δ1, δ2), (3.7)

and far from E, I2:=

Z

{|Un|≤k}

(1−Φδ1,δ2)A(x, t,∇un).∇(Un−hTk(U)iν)≤ω(n, ν, δ1, δ2). (3.8)

Then it follows that

limn,ν

Z

{|Un|≤k}

A(x, t,∇un).∇(Un−hTk(U)i_ν)≤0, (3.9) which implies

limn→∞

Z

{|Un|≤k}

A(x, t,∇un).∇(Un−Tk(U))≤0, (3.10)

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since{hTk(U)i_ν} converges toTk(U) inX. On the other hand, from the weak convergence of{Tk(Un)} to Tk(U) in X,we verify that

Z

{|Un|≤k}

A(x, t,∇(Tk(U) +hn)).∇(Tk(Un)−Tk(U)) =ω(n).

Thus we get Z

{|Un|≤k}

(A(x, t,∇un)−A(x, t,∇(Tk(U) +hn))).∇(un−(Tk(U) +hn)) =ω(n).

Then, it is easy to show that, up to a subsequence,

{∇un} converges to∇u, a.e.inQ. (3.11)

Therefore,{A(x, t,∇un)} converges toA(x, t,∇u) weakly in (L^p^′(Q))^N ; and from (3.10) we find limn→∞

Z

Q

A(x, t,∇un).∇Tk(Un)≤ Z

Q

A(x, t,∇u)∇Tk(U).

Otherwise,{A(x, t,∇(Tk(Un) +hn))}converges weakly in (L^p^′(Q))^Nto someFk,from Proposition 2.10, and we obtain thatFk =A(x, t,∇(Tk(U) +h)). Hence

limn→∞

Z

Q

A(x, t,∇(Tk(Un) +hn)).∇(Tk(Un) +hn)

≤limn→∞

Z

Q

A(x, t,∇un).∇Tk(Un) + limn→∞

Z

Q

A(x, t,∇(Tk(Un) +hn)).∇hn

≤ Z

Q

A(x, t,∇(Tk(U) +h)).∇(Tk(U) +h).

As a consequence

{Tk(Un)} converges toTk(U), strongly inX, ∀k >0. (3.12) Thento finish the proof we have to check that uis a solution of (1.1).

In order to prove (3.7) we need a first Lemma, inspired of [14, Lemma 6.1]. It extends the results of [22, Lemma 6 and Lemma 7] relative to sequences of solutions with smooth data:

Lemma 3.3 Let ψ1,δ, ψ2,δ ∈C¹(Q) be uniformly bounded in W^1,∞(Q) with values in [0,1], and such that R

Qψ1,δdµ⁻_s ≤δandR

Qψ2,δdµ⁺_s ≤δ. Let{un}satisfying the assumptions of Theorem 1.1, andUn=un−hn.

Then 1

m Z

{m≤Un<2m}

|∇un|^pψ2,δ=ω(n, m, δ), 1 m

Z

{m≤Un<2m}

|∇Un|^pψ2,δ=ω(n, m, δ), (3.13) 1

m Z

−2m<Un≤−m

|∇un|^pψ1,δ=ω(n, m, δ), 1 m

Z

−2m<Un≤−m

|∇Un|^pψ1,δ=ω(n, m, δ), (3.14) and for anyk >0,

Z

{m≤Un<m+k}

|∇un|^pψ2,δ=ω(n, m, δ),

Z

{m≤Un<m+k}

|∇Un|^pψ2,δ=ω(n, m, δ), (3.15)

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Z

{−m−k<Un≤−m}

|∇un|^pψ1,δ=ω(n, m, δ),

Z

{−m−k<Un≤−m}

|∇Un|^pψ1,δ=ω(n, m, δ). (3.16)

Proof. (i) Proof of (3.13), (3.14). Set for anyr∈Rand anym, ℓ≥1 Sm,ℓ(r) =

Z r 0

−m+τ

m χ[m,2m](τ) +χ(2m,2m+ℓ](τ) +4m+ 2h−τ

2m+ℓ χ(2m+ℓ,4m+2h](τ)

dτ,

Sm(r)=

Z r 0

−m+τ

m χ[m,2m](τ) +χ(2m,∞)(τ)

dτ .

Note thatS_m,ℓ^′′ =χ[m,2m]/m−χ[2m+ℓ,2(2m+ℓ)]/(2m+ℓ).We choose (ξ, J, S) = (ψ2,δ, T1, Sm,ℓ) as test functions in (2.15) forun,and observe that, from (3.5),

d

µn,0=µn,0−(hn)t=λdn,0+ρn,0−ηn,0=fn−divgn+ρn,0−ηn,0. (3.17) Thus we can writeP6

i=1Ai≤P12

i=7Ai,where A1=−

Z

Ω

ψ2,δ(0)T1(Sm,ℓ(u0,n))Sm,ℓ(u0,n)dx, A2=− Z

Q

(ψ2,δ)_tT1(Sm,ℓ(Un)), A3=

Z

Q

S_m,ℓ^′ (Un)T1(Sm,ℓ(Un))A(x, t,∇un)∇ψ2,δ, A4= Z

Q

(S_m,ℓ^′ (Un))²ψ2,δT^′₁(Sm,ℓ(Un))A(x, t,∇un)∇Un, A5= 1

m Z

{m≤Un≤2m}

ψ2,δT1(Sm,ℓ(Un))A(x, t,∇un)∇Un,

A6=− 1 2m+ℓ

Z

{2m+ℓ≤Un<2(2m+ℓ)}

ψ2,δA(x, t,∇un)∇Un,

A7= Z

Q

S_m,ℓ^′ (Un)T1(Sm,ℓ(Un))ψ2,δfn, A8= Z

Q

S_m,ℓ^′ (Un)T1(Sm,ℓ(Un))gn.∇ψ2,δ, A9=

Z

Q

S_m,ℓ^′ (Un)2

T₁^′(Sm,ℓ(Un))ψ2,δgn.∇Un, A10= 1 m

Z

m≤Un≤2m

T1(Sm,ℓ(Un))ψ2,δgn.∇Un,

A11=− 1 2m+ℓ

Z

{2m+ℓ≤Un<2(2m+ℓ)}

ψ2,δgn.∇Un, A12= Z

Q

S_m,ℓ^′ (Un)T1(Sm,ℓ(Un))ψ2,δd(ρn,0−ηn,0).

Since||Sm,ℓ(u0,n)||1,Ω≤ R

{m≤u0,n}

u0,ndx, we findA1=ω(ℓ, n, m). Otherwise

|A2| ≤ kψ2,δk_W1,∞(Q)

Z

{m≤Un}

Un, |A3| ≤ kψ2,δk_W1,∞(Q)

Z

{m≤Un}

|a|+ Λ2|∇un|^p−1 ,

which implyA2=ω(ℓ, n, m) andA3=ω(ℓ, n, m).Using (2.3) forun, we have A6=−

Z

Q

ψ2,δd(ρn,s−ηn,s)⁺+ω(ℓ) =ω(ℓ, n, m, δ).

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Hence A6 = ω(ℓ, n, m, δ), since (ρn,s−ηn,s)⁺ converges to µ⁺_s as n → ∞ in the narrow topology, and R

Qψ2,δdµ⁺_s ≤δ.We also obtainA11=ω(ℓ) from (2.10).

Nown

S_m,ℓ^′ (Un)T1(Sm,ℓ(Un))o

ℓconverges toS_m^′ (Un)T1(Sm(Un)),{S_m^′ (Un)T1(Sm(Un))}_nconverges toS_m^′ (U) T1(Sm(U)),{S_m^′ (U)T1(Sm(U))}_mconverges to 0, weak-^∗inL^∞(Q) and{fn}converges tof weakly inL¹(Q), {gn} converges tog strongly in (L^p^′(Q))^N. From Remark 3.2, we obtain

A7= Z

Q

S_m^′ (Un)T1(Sm(Un))ψ2,δfn+ω(ℓ) = Z

Q

S_m^′ (U)T1(Sm(U))ψ2,δf+ω(ℓ, n) =ω(ℓ, n, m), A8=

Z

Q

S_m^′ (Un)T1(Sm(Un))gn.∇ψ2,δ+ω(ℓ) = Z

Q

S^′_m(U)T1(Sm(U))g∇ψ2,δ+ω(ℓ, n) =ω(ℓ, n, m).

Otherwise,A12≤R

Qψ2,δdρn, andnR

Qψ2,δdρn

oconverges toR

Qψ2,δdµ⁺_s,thusA12≤ω(ℓ, n, m, δ).

Using Holder inequality and the condition (1.2), we have gn.∇Un−A(x, t,∇un)∇Un≤c1

|gn|^p^′+|∇hn|^p+|a|^p^′

withc1=c1(p,Λ1,Λ2), which implies A9−A4≤c1

Z

Q

S_m,ℓ^′ (Un)2

T^′₁(Sm,ℓ(Un))ψ2,δ

|gn|^p^′+|hn|^p+|a|^p^′

=ω(ℓ, n, m).

Similarly we also show thatA10−A5/2≤ω(ℓ, n, m). Combining the estimates, we getA5/2≤ω(ℓ, n, m, δ).

Using Holder inequality we have

A(x, t,∇un)∇Un≥Λ1

2 |∇un|^p−c2(|a|^p^′+|∇hn|^p).

withc2=c2(p,Λ1,Λ2), which implies 1

m Z

{m≤Un<2m}

|∇un|^pψ2,δT1(Sm,ℓ(Un)) =ω(ℓ, n, m, δ).

Note that for allm >4,Sm,ℓ(r)≥1 for anyr∈[³₂m,2m]; henceT1(Sm,ℓ(r)) = 1.So, 1

m Z {³2m≤Un<2m}

|∇un|^pψ2,δ =ω(ℓ, n, m, δ).

Since|∇Un|^p≤2^p−1|∇un|^p+ 2^p−1|∇hn|^p, there also holds 1

m Z {³2m≤Un<2m}

|∇Un|^pψ2,δ=ω(ℓ, n, m, δ).

We deduce (3.13) by summing on each set

(⁴₃)ⁱm≤Un ≤(⁴₃)ⁱ⁺¹m fori= 0,1,2.Similarly, we can choose (ξ, ψ, S) = (ψ1,δ, T1,S˜m,ℓ) as test functions in (2.15) forun,where ˜Sm,ℓ(r) =Sm,ℓ(−r),and we obtain (3.14).

(ii) Proof of (3.15), (3.16). We set, for anyk, m, ℓ≥1, Sk,m,ℓ(r) =

Z r 0

Tk(τ−Tm(τ))χ_[m,k+m+ℓ]+k2(k+ℓ+m)−τ

k+m+ℓ χ(k+m+ℓ,2(k+m+ℓ)]

dτ