On uniqueness and existence of entropy solutions for a nonlinear ...

J.evol.equ. 8 (2008) 449–490 © 2008 Birkhäuser Verlag, Basel 1424-3199/08/030449-42, published online June 03, 2008 DOI 10.1007/s00028-008-0365-8

On uniqueness and existence of entropy solutions for a nonlinear parabolic problem with absorption Boris Andreianov, Karima Sbihi and Petra Wittbold

Abstract. The aim of this paper is to study, for L1 -data, the absorption problem of parabolic type : ut − div a(u, Du) + β(x, u) f with Dirichlet boundary conditions and initial conditions. Here a satisfies the classical Leray-Lions hypotheses and β(x, ·) is the subdifferential ∂j(x, ·), where j is a convex function such that j(·, 0) = 0. Existence and uniqueness of an entropy solution is established.

1. Introduction Let be a bounded open set in RN , 1 < p < N. Consider the nonlinear parabolic problem with absorption   ut − div a(u, Du) + β(x, u) f in Q := (0, T) × (P)(u0 , f) u = 0 on := (0, T) × ∂  u(0, ·) = u0 in , where T > 0, f ∈ L1 (Q), u0 ∈ L1 (), Du denotes the gradient of u and, for a.e. x ∈ , β(x, ·) = ∂j(x, ·) is the subdifferential of a function j : × R → [0, ∞] which is convex, lower semicontinuous (l.s.c. for short) in r ∈ R for a.e. x ∈ with j(·, 0) = 0, and the vector field a : R × RN → RN is continuous satisfying the following classical LerayLions-type conditions : (H1 )- monotonicity in ξ ∈ RN : (a(r, ξ) − a(r, η)) · (ξ − η) ≥ 0 ∀r ∈ R, ∀ξ, η ∈ RN ; (H2 )- coerciveness : there exist λ0 > 0, p > 1 such that (a(r, ξ) − a(r, 0)) · ξ ≥ λ0 |ξ|p ∀r ∈ R, ∀ξ ∈ RN ; (H3 )- growth restriction : there exists a function : R+ → R such that |a(r, ξ)| ≤ (|r|)(1 + |ξ|p−1 ) ∀r ∈ R, ∀ξ ∈ RN ; Mathematics subject classification 2000: 35K55, 31C15, 35B20, 47H06. Key-words: absorption problem, weak and entropy solution, capacity, L1 -data, nonlinear semigroup theory, doubling of variables.

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and, moreover, (H4 )- there exists C : R × R → R continuous such that |a(r, ξ) − a(s, ξ)| ≤ C(r, s)|r − s|(1 + |ξ|p−1 ) ∀r, s ∈ R. The graph β may be multivalued. In particular, one obtains the obstacle problem by taking j(x, r) = 0 if ψ1 (r) ≤ r ≤ ψ2 (r), = +∞ otherwise, with ψ1 : → [−∞, 0], ψ2 : → [0, +∞] measurable. The study of the obstacle problem has received a great deal of attention. We refer to [8, 32, 33, 34] for the linear case, and to [14, 15, 28, 29] for the nonlinear case. Problems of type (P) have already been studied in the literature, see for instance [6, 30, 18, 11, 36, 39, 40]. In [40] the elliptic diffusion-absorption problem u − div a(·, Du) + β(·, u) f on , u|∂ = 0

(1.1)

is considered. Existence, uniqueness and continuous dependence on the absorption term β are shown within a class of generalized solutions for data f ∈ L1 () ∩ L∞ (). Following [40], by a generalized solution of (1.1) we mean a function u ∈ L1 () ∩ L∞ () such that there exists a measure µ not charging sets of null p-capacity, i.e., the capacity defined from the Sobolev space W 1,p (RN ), such that u−div a(·, Du) + µ = f in D () and µ(x) ∈ 1,p ∂J (u(x)) for a.e. x ∈ , where J : W0 () ∩ L∞ () → [0, +∞] is the functional defined by J (u) = j(·, u)dx. Thus, the measure µ realizes the formal absorption term β(x, u(x)). Thanks to Bouchitté’s result [17], the elements in this subdifferential can be characterized as follows : µr (x) ∈ ∂j(x, u(x)) + ∂I[γ− (x),γ+ (x)] (u(x)) a.e. x ∈ (1.2) µ ∈ ∂J (u) ⇐⇒ ˜ = γ+ µ+ u˜ = γ− µ− s -a.e. on , u s -a.e. on , where u˜ is the quasi-continuous representative of u, γ+/− are quasi-continuous functions delimiting the domain of the functional j and µr +µs is the Radon-Nikodym decomposition of the measure µ (the notion of quasi-continuity with respect to the p-capacity is recalled in Section 2). As a consequence of the result of [40], we can recast in the abstract form ut + Au f, u(0) = u0 the corresponding parabolic problem (P)(u0 , f) and solve it from the point of view of nonlinear semigroup theory. Therefore, for every initial datum in L1 () there exists a unique mild solution of (P) (see [10, 12] for the definition of mild solutions). However, in principle, it is not clear how these mild solutions have to be interpreted. The purpose of the present paper is to characterize intrinsically these mild solutions, for which the problem is well posed. The main interest in our work is that we are dealing with the general operator −div a(u, Du). We need to prove that a solution of (P) exists in a sense which should ensure uniqueness; this machinery was developed, in both the elliptic and the parabolic cases, with the use of the notion of entropy solutions. Such solutions were

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first introduced in [11] where the authors studied existence of solutions of elliptic equations −div a(·, Du) = f . Since them, many results on entropy solutions of various problems were obtained; see, among others, [3, 16, 29, 37, 5, 19, 36]. Our proof of the existence result of entropy solutions consists of three steps. In the first step, we study the approximate problem (Pλ ) : ut − div a(u, Du) + βλ (·, u) = f on Q (+Dirichlet boundary condition and initial condition), where βλ is, roughly speaking, the Yosida approximation of the graph β, and prove that the mild solution of the corresponding Cauchy problem is a weak solution of Problem (Pλ ). In the second step, we pass to the limit with λ. In order to deal with, we need L∞ estimates of the sequence (βλ (·, uλ ))λ , which are not easy to obtain. To overcome this difficulty we add to Problem (Pλ ) a monotone function ψm,n (u). This type of perturbations was already used in [3, 2, 4]. Due to the strongly monotone perturbation term, one can prove an L∞ -estimate and, in particular, the strong compactness of the sequence of solutions uλ and also its strong convergence in L1 to a measurable function u. This allows to pass to the limit as λ → 0 in Problem (Pλ ) with a fixed perturbation ψm,n ; in particular in the nonlinearity a(u, Du) we pass to the limit using the standard pseudo-monotonicity argument (see for instance [13]). Another obstacle that we encounter in this approach is that a measure µ appears as the limit as λ → 0 of these measures βλ (·, uλ ) and which verifies ut + µ ∈ Lp (0, T ; W −1,p ()) + L1 (Q). It is not easy to separate the time derivative and the measure and, in particular, to characterize this measure as in [40, 38]. To this aim we first prove that this measure does not charge sets of of null p-capacity for almost every t, and verifies (1.2) for a.e. t. Secondly, we prove that it belongs to the space L10 (0, T ; w∗−Mb ())a which (see Lemma 2.1 below) is continuously embedded in the dual 1,p space of Lp (0, T ; W0 ()) ∩ L∞ (Q). A kind of "maximal regularity" property follows: each of the terms ut , µ belongs to the same space (Lp (0, T ; W 1,p ()) ∩ L∞ (Q))∗ as the sum ut + µ ∈ Lp (0, T ; W −1,p ()) + L1 (Q) ⊂ (Lp (0, T ; W 1,p ()) ∩ L∞ (Q))∗ . In the third step, using bi-monotone approximations u0m,n , fm,n of the data u0 , f in the way of [3], we obtain a monotone sequence of weak solutions um,n of Problem (Pm,n ) : ut − div a(u, Du) + β(·, u) + ψm,n (u) fm,n . Moreover, a comparison principle holds for weak solutions corresponding to different penalization parameters: for any m, n, m, ˜ n˜ ∈ N ∗ with ± ± ± m ˜ > m, n > n˜ we get ±um,n˜ ≤ ±um,n ≤ ±um,n a.e. on Q. By means of the monotone ˜ convergence theorem, this comparison result ensures the strong convergence in L1 (Q) of the sequence (um,n )m,n to a function u as m, n → ∞. In the same way, we also prove the strong monotone convergence of measures µm,n in L10 (0, T ; w∗− Mb ()), and also, by 1,p Lemma 2.1, in the dual space of Lp (0, T ; W0 ()) ∩ L∞ (Q). This strong convergence allows the same characterization (1.2) for the limit measure for almost every t. To pass to a L1 (0, T ; w∗− M ()) := {µ : (0, T) → M () weak-∗ measurable; b b

not charge sets of null capacity for a.e. t}

T

0 µ(t)Mb () dt < ∞, µ(t) does

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the limit in the nonlinearity a(um,n , Dum,n ), essential tools are the regularization method of Landes [27] and the strong convergence of the sequences of solutions and measures. Finally, we deduce that u is an entropy solution of the limit problem. The main difficulty when dealing with the general operator −div a(u, Du) is the uniqueness of solutions. This question is more difficult and most arguments used in the literature are based on doubling of variables method (see for instance [26, 19, 24, 20]). In this paper, the uniqueness of entropy solutions is proved through this method, and the proof goes essentially as follows: by using the doubling of variables in time technique, we prove that, if u is an entropy solution of (P), then a weak solution um,n of (Pm,n ) converges, as m, n → ∞, strongly in L1 to u for almost every t. Using the method of doubling of variables again, we derive a contraction principle of weak solutions of Problem (Pm,n ) : 0 0 |u − v |(t) ≤ |u m,n m,n m,n − vm,n | + Q |fm,n − gm,n |. Thereby we obtain a contraction principle for entropy solutions for almost every t. It is an open problem how to prove directly a comparison principle for entropy solutions without using the approximation method. Before concluding this introduction, let us mention the result of Droniou et al [24] on the existence of solutions to the problem ut − div a(t, x, Du) = µ, where µ is a measure not charging sets of Q of null capacity. In [24] the authors have developed a theory of capacity related to the parabolic operator ut − div a(t, x, Du) and proved a representation theorem for measures that are zero on subsets of Q of null capacity. It would be natural, in order to study the absorption problem (P), to use the parabolic capacities introduced in [24] in order to characterize the measure µ which realizes the formal absorption term β(x, u(t, x)). To this end, one should prove that solutions u belong to the 1,p space {u ∈ Lp (0, T ; W0 ())∩L∞ (0, T ; L2 ()), ut ∈ Lp (0, T ; W −1,p ())+L1 (Q)}. As mentioned above, the equation only yields ut +µ ∈ Lp (0, T ; W −1,p ())+L1 (Q), and in order to use the theory of [24] one needs to show that each of ut , µ belongs to this same space. Proving somewhat stronger (see Lemma 2.1) property µ ∈ L10 (0, T ; w∗−Mb ()), we were able to bypass the use of parabolic capacity, characterizing µ(t) by (1.2) for a.e. t. The paper is structured as follows: in the next section, we briefly set up notations and terminology used throughout this paper. Section 3 is devoted to the study of a perturbed problem obtained by adding a monotone bi-Lipschitz absorption term. Existence of a weak solution is proved for L∞ -data. In Section 4, under suitable hypotheses on the data u0 , we give the proof of existence and uniqueness of an entropy solution for Problem (P)(u0 , f). It is shown that a weak solution of the perturbed problem converges to an entropy solution. Finally, in the Appendix we give the proof of the technical Lemma 2.1.

2. Notations and some tools In this section, we introduce some notations and definitions used in this paper. We denote by |A| the Lebesgue measure of a set A ⊂ RN and by χA the characteristic function of

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1,p

A. For 1 ≤ p < ∞, Lp () and W0 () denote respectively the standard Lebesgue and Sobolev spaces. For k ≥ 0, we denote by Tk the truncation function at the level k, defined by ksign0 (u) if |u| > k Tk (u) = u if |u| ≤ k, where sign0 (·) denotes the single-valued function defined by sign0 (r) = −1 if r < 0, − sign0 (r) = 1 if r > 0, sign0 (r) = 0 if r = 0. We denote by sign+ 0 (·) and sign0 (·) − + the functions defined by sign0 (r) = 1 if r > 0, = 0 otherwise, and sign0 (r) = −1 if r < 0, = 0 otherwise. + − − For ε > 0 we denote by p+ ε (·), pε (·) the approximate functions of sign0 (·), sign0 (·), respectively, defined as follows:    1 if r > ε  0 if r > 0 − r r p (r) = p+ if 0 ≤ r ≤ ε if −ε ≤ r ≤ 0 ε (r) = ε  ε  ε 0 if r < 0 −1 if r < −ε. Throughout the paper, for the sake of simplicity, for u a function of (t, x) and for k a real number, we denote, for example, {|u| ≤ k} the set {(t, x) ∈ Q; |u(t, x)| ≤ k}.1We also write u for Q Q u(t, x)dtdx, etc... We denote by P the set of functions {S ∈ C (R)/ S(0) =

0, 0 ≤ S ≤ 1, Supp(S ) is compact}. In the sequel C denotes a constant that may change from line to line. As we said in the introduction, our framework is the theory of nonlinear semigroups. We refer to [10, 12] for most basic definitions and tools; however, among them, let us recall explicitly some that will play a crucial role in the methods we use. Let A be a multivalued operator in L1 (). Recall that A is said to be accretive in L1 () if u − u ˜ 1 ≤ u − u˜ + α(v − v˜ )1 for any (u, v), (u, ˜ v˜ ) ∈ A and α > 0. That is, for any α > 0, the resolvent JαA = (I + αA)−1 of A is a single-valued operator and a contraction in L1 -norm. A is called T -accretive if (u − u) ˜ + 1 ≤ u − u˜ + α(v − v˜ )+ 1 for any (u, v), (u, ˜ v˜ ) ∈ A and α > 0. Finally, A is called m-accretive (resp. m-T -accretive) in L1 () if A is accretive (resp. T -accretive) and moreover, R(I + αA) = L1 () for any α > 0. For a monotone graph β in R × R and λ ∈ N we denote by βλ the Yosida approximation of β, given by βλ = λ1 (I − (I + λβ)−1 ). The function βλ is monotone and Lipschitz. We recall the definition of the main section β0 of β :   inf β(r) if r > 0 β0 (r) = 0 if r = 0  sup β(r) if r < 0, with the usual convention inf ∅ = +∞ and sup ∅ = −∞. We need also the following definition β(r + ) = inf β(]r, +∞[), β(r − ) = sup β(] − ∞, r[).

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Now, let us introduce some notations about capacities and measures used throughout this paper (we refer the reader to [41, 25, 31] for a background material on the theory of capacities and measures). Given B ⊂ , cap(B) denotes the p-capacity of B with respect 1,p to the norm of W0 () and it is defined in the following way : If K ⊂ is compact, then 1,p

cap(K) = inf{ϕ1,p ; ϕ ∈ W0 (), ϕ ≥ 1 a.e. on a neighbourhood of K}. The p-capacity of an open subset O ⊂ is defined by cap(O) = sup{cap(K), K compact, K ⊆ O}, and that of an arbitrary subset B ⊂ is cap(B) = inf{cap(O), O open, B ⊆ O}. A function u defined on is said to be cap-quasi continuous if for every ε > 0 there exists an open set O ⊆ with cap(O) < ε such that the restriction of u to \O is continuous. 1,p It is well-known that every function in W0 () has a cap-quasi continuous representative, i.e., a function u˜ : → R such that u = u˜ a.e. on and u˜ is cap-quasi-continuous. As usual, a property will be said to hold cap-quasi everywhere (q.e. for short) if it holds everywhere except on a set of null p-capacity. Let Mb () be the space of the Radon measures on with bounded total variation. For µ ∈ Mb () denote by µ+ , µ− and |µ| the positive part, negative part and the total variation of the measure µ, respectively, and denote by µ = µr dx+µs the Radon-Nikodym decomposition of µ relatively to the Lebesgue measure dx. We denote by M0 () the set of Radon measures µ which satisfy µ(B) = 0 for every Borel set B ⊆ such that cap(B) = 0, i.e., the Radon measures which do not charge sets of null p-capacity. We denote J0 () = {j/j : × R → [0, ∞], j(·, r) measurable in x ∀r ∈ R, j(x, ·) convex, l.s.c. satisfying j(x, 0) = 0 for a.e. x ∈ }. Given j ∈ J0 (), for a.e. x ∈ , we define β(x, r) = ∂j(x, r) and J : W0 () ∩ L∞ () −→ [0, ∞] u −→ j(·, u)dx. 1,p

·

Note that J is convex, l.s.c. and J (0) = 0. Moreover C := D(J ) 1,p is a convex bilateral 1,p set, i.e., C is a closed convex subset of W0 () satisfying u, w ∈ C ⇒ u ∧ w ∈ C and u ∨ w ∈ C. According to [7], there exist unique (in the sense q.e.) functions γ+ , γ− which are cap-quasi-l.s.c. and cap-quasi-u.s.c., respectively, such that ·1,p

D(J )

1,p

= {u ∈ W0 (); γ− (x) ≤ u(x) ˜ ≤ γ+ (x) q.e. on }.

(2.1)

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Moreover, γ− (x) = inf n u˜ n (x) = limn inf 1≤k≤n u˜ k (x) for q.e. x ∈ for any · 1,p -dense sequence (un )n in D(J ). An analogous property holds for γ+ . Recall that the subdifferential 1,p operator ∂J ⊆ (W0 () ∩ L∞ ()) × (W −1,p () + (L∞ ())∗ ) is monotone and is given by 1,p u ∈ W0 () ∩ L∞ (); M ∈ W −1,p () + (L∞ ())∗ M ∈ ∂J (u) ⇐⇒ 1,p and J (w) ≥ J (u) + M, w − u ∀w ∈ W0 () ∩ L∞ (), where, here and on the sequel, if not explicitly stated otherwise, ·, · denotes the duality 1,p between W0 () ∩ L∞ () and its dual. Since Mb () = (C0 ())∗ and C0 () is separable, if µ : (0, T) → Mb () is weak-∗ measurable, then t → µ(t)Mb () is measurable (see [23, Lemme 1]). Consequently, we can define the following spaces which appear in our study of the absorption problem L1 (0, T ; w∗− Mb ()) ∗

T

:= {µ : (0, T) → Mb () weak- measurable;

µ(t)Mb () dt < ∞} 0

and L10 (0, T ; w∗− Mb ()) := {µ ∈ L1 (0, T ; w∗− Mb ()); µ(t) ∈ M0 () for a.e. t}. (2.2) As we said before, an important tool in order to prove the existence of solutions is the following technical lemma. A proof of this result will be given in the Appendix. T 0

LEMMA 2.1. The space L10 (0, T ; w∗ − Mb ()), endowed with its natural norm ∗ 1,p .Mb () dt, is continuously embedded in Lp (0, T ; W0 ()) ∩ L∞ (Q) .

3. Existence of weak solutions To prove existence of entropy solutions, we will proceed by approximation. We need first to prove, for bounded data f ∈ L∞ (Q) and u0 ∈ L∞ (), existence of a weak 1,p solution u ∈ Lp (0, T ; W0 ())∩L∞ (Q) of the parabolic problem with additional strongly monotone perturbation ψm,n , where ψm,n (r) = m1 r + − n1 r − , m, n ∈ N:   ut − div a(u, Du) + β(·, u) + ψm,n (u) f in Q (Pm,n )(u0 , f) u(0, ·) = u0 in  u=0 on .

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This is done via approximation by a sequence of regularized parabolic problems   ut − div a(u, Du) + βλ,ν (·, u) + ψm,n (u) = f in Q (Pλ,ν,m,n )(u0 , f) u(0, ·) = u0 in  u=0 on , where λ, ν ∈ N and for all x ∈ , βλ,ν (x, r) = βλ (x, r + ) + βν (x, −r − ). For these regularized problems we obtain existence of weak solutions with appropriate estimates and monotonicity properties, which allow us to pass to the limit. According to a preceding work [3], by nonlinear semigroup theory, for any λ, ν, m, n ∈ N, for all u0 ∈ L1 (), for all f ∈ L1 (Q), there exists a unique mild solution u ∈ C([0, T ]; L1 ()) of the abstract Cauchy problem : ut + Aλ,ν m,n u f, u(0) = u0 ,

(3.1)

λ,ν where the m-T -accretive operator Aλ,ν m,n is defined by (u, f) ∈ Am,n if and only if u ∈

W0 () ∩ L∞ (), f ∈ L1 () and for all φ ∈ W0 () ∩ L∞ () we have a(u, Du) · Dφ + βλ,ν (·, u)φ + ψm,n (u)φ = fφ. 1,p

1,p

For all bounded data, we prove that the mild solution u of the Cauchy problem (3.1) 1,p is a weak solution, i.e., u ∈ Lp (0, T ; W0 ()) ∩ L∞ (Q) and verifies for all φ ∈ 1,p Lp (0, T ; W0 ()) ∩ L∞ (Q) −

(u − u0 )φt +

Q

a(u, Du) · Dφ +

Q

βλ,ν (·, u)φ +

Q

ψm,n (u)φ =

Q

fφ. Q

PROPOSITION 3.1. For f ∈ L∞ (Q) and u0 ∈ L∞ (), let u be the mild solution of Problem (3.1). Then u is a weak solution of Problem (Pλ,ν,m,n )(u0 , f). Proof. Consider the implicit ε-time discretization scheme uεi − uεi−1 ti − ti−1

ε ε + Aλ,ν m,n ui fi ,

(3.2)

where • 0 = t0< t1 < · · · < tl ≤ T such that ti − ti−1 < ε ∀i = 1, · · · , l and T − tl < ε

l ti ε ε • i=1 ti−1 f(t) − fi 1 dt ≤ ε with fi L∞ () ≤ f L∞ (Q) for all i = 1, · · · , l

l T ε • i=1 (ti − ti−1 )fi L∞ () ≤ 0 f(t, ·)L∞ () dt.

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We denote by uε the piecewise constant function defined by uε (t) = uεi for all t ∈ ]ti−1 , ti ], uε (0) = u0 . We recall that, by nonlinear semigroup theory, uε converges in L∞ (0, T ; L1 ()) to the mild solution u of the Cauchy problem (3.1). To pass to the limit with ε → 0 we need first L∞ -estimates on uε . To this end, take ± ε pδ (ui − k), δ > 0, with k ≥ uεi−1 ∞ + (ti − ti−1 )fiε ∞ , as test functions in (3.2), where − the functions p+ δ , pδ are recalled in the above section. Using Assumptions (H1 ), (H4 ) and passing to the limit with δ → 0 in both equations we get uεi ∞ ≤ uεi−1 ∞ + (ti − ti−1 )fiε ∞ . Thus, uεi ∞ ≤ u0 ∞ +

i j=1

Consequently

(3.3)

(tj − tj−1 )fjε ∞ .

uε L∞ (Q) ≤ C f L∞ (Q) , u0 L∞ () .

We can now follow the arguments of [13] to prove that, as ε → 0, (a subsequence of) uε 1,p 1,p converges weakly in Lp (0, T ; W0 ()) to a measurable function u ∈ Lp (0, T ; W0 ())∩ L∞ (Q) and uL∞ (Q) ≤ C(f L∞ (Q) , u0 L∞ () ).

(3.4)

Since (Duε )ε is bounded in Lp (Q) and (uλ )λ is bounded in L∞ (Q), then, by Assumption (H3 ), (a(uε , Duε ))ε is bounded in (Lp (Q))N , thus after passing to a suitable subsequence, we can assume that a(uε , Duε ) converges weakly to some χ ∈ (Lp (Q))N . Due to the strong L∞ (0, T ; L1 ()) convergence of uε as ε → 0, using uεi −u as atest function in (3.2), by the Alt-Luckhaus chain rule lemma (see [1]), we obtain lim supε→0 Q a(uε , Duε )·D(uε −u) ≤ 0. Now it follows easily from the usual Minty-Browder argument that div χ=div a(u, Du) in D (Q). Combining all estimates, for some appropriately chosen sequence (still denoted by uε for simplicity), we can pass to the limit with ε → 0 in the weak formulation of (Pλ,ν,m,n ) (u0 , f). As we said in the introduction, our first aim is to study the existence of a weak solution for Problem (Pm,n )(u0 , f). The concept of weak solutions we have in mind is the following. Recall that, given an interval [a, b] ⊂ R, I[a,b] denotes the convex l.s.c. functional on R defined by 0 on [a, b], +∞ otherwise. DEFINITION 3.1. A measurable function u : Q → R is a weak solution of Problem 1,p (Pm,n )(u0 , f) if u ∈ C([0, T ]; L1 ()) ∩ Lp (0, T ; W0 ()) ∩ L∞ (Q) and there exists a measure µ ∈ L10 (0, T ; w∗−Mb ()) with

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µr (t) ∈ ∂j(·, u(t)) + ∂I[γ− (·),γ+ (·)] (u(t)) for almost every t and

+/−

u(t) = γ+/− µs

(t) a.e. on for almost every t

such that for all φ ∈ Lp (0, T ; W0 ()) ∩ L∞ (Q) − (u − u0 )φt + a(u, Du) · Dφ + ψm,n (u)φ = fφ − φ(t)dµ(t)dt. 1,p

Q

Q

Q

Q

Q

REMARK 3.1. Note that the last integral in the preceding definition is well defined since 1,p the function t → φ(t)dµ(t) is measurable, and φ(t) ∈ W0 () ∩ L∞ () for a.e. t and admits a quasi-continuous representative for almost every t. Example below demonstrates rather strikingly that the second term in the characterization of the regular measure is necessary. EXAMPLE 3.1. Let = (0, 1), Q = {ri , i ∈ N} be the countable set of rational numbers and R := {r; ∃i ∈ N such that |r − ri | ≤ 1/2i+2 } and f ∈ L1 (Q) with f < 0 a.e. on Q. Consider the case where a(r, ξ) = ξ, i.e., Au = −uxx for u ∈ D(A), and j(x, r) = χR (x)j1 (r) with j1 (r) = ∞ if r < 0, = 0 if r > 0. Clearly 1,p

˜ < +∞ q.e. x ∈ (0, 1)}. D(J )1,p = {u ∈ W0 (0, 1); 0 ≤ u(x) The comparison principle (see [40]) implies that u(t) ≤ 0 a.e. on . As u(t) ∈ D(J )1,p , it follows in particular that u(t) ≥ 0 a.e. on for almost everywhere t. As a consequence u = 0 is the unique solution. Thus, by the definition of weak solutions there exists a unique measure such that µ = f in D (Q), i.e., µ is a regular measure. Let us note that in this case it is not true that µ(t) = µr (t) ∈ ∂j(·, u(t)) for a.e. t, because this would imply that f = µr = 0 a.e. on (0, T) × (\R) which contradicts the fact that f < 0 a.e. on Q. Thus, the addition of the term ∂I[0,+∞] (u(t)) is necessary. The passage to the limit in Problem (Pλ,ν,m,n )(u0 , f) with λ, ν → 0 requires some additional assumptions. To introduce them we need some notations and definitions. Let · A = A0 L1 , where A0 denotes the operator in L1 () defined on D(A0 ) := {h ∈ 1,p W0 () ∩ L∞ (); −div a(h, Dh) ∈ L∞ ()} by A0 u := −div a(u, Du), and B is the operator defined by w ∈ Bu ⇔ w, u ∈ L1 (), w(x) ∈ ∂j(x, u(x)) a.e. x ∈ ,

(3.5)

while B+ is defined by w ∈ B+ u ⇔ w, u ∈ L1 (), w(x) ∈ ∂j(x, u+ (x)) a.e. x ∈ , and the operator B− corresponds to j(·, −r − ).

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459

Let Bλ be the Yosida approximation of B. Note that Bλ corresponds to ∂jλ (via the analogue of (3.5)), where jλ ∈ J0 (), jλ (x, r) = inf {1/(2λ)|r − s|2 + j(x, s)}. Obviously s∈R

Bλ = (B+ )λ + (B− )λ and B = B+ + B− . Notice that A is an m-T -accretive operator in L1 (). In general A + B fails to be an m-T -accretive operator in L1 (). However, it is shown in [37, Théorème 3.5.2] that A + B always admits a natural m-T -accretive extension Aj := lim inf A + (B+ )λ + (B− )ν . λ,ν→0

We need to recall the definition of sub/super-solution associated to an m-T -accretive operator. DEFINITION 3.2. [9] Let A be an m-T -accretive operator in L1 (). We say that v is a sub-solution (respectively a super-solution) of y ∈ Av if and only if v ≤ JA (v + y) (respectively v ≥ JA (v + y)) for all > 0, where JA denotes the resolvent of the operator A. We need to introduce the following sets.

, v ∈ L∞ () such that ± v+/− is a sub/super-solution of ∃y Cλ,ν := u ∈ L∞ (); +/− (A + (B+ )λ + (B− )ν )(±v+/− ) y+/− and − v− ≤ u ≤ v+

and

∃y+/− , v ∈ L∞ () such that ± v+/− is a sub/super-solution of C := u ∈ L∞ (); . Aj (±v+/− ) y+/− and − v− ≤ u ≤ v+

The following lemma gives a characterization of the closure of the domain of Aj and also properties on the sets C and Cλ,ν . LEMMA 3.1. Let ν(·) be a nondecreasing sequence. The set C is bilateral (i.e., stable · by the sup and the inf operations), Cλ,ν(λ) ⊂ Cµ,ν(µ) for all 0 < µ < λ, C = ∪λ Cλ,ν(λ) L1 and b b Recall that (see e.g. [12]) given an operator A, the generalized domain of A is defined by D (A) := {u; |u|A
0

λ→0

460

J.evol.equ.

B. Andreianov, K. Sbihi and P. Wittbold ·L1 ()

D(Aj )

(A) ∩ D(B) =D

·L1 () ·L1 ()

1,p

= {u ∈ W0 () ∩ L∞ (); j(·, u) ∈ L1 ()}

·L1 ()

1,p

= {u ∈ W0 () ∩ L∞ (); γ− (x) ≤ u(x) ˜ ≤ γ+ (x) q.e. x ∈ }

(3.6) and

·L1 ()

D(Aj )

⊆C

·L1 ()

,

+ Proof. Let u1 , u2 ∈ C, then u1 ∨ u2 ∈ C, because u1 ∨ u2 ≤ (u+ 1 ) ∨ (u2 ) and the maximum of sub-solutions is still a sub-solution. Similarly, u1 ∧ u2 ∈ C. Using the properties of the resolvent, clearly we have if 0 < µ < λ, then Cλ,ν(λ) ⊂ Cµ,ν(µ) , and

C = ∪λ Cλ,ν(λ)

·L1

. We may adapt the methods of [39] to prove the first equality in (3.6). For · (A) L1 ⊂ D(A)·L1 = L1 (), then for the direct inclusion the second one, recall that D ·

·L1 ()

1,p

. it suffices to prove that D(B) L1 () = {u ∈ W0 () ∩ L∞ (); j(·, u) ∈ L1 ()} 1,p ∞ ∞ −1 Let u ∈ D(B) ∩ L () and uδ := (I + δA) u ∈ L () ∩ W0 (). We have uδ ∞ ≤ u∞ , uδ → u in L1 (), uδ ∈ D(A0 ) and A0 uδ is bounded in L1 (). Then uδ is bounded 1,p in W0 () and admits a weakly convergent subsequence. The limit of it is necessary equal 1,p to u, thus u ∈ W0 (). On the other hand, as u ∈ D(B) ∩ L∞ (), then j(·, u) ∈ L1 (), because j(x, u(x)) ≤ j(x, 0) + w(x)u(x) ≤ u(x)∞ |w(x)| for w ∈ ∂j(·, u). Consequently D(B) ∩ L∞ () ⊂ 1,p {u ∈ W0 () ∩ L∞ (); j(·, u) ∈ L1 ()} and the first inclusion is proved. 1,p Reciprocally, let u ∈ W0 () ∩ L∞ () with j(·, u) ∈ L1 () and consider uδλ,ν = (I + δA + δ(B+ )λ + δ(B− )ν )−1 u. We have uδλ,ν ∈ D(A0 ) and uδλ,ν − u22 + δ a(uδλ,ν , Duδλ,ν ) · D(uδλ,ν − u) ≤ δ βλ (·, uδλ,ν )(u − uδλ,ν ) ≤ δ j(·, u).

Using Hypothesis (H1 ), (H3 ) and Hölder’s inequality we get δ 2 p uλ,ν − u2 ≤ Cδ + δ j(·, u) → 0 as δ → 0 (uniformly in λ, ν).

Thus uδ := lim

λ,ν→0

uδλ,ν

converges to u in L1 () as δ, ν → 0, which proves the reciprocal

inclusion. The last equality is a straightforward consequence of (2.1). It remains to prove that D(Aj ) ⊆ C. Let u ∈ D(Aj ) ∩ L∞ (). There exists y ∈ L∞ () such that Aj u y, then Aj (u ∨ 0) ≤ y ∨ 0, i.e., u+ is a sub-solution of Aj (u+ ) y+ . The same reasoning is applied to prove that −u− is a super-solution of Aj (−u− ) y− , and we have of course −u− ≤ u ≤ u+ .

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461

We are now ready to state the main result of this section : PROPOSITION 3.2. For any f ∈ L∞ (Q) and u0 ∈ ∪λ Cλ,ν(λ) , there exists a weak solution of Problem (Pm,n )(u0 , f). Proof. From now on and until Section 4, we omit the index m, n to lighten the notations. Let λ, ν > 0, and uλ,ν be a weak solution of Problem (Pλ,ν,m,n )(u0 , f). We deduce from (3.4) a uniform estimate on λ and ν : uλ,ν L∞ (Q) ≤ C.

(3.7)

Taking ξ = uλ,ν as a test function in (Pλ,ν,m,n )(u0 , f), using the Alt-Luckhaus chain rule lemma, by Assumptions (H2 ) and (H3 ), theorem of Gauss-Green, monotonicity of ψm,n and (3.7), we deduce |Duλ,ν |p ≤ C, Q

where C is a constant independent of λ and ν. Thus, the sequence (uλ,ν )λ,ν is bounded in 1,p Lp (0, T ; W0 ()). In order to pass to the limit as λ, ν → 0 in Problem (Pλ,ν,m,n ) we need to show the strong convergence of uλ,ν . To this end, let λ > λ > 0, ν > 0 and consider the test function p+ (u − u ) in the equations corresponding to the solutions uλ,ν , u λ,ν , adding λ,ν ε λ,ν both equations and using the Alt-Luckhaus chain rule lemma, we get

(uλ,ν −u λ,ν )(t)

p+ ε (r)dr +

0

(a(uλ,ν , Duλ,ν ) − a(u λ,ν , Du λ,ν )) · Dp+ ε (uλ,ν − u λ,ν )

Q

+

(βλ,ν (uλ,ν ) − β λ,ν (u λ,ν ))p+ ε (uλ,ν − u λ,ν )

Q

+

(ψm,n (uλ,ν ) − ψm,n (u λ,ν ))p+ ε (uλ,ν − u λ,ν ) = 0.

Q

By Assumptions (H1 ) and (H4 ), it is clear that lim inf (a(uλ,ν , Duλ,ν ) − a(u λ,ν , Du λ,ν )) · Dp+ ε (uλ,ν − u λ,ν ) ≥ 0. ε→0

Q

Dropping nonnegative terms in (3.8) and letting ε → 0, we get (ψm,n (uλ,ν ) − ψm,n (u λ,ν ))+ ≤ 0. Q

(3.8)

462


J.evol.equ.

As ψm,n is strictly nondecreasing, uλ,ν ≤ u λ,ν a.e. on Q. In the same way we can prove, for all ν > ν > 0, and λ > 0, that uλ,ν ≤ uλ, ν a.e. on Q. Estimate (3.7) implies, thanks to the monotone convergence theorem, uλ,ν ↓λ u0,ν ↑ν u in L1 (Q).

(3.9)

Here and in the sequel, we use the notation ↑ν , respectively ↓ν , to denote convergence of a sequence which is monotone increasing, respectively decreasing, in ν. Applying the diagonal procedure, we may assume that, for some sequence ν(λ), uλ := uλ,ν(λ) → u in L1 (Q). Extracting a subsequence if necessary, we may therefore assume that 1,p uλ u weakly in Lp (0, T ; W0 ()), uλ → u in C([0, T ]; L1 ()) and uλ → u a.e. on Q. The task now is to pass to the limit in the term βλ,ν(λ) (·, uλ ). Choosing 1k Tk (uλ ) as a test function in Problem (Pλ,ν(λ),m,n )(u0 , f), using Assumption (H2 ) and the Alt-Luckhaus chain rule lemma, we deduce uλ (t)

u0

1 ≤ k

1 1 Tk (r)drdx + k k

Q

a(uλ , 0) · DTk (uλ ) + Q

1 fTk (uλ ) − k

1 k

βλ,ν(λ) (·, uλ )Tk (uλ ) Q

ψm,n (uλ )Tk (uλ ). Q

From Gauss-Green’s theorem and monotonicity of the function ψm,n , letting k → 0, we get |βλ,ν(λ) (·, uλ )| ≤ C, Q

where C is a constant independent of λ. Thus (βλ,ν(λ) (·, uλ ))λ is bounded in Mb (Q). In the sequel, we denote βλ := βλ,ν(λ) and Bλ = (B+ )λ + (B− )ν(λ) . The continuation of the proof is divided into three lemmas. We first begin by giving strong properties on the sequence (βλ (·, uλ ))λ , LEMMA 3.2. The sequence (βλ (·, uλ ))λ verifies the following properties : (βλ (·, uλ ))λ is bounded in L∞ (0, T ; Mb ()),

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463

βλ (·, uλ ) µ weakly in Lp (0, T ; W −1,p ()) and µ ∈ L10 (0, T ; w∗−Mb ()) Proof. Let k ≥ max{m, n} and φm,n (r) := ψm,n (r) + 1k r. We have u0 ∈ Cλ,ν(λ) , therefore there exist y+ , v ∈ L∞ () such that u0 ≤ v+ , while + v is a sub-solution of (A + Bλ )(v+ ) y+ . Thus v+ ≤ JkA+Bλ (v+ + ky+ ) =: w. The function w ∈ D(A + Bλ ) ∩ L∞ () is a solution of the problem

1 k w − div

w=0

a(w, Dw) + βλ (·, w) + φm,n (w) = 1k v+ + y+ + φm,n (w) in on ∂.

Let vλ be a solution of the following stationary problem:

1 k vλ − div vλ = 0

a(vλ , Dvλ ) + βλ (·, vλ ) + φm,n (vλ ) = g in on ∂,

(3.10)

where g ∈ L∞ () is such that g ≥ max{ 1k v+ + y+ + φm,n (w), f ∞ }. As in (3.3) we prove that the sequence (vλ )λ is uniformly bounded in L∞ () in λ, and from (3.10) we deduce that

βλ (·, vλ ) ∈ (L∞ () + W −1,p ()) ∩ Mb () = W −1,p () ∩ Mb (),

(3.11)

moreover, βλ (·, vλ )W −1,p () ≤ C and βλ (·, vλ )Mb () ≤ C (uniformly in λ).

(3.12)

We claim that vλ ≥ u0 . Indeed, by means of the comparison principle applied to the solutions vλ and w, we have

(w − vλ )+ ≤ k

1 + v + y+ + φm,n (w) − g k

+ .

By the choice of g, vλ ≥ w ≥ v+ ≥ u0 , thus the result follows. Remark that vλ is also a solution of the problem   vλ,t − div a(vλ , Dvλ ) + βλ (·, vλ ) + ψm,n (vλ ) = g in Q (I) vλ (0, ·) = vλ in  vλ = 0 on .

464

J.evol.equ.


Using in (Pλ,ν(λ),m,n )(u0 , f) and in (I) the test function p+ ε (uλ −vλ ), adding both equations, we get after dropping some nonnegative terms

−vλ )(t) (uλ

p+ ε (r)dr

p+ ε (r)dr

−

0

u 0 −vλ

0

(a(uλ , Duλ ) − a(vλ , Dvλ )) · D(uλ − vλ )p ε (uλ − vλ )

+ Q

(βλ (·, uλ ) − βλ (·, vλ ))p+ ε (uλ

+

− vλ ) ≤

(f − g)p+ ε (uλ − vλ ).

(3.13)

Q

Q

Thanks to Assumptions (H1 ) and (H4 ) lim inf (a(uλ , Duλ ) − a(vλ , Dvλ )) · D(uλ − vλ )p ε (uλ − vλ ) ≥ 0. ε→0

Q

Letting ε → 0 in inequality (3.13), we get + + (uλ − vλ ) (t) − (u0 − vλ ) + (βλ (·, uλ ) − βλ (·, vλ ))χ{uλ >vλ } ≤ 0, Q

which implies uλ (t) ≤ vλ

and βλ (·, uλ (t)) ≤ βλ (·, vλ ) a.e. on for a.e. t.

In the same way we reason on super-solutions to get uλ (t) ≥ −vλ a.e. on for a.e. t. Thus, using (3.11) and (3.12), we get (βλ (·, uλ ))λ is bounded in L∞ (0, T ; Mb ()).

(3.14)

Now let us study separately βλ (·, r + ) and βλ (·, −r − ). Let φ ∈ Lp (0, T ; W0 ()). We have from (3.12), + φβλ (·, uλ ) ≤ |φ||βλ (·, u+ λ )| Q Q ≤ |φ||βλ (·, v+ λ )| 1,p

Q

T =

βλ (·, v+ λ ), |φ|

0

≤ CφLp (0,T ;W 1,p ()) . 0

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This means that the linear functional φ∈L

p

1,p (0, T ; W0 ())

→

465

φβλ (·, u+ λ)

Q 1,p Lp (0, T ; W0 ()).

is continuous for the norm of It has a unique extension Fλ ∈ Lp (0, T ; W −1,p ()) with Fλ Lp (0,T ;W −1,p ()) ≤ C. After extracting a subsequence

if necessary, we may assume that Fλ F in Lp (0, T ; W −1,p ()). We have 0 ≤ lim

λ→0

βλ (·, u+ λ )φ

T = lim

Fλ (t), φ(t) =

λ→0 0

Q

for all φ and ξ ∈

T

1,p ∈ Lp (0, T ; W0 ())∩L∞ (Q), 1,p W0 (), ξ ≥ 0, we deduce

F(t), φ(t) 0

φ ≥ 0. Considering φ = κξ, where κ ∈ D+ [0, T)

F(t), ξ ≥ 0 for almost every t, which implies that F(t) is a positive linear form on D(), thus it is a Radon measure for almost every t, noted by µ+ (t). On the other hand, for all κ ∈ D[0, T) and ζ ∈ D() we have T

T

+

ζκdµ (t)dt = 0

κ(t)F(t), ζW −1,p (),W 1,p () dt. 0

0

Thus µ+ (t) ∈ W −1,p () ∩ Mb () ⊂ M0 () for almost every t (see [16, Theorem 2.1]). − Analogous reasoning for βλ (·, −u− λ ) yields a measure −µ (t) with the same properties. ∗ Note that L∞ (0, T ; Mb ()) is a subspace of (L1 (0, T ; Cc ())) , identified with L∞ (0, T ; w∗ − Mb ()) (see [22, Theorem 1.4.5]), which is a subspace of L1 (0, T ; w∗− Mb ()). Hence, after passing to a subsequence, we may assume that ∗

βλ (·, uλ ) µ in L∞ (0, T ; w∗− Mb ()),

(3.15)

µ ∈ L1 (0, T ; w∗− Mb ()) with µ(t) ∈ M0 () for almost every t, i.e., µ ∈ L10 (0, T ; w∗−Mb ()).

Since ut + µ ∈ Lp (0, T ; W −1,p ()) + L1 (Q), we deduce from Lemma 2.1 that ∗ 1,p ut , µ ∈ Lp (0, T ; W0 ()) ∩ L∞ (Q) .

466

J.evol.equ.


LEMMA 3.3. There exists a field χ ∈ (Lp (Q))N such that

a(uλ , Duλ ) χ weakly in (Lp (Q))N and

div χ = div a(u, Du) in D (Q).

Proof. Since (Duλ )λ is bounded in Lp (Q) and (uλ )λ is bounded in L∞ (Q), then, by Assumption (H3 ), (a(uλ , Duλ ))λ is bounded in (Lp (Q))N , thus after passing to a suitable subsequence, we can assume that a(uλ , Duλ ) χ weakly in (Lp (Q))N . The aim is to show, via the pseudo-monotonicity argument, that div χ = div a(u, Du) in D (Q). To this end, we will show that (3.16) lim sup a(uλ , Duλ ) · D(uλ − u) = 0. λ→0

Q

We have lim a(uλ , Duλ ) · D(uλ − u) = lim a(uλ , Duλ ) · Duλ − χ · Du λ→0

λ→0

Q

Q



 = lim 

Q

f(uλ − u)−

λ→0

Q

ψm,n (uλ )uλ −ψm,n (u)u Q

−uλ,t , uλ + ut , u − +

βλ (·, uλ )uλ Q

  u(t)dµ(t)dt 

Q

=: lim I1 − I2 − I3 + I4 − I5 + I6 , λ→0

where ·, · is the duality paring between (Lp (0, T ; W0 ()) ∩ L∞ (Q))∗ and 1,p Lp (0, T ; W0 ()) ∩ L∞ (Q). It is clear that lim I1 = 0. As (uλ )λ is uniformly bounded in L∞ (Q), uλ → u in 1,p

λ→0

C([0, T ]; L1 ()) and the function ψm,n is continuous, then by the dominated convergence theorem, we get lim I2 = 0. λ→0

∞ The Alt-Luckhaus chainrule lemma implies, thanks to the uniform L estimates on uλ

and the fact that uλ,t , ut ∈ Lp (0, T ; W0 ()) ∩ L∞ (Q) 1,p

C([0, T ]; L1 ()),

∗

and that uλ converges to u in

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lim (I3 − I4 ) = lim

λ→0

λ→0

1 2 uλ (T) − u20 − 2

467

1 2 u (T) − u20 = 0. 2

As βλ is monotone, lim (I5 − I6 ) = lim

λ→0

(βλ (·, uλ ) − βλ (·, u))(uλ − u) + βλ (·, u)(uλ − u)

λ→0 Q

+ βλ (·, uλ )u − Q

≥ lim

u(t)dµ(t)dt ˜

βλ (·, u)(uλ − u) + βλ (·, uλ )u −

λ→0 Q

u(t)dµ(t)dt = 0, Q

because βλ (·, u) → β◦ (·, u) in L1 (Q), uλ → u in L1 (Q), uλ − u∞ ≤ C and βλ (·, uλ ) µ weakly in Lp (0, T ; W −1,p ()). From these limits, it follows that lim sup a(uλ , Duλ ) · D(uλ − u) ≤ 0. λ→0

Thanks to the monotonicity of a and the almost everywhere convergence of uλ to u as λ → 0, (3.16) holds. Let φ ∈ D(Q) and α ∈ R. We have α χ · Dφ ≥ lim α a(uλ , Duλ ) · Dφ λ→0

Q

Q

≥ lim sup λ→0

a(uλ , Duλ ) · D(uλ − u + αφ) Q

≥ lim sup λ→0

= α

a(uλ , D(u − αφ)) · D(uλ − u + αφ) Q

a(u, D(u − αφ)) · Dφ. Q

Dividing by α > 0, resp. α < 0, and passing to the limit with α → 0, we get div a(u, Du) = div χ in D (Q). In order to characterize the measure obtained, we will essentially use the following result which gives a characterization of the elements in the subdifferential ∂J which are in M0 ().

468


J.evol.equ.

PROPOSITION 3.3. [17, Proposition 20] Let µ ∈ M0 (), then µ ∈ ∂J (u) ⇐⇒

µr (x) ∈ ∂j(x, u(x)) + ∂I[γ− (x),γ+ (x)] (u(x)) a.e. x ∈ ˜ = γ+ µ+ and u˜ = γ− µ− s -a.e. on , u s -a.e. on .

LEMMA 3.4. Let µ be the measure obtained in Lemma 3.2, then µr (t) ∈ ∂j(·, u(t)) + ∂I[γ− (·),γ+ (·)] (u(t)) a.e. on for a.e. t −/+ and u(t) = γ−/+ µs -a.e. on for a.e. t. Proof. Notice that βλ = ∂jλ , where jλ ∈ J0 () and for a.e. x ∈ and for all r ∈ R, jλ (x, r) ↑ j(x, r) as λ ↓ 0. Thus, by definition of the subdifferential, for all > λ > 0, for a.e. x ∈ and for a.e. t, j(x, r) ≥ jλ (x, r) ≥ jλ (x, uλ (t, x)) + ∂jλ (x, uλ (t, x))(r − uλ (t, x)) ≥ j (x, uλ (t, x)) + ∂jλ (x, uλ (t, x))(r − uλ (t, x)) ∀r ∈ R. Let ξ ∈ W0 () ∩ L∞ () and ζ ∈ L∞ (0, T), ζ ≥ 0. Multiplying the above inequality by ζ and integrating on , then on (0, T), we get 1,p

T 0

 ζ

j(·, ξ) −



T j (·, uλ (t)) ≥ ζ βλ (·, uλ (t))(ξ − uλ (t)). 0

Passing to the limit with λ → 0, then with → 0 and using Fatou’s lemma, monotone → u a.e. as λ → 0 and that convergence theorem and the fact that u λ Q ζβλ (·, uλ )(ξ − u) → Q ζ(ξ − u)dµ, we deduce T ζ 0

j(·, ξ) −

j(·, u(t)) ≥

ζ(ξ˜ − u(t))dµ(t)dt

Q

+ lim inf

ζβλ (·, uλ )(u − uλ ).

λ→0

(3.17)

Q

Assume for the moment that the following estimate holds: lim inf ζβλ (·, uλ )(u − uλ ) ≥ 0. λ→0

Q

(3.18)

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469

From inequality (3.17) T

 J (ξ) − J (u) −

0

  ζ(t) ≥ 0 ∀ζ ∈ L∞ (0, T), ζ ≥ 0, (ξ˜ − u(t))dµ(t) ˜

which implies J (ξ) − J (u(t)) − µ(t), ξ − u(t) ≥ 0 for almost every t

(3.19)

for all ξ ∈ W0 () ∩ L∞ (). We deduce from (3.19) that µ(t) ∈ ∂J (u(t)) for almost every t; thus the result of Lemma 3.4 follows by Proposition 3.3. It remains to prove (3.18). Choosing (uλ − u)ζ as a test function in Problem (Pλ,ν(λ),m,n )(u0 , f) and using the monotonicity of ψm,n , we obtain ζβλ (·, uλ )(u − uλ ) ≥ f(u − uλ )ζ + a(uλ , Duλ ) · D(uλ − u)ζ 1,p

Q

Q

+

Q

ψm,n (u)(uλ −u)ζ + ut , (uλ −u)ζ + uλ,t − ut , (uλ −u)ζ, Q

where ·, · is the duality paring between (Lp (0, T ; W0 ()) ∩ L∞ (Q))∗ and 1,p Lp (0, T ; W0 ()) ∩ L∞ (Q). Thanks to the L∞ estimates on uλ and u, the almost every1,p where convergence of uλ to u on Q, the weak convergence of uλ to u in Lp (0, T ; W0 ()), the limit (3.16) and the Alt-Luckhaus chain rule lemma, it is easy to pass to the limit in the above inequality. 1,p

The proof of existence of weak solutions easily follows from the aforementioned estimates proved and Lemmas 3.2, 3.3 and 3.4. 4. Entropy solutions and well-posedness Let us define entropy solutions for Problem (P)(u0 , f). 1,p

REMARK 4.1. Let u : → R be a measurable function such that Tk (u) ∈ W0 () for all k ∈ N. Then the function u˜ := lim ( max Tk (u+ ) − Tk (u− )) is a measurable k→∞ 1≤n≤k

˜ = T function and Tk (u) k (u) is a quasi-continuous function for all k. Moreover, by choosing ∞ () weakly measurable, then T (u) ∞ T (u) : (0, T) → L k k ˜ : (0, T) → L () is weakly measurable for all k ∈ N.

470


J.evol.equ.

DEFINITION 4.1. A measurable function u : Q → R is an entropy solution for Problem P(u0 , f) if u ∈ C([0, T ]; L1 ()) and 1,p

Tk (u) ∈ Lp (0, T ; W0 ()) for all k > 0 and if there exists a measure µ ∈ L10 (0, T ; w∗−Mb ()) such that µr (t) ∈ ∂j(·, u(t)) + ∂I[γ− (·),γ+ (·)] (u(t)) for almost every t, +/−

u(t) = γ+/− µs

(t) − a.e. on for almost every t

and for all φ ∈ W0 () ∩ L∞ () and ξ ∈ D+ [0, T) we have 1,p

T u(t) ξt Tk (r − φ)drdxdt + ξa(u, Du) · DTk (u − φ)dxdt

T −

u0

0

T

0

T

≤

fTk (u − φ)ξdxdt − 0

ξTk ( u(t) − φ)dµ(t)dt.

0

REMARK u(t) 4.2. Note that each integral in the preceding definition is well defined. Indeed, | u0 Tk (r − φ)dr| ≤ k|u(t) − u0 | ∈ L1 () and is integrable in t. The second term can be understood as ξa(Tl (u), DTl (u)) · DTk (u − φ), Q

where l ≥ k + φ∞ . Finally, the last integral in the above definition is well defined 1,p since Tk (u(t) − φ) ∈ W0 () ∩ L∞ () and admits Tk ( u(t) − φ) for a quasi-continuous representative for almost every t. 4.1. Existence of entropy solutions The main result of this section is 1,p

THEOREM 4.1. For all u0 ∈ {v ∈ W0 () ∩ L∞ (); γ− (x) ≤ v˜ (x) ≤ γ+ (x) q.p. ·L1 ()

x ∈ }

and f ∈ L1 (Q), Problem (P)(u0 , f) admits an entropy solution.

REMARK 4.3. Recall that the preceding result of existence of a weak solution remains true for all u0 ∈ ∪λ Cλ,ν(λ) . We are going to prove that Theorem 4.1 is valid for all u0 in the set C

·L1

1,p

= ∪λ Cλ,ν(λ) , which contains {v ∈ W0 () ∩ L∞ (); γ− (x) ≤ v˜ (x) ≤ γ+ (x) ·L1 ()

q.p. x ∈ }

by Lemma 3.1.

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We first need the following result: LEMMA 4.1. Let E, F be two subsets of L1 () such that E is a dense set in F , for · L1 , and E is stable by sup operation. Then for all e ∈ F , there exist (em,n )m,n ⊂ E and (en )n ⊂ L1 () such that em,n ↑m→∞ en ↓n→∞ e in L1 (). Proof. We choose a subsequence (ek )k ⊂ E which converges to e ∈ F in L1 (), almost everywhere on , and such that |ek | ≤ h ∈ L1 (). We set em,n = supn≤k≤m ek . Then |em,n | ≤ h and em,n is nondecreasing in m, thus has a limit en ∈ L1 (), and |en | ≤ h. In the same way, we find en ↓n→∞ eo in L1 (). Finally, eo = e because for almost all x ∈ , |em,n (x) − e(x)| ≤ supn≤k |ek (x) − e(x)| → 0 as n → ∞, thanks to almost everywhere convergence of ek . We can now start the proof of the existence result for Problem (P)(u0 , f). Following a standard approach, we obtain the existence of a solution as limit of approximating problems. To this purpose let u0m,n ∈ C be an approximation of u0 given by Lemma 4.1 (recall that C is a bilateral set) and let fm,n ∈ L∞ (Q) be a bi-monotone approximation of f in L1 (Q). Then, by Proposition 3.2 there exists a weak solution 1,p um,n ∈ Lp (0, T ; W0 ()) ∩ L∞ (Q) of the Cauchy problem (Pm,n )(u0m,n , fm,n ). Moreover, there exists a measure µm,n ∈ L10 (0, T ; w∗−Mb ()) with (µm,n )r (t) ∈ ∂j(·, um,n (t)) + ∂I[γ− (·),γ+ (·)] (um,n (t)) for almost every t and

+/−

um,n (t) = γ+/− (µm,n )s

(t)-a.e. on for almost every t,

such that for all φ ∈ Lp (0, T ; W0 ()) ∩ L∞ (Q) we have 1,p

T

T (um,n )t φ +

0

T a(um,n , Dum,n ) · Dφ +

0

0

=

ψm,n (um,n )φ

T

fm,n φ −

φ(t)dµm,n (t)dt.

0

Q

From now on we fix k > 0. Let us begin by getting a priori estimates on um,n . LEMMA 4.2. Let um,n be a solution of (Pm,n )(u0m,n , fm,n ). Then |DTk (um,n )|p ≤ Ck,

λ0 Q

(4.1)

472

J.evol.equ.


where C is a constant independent of m, n, and (um,n )m,n is a Cauchy sequence in C([0, T ]; L1 ()). Proof. Choosing φ = Tk (um,n ) as a test function in equation (4.1), using Assumption (H2 ), the monotonicity of ψm,n and the Alt-Luckhaus chain rule lemma we get

0

um,n Tk (r)dr + Tk (r)dr + λ0 |DTk (um,n )|p + Tk (um,n )ψm,n (um,n )

u m,n (t)

0

≤

0

Q

Tk (um,n )fm,n −

Tk (u˜ m,n (t))dµm,n (t)dt −

a(um,n , 0) · DTk (um,n ).

(4.2)

Q

Q

Q

Q

Tk (u˜ m,n (t))dµm,n (t)dt

By Gauss-Green theorem, the last integral vanishes. The integral Q

is nonnegative since it is decomposed into three nonnegative terms : Tk (um,n (t))(µm,n )r (t)dt + Tk (γ + )d(µm,n )+ (t)dt − Tk (γ − )d(µm,n )− s s (t)dt. (4.3) Q

Q

Q

From inequality (4.2) we obtain

u m,n (t)

Tk (r)dr + λ0

0

|DTk (um,n )|p ≤ Ck,

(4.4)

Q

where C is a constant independent of m, n. Thus (DTk (um,n ))m,n is bounded in (Lp (Q))N . 1,p

Hence (Tk (um,n ))m,n is bounded in Lp (0, T ; W0 ()), and thus Tk (um,n ) vk weakly 1,p (up to a subsequence) in Lp (0, T ; W0 ()) as m, n → ∞. We deduce also from (4.4) that ∞ (um,n )m,n is bounded in L (0, T ; L1 ()). Now let us prove that (um,n )m,n is a Cauchy sequence in C([0, T ]; L1 ()). Let m > m , n > n . Considering pε (um,n − um ,n ), ε > 0 (where pε (·) is an approximation of sign(·)) as a test function in equations corresponding to the solutions um,n and um ,n , adding both equations, using the Alt-Luckhaus chain rule lemma, Assumptions (H1 )-(H4 ) and the monotonicity of ψm,n , passing to the limit with ε go to zero we get |um,n (t) − um ,n (t)| ≤ |fm,n − fm ,n | + |u0m,n − u0m ,n | for a.e. t.

Q

Since fm,n , fm ,n → f in L1 (Q) and u0m,n , u0m ,n → u0 in L1 (), then (um,n )m,n is a Cauchy sequence in C([0, T ]; L1 ()) and converges to u ∈ C([0, T ]; L1 ()).

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The purpose now is to prove that, up to subsequence, um,n and µm,n converge strongly in the corresponding spaces. LEMMA 4.3. There exist a measurable function u and a Radon measure µ on Q such that um,n ↑m→∞ ↓n→∞ u strongly in L1 (Q), µm,n ↑m→∞ ↓n→∞ µ strongly in L1 (0, T ; w∗− Mb ()) and µ(t) ∈ M0 () for almost everywhere t. 1,p

In particular, the weak limit vk of Tk (um,n ) in Lp (0, T ; W0 ()) equals to Tk (u). Proof. Let uλm,n be a weak solution for Problem (Pλ,ν(λ),m,n )(u0m,n , fm,n ). From (3.9) ∗

+/−

uλm,n → um,n strongly in L1 (Q) and from (3.15) βλ (·, uλm,n )+/− µm,n weakly star in L∞ (0, T ; w∗− Mb ()) as λ → 0. We need the strong convergence of these sequences. We first prove the following comparison lemma LEMMA 4.4. Let m ˜ > m > 0, n > n > 0, then uλm, n ≤ uλm,n ≤ uλm ,n

a.e. on Q

and βλ (·, uλm, n ) ≤ βλ (·, uλm,n ) ≤ βλ (·, uλm ,n )

a.e. on Q.

Proof. The proof is adapted from the proof of Proposition 3.2. It suffices to take in the λ ) + λ equations corresponding to the solutions uλm,n et uλm ,n the test functions pε (um,n − um ,n λ + λ and pε (um, n − um,n ). The above result also holds for the positive and negative parts of the solutions, i.e., ± ±(uλm, n )± ≤ ±(uλm,n )± ≤ ±(uλm a.e. on Q ,n )

and

± ±βλ (·, uλm, n )± ≤ ±βλ (·, uλm,n )± ≤ ±βλ (·, uλm a.e. on Q. ,n )

Letting λ → 0, we get ± ± ±u± m, n ≤ ±um,n ≤ ±um ,n a.e. on Q,

and by means of the monotone convergence theorem we have 1 ± ±u± m,n ↑ ±un in L (Q) as m → ∞.

474

J.evol.equ.


± 1 The same reasoning implies ±u± n ↓ ±u in L (Q) as n → ∞. λ On the other hand, choosing the test function pε (uλm ,n − um,n ) in the equations corλ λ responding to the solutions um ,n and um,n , letting ε → 0 and neglecting all nonnegative terms, we get λ λ 0 βλ (·, um |fm |u0m ,n − fm,n | + ,n ) − βλ (·, um,n ) ≤ ,n − um,n |, Q

Q

which implies for all ϕ ∈ L1 (0, T ; C0 ()) with 0 ≤ ϕ ≤ 1 that T

+ ϕ(βλ (·, uλm ,n )

− βλ (·, uλm,n )+ ) +

0

≤

T 0

|fm ,n − fm,n | +

Q

− λ − ϕ(−βλ (·, uλm ,n ) + βλ (·, um,n ) )

|u0m ,n

− u0m,n |.

Passing to the limit with λ → 0 we deduce for all 0 ≤ ϕ ≤ 1 T

ϕ(dµ+ m ,n

− dµ+ m,n ) +

0

≤

T

− ϕ(−dµ− m ,n + dµm,n )

0

|fm ,n − fm,n | + Q

0 |u0m ,n − um,n |.

Thus  T 

T

 

− + µ+ µ− m,n (t) − µm m ,n (t) − µm,n (t)Mb () , ,n (t)Mb ()   0 0 0 ≤ |fm |u0m ,n − fm,n | + ,n − um,n |.

max

Q

± 1 ∗ Consequently, ±µ± m,n ↑m ±µn strongly in L (0, T ; w − Mb ()). The same arguments ± ± 1 can be applied to prove that ±µn ↓n ±µ in L (0, T ; w∗− Mb ()). We conclude that

um,n ↑m→∞ ↓n→∞ u strongly in L1 (Q), µm,n ↑m→∞ ↓n→∞ µ strongly in L1 (0, T ; w∗− Mb ()) and µ(t) ∈ M0 () for almost everywhere t.

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Thanks to Lemma 2.1, we deduce also the strong convergence of the sequence (µm,n )m,n ∗ 1,p p in L (0, T ; W0 ()) ∩ L∞ (Q) . Finally, from the Sobolev embedding theorem, Tk (um,n ) → vk strongly in L1 (Q). On the other hand, Tk (um,n ) → Tk (u) in L1 (Q), whence vk = Tk (u). We need to recall the following definition of a time regularization of Tk (u), which was first introduced in [27], and used in several papers afterward (see e.g. [3, 24]). Let ν > 0 and (u0ν )ν be a sequence of functions such that  0 1,p uν ∈ W0 () ∩ L∞ ()     u0 ∞ ν L () ≤ k 0 → T (u ) a.e. on as ν → ∞ u  ν k 0    1 u0 1,p → 0 as ν → ∞. ν ν W () 0

Then, for all k, ν > 0, we denote by (Tk (u))ν the unique solution of the problem   ∂(Tk (u))ν = ν((Tk (u) − (Tk (u))ν ) on Q ∂t  (T (u)) (0, ·) = u0 on . k

ν

ν

Then (Tk (u))ν ∈ Lp (0, T ; W0 ()) ∩ L∞ (Q), ∂(Tk∂t(u))ν ∈ Lp (0, T ; W0 ()) ∩ L∞ (Q), and up to a subsequence, we can assume that 1,p

1,p

1,p

(Tk (u))ν → Tk (u) strongly in Lp (0, T ; W0 ()), (Tk (u))ν (t) → Tk (u)(t) a.e. on for a.e. t and (Tk (u))ν L∞ (Q) ≤ k ∀ν > 0. Let κ ∈ D+ (0, T) and hl (r) = (l + 1 − |r|)+ ∧ 1, l ∈ N, l > k. LEMMA 4.5. Let um,n be a solution of (Pm,n )(u0m,n , fm,n ), and let u be given by Lemma 4.3. Then, lim sup lim sup κa(um,n , Dum,n ) · D(hl (um,n )(Tk (um,n ) − (Tk (u))ν )) ≤ 0. ν→∞ m,n→∞

Q

Furthermore, there exists χk ∈ (Lp (Q))N such that

a(Tk (um,n ), DTk (um,n )) χk weakly in (Lp (Q))N and div a(Tk (u), DTk (u)) = div χk in D (Q) ∀k > 0.

(4.5)

476

J.evol.equ.

B. Andreianov, K. Sbihi and P. Wittbold 1,p

Proof. Since (Tk (um,n ))m,n is bounded in Lp (0, T ; W0 ()), then thanks to Assump

N

tion (H3 ), (a(Tk (um,n ), DTk (um,n )))m,n is bounded in (Lp (Q)) , thus after passing to a

N

suitable subsequence, it converges weakly in (Lp (Q)) to some χk as m, n → ∞. The aim is to show, via the pseudo-monotonicity argument, that div χk = div a(Tk (u), DTk (u)) in D (Q) ∀k > 0. To this end, consider κhl (um,n )(Tk (um,n ) − (Tk (u))ν ) as a test function in (4.1) and pass to the limit with m, n → ∞ in each term. We use the same techniques as in [3, Proof of Theorem 2.4] to prove that lim inf

lim (um,n )t , κhl (um,n )(Tk (um,n ) − (Tk (u))ν ) ≥ 0.

ν→∞ m,n→∞

(4.6)

It is clear that lim

ψm,n (um,n )κhl (um,n )(Tk (um,n ) − (Tk (u))ν ) = 0

lim

ν→∞ m,n→∞

(4.7)

Q

and lim

fm,n κhl (um,n )(Tk (um,n ) − (Tk (u))ν ) = 0.

lim

ν→∞ m,n→∞

(4.8)

Q

Let us estimate the term lim

lim

um,n (t))− (Tk (u))ν (t))dµm,n (t)dt. κhl ( um,n (t))(Tk ( ν→∞ m,n→∞ Q

For this purpose, we split it as follows: κhl ( um,n (t))(Tk ( um,n (t)) − (Tk (u))ν (t))(dµm,n (t) − dµ(t))dt Q

+

κhl ( um,n (t))(Tk ( um,n (t)) − (Tk (u))ν (t))dµ(t)dt =: I1 + I2 .

Q

Thanks to the strong convergence of µm,n to µ in L1 (0, T ; w∗−Mb ()) as m, n → ∞, we have |I1 | ≤ 2kκ∞ µm,n − µL1 (0,T ;w∗−Mb ()) → 0. On the other hand, I2 = κhl ( um,n (t))(Tk ( um,n (t)) − Tk ( u(t)))dµ(t)dt Q

+ Q

κhl ( um,n (t))(Tk ( u(t)) − (Tk (u))ν (t))dµ(t)dt =: I21 + I22 .

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Moreover, since Tk (um,n (t)) → Tk (u(t)) a.e. on and Tk ( um,n (t)), Tk ( u(t)) are quasicontinuous for a.e. t, then Tk ( um,n (t)) converges to Tk ( u(t)) q.e. on for almost everywhere t. Similarly, hl ( um,n (t)) converges to hl ( u(t)) q.e. on for a.e. t. We get 1 2 u(t))(Tk (u(t)) ˜ − (Tk (u))ν (t))dµ(t)dt. lim I2 = 0 and lim I2 = κhl ( m,n→∞

m,n→∞

Q

Since (Tk (u))ν (t) → Tk (u)(t) ˜ q.e. on for a.e. t as ν → ∞, so that lim

lim I 2 ν→∞ m,n→∞ 2

Therefore, we conclude that lim lim κhl ( um,n (t))(Tk ( um,n (t)) − (Tk (u))ν (t))dµm,n (t)dt = 0. ν→∞ m,n→∞

= 0.

(4.9)

Q

Putting together all limits (4.6)-(4.9), we get lim sup lim sup κa(um,n , Dum,n ) · D(hl (um,n )(Tk (um,n ) − (Tk (u))ν )) ≤ 0.

(4.10)

ν→∞ m,n→∞

Q

An equivalent formulation of (4.10) is lim sup lim sup κhl (um,n )a(um,n , Dum,n ) · D Tk (um,n ) − (Tk (u))ν ν→∞ m,n→∞

Q

κh l (um,n )(Tk (um,n )

+ {l