Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces

The existence of a capacity solution to a coupled nonlinear parabolic–elliptic system is analyzed, the elliptic part in the parabolic equation being of the form $$-\,\mathrm{div}\, a(x,t,u,\nabla u)$$-diva(x,t,u,∇u). The growth and the coercivity conditions on the monotone vector field a are prescribed by an N-function, M, which does not have to satisfy a $$\Delta _2$$Δ2 condition. Therefore we work with Orlicz–Sobolev spaces which are not necessarily reflexive. We use Schauder’s fixed point theorem to prove the existence of a weak solution to certain approximate problems. Then we show that some subsequence of approximate solutions converges in a certain sense to a capacity solution.


Introduction
In recent years, there has been an increasing interest in the study of various mathematical problems involving the operators satisfying non-polynomial growth conditions instead of having the usual p-structure which employ the standard theory of monotone operators relying on the Sobolev space W 1,p (Ω), the origins of which can be traced back to the work of Orlicz in the 1930s. Later on, Polish and Czechoslovak mathematicians investigated the modular function spaces (see, for example, Musielak [19] and Krasnoselskii and Rutickii [18]). Many properties of Sobolev spaces have been extended to Orlicz-Sobolev spaces, mainly by Dankert [7] Donaldson and Trudinger [9] and O'Neil [20] (see also [1] for an excellent account of those works). At present, the operators satisfying non-polynomial growth arouse much interest with the development of elastic mechanics, electro-rheological fluids as an important class where Ω ⊂ R d , d ≥ 2, is the space region occupied by the semiconductor, Au = − div a(x, t, u, ∇u) is a Leray-Lions operator defined on W 1,x 0 L M (Q T ), M is an appropriate N -function, and the functions ϕ 0 and u 0 are given.
The functional spaces to deal with these problems are Orlicz-Sobolev spaces. In general, Orlicz-Sobolev spaces are neither reflexive nor separable.
This problem may be regarded as a generalization of the so-called thermistor problem arising in electromagnetism [3,13,14].
Since we are dealing with a nonuniformly elliptic problem (see assumption (3.6) on ρ(s) below), one readily realizes that the search of weak solutions to problem (1.1) are not well suited. Indeed, ρ(s) may converge to zero as |s| tends to infinity and as a result, if u is unbounded in Q T , the elliptic equation becomes degenerate at points where u is infinity and, therefore, no a priori estimates for ∇ϕ will be available and thus, ϕ may not belong to a Sobolev space. Instead of ϕ, we may consider the function Φ = ρ(u)|∇ϕ| 2 as a whole and then show that belongs to L 2 (Q T ) d . This means that a new formulation of the original system is possible and the solution to this new formulation will be called capacity solution.
The concept of capacity solution was first introduced by Xu in [25] in the analysis of a modified version of the thermistor problem. The same author applied this concept to more general settings where weaker assumptions [24] or mixed boundary conditions [26] are considered.
The existence of a capacity solution of (1.1) in the classical Sobolev spaces has been proved by González Montesinos and Ortegón Gallego in [14].
Notice that L M (Ω) is a Banach space under the so-called Luxemburg norm, namely Convergence in norm in Orlicz or Orlicz-Sobolev spaces is rather strict when M does not satisfies the Δ 2 -condition. To this end, it is very convenient to introduce the concept of modular convergence. Definition 2.1. Let (u n ) ⊂ L M (Ω) and u ⊂ L M (Ω). We say that u n converges to u for the modular convergence in L M (Ω) if for some λ > 0, . We say that (u n ) converges to u for the modular convergence in If M satisfies the Δ 2 -condition on (near infinity only when Ω has finite measure), then modular convergence coincides with norm convergence. This is not true in the general case. For instance, consider the following 1D example: x we have lim n→∞ 1 0 M (u n ) = 0 and thus u n → 0 in L M (0, 1) for the modular convergence. On the other hand, for ε > 0 we obtain and, consequently, (u n ) does not converge in the norm of L M (0, 1).
The following result shows that the modular convergence in L M implies, in particular, the convergence in the weak- * topology σ(L M , LM ). Lemma 2.2. ( [11,16]) Let (u n ) ⊂ L M (Ω), u ∈ L M (Ω) and v ∈ LM (Ω) such that u n → u with respect to the modular convergence. Then, and thus, from Lebesgue's dominated convergence theorem it yields u n k v → uv strongly in L 1 (Ω). Since the limit uv does not depend on the subsequence (u n k ) k , it is the whole sequence (u n ) that converges strongly in L 1 (Ω). In order to show the second assertion, we have and using the first assertion, we deduce the desired result.

Lemma 2.3.
Assume that the open set Ω ⊂ R d has finite measure. If P M and u n → u for the modular convergence in L M (Ω), then u n → u strongly in E P (Ω).
Proof. By Theorem 2.1 in [23] we have u n , u ∈ E P (Ω). Let > 0 be arbitrary. There exists λ > 0 such that for a subsequence still denoted u n . Now choose t 0 > 0 such that Then, Since h + P ( t0 ) ∈ L 1 (Ω), we have by Lebesgue's dominated convergence theorem. As > 0 is arbitrary, we have u n → u in E P (Ω).
Let W −1 LM (Ω) (resp. W −1 EM (Ω)) denote the space of distributions on Ω which can be written as sums of derivatives of order up to one of functions in LM (Ω) (resp. EM (Ω)). It is a Banach space under the usual quotient norm.
If the open set Ω has the segment property, then the space D(Ω) is dense in W 1 0 L M (Ω) for the modular convergence and for the topology σ(ΠL M , ΠLM ) (see [15]). Consequently, the action of a distribution in W −1 LM (Ω) on an element of W 1 0 L M (Ω) is well defined. For more details the reader is referred to [1,18].
For K > 0, we define the truncation at height K, T K : R → R by The following abstract lemmas will be applied to the truncation operators.
x the distributional derivative on Q T of order α with respect to the variable x ∈ Ω, and |α| = α 1 + · · · + α d . The inhomogeneous Orlicz-Sobolev spaces are defined as follows, The last space is a subspace of the first one, and both are Banach spaces under the norm, We can easily show that they form a complementary system when Ω satisfies the segment property [15]. These spaces are considered as subspaces of the product space ΠL M (Q T ) = L M (Q T ) d+1 . We shall also consider the weak- * topologies σ(ΠL M , ΠEM ) and σ(ΠL M , ΠLM ).
If, further, u ∈ W 1,x E M (Q T ) then the concerned function is a W 1 E M (Ω)valued and is strongly measurable. Furthermore the following embedding holds: . We can easily show as in [16] that when Ω has the segment property, then each element u of the closure of D(Q T ) with respect of the weak- * topology σ(ΠL M , ΠEM ) is a limit, in W 1,x L M (Q T ), of some sequence (u n ) ⊂ D(Q T ) for the modular convergence, i.e., there exists λ > 0 such that for all |α| ≤ 1, . This space will be denoted by Thus both sides of the last inequality are equivalent norms on W 1,x 0 L M (Q T ). We have then the following complementary system . It is also, except for an isomorphism, the quotient of ΠLM by the polar set W 1,x 0 E M (Q T ) ⊥ , and will be denoted by F = W −1,x LM (Q T ) and it can be shown that, This space will be equipped with the usual quotient norm where the infimum is taken over all possible decompositions f = |α|≤1 ∇ α x f α , f α ∈ LM (Q T ). The space F 0 is then given by, and is denoted by Remark 2.6. We can easily check, using Lemma 2.4, that each Lipschitz continuous mapping F , with F (0) = 0, acts in the inhomogeneous Orlicz-Sobolev space of order one In the sequel, we will make use of the following results which concern mollification with respect to time and space variables and some trace results. For a function u ∈ L 1 (Q T ) we introduce the functionũ ∈ L 1 (Ω × R) as u(x, s) = u(x, s)χ (0,T ) and define, for all μ > 0, t ∈ [0, T ] and a.e. x ∈ Ω, the function u μ given as follows Capacity solution in Orlicz-Sobolev spaces Page 9 of 37 14 for the modular convergence).

Lemma 2.9. ([12]) Let Ω be a bounded open subset of R d with the segment property. Consider the Banach space
Then the embedding W ⊂ C([0, T ]; L 1 (Ω)) holds true and is continuous.

Assumptions and statement of the main results
In the sequel, Ω is a bounded open set in R d , d ≥ 2 an integer, T > 0 is given and Q T = Ω × (0, T ). We consider the Banach space W given as follows provided with its standard norm Throughout this paper ·, · stands for the duality pairing between the spaces , and we assume the following assumptions: Consider a second order partial differential operator where c ∈ EM (Q T ), e ∈ E P (Q T ) and α, ζ, k > 0 are given real numbers.
ρ ∈ C(R) and there existsρ ∈ R such that 0 < ρ(s) ≤ρ, for all s ∈ R, Taking v = u/ u (P ) with u = 0 in (3.9) and using (2.8) it yields and the first assertions of this Lemma are readily deduced. Now let P M . Owing to the convexity of P and M we can derive the following estimates Making v = u/ u (M ) , u = 0, in this last inequality and using (2.8) we finally deduce

Remark 3.2.
Under the assumptions of Lemma 3.1, we have

Definition of a capacity solution
The definition of a capacity solution for problem (1.1) can be stated as follows.
is called a capacity solution of (1.1) if the following conditions are fulfilled: Notice that, thanks to Lemma 2.9 and the regularity of u, we obtain in particular u ∈ C([0, T ]; L 1 (Ω)) and thus the initial condition (C 4 ) makes sense at least in L 1 (Ω).

An existence result
This section is devoted to the proof of the following existence theorem which is the main result of this work. In order to prove this result, we will need to show the existence of a weak solution to a similar problem but with stronger assumptions, namely, there exists c ∈ EM (Q T ), and two real numbers ζ > 0 and k ≥ 0, such that for almost every (x, t) ∈ Q T and for all s ∈ R, ξ ∈ R d , Proof. So as to prove the existence of a weak solution, Schauder's fixed point theorem will be applied together with the existence and uniqueness result of a weak solution to a parabolic equation. For every ω ∈ E P (Q T ) and almost everywhere t ∈ (0, T ), we consider the elliptic problem div(ρ(ω)∇ϕ) = 0 in Ω, Thanks to Lax-Milgram's theorem, (5.3) has an unique solution ϕ(t) ∈ H 1 (Ω), in fact, ϕ is measurable in t with values in H 1 (Ω) [3]. In that case, it is ϕ ∈ L ∞ (0, T ; H 1 (Ω)). Indeed, by the maximum principle we have Capacity solution in Orlicz-Sobolev spaces Page 13 of 37 14 Using ϕ − ϕ 0 ∈ H 1 0 (Ω) as a test function in (5.3) we get, By the Cauchy-Schwarz inequality, we obtain Notice that the right hand side in the original parabolic equation is ρ(u)|∇ϕ| 2 ∈ L 1 (Ω × (0, T )). Thanks to the elliptic equation, this term also belongs to the space L 2 (0, T ; H −1 (Ω)). Indeed, let φ ∈ D(Ω) and take ξ = φϕ as a test function in (5.3). We have, for a.e. t ∈ [0, T ], This means that ρ(ω)|∇ϕ| 2 = div(ρ(ω)ϕ∇ϕ) in D (Ω) and a.e. in [0, T ]. (5.6) Since ρ(ω)ϕ∇ϕ ∈ L 2 (Q T ) d we finally deduce the regularity div(ρ(ω)ϕ∇ϕ) ∈ L 2 (0, T ; H −1 (Ω)).
The identity (5.6) is one of the keys that allows us to solve the classical thermistor problem and the introduction of the notion of a capacity solution as well. Now we introduce the following parabolic problem The variational formulation of the parabolic equation is given as follows.
for all φ ∈ W 1,x 0 L M (Q T ), for all t ∈ [0, T ], u(·, 0) = u 0 in Ω. The existence of a solution to (5.8) is obtained by a straightforward application of Theorem 1, p. 107 in [10]. Also we can easily check that the solution of (5.8) is unique [2] Now, we show that |∇u| ∈ L M (Q T ), and the estimates where C 1 only depends on data, but not on ω. Indeed, let λ > 0 such that . By taking φ = u as a test function in (5.8), from (3.4), (3.5), (5.2), (5.4) and Young's inequality, we obtain α λμ This shows that |∇u| ∈ L M (Q T ) and, consequently, the estimate (5.9) is derived by just taking λ = 1 in this last inequality. In order to obtain (5.10), first notice that from the last inequality we also have (a(x, t, ω, ∇u) − a(x, t, ω, ∇φ))(∇u − ∇φ) dx dt, and thus, using (5.11) and Young's inequality, where ζ is the constant appearing in (5.1). Sincē then, using (2.7) Notice that C 2 only depends on data (but not on ω). Therefore, gathering all these estimates, we deduce for all which finally yields the estimate (5.10) by considering the dual norm on LM (Q T ).
We may define the operator G : ω ∈ E P (Q T ) −→ G(ω) = u ∈ W, with u being the unique solution to (5.8). From Lemma 2.10, and Lemma 2.11 with Y = L 1 (Ω), we have that W → E P (Q T ) with compact embedding. Consequently, G maps E P (Q T ) into itself and, due to the estimates (5.9) and (5.12), G is a compact operator. Moreover, from (5.9) we have, for R > 0 large enough To complete the proof, it remains to show that G is a continuous operator. Thus, let (ω n ) ⊂ B R be a sequence such that ω n → ω strongly in E P (Q T ) and consider the corresponding functions to ω n , that is, u n = G(ω n ) and ϕ n and put F n = ρ(ω n )ϕ n ∇ϕ n and F = ρ(ω)ϕ∇ϕ. We have to show that Owing to P M and (5.9), we have ∇u ∈ E P (Q T ) d . Since the inclusion L P (Q T ) ⊂ L 2 (Q T ) is continuous, we also have ω n → ω strongly in L 2 (Q T ) and thus, we may extract a subsequence, still denoted in the same way, such that ω n → ω a.e. in Q T . Then, it is an easy task to show that ϕ n → ϕ strongly in L 2 (0, T ; H 1 (Q T )) and, consequently, also for another subsequence denoted in the same way, F n → F strongly in L 2 (Q T ).
On the other hand, since (ω n ) ⊂ L P (Q T ) is bounded, in virtue of the estimates obtained above, we deduce, again modulo a subsequence, By subtracting the respective equations of (5.8) for u n and u, and taking φ = u n − u as a test function, for all t ∈ [0, T ], we obtain 1 2 ||u n (t) − u(t)|| 2 For the first term of the right hand side of (5.16), we have The first of these last integrals converges trivially to zero. As for second one, using the fact that |hn| 4ζ > 1 on the set {|h n | > 4ζ} and (3.9), it yields In virtue of (3.3), we deduce P |h n | 4ζ ≤ 1 4 (P (e) + P (ω n ) + P (ω) + kM (|∇u|)) , and since P (ω n ) → P (ω) strongly in L 1 (Q T ), by Lebesgue's dominated theorem it yields that For the second term of the right hand side of (5.16), we use Young's inequality and (3.9). It yields, It has been already shown that the first of these terms converges to zero. As for the second one, since P M , we can take λ 0 large enough such that P (s) ≤ M (s) for |s| > λ 0 , and then, Consequently, for some sequence ( n ) ⊂ R, n → 0, we have the following estimate and integrating this inequality over [0, T ], we have The first term of right hand side in (5.17) converges to zero since F n → F strongly in L 2 (Q T ) d and (T − t)(∇u n − ∇u) is bounded in L 2 (Q T ) d . In conclusion, u n → u strongly in L 2 (Q T ). Since this limit does not depend upon the subsequence one may extract, it is in fact the whole sequence (u n ) which converges to u strongly in L 2 (Q T ). On the other hand, in virtue of (5.13), we also have u n → U strongly in L 2 (Q T ), so that u = U and we can rewrite (5.13) to give u n → u strongly in E P (Q T ). This shows that G is continuous and this ends the proof of Theorem 5.2.

Remark 5.3.
It can be easily shown that we can rid of the assumption (3.3) in Theorem 5.2 whenM verifies the Δ 2 -condition. Also in the case a(x, t, s, ξ) = a(x, t, ξ).

Proof of Theorem 5.1
The proof is divided into several steps, first we introduce a sequence of approximate problems and derive a priori estimates for the approximate problem and we show two intermediate results, namely the strong convergence in L 1 (Q T ) of both ∇u n and ϕ n , where (u n , ϕ n ) is a weak solution to the approximate problem of (1.1).
Step 1. For every n ∈ N, we introduce the following regularization of the data, ρ n (s) = ρ(s) + 1 n , (5.18) a n (x, t, s, ξ) = a(x, t, T n (s), ξ), (5.19) and consider the approximate system given as where c n ∈ EM (Q T ) is given by c n (x, t) = c(x, t) +M −1 (P (kn)). Also, in view of (3.6), we have that Thus, we can apply Theorem 5.2 to deduce the existence of a weak solution (u n , ϕ n ) to the system (5.20)-(5.24).
By the maximum principle we have hence there exists a function ϕ ∈ L ∞ (Q T ) and a subsequence, still denoted in the same way, such that Now let multiply (5.21) by ϕ n − ϕ 0 ∈ L 2 (0, T ; H 1 0 (Ω)) and integrate over Q T . We get where C 1 = C 1 (ρ, ϕ 0 L 2 (0,T ;H 1 (Ω)) ). Consequently, the sequence (ρ n (u n )∇ϕ n ) is bounded in L 2 (Q T ). Thus, there exists a function Φ ∈ L 2 (Q T ) d and a subsequence, still denoted in the same way, such that ρ n (u n )∇ϕ n → Φ weakly in (L 2 (Q T )) d . (5.29) This weak limit function Φ ∈ (L 2 (Q T )) d is in fact the third component of the triplet appearing in the Definition 4.1 of a capacity solution.
Taking u n as a test function in (5.20), for all t ∈ [0, T ], we obtain ρ n (u n )ϕ n ∇ϕ n ∇u n dx dt. where C is a positive constant not depending on n. It follows that the sequence (u n ) is bounded in W 1,x 0 L M (Q T ). Consequently, there exist a subsequence of (u n ), still denoted in the same way, and a function u ∈ W 1,x 0 L M (Q T ) such that: On the other hand, Let φ ∈ W 1,x 0 E M (Q T ) d be arbitrary with ∇φ (M ) = 1/(k + 1). In view of the monotonicity of a n , one easily has ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ QT a n (x, t, u n , ∇u n )∇φ ≤ QT a n (x, t, u n , ∇u n )∇u n − QT a n (x, t, u n , ∇φ)(∇u n − ∇φ) ≤ C + QT |a n (x, t, u n , ∇φ)∇u n | + QT a n (x, t, u n , ∇φ)∇φ.

(5.34)
We can show that the two last integrals in (5.34) are bounded with respect to n. Indeed, for the first one, by Young's inequality , and owing to Poincare's inequality, there exists λ > 0 such that QT M (u n /λ) ≤ 1 for all n ≥ 1. Also, since P M , there exists s 0 > 0 such that P (ks) ≤ P (ks 0 ) + M (s/λ) for all s ∈ R. Consequently, and thus QT |a n (x, t, u n , ∇φ)∇u n | ≤ C, for all n ≥ 1 and φ ∈ W 1,x 0 E M (Q T ) d such that ∇φ (M ) = 1/(k + 1). On the other hand, the second integral in (5.34), namely QT a n (x, t, u n , ∇φ)∇φ can be dealt in the same way so that it is easy to check that it is also bounded. Gathering all these estimates, and using the dual norm, one easily deduce that (a n (x, t, u n , ∇u n )) is bounded in LM (Q T ) d . Finally, since both sequences (div a n (x, t, u n , ∇u n )) and (div(ρ n (u n )ϕ n ∇ϕ n )) are bounded in the space W −1,x LM (Q T ) then, according to (5.20), we have Consequently, (u n ) ⊂ W is bounded and, since the embedding W → E P (Q T ) is compact, for a subsequence, still denoted in the same way, we have u n → u strongly in E P (Q T ) and a.e. in Q T , (5.38) where u ∈ W 1,x 0 L M (Q T ) is also the limit function appearing in (5.33).
Step 2. Introduction of regularized sequences and the almost everywhere convergence of the gradients.
We first introduce two smooth sequences, namely, (v j ) ⊂ D(Q T ) and (ψ i ) ⊂ D(Ω) such that for the modular convergence; 2. v j → u and ∇v j → ∇u and almost everywhere in For a fixed positive real number K, we consider the truncation function at height K, T K , defined in (2.11). Then, for every K, μ > 0 and i, j ∈ N, we introduce the function w i μ,j ∈ W 1,x 0 L M (Q T ) (to simplify the notation, we drop out the index K) defined as w i μ, for the modular convergence as j → ∞.
for the modular convergence as μ → ∞. Since we may consider subsequences in (5.39)-(5.41), we will assume without loss of generality that the convergences (5.40) and (5.41) also hold almost everywhere in Q T . We will establish the following proposition. as n tends to +∞.
Proof. In the sequel and throughout the paper, χ j s and χ s will denote, respectively, the characteristic functions of the sets We also introduce the primitive of the truncation function T K vanishing at the origin, Θ K , that is It is straightforward to show that 0 ≤ Θ K (t) ≤ K|t| for all t ∈ R. We will also make use of the following notation for vanishing sequences: (n) means a sequence such that lim n→∞ (n) = 0 or lim sup n→∞ (n) = 0; (n, j) is a term such that lim j→∞ lim n→∞ (n, j) = 0 where any occurrence of lim may be substituted by lim sup. And so on for (n, j, μ), etc.
For any μ, ν > 0 and i, j, n ≥ 1 we may use the admissible test function ϕ μ,i n,j,ν = T ν (u n − w i μ,j ) in (5.20). This leads to (5.44) By using (5.28), we get As far as the parabolic term is concerned, we have The first term of the right hand side in (5.46) can be written as we deduce that, for all i, j, n ≥ 1 and μ, n, K > 0, it is As for the second term of the right hand side in (5.46) we have Owing to (5.39) and (5.40) we have |w i μ | ≤ K almost everywhere in Q T . Also, since sT ν (s) ≥ 0 for all s ∈ R, we deduce, for all μ, ν, K > 0 and i ≥ 1, It remains to analyze the diffusion term of (5.44). We have (5.51) On one hand, let us observe that for any K > 0, and for n large enough, namely n > K + ν ≥ K, we have, a n (x, t, T K (u n ), ∇T K (u n )) = a(x, t, T K (u n ), ∇T K (u n )). (5.52) On the other hand, from (5.39), we have |w i μ,j | ≤ K a.e. in Q T , then in the (5.53) From (5.52) and (5.53), (5.51) becomes a(x, t, T ν+K (u n ), ∇T ν+K (u n ))∇w i μ,j .
(5.54) We put Since On the other hand, note that (5.57) The integral term J 3 tends to 0 as first n, then j, μ, i and s go to ∞. Indeed, since, and since,

NoDEA
Capacity solution in Orlicz-Sobolev spaces Page 29 of 37 14 function ψ ∈ L 1 (0, T ; W 1,1 (Ω)) satisfying the following properties: Proof. According to lemmas 2.3 and 3.1 we deduce the the following continuous inclusions: Since (u n ) is relatively compact in E P (Q T ), we can extract a subsequence (u n(k) ) ⊂ (u n ) such that : Fix K > 0 to be chosen later big enough and introduce the function γ given by Summing up these inequalities, bearing in mind that (u n(k) ) and (v k ) are bounded in W 1,x L M (Q T ) and (5.76), we deduce Hence ||γ|| L 1 (0,T ;W 1,1 (Ω)) ≤ C K .
The next two results analyze the behavior of certain subsequences of (ϕ n ). They will allow us, together with the convergences deduced in the previous steps, to pass to the limit in the approximate problems (5.20)-(5.24) in order to show the existence of a capacity solution to the system (1.1).
Lemma 5.7. ( [14]) Let (u n , ϕ n ) be a weak solution to the system (5.20)-(5.24), u ∈ E P (Q T ) and ϕ ∈ L ∞ (Q T ) the limit functions appearing, respectively, in (5.27) and (5.38). Then, for any function S ∈ C 1 0 (R), there exists a subsequence, still denoted in the same way, such that S(u n )ϕ n S(u)ϕ weakly in L 2 (0, T ; H 1 (Ω)). Proof. The proof of this result is almost identical to that of Lemma 4.8 in [14]. For the sake of completeness, we include it here. Since the conditions of Lemma 5.6 are fulfilled by a suitable subsequence (u n(k) ), we have for every > 0 there exists M > 0 and ψ ∈ L 1 (0, T ; W 1,1 (Ω)) such that ( and using Poincaré's inequality, we obtain  And since > 0 and m ≥ 1 are arbitrary, we derive the desired result. Step 5. Passing to the limit. According to (5.27), (5.29), (5.33), (5.35) and (5.37), it is straightforward that the condition (C 1 ) of Definition 1 is fulfilled. The convergences in Proposition 5.4 and Lemma 5.8 lead us to (C 2 ) of Definition 1, and in order to obtain the condition (C 3 ), using Proposition 5.4 and Lemma 5.8 again with (5.77), it is enough to let k goes to infinity in the following expression S(u n(k) )ρ n(k) (u n(k) )∇ϕ n(k) = ρ n(k) (u n(k) )[∇(S(u n(k) )ϕ n(k) ) − ϕ n(k) ∇S(u n(k) )] Step 6. Regularity of u.
Finally, it remains to establish the regularity u ∈ C([0, T ]; L 1 (Ω)). Though this is a straightforward consequence of Lemma 2.9, since u ∈ W ⊂ W ⊂ C([0, T ]; L 1 (Ω)), it is interesting to show this property from the results deduced on the previous steps about the (sub)sequences (u n ) and (ϕ n ) and how the notion of capacity solution is used along this proof. To this end, we go back to the expression (5.44) but the integration in time happens in the interval (0, τ) for any τ ∈ (0, T ], namely (see [12]) ∂u n ∂t , T ν (u n − w i μ,j ) Qτ = Qτ a n (x, t, u n , ∇u n )(∇w i μ,j − ∇u n )χ {|un−w i μ,j |≤ν} − Qτ ρ n (u n )ϕ n ∇ϕ n ∇T ν (u n − w i μ,j ). (5.83) where ν ∈ (0, 1], Q τ = (0, τ) × Ω and ·, · Qτ is the duality product between W −1,x LM (Q τ ) and W 1,x 0 L M (Q τ ). We will consider the necessary subsequences to assure the almost everywhere convergence in Q T of ϕ n → ϕ, u n → u, ∇u n → ∇u, and also for (T ν (u n − w i μ,j )), etc. From Then, passing to the limit in these two expressions, first in j, then in μ, i and K, we deduce, uniformly in τ , that Qτ a n (x, t, u n , ∇u n )(∇w i μ,j − ∇u n )χ {|un−w i μ,j |≤ν} ≤ (n, j, μ, i, K). (5.84) The analysis of the term Qτ ρ n (u n )ϕ n ∇ϕ n ∇T ν (u n − w i μ,j ) dx dt is more involved. Here the difficulty relies on the fact that the sequence (ρ n (u n )|∇ϕ n | 2 ) does not converge, in general, strongly in L 1 (Q T ). In order to deal with this situation, we are going to make use of the properties already shown for a capacity solution. Indeed, we first notice that ∇T ν (u n − w i μ,j ) = 0 in the set {|u n | ≤ K + ν} ⊂ {|u n | ≤ K + 1}. Then we consider a sequence of functions S K ⊂ C 1 0 (R) such that 0 ≤ S K ≤ 1, S K = 1 in [−(K + 1), K + 1], for all K > 0.