Solution of degenerate parabolic variational inequalities with convection

Mod´elisation Math´ematique et Analyse Num´erique Vol. 37, N 3, 2003, pp. 417–431 DOI: 10.1051/m2an:2003035

SOLUTION OF DEGENERATE PARABOLIC VARIATIONAL INEQUALITIES WITH CONVECTION

Jozef Kacur

and Roger Van Keer

Abstract. Degenerate parabolic variational inequalities with convection are solved by means of a combined relaxation method and method of characteristics. The mathematical problem is motivated by Richard’s equation, modelling the unsaturated – saturated flow in porous media. By means of the relaxation method we control the degeneracy. The dominance of the convection is controlled by the method of characteristics.

Mathematics Subject Classification. 65M25, 65M12.

Received: February 1st, 2002.

1. Introduction

In this paper we consider the parabolic convection-diffusion variational inequality

(∂_tb(u), v−u)+(A∇u,∇(v−u))+(div ¯F(x, u), v−u)+(g(t, u), v−u)_Γ2≥(f(t, u), v−u) ∀v∈L₂(I, K) (1) withu(x,0)∈K.

Here, (., .) is the scalar product inL₂(Ω), Ω⊂R^N is a bounded domain with Lipschitz continuous boundary

∂Ω, I≡(0, T),W₂¹(Ω) is the standard first order Sobolev space, V ≡ {v∈ W₂¹ :v = 0 on Γ₁}, K is a closed convex set inV andu:I→V is an abstract function. Let (u, v)_Γ2 =

Γ2uvdx; Γ₁ and Γ₂⊂∂Ω are open with Γ₁∩Γ₂=∅and meas_N−1Γ₁+ meas_N−1Γ₂= meas_N−1∂Ω.

We assume thatb(s) is increasing,A(x) is a possitive definite symmetric matrix,∂_sF¯(x, s) is bounded andg, f are sublinear inu. The problem (1) includes many practical problems governed by convection-diffusion phe- nomena with adsorption, with unilateral conditions (in Ω or on∂Ω). IfK≡V then (1) represents the variational formulation of degenerate convection-diffusion problem, which also includes “porous media type equation” with convection, which can be dominant. It is well known that the numerical approximation (convergence analysis and implementation) of these type of problems is very difficult, since the solution can possess the finite support with sharp fronts on the boundary of the support. We present a motivating example concerning the flow in

Keywords and phrases. Richard’s equation, convection-diffusion, parabolic variational inequalities.

1 Faculty of Mathematics, Physics and Informatics, Comenius University Mlynsk´a dolina, 84248 Bratislava, Slovakia.

e-mail:kacur@fmph.uniba.sk

2 Ghent University, Department of Mathematical Analysis, Galglaan 2, 9000 Gent, Belgium.

e-mail:rvk@cage.rug.ac.be

c EDP Sciences, SMAI 2004

unsaturated-saturated porous media governed by Richard’s equation

∂_tθ= div (k(h)A(x)∇(h+z)). (2)

Here,θis the volumetric water content,his the pressure head,kis the hydraulic permeability. We use the Van Genuchten–Mualem model, describing the retention and permeability curves

θ=θ(h) =θ_r+ θ_s−θ_r

(1 + (αh)ⁿ)^m, k(S) =S^1/2 1−

1−S^1/m_m2

, where S = _θ^θ−θr

s−θr is the effective saturation and k(h) = k(θ(h)) forh < 0,k(h) = 1 for h >0. Here, θ_r, θ_s, 1< nandm, (m= 1−_n¹), are so called soil parameters. We verify that (2) is a degenerate parabolic equation where the degeneracy occurs in both elliptic and parabolic (storativity) terms. This generates two interfaces in the flow: the interface between dry and wet regions and the interface between unsaturated (h < 0) and saturated (h >0) zones. These degeneracies can be transformed only to the parabolic term, using Kirchhoff’s transformation

u:=β(h) = _h

0

k(z)dz, b(u) :=θ(β⁻¹(u)).

Then, forh <0 (unsaturated zone) we obtain

∂_tb(u)−div (A(x)∇u)−∂_zK(x, b(u)) = 0, (3) where K(x, b(u)) =A(x)¯e˜k(b(u)), ¯e being the unit vector in directionz. The unknown uvaries in the inter- val (u^∗,0). Moreover, we can verify that b(u^∗) =∞. The flow in the saturated zone h≥ 0 is governed by Darcy’s law, which leads to the mathematical model

S_e∂_th−div (A(x)∇(h+z)) = 0, (4)

where S_e is the specific (elasticity) storativity coefficient,k(h)≡1 forh≥0 and θ=S_eh– see [6]. Also here we can use Kirchhoff’s transformation and obtain u=hforh >0,b(u) =S_euandK(x, b(u)) =A(x)¯e. If we prolonguandb(u) foru >0, then (3) describes unsaturated-saturated flow in porous media. We can shift the unknown so thatu^∗ will be translated to 0. Then, we are looking for a nonegative solution of (1). Generally, we are not able to guarantee that the numerical approximations of such a solution will be nonnegative too.

Therefore, it is natural to reformulate (3) in terms of a variational inequality and to look for the solution in a closed convex set K = {v ∈ V : v ≥ 0}. Thus, we arrive at a variational inequality of the type (1) with degenerated b(u). The degeneracyb(u^∗) = ∞ gives rise to sharp fronts between the wet region and the dry region (h= −∞, i.e. u= u^∗) in the unsaturated part of the porous media (infiltration phenomenon). The solution has a sharp front there. In this case the convective termK(x, b(u)) is effected by the gravitation. The boundedness of ∂_uK(x, b(u)) is required in our numerical approximation. We can verify that∂_uK(x, b(u)) is bounded (uniformly inx∈Ω foru≥u^∗) provided that m <1−²_n. If we acceptm= 1−_n¹, then we have the requirementn≥2, which is satisfied for a large scale of porous media.

The mathematical formulation of (3) in terms of variational inequalities also allows us to consider unilateral boundary conditions which are quite natural for the unsaturated-saturated flow – see [2]. For an illustration we present the following model. Let us assume that on the part Γ₂⊂∂Ω the medium is in contact with water, or air, or with both of them. When it is in contact with water, then a positive pressure has to be prescribed. When the medium is in contact with air, then either the flux is zero and the pressure is nonpositive (not prescribed), or the flux is positive and the pressure is zero (overflow). The mathematical formulation is as follows – see [2]

(i) h⁺=p∗ (data) on Γ₂where h >0;

(ii) k(h)A(∇h+ ¯e).ν≤0 on Γ₂ where h= 0;

(iii) k(h)A(∇h+ ¯e).ν= 0 on Γ₂ where h <0.

Here,h⁺ := max{h,0}. The vectorν is the outward unit normal vector to∂Ω. Both conditions (ii) and (iii) can be formulated in the form

k(h)A(∇h+ ¯e).ν(h−w)≤0 on Γ₂, ∀w:w⁺=p∗.

We assume thatp∗ ∈L₂(I, W₂¹) andp∗ ≥0. The mathematical formulation of this type of boundary conditions leads again to a variational inequality of the type (1). Using Kirchhoff’s transformation (including the shifting mentioned above) we take the convex setL₂(I, K) in the form{v∈L₂(I, V) :v≥0,(v+u^∗)⁺=β(p∗) on Γ₂}. Integrating by parts in (1) for a smooth solutionuwe obtain

A

∇u+ ¯e˜k(b(u))

.ν, u−v

Γ2 ≤0, from which the conditions (ii) and (iii) follow since sgn (u−v) = sgn (h−w).

The mathematical model (1) has been analysed with respect to existence and uniqueness of the solution in [1]

and [27]. Richard’s equation (3) has been solved (by finite volume approximations) in [12] under the assumption that b is Lipschitz continuous. The variational inequality (1) has been solved by the relaxation method in [3]

without convective term and with a Lipschitz continuousb.

The aim of our paper is to approximate the solution to (1) by the method of semi-discretization in time and by full discretization techniques and to prove their convergence. The approximation method is based on the relaxation method (to control the degeneracy ofb) – see [3, 15, 17–19, 25], and on a modification of the method of characteristics – see [4, 5, 7–11, 20, 22, 28, 29] among others. We follow the idea of regularized characteristics analysed in [20]. This allows us to include the dominance of the convective term ¯F, which, in addition, may depend on the unknown u.

In Section 2 we present the approximation scheme and we specify the assumptions on the data. The conver- gence of the semi-discretization scheme is proved in Section 3. The full discretization method is discussed in Section 4.

2. Variational solution and the approximation scheme

ByC we denote a generic positive constant. ByL_∞, L₂,W₂¹,L₂(I, L₂),L₂(I, V),L₂(I, V^∗), (V ⊂W₂¹is a subspace), we denote standard functional spaces (see [23]). Let .∞, .0, . and.∗ be the norms inL_∞, L₂,V andV^∗, respectively.

We shall introduce the following 5 hypotheses:

(H₁) b(s) ≥γ >0 and |b(s)| ≤ C(1 +|s|) ∀s∈ R, b(0) = 0. There exist regularizationsb_n(s) with the properties:

(i) b_n(s)→b(s) forn→ ∞locally uniformly;

(ii) |b_n(s)| ≤C₁+C₂|s|;

(iii) min{b(s), µ} ≤b_n(s)≤ K_n ) ∀s∈R, with K_n→ ∞for n→ ∞and with 0< µ < C_e. (H₂) A(x) is symmetric and uniformly positive definite,

i.e. (A(x)ξ, ξ)≥C|ξ|² ∀x∈Ω,∀ξ∈R^N. (H₃) ∂_sF¯(x, s)_∞≤C,|div_xF¯(x, s)| ≤C₁(1 +|s|).

(H₄) |f(t, x, s)| ≤C(1 +|s|), |∂_tg(t, x, s)| ≤C(1 +|s|) and

|∂_sg(t, x, s)| ≤C, ∀(x, t)∈Ω×I,∀s∈R.

(H₅) u₀∈K (K being a closed convex set inV).

Remark 1. The regularization b_n(s) of b(s) can be constructed as follows. Put _n(z) := min{b(z), K_n} and b_n(s) :=

_s

0

_n(z)dz, whereK_n→ ∞forn→ ∞.

If b(u) is not Lipschitz continuous, then we are not able to guarantee that∂_tb(u)∈ L₂(I, V^∗). If∂_tb(u)∈ L₂(I, V^∗) andv∈L₂(I, K) with∂_tv∈L₂(I, L₂), then

_t

0 ∂_tb(u), v−udt= (b(u(t)−b(u₀), v(t)−u(t))− _t

0

((b(u)−b(u₀)), ∂_t(v−u)).

Thus, in the general case we weaken the variational solution to (1) using integration by parts in thet-variable.

We define the weak variational solution to (1) as follows

Definition 1. u∈L₂(I, V), withb(u)∈L₂(I×Ω), is a weak variational solution to (1) iff

− _t

0

(b(u)−b(u₀), ∂_t(v−u)) + (b(u(t)−b(u₀), v(t)−u(t)) +

_t

0 (A∇u,∇(v−u))dx+ _t

0 (div ¯F(x, u), v−u)dt≥ _t

0

(f(t, u), v−u)dx, ∀v∈L₂(I, K) with ∂_tv∈L₂(I, L₂).

Letu_i be an approximation ofu(x, t) at time pointst=t_i,t_i=iτ ,i= 1, . . . , n, whereτ= ^T_n is the time step.

To control the convective term generated by ¯F(x, u) we use the method of characteristics in the following way. Letϕⁱ(x) be the approximate characteristic map for the time interval (t_i−1, t_i) defined by

ϕⁱ(x) :=x−τω_h∗∂_uF¯(x, u_i−1), (ω_h∗z)(x) =

R^N

ω_h(x−ξ)z(ξ)dξ, whereω_his a modifier:

ω_h(x) =h^−Nω₁(x h), with

ω₁(x) =κexp

|x|²

|x|²−1

for|x| ≤1, ω₁= 0 for|x| ≥1,

R^Nω₁(x)dx= 1.

We shall assume that h = τ^ω for ω ∈ (0,1). The map ϕⁱ(x) is a regularization of the map ¯ϕ(x) := x− τ∂_uF¯(x, u_i−1), which is the Euler-backwards approximation of theϕⁱ(x)-characteristic defined by the ODE

dX(s;t_i, x)

ds = ¯F(X(s;t_i, x), u_i−1(X(s;t_i, x)) for s∈(t_i−1, t_i) (5) with the initial conditionX(t_i;t_i, x) =x. Thenϕⁱ(x)≡X(t_i−1;t_i, x).

To control the dominance of the convective term and the degeneracy of the parabolic term we suggest the following approximation scheme. We determineu_iat the time pointt_i (assuming thatu_i−1 is known) from the elliptic variational inequality

1

τ(λ_i(u_i−u_i−1), v−u_i) + (A∇u_i,∇(v−u_i)) + (g(t, u_i), v−u_i)_Γ2 ≥ (H(t, u_i−1), v−u_i)−1

τ(u_i−1−u_i−1◦ϕⁱ, v−u_i), ∀v∈K, (6) where

H(t, u_i−1) :=f(t_i, x, u_i−1)−div_xF¯(x, u_i−1)

and whereλ_i∈L_∞(Ω) is a relaxation function which has to satisfy the “convergence condition”

λ_i−b_n(u_i)−b_n(u_i−1) u_i−u_i−1

∞

< τ. (7)

Here, the convective term is included in the source term. In fact, the scheme (6, 7) is implicit with respect to (λ_i, u_i). To determineλ_i we propose the iteration procedure

(λ_i,k−1(u_i,k−u_i−1), v−u_i,k) +τ(A∇u_i,k,∇(v−u_i,k)) +τ(g_i, v−u_i,k)_Γ2 ≥

τ(H(t_i, u_i−1), v−u_i,k)−(u_i−1−u_i−1◦ϕⁱ, v−u_i,k) ∀v∈K, (8) with

λ_i,k:= b_n(u_i,k)−b_n(u_i−1)

u_i,k−u_i−1 · (9)

This concept of relaxation of the parabolic term has been used for variational equations in [17–19]. In fact, updating the valueλ_i,k, we can simultaneously update the convective term, where in the place of∂_uF(x, u¯ _i−1) we can take∂_uF(x, u¯ _i,k−1) to evaluateϕⁱ(x).

Another practical improvement can be realized by shortening the time step in the convective process and thus increasing the order of approximation. In this case we use Euler’s backwards approximation in (5) with smaller time steps. LetI_i^l= (t^(l)_i−1−t^(l−1)_i−1 ), l= 1, . . . , m, where t⁽⁰⁾_i−1=t_i−1 andt^(m)_i−1≡t_i. We denote

v(x) :=∂_uF¯(x, u_i−1), v^h:=ω_h∗v≡v^h₀, v^h₁(x) :=x−

t⁽¹⁾_i−1−t_i−1 v₀^h(x) and

v_l^(h)(x) :=v_l−1^h (x)−

t^(l)_i−1−t^(l−1)_i−1

v^h v^h_l−1(x) . Then we put

ϕⁱ(x) :=v^h_m−1(x)−

t_i−t^(m−1)_i−1

v^h v_m−1^h (x)

. (10)

Here, v^h_l(x) represents the position of the initial point x after l smaller time steps along the approximated characteristic in the time interval

t⁽⁰⁾_i−1, t^(l)_i−1 .

Thus, our approximation scheme (6, 7), with the mapϕⁱ taken from (10), represents the approximation of the solution in the time interval (t_i−1, t_i), based on superposition of the convective part, represented by the last term of (6) (realized by shorter time steps I_i^l) and the diffusive part (realized by the shorter time stepτ).

The application of the method of characteristics requires the one to one property of the map ϕⁱ. The convergence analysis requires the Lipschitz-continuity ofϕⁱ and its inverse. Since the velocity field represented byv=∂_uF¯(x, u) depends on the unknownu, it is difficult to guarantee the one to one property for the mapϕⁱ (resp. ϕⁱ) since it would lead to the boundedness of∇¯vand consequently to the boundedness of∇u. But this is, generally, not true (especially when a porous media type phenomenon occurs). That’s why we are regularizing ¯v by ¯v^h. This allows us to guarantee the one to one property ofϕⁱ (see [20]).

Lemma 1. There existsτ₀, τ₁,(τ₁< τ₀), such that (i) ¹₂|x−y| ≤ |ϕⁱ(x)−ϕⁱ(y)| ≤2|x−y| forτ ≤τ₀ (ii) ¹₂|x−y| ≤ |ϕⁱ(x)−ϕⁱ(y)| ≤2|x−y| forτ ≤τ₁ for allx, y ∈Ω, and¯ i= 1, . . . , n, whereϕⁱ is from (10).

Proof. Assertion (i) is proved in [20]. To proove (ii) we use (10) and the fact that∇v^h_l =∇v^h_l−1(1 +τ_l∇y(v^h)), (τ_l≡ |I_lⁱ|, y=∇v_l−1^h (x)). We use|∇ω_h∗g| ≤ ^C_hg∞≤ ^C_h.Then we obtain

|∂_xj{ϕⁱ(x)}j−1| ≤τ

hC(1 + τ hC+ (τ

h)²C²+· · ·+ (τ

h)^m−1C^m−1)≤Cτ h

1 1−(τ/h)C,

|∂_xj{ϕⁱ(x)}_k| ≤Cτ h

1 1−(τ/h)C,

for anyj andk= 1, . . . , N, j=k,. Here{¯v}j is thejth component of ¯v.

Sinceh=τ^ω, withω∈(0,1), we obtain the required estimate (ii).

The function ϕⁱ maps Ω into Ω_i ⊂ Ω^∗ for i = 1, . . . , n, where Ω^∗ ⊃Ω is a small neighbourhood provided thatτ ≤τ₀. Ifϕⁱ(x)∈/Ω then we extendu_i−1fromW₂¹(Ω) tou_i−1∈W₂¹(Ω^∗) so that (see [26], prolongation of Nikolskij),

u_i−1W2¹_(Ω∗)≤Cu_i−1W2¹_(Ω). (11) Remark 2. The existence of the function u_i ∈ K satysfying (6), resp. (8), is guaranteed by [24] and by the fact that u_i−1 ∈ L₂(Ω) implies that u_i−1◦ϕⁱ ∈ L₂(Ω) because of Lemma 1. The convergence of the iterations (8) and (9) has been analysed in [19] for variational equalities. The analysis can be adopted for variational inequalities.

3. Convergence of the scheme (6, 7)

First, we prove some a priori estimates for {u_i}ⁿ₁ and then we show the compactness of {¯uⁿ}^∞_n=1 in the corresponding functional spaces. Here, we define the Rothe’s functions

u¯ⁿ(t) :=u_i anduⁿ(t) :=u_i−1+t−t_i−1

τ (u_i−u_i−1) (12)

fort∈(t_i−1, t_i), i= 1, . . . , n.

Lemma 2. Under the assumptions (H₁)–(H₅) it holds that

1≤j≤nmax

Ω

B_n(u_j)dx≤C, ⁿ

i=1

u_i²τ≤C, maxu_j0≤C, j

i=1

b_n(u_i)−b_n(u_i−1)

τ ,u_i−u_i−1 τ

τ≤ε

j i=1

δu_i²₀τ+C_ε,

whereε >0 is any small, fixed number.

Proof. We split

λ_i(u_i−u_i−1) =b_n(u_i)−b_n(u_i−1) +χ_i(u_i−u_i−1)τ, (13)

where χ_i∞ ≤1 and next we put v =u^∗ (u^∗ ∈ K is fixed) into (6). We multiply the resulting inequality withτ and sum up for i= 1, . . . , j. We obtain

j i=1

(b_n(u_i)−b_n(u_i−1), u_i) + j

i=1

(A∇u_i,∇u_i)τ ≤ j

i=1

(b_n(u_i)−b_n(u_i−1), u^∗) + j i=1

(A∇u_i,∇u^∗)τ

+ j i=1

(χ_i(u_i−u_i−1), u_i−u^∗)τ+ j i=1

(g(t_i, u_i−1), u_i−u^∗)_Γ2τ

+ j i=1

(H(t_i, u_i−1), u_i−u^∗)τ+ j

i=1

(u_i−1−u_i−1◦ϕⁱ, u_i−u^∗).

This inequality is briefly denoted asJ₁+J₂≤J₃+· · ·+J₈. We successively estimate all terms. First we have J₁=

j i=1

(b_n(u_i)−b_n(u_i−1), u_i)

= (b_n(u_j), u_j)−(b_n(u₀), u₀)− j i=1

(b_n(u_i−1), u_i−u_i−1)

≥(b_n(u_j), u_j)−(b_n(u₀), u₀)− j i=1

Ω

_uj

uj−1

b_n(z)dz

≥

Ω

B_n(u_j)dx−

Ω

B_n(u₀)dx, (14)

whereB_n(s) =sb_n(s)−_s

0 b_n(z)dz. From hypotesis (H₂) and from Young’s inequality, (ab≤δ^a₂²+_2δ¹b²,δ >0 arbitrary), we can estimate

J₂≥C j i=1

∇u_i²₀τ, |J₃| ≤βu_j²−C_β,

|J₄| ≤β j i=1

∇u_i²₀+C_β, |J₅| ≤C₁+C₂ j i=1

u_i²₀τ.

To estimate the boundary term we use (H₄), the continuous imbedding V → L₂(∂Ω) and the inequality, (see [26]),

v²_∂Ω≤ε∇v²₀+C_εv²₀, ∀v∈W₂¹, ε >0 arbitrary. (15) Then we obtain

|J₆| ≤ε j i=1

∇u_i²₀τ+C_ε j i=1

u_i²₀τ+C.

Moreover, we readily get

|J₇| ≤C₁+C₂ j

i=1

u_i²₀τ.

The critical point is to estimateJ₈. For this purpouse we use the formula u_i−1−u_i−1◦ϕⁱ

τ =

₁

0 ∇u_i−1(x+s(ϕⁱ(x)−x))ds·ω_h∗∂_uF(x, u¯ _i−1). (16)

Since ∂_uF¯(x, u_i−1) is bounded, (see (H₃)), ω_h∗∂_uF¯(x, u_i−1) is bounded too. Now we use (11) and Young’s inequality to obtain

|J₈| ≤β j i=1

∇u_i²₀τ+C_β j i=1

u_i²₀τ+C.

The asymptotic properties ofb_n(s) guarantee that

γs²−C₂≤B_n(s)≤C₁s²+C₂, (17) whereγ is from hypotesis (H₁). Consequently we get from (14)

J₁≥εu_j²₀−C. (18)

Substituting the estimates forJ₁, ..., J₈in the inequalityJ₁+J₂≤J₃+...+J₈ and using Gronwall’s argument we conclude that

1≤i≤nmax u_i0≤C, max

1≤i≤n

Ω

B_n(u_i)dx≤C, j i=1

u_i²τ ≤C. (19) Next we putv=u_i−1into (6) and sum up fori= 1, . . . , j. We use the decomposition (13) and (16). In addition we apply thea priori etimates (19). This leads to

j i=1

b_n(u_i)−b_n(u_i−1)

τ ,u_i−u_i−1 τ

τ+

j i=1

(A∇u_i,∇(u_i−u_i−1)) + j i=1

(g(t, u_i−1), u_i−u_i−1)_Γ2

≤ε j i=1

δu_i²₀τ+C_ε j i=1

u_i²τ+τ j i=1

δu_i²₀τ. (20)

For the 2nd term of the LHS, we use the symmetry ofAto get j

i=1

(A∇u_i,∇(ui−u_i−1))≥ 1

2(A∇u_j,∇u_j)−1 2

j i=1

(A∇u₀,∇u₀) +1

2(A∇(ui−u_i−1),∇(ui−u_i−1))

≥C_e(∇u_j²₀+ j i=1

∇(u_i−u_i−1)²₀)−C₂. (21)

There existsL >0 such that ˜g(t, x, s) :=g(t, x, s)−Ls is decreasing ins. Then due to hypothesis (H₄) we can estimate

j i=1

(˜g(t, u_i−1), u_i−u_i−1)_Γ2 ≥

Γ2

[G(t_j, u_j)−G(0, u₀)]dx−C j i=1

u_i²₀τ, whereG(t, x, s) =_s

0 ˜g(t, x, z)dz. Moreover, we have j

i=1

(u_i−1, u_i−u_i−1)_Γ2 =1

2u_j²_Γ2−1

2u₀²_Γ2−1 2

j i=1

u_i−u_i−1²_Γ2.

Estimating the boundary terms we further use (15) and (19) to obtain j

i=1

(g(t, u_i−1), u_i−u_i−1)_Γ2≥

ε∇u_j²₀−C_εu_j²₀−C j i=1

u_i²₀τ−C j i=1

∇u_i²₀τ−ε j i=1

∇(u_i−u_i−1²₀−C_ετ j i=1

δu_i²₀τ−C_ε≥ ε∇u_j²₀−C_ετ

j i=1

δu_i²₀τ−C_ε. (22) We invoke (21) and (22) in (20). Then from (H₁) and from Gronwall’s argument we deduce the last estimate

in Lemma 2.

A subsequence of{n} is denoted again by{n}.

Lemma 3. {u¯ⁿ}^∞₁ and {b_n(¯uⁿ)}^∞₁ are compact in L₂(Q_T). There exists u ∈ L₂(I, K) such that u¯ⁿ → u, b_n(¯uⁿ)→b(u)inL₂(I, L₂) andu¯ⁿ uin L₂(I, V). Moreover, ∂_tu∈L₂(I, L₂)and∂_tuⁿ ∂_tuinL₂(I, L₂).

Proof. We define

ρ(s) = min{b(s), µ} and W(s) = _s

0

ρ(z)dz.

From (H₁) we have thatb_n(s)≥γ >0 and

b_n(u_i)−b_n(u_i−1)

τ ,u_i−u_i−1 τ

≥γδ(u_i)²₀.

Then, choosingεsufficiently small in the last estimate in Lemma 2, we get thea priori estimate n

i=1

δu_i²₀τ ≤C.

This estimate and (19)₃can be respectively rewritten in the form

I∂_tuⁿ²₀dt≤C,

I¯uⁿ²dt≤C, (23)

which implies the compactness of {¯uⁿ} in L₂(I, L₂) because of Rellich’s compactness argument (W₂¹→L₂) – see [26]. Consequently, there exists a subsequence of{u¯ⁿ}^∞_n=1 and a functionu∈L₂(I, L₂) such that ¯uⁿ →u in L₂(I, L₂) (a subsequence of{u¯ⁿ} is denoted again by{¯uⁿ}). From the first estimate in (23) we deduce that

Iuⁿ−u¯ⁿ²₀dt≤ C

n and ∂_tuⁿ χ inL₂(I, L₂).

The 1st inequality implies that alsouⁿ → uin L₂(I, L₂). Thus we find that χ≡∂_tu. From the asymptotic properties ofb_n, (see (H₁), (part (ii)) we have

C₂(|¯uⁿ| −1)≤b_n(¯uⁿ)≤C₁(1 +|¯uⁿ|),

which implies that b_n(¯uⁿ)→b(u) inL₂(I, L₂). The weak convergence ¯uⁿ u inL₂(I, V) is a consequence of Lemma 2. SinceL₂(I, K) is a closed convex subset ofL₂(I, V), we have thatu∈L₂(I, K). Thus Lemma 3 is

proved.

Now we can formulate our main result. By{u¯^n¯}we denote a suitable subsequence of{u¯ⁿ}.

Theorem 1. If the assumptions (H₁)–(H₅) are satisfied, then there exists a weak variational solutionuto the problem (1) in the sense of Definition 1. Moreover, u¯^¯n →u in L₂(I, L₂), u¯^¯n u in L₂(I, V), where {¯uⁿ} is from (6, 7) and (12). If the weak variational solution is unique, then the original sequence{¯uⁿ} is convergent.

Proof. We rewrite (6) in the form

(∂_tb_n(¯uⁿ), v−u¯ⁿ) + ( ¯χⁿ(¯uⁿ−u¯ⁿ_τ), v−u¯ⁿ) + (A∇u¯ⁿ,∇(v−u¯ⁿ)) + (¯gⁿ(t,u¯ⁿ_τ), v−u¯ⁿ)_Γ2 ≥ ( ¯Hⁿ(t,u¯ⁿ_τ), v−u¯ⁿ)−

1

τ(¯uⁿ_τ−u¯ⁿ_τ◦ϕ¯ⁿ), v−u¯ⁿ

∀v∈L₂(I, K), (24) where

u¯ⁿ_τ := ¯uⁿ(t−τ) (¯uⁿ(s) =u₀ fors∈(−τ,0)), and where

˜b_n(¯uⁿ) =b_n(¯uⁿ) +t−t_i−1

τ (b_n(¯uⁿ)−b_n(¯uⁿ_τ)) fort∈(t_i−1, t_i), i= 1, . . . , n, and

g¯ⁿ(t, s) =g(t_i, s) fort∈(t_i−1, t_i), i= 1, . . . , n.

Similary, we introduce ¯Hⁿ,F¯ⁿ(t,u¯ⁿ_τ) and ¯ϕⁿ. We first integrate (24) over (0, t). The first term leads to J_n(t)≡

_t

0

(∂_t(˜b_n(¯uⁿ)−˜b_n(u₀)), v−uⁿ)dz

= (˜b_n(¯uⁿ(t)−b_n(u₀), v(t)−u(t))

− _t

0 (˜b_n(¯uⁿ(z))−b_n(u₀), ∂_t(v−uⁿ)dz.

From Lemma 3 we deduce that

J_n(t)→(b(u(t))−b(u(0)), v(t)−u(t))− _t

0

(b(u)−b(u₀), ∂_t(v−u))dt. (25) To this end notice that

_t

0

(∂_t(˜b_n(¯uⁿ)−˜b_n(u₀),u¯ⁿ−uⁿ)dt ≤2

Ib_n(¯uⁿ)−b_n(¯uⁿ_τ)₀∂_tuⁿ₀dt

≤C

Ib_n(¯uⁿ)−b_n(¯uⁿ_τ)²₀dt _1/2

→0 forn→ ∞, since ¯uⁿ →uand ¯uⁿ_τ →uinL₂(I, L₂).

Similarly we obtain for the 2nd term in (24) _t

0 ( ¯χⁿ(¯uⁿ−u¯ⁿ_τ),u¯ⁿ−u¯ⁿ_τ)dt→0 forn→ ∞. (26) From ¯uⁿ uinL₂(I, V) it follows that

_t

0 (A∇u¯ⁿ,∇v)dt→ _t

0(A∇u,∇v)dt (27)

and also

lim _t

0 (A∇¯uⁿ,∇¯uⁿ)dt≥ _t

0 (A∇u,∇u)dt, (28)

since_t

0(A∇w,∇w)dt is convex in wand consequently lower semicontinuous.

From ¯uⁿ_τ →uin L₂(I, L₂) and from ¯uⁿ_τ uin L₂(I, V) we have that ¯uⁿ_τ →uin L₂(I, L₂(∂Ω)) – see (14) or [21]. Then, we have for the boundary term in (24)

_t

0 (¯gⁿ(t,u¯ⁿ_τ), v−u¯ⁿ)_Γ2dt→ _t

0

(g(t, u), v−u)_Γ2dt. (29)

Similarly, we obtain _t

0

( ¯Hⁿ(t,u¯ⁿ), v−u¯ⁿ)dt→ _t

0

(f(t, u)−div_x F¯(x, u), v−u)dt. (30)

To take the limitn→ ∞in the last term of (24) we use the formula (16) and denote

∇w¯_τⁿ:=

₁

0 ∇uⁿ_τ(x+s( ¯ϕⁿ(x)−x))ds.

Due to (11) and Lemma 2,{∇w¯ⁿ_τ}^∞₀ is bounded inL₂(I, V). Thus ∇w¯ⁿ_τ ψinL₂(I, L₂). From

w¯ⁿ_τ −u¯ⁿ_τ = ₁

0 (uⁿ_τ(x+s( ¯ϕⁿ(x)−x)−u¯ⁿ_τ)ds

= ₁

0

₁

0 ∇uⁿ_τ(x+sr( ¯ϕⁿ(x)−x)dsdr. ω_h∗∂_uF¯ⁿ(x,u¯ⁿ_τ)τ (for a.e. x) we deduce that

Iw¯ⁿ_τ −u¯ⁿ_τ²₀dt≤Cu¯ⁿ²_L2(I,V)τ ≤Cτ →0 for n→ ∞ since ω_h∗∂_uF¯(x,u¯ⁿ_τ)∞≤C.

Thus, ¯wⁿ_τ →uin L₂(I, L₂). Recalling that ∇w¯ⁿ_τ ψ in L₂(I, L₂) we haveψ ≡ ∇u. From Lemma 3 and hypothesis (H₃) we obtain ∂_uF¯(x,u¯ⁿ_τ)→∂_uF(x, u) inL₂(I, L₂) and, consequently,

ω_h∗∂_uF¯(x,u¯ⁿ_τ)→∂_uF¯(x, u) a.e in Ω×I.

Therefore,

ω_h∗∂_uF¯(x,u¯ⁿ_τ)

(v−u¯ⁿ)→∂_uF¯(x, u)(v−u) in L₂(I, L₂)

and _t

0 (¯uⁿ_τ −u¯ⁿ_τ ◦ϕⁿ, v−u¯ⁿ)→ _t

0

(∂_uF(x, u)¯ · ∇u, v−u)dt. (31)

Combining the limit results (25)–(31) in the inequality (24) we conclude that the function u from Lemma 3 satisfies the inequality

(b(u(t))−b(u₀), v(t)−u(t))− _t

0

(b(u)−b(u₀), ∂_t(v−u))dz +

_t

0 (A∇u,∇(v−u)) + (div ¯F(x, u), v−u)≥ _t

0 (f(s, u(s)), v(s))−u(s))}ds a.e.t∈I and ∀v∈L₂(I, K).

Hence we conclude thatuis a weak variational solution to (1). To show that ¯uⁿ →uin L₂(I, V) we proceed in the following way. We putv=u(t) in (24) and integrate over (0, t). Since∂_tuⁿ ∂_tuand ˜b_n(¯u_n)→b(u) in L₂(I, L₂) we find by the same arguments as before (see (25)) thatJ_n →0 forn→ ∞. Next, the elliptic part gives

_t

0 (A∇¯uⁿ,∇(¯uⁿ−u)) = _t

0(A∇(¯uⁿ−u),∇(¯uⁿ−u))−C_n≥C_{e t}

0 ∇(¯uⁿ−u)²₀dt−C_n, with

C_n= _t

0 (A∇u,∇(¯uⁿ−u))→0 forn→ ∞ since ¯uⁿ uin L₂(I, V).

The remaining terms in (24) converge to 0 since ¯uⁿ,u¯ⁿ_τ →uinL₂(I, L₂))and also inL₂(I, L₂(Γ₂)). Sumarizing the above arguments, we obtain ¯uⁿ →uin L₂(I, V) and the proof is complete.

A stronger variational solution can be obtained when b is Lipschitz continuous. In that case we do not regularizeb byb_n and in (6) and (7) we considerb_n ≡b.

Theorem 2. Let the assumptions of Theorem 1, except of parts (i)–(iv) in hypothesis (H₁), be satisfied. If, additionally, b is Lipschitz continuous, then there exists a variational solution u ∈ L₂(I, V), with ∂_tb(u) ∈ L₂(I, L₂) which satisfies (1). The convergence results of Theorem 1 hold and, moreover, ∂_tb(¯uⁿ(t)) b(u)in L₂(I, L₂) where u¯ⁿ is from (6, 7) and (12), with b_n(s)≡ b(s). If the variational solution is unique, then the original sequences{u¯ⁿ},{b_n(¯uⁿ)} are convergent.

Proof. We obtain the samea priori estimates as in Lemma 2 and compactness result as in Lemma 3. From the last estimate of Lemma 2 we obtain

n i=1

b(u_i)−b(u_i−1)

τ ²₀τ≤C n i=1

b(u_i)−b(u_i−1)

τ ,u_i−u_i−1 τ

τ≤C.

Then, ∂_tb_n(¯uⁿ) χ inL₂(I, L₂). By the same arguments as in Lemma 3 we haveb_n(¯uⁿ)→b(u), ¯uⁿ →uin L₂(I, L₂). From the estimate

n i=1

b_n(¯uⁿ)−b_n(¯uⁿ)²

0τ≤2τ n i=1

b(u_i)−b(u_i−1) τ

²

0

τ →0

we deduce thatb_n(¯uⁿ)→b(u), which, together with the weak convergence of∂_tb_n(¯uⁿ), implies thatχ≡∂_tb(u).

This readily leads to

_t+z

t

∂_tb_n(¯uⁿ), v−u¯ⁿ dt→

_t+z

t

(∂_tb(u), v−u)dt ∀t∈I