Convergence of a full discretization for a second-order nonlinear elastodynamic equation in isotropic and anisotropic Orlicz spaces

Nonlinear Elastodynamic Equation in Isotropic and Anisotropic Orlicz Spaces

A. M. Ruf

Abstract. In this paper, we study a second-order, nonlinear evolution equation with damping aris- ing in elastodynamics. The nonlinear term is monotone and possesses a convex potential but ex- hibits anisotropic and nonpolynomial growth. The appropriate setting for such equations is that of monotone operators in Orlicz spaces. Global existence of solutions in the sense of distributions is shown via convergence of the backward Euler scheme combined with an internal approximation.

Moreover, we show uniqueness in a class of sufficiently smooth solutions and provide an a priori error estimate for the temporal semidiscretization.

Mathematics Subject Classification (2010).35L20; 47J35; 47H05; 65M12; 65M06; 35M60.

Keywords. Nonlinear evolution equation of second order in time with damping; elastodynamics;

monotone operator; nonpolynomial growth; anisotropic Orlicz space; existence; full discretization;

convergence.

1. Introduction

In this paper, we study the following second-order nonlinear hyperbolic elastodynamic equation

∂ttu−∆∂tu− ∇ ·σ(∇u) =f in Q:= Ω×(0, T) (1.1a)

u= 0 on∂Ω×(0, T) (1.1b)

u(·,0) =u₀, ∂_tu(·,0) =v₀ in Ω. (1.1c)

We want to show existence of solutionsu : Ω×[0, T] → R, where Ω ⊂ R^d is a bounded Lipschitz domain, andT >0. The stress σ:R^d →R^d is assumed to have the potential φ:R^d →R. Such an equation arises (although as a system) in solid mechanics describing viscoelastic material [20, 21, 30].

In this paper, we will assume that the potential φ is anN-function (see definition 2.1) and is thus convex. Note that the nonlinearity then is monotone such that

(σ(ξ)−σ(η))·(ξ−η)≥0 for allξ, η∈R^d.

Moreover, we assume thatσsatisfies the following growth condition in terms of theN-functionφand its conjugate:

φ^∗(σ(ξ))≤C(1 +φ(ξ)) for allξ∈R^d. (1.2) This constitutes a generalization of the current results concerning equation (1.1) since we do not need to assume polynomial growth ofφ.

This project has received funding from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Sk lodowska-Curie Grant Agreement No. 642768.

Previous results regarding equation (1.1) in higher dimensions were obtained by Gajewski, Gr¨oger, and Zacharias [17], Clements [6], Friedman and Ne˘cas [13], Engler [12], and Rybka [30], all of which rely either on monotonicity or global Lipschitz continuity ofσ. More recent contributions from Friesecke and Dolzmann [14] and Emmrich and ˇSiˇska [9] have generalized these results in the sense that they only assume thatσ satisfies the Andrews-Ball condition, i.e. there existsλ >0 such that

(σ(ξ)−σ(η))·(ξ−η)≥ −λ|ξ−η|² for allξ, η∈R^d, which is fulfilled ifσis monotone or globally Lipschitz continuous.

On the other hand, all of the contributions listed above critically rely uponσand its potential φ satisfying a polynomial growth and coercivity condition, i.e. there exists p ≥ 2 and constants C1, C2, C3≥0 such that

φ(ξ)≥C1|ξ|^p−C2

and |σ(ξ)| ≤C3(1 +|ξ|)^p−1

which, in essence, is the growth condition (1.2) forφ∼ | · |^p andφ^∗=C| · |^p−1p .

However, in this paper we want to generalize the growth condition and allow for anisotropic and nonpolynomial growth. The appropriate setting is that of Orlicz spaces, where we demand that the potentialφis anN-function and therefore convex. Hence, we obtain monotonicity of the nonlinearity σ. We are aware of the fact that it would be desirable to weaken the monotonicity assumption and only demand the Andrews-Ball condition, although we are not yet able to prove convergence under those assumptions.

The polynomial growth and coercivity assumption leads to an L^p-setting where the Lebesgue space L^p(Q) over the space-time cylinder is isometrically isomorphic to the Bochner-Lebesgue space L^p(0, T; L^p(Ω)) for p < ∞. This assumption allows us to reduce the partial differential equation to an operator differential equation for functions in time taking values in an appropriate Banach space of functions in space. However, the Orlicz space L_φ(Q), generated by theN-functionφ, is only isometrically isomorphic to the Orlicz space L_φ(0, T; L_φ(Ω)) ifφis equivalent to some power function (see [7, Proposition 1.3 on p. 218]). This fact poses a main difficulty in our approach.

Our main result, which will be presented in Theorem 4.1, provides global existence of a solution.

The proof shows the convergence of a subsequence of the sequence of approximate solutions, generated by a discretization in time by the backward Euler scheme and in space by a suitable Galerkin scheme.

Qualitative studies and numerical results in the case of polynomial growth conditions have been performed by Ball, Holmes, James, Pego and Swart [3], Friesecke and McLeod [15, 16], and Carstensen and Dolzmann [5]. Additionally, Prohl considered a finite element based full discretization of the equation

∂_ttu−ε∆∂_tu− ∇ ·σ(∇u) = 0

forε >0, as well as forε= 0 and presents numerical experiments [28]. The limit caseε= 0 constitutes the elastodynamic equation

∂ttu− ∇ ·σ(∇u) =f (1.3) for which one cannot expect smooth solutions even for smooth initial data (see [2, 26]). For an excellent survey of the literature concerning equation (1.3) see [8].

The remainder of this paper is structured as follows: In Section 2, we introduce the necessary notation, give a brief introduction to Orlicz spaces and compare the growth condition (1.2) with the restrictive ∆₂-condition. The description of the numerical method we employ, the construction of the Galerkin scheme, the proof of existence and uniqueness of the numerical solution, and the derivation of a priori estimates for the fully discrete solution and the discrete time derivative follow in Section 3. Finally, in Section 4 we show convergence towards and, thus, existence of an exact solution, as well as its uniqueness (under additional regularity assumptions) and an error estimate for the temporal semidiscretization. The appendix contains an elementary lemma concerning the separability of the space for wich we want to construct the Galerkin scheme.

2. Notation and Preliminaries

After a brief survey of the notation we employ, this section provides a quick introduction to the theory of Orlicz spaces and the specific results that are needed for the rest of this paper.

2.1. General Notation

We keep the usual notation for function spaces. Let Ω⊂R^d be a bounded domain. By L^p(Ω) (p∈ [1,∞]), we denote the Lebesgue space, for R^d-valued functions, we write L^p(Ω;R^d), both equipped with the standard norm k·k_p,Ω. Moreover, we rely upon the usual notation for Sobolev spaces. In particular, we have W^1,p(Ω) ={w∈L^p(Ω)|∇w∈L^p(Ω;R^d)}(with H¹(Ω) = W^1,2(Ω)), and W^1,p₀ (Ω) (p∈[1,∞)) denotes the closure of Cc^∞(Ω) with respect to the W^1,p-norm (with H¹₀(Ω) = W^1,2₀ (Ω)).

HereCc^∞(Ω) denotes the space of infinitely times differentiable functions with compact support in Ω.

Similarly, by H^r(Ω) we denote higher Sobolev spaces consisting of elements of L²(Ω) for which the derivatives up to orderrare again in L²(Ω).

With L^p(0, T;X) (p ∈ [1,∞]), we denote the usual Bocher-Lebesgue space, where X denotes a Banach space. We recall that L^p(0, T; L^p(Ω)) = L^p(Q) if p < ∞. Here, we identify the abstract functionu: [0, T]→L^p(Ω) with the functionu:Q→Rvia [u(t)](x) =u(x, t). The standard norm is then denoted byk·k_p,Q. The space of functions in L¹(0, T;X) whose distributional time derivative is again in L¹(0, T;X) is denoted by W^1,1(0, T;X) and equipped with the standard norm. Analogously we define W^1,2(0, T;X). ByC([0, T];X),A C([0, T];X) andCw([0, T];X), we denote the usual spaces of uniformly continuous, absolutely continuous and demicontinuous (i.e. continuous with respect to the weak topology inX) functionsu: [0, T]→X, respectively (see also [17] for details). Byh·,·i, we denote the duality pairing. We will use the notationX ,→Y andX ,→^c Y to indicate that a Banach spaceX is continuously respectively compactly embedded in a Banach spaceY. Finally,Cdenotes a generic positive constant.

2.2. Orlicz Spaces

In this section, we provide the definition and basic properties of Orlicz spaces. For an introduction to Orlicz spaces, refer to [23], as well as [1, 18, 22, 31, 32, 34]. Since we want to include nonlinearities with anisotropic growth, we rely upon anisotropic Orlicz classes and spaces defined by N-functions with vector-valued arguments, as presented in [31, 32, 7].

Definition 2.1 (N-function). A function φ : R^d → R is said to be an N-function if it satisfies the following conditions:

(i) φis continuous,φ(ξ) = 0 if and only ifξ= 0, φ(−ξ) =φ(ξ) for allξ∈R^d; (ii) φis convex;

(iii) φhas superlinear growth such that lim_|ξ|→∞^φ(ξ)_|ξ| =∞, and lim_|ξ|→0^φ(ξ)_|ξ| = 0.

Some authors prefer the term generalizedN-function in order to emphasize the dependence on ξand not only on|ξ|. Note that (i) and (ii) implyφ(ξ)≥0 for allξ∈R^d. Because of the anisotropic character, the functionφneed not be increasing with respect to the components of its vector-valued argument, e.g.φ(ξ₁, ξ₂) =ξ₁²+ξ²₂+ (ξ₁−ξ₂)². For more examples, refer to [11].

For anN-functionφ,φ^∗denotes the conjugate function given by the Legendre-Fenchel transform φ^∗(η) = sup_ξ∈Rd(ξ·η−φ(ξ)),η∈R^d. According to [31], the conjugate function is again anN-function, andφ^∗∗=φ. Let us recall the Fenchel-Young inequality

|ξ·η| ≤φ(ξ) +φ^∗(ξ) for allξ, η∈R^d.

The anisotropic Orlicz class Lφ(Ω;R^d) is the set of all (equivalence classes of almost everywhere equal) measurable functionsξ: Ω→R^d such that

ρ_φ,Ω(ξ) :=

Z

Ω

φ(ξ(x)) dx<∞.

Although Lφ(Ω;R^d) is a convex set, it may not be a linear space, e.g. if d = 1, Ω = (0,1) and φ(ξ) =e^|ξ|−1, thenξ=−¹₂ln∈Lφ(Ω;R^d) butζ= 2ξ6∈Lφ(Ω;R^d). The mappingρφ,Ωis a modular in the sense of [23, p. 208].

Since the functionφis continuous, ξ=ξ(x)∈L^∞(Ω;R^d) impliesx7→φ(ξ(x))∈L^∞(Ω), which shows that L^∞(Ω;R^d)⊆Lφ(Ω;R^d).

Theanisotropic Orlicz space L_φ(Ω;R^d) is defined as the linear hull ofLφ(Ω;R^d). It is a Banach space with respect to the Luxemburg norm

kξk_φ,Ω:= inf

λ >0 Z

Ω

φ ξ(x)

λ

dx≤1

,

where the infimum is attained ifξ6= 0. Let us emphasize that, in general, Lφ(Ω;R^d) is neither separable nor reflexive. Note that ρφ,Ω(ξ) ≤ kξk_φ,Ω if kξk_φ,Ω ≤ 1 and ρφ,Ω(ξ)≥ kξk_φ,Ω ifkξk_φ,Ω >1 for all ξ∈Lφ(Ω;R^d). Thus

kξk_φ,Ω≤ρφ,Ω(ξ) + 1.

Because of the superlinear growth ofφwe have L_φ(Ω;R^d)⊆L¹(Ω;R^d) as shown in [11, p. 1167].

By definition, the anisotropic Orlicz class and space coincide with the isotropic Orlicz class and space, respectively, if theN-functionφis a radial function.

Let us denote by E_φ(Ω;R^d) the closure with respect to the Luxemburg norm of the set of bounded measurable functions defined on Ω. It turns out that Eφ(Ω;R^d) is the largest linear space contained in the Orlicz classLφ(Ω;R^d) such that

E_φ(Ω;R^d)⊆Lφ(Ω;R^d)⊆L_φ(Ω;R^d)

with, in general, strict inclusions. From the equivalence of the Luxemburg and the Orlicz norm kξk^O_φ,Ω:= sup

Z

Ω

ξ·ηdx

η ∈Lφ^∗(Ω,R^d) withρφ^∗,Ω(η)≤1

,

one findes that L^∞(Ω;R^d) is continuously embedded in Eφ(Ω;R^d).

The space Eφ(Ω;R^d) is separable andCc^∞(Ω;R^d) is dense in Eφ(Ω;R^d). The space Lφ(Ω;R^d) is the dual of E_φ^∗(Ω;R^d), and the duality pairing is given by

hξ, ηi= Z

Ω

ξ·ηdx ξ∈Lφ(Ω;R^d), η∈Eφ^∗(Ω;R^d).

At this point, we may recall the generalized H¨older inequality Z

Ω

ξ·ηdx≤2kξk_φ,Ωkηk_φ∗,Ω for allξ∈Lφ(Ω;R^d), η∈Lφ^∗(Ω;R^d),

which shows that ξ·η ∈ L¹(Ω) if ξ ∈ Lφ(Ω;R^d) and η ∈ Lφ^∗(Ω;R^d). The factor 2 in the H¨older inequality is due to the use of the Luxemburg norm instead of the Orlicz norm.

2.3. Growth Conditions in Orlicz Spaces

If theN-functionφsatisfies the so-called ∆2-condition, i.e., if there existsC >0 such that φ(2ξ)≤Cφ(ξ) for allξ∈R^d,

then Eφ(Ω;R^d) =Lφ(Ω;R^d) = Lφ(Ω;R^d) (see [32, Theorem 2.2]). The ∆2-condition, however, restricts the growth significantly. For the isotropic case, it is known that the ∆2-condition is not fulfilled if the N-functionφgrows faster than a polynomial as shown in [23, Remark 3.4.6].

The following proposition illustrates the connection between the ∆₂-condition and other growth conditions.

Proposition 2.2. Letφbe a differentiableN-function. Then the following two statements are equivalent:

(i) φsatisfies the∆2-condition.

(ii) There exists a constantC >0 such that

φ^∗(φ⁰(ξ))≤Cφ(ξ) for allξ∈R^d.

Proof. According to [31, Theorem 5.1], we have equality in the Fenchel-Young inequality ifη=φ⁰(ξ).

Thus φ^∗(φ⁰(ξ)) = φ⁰(ξ)·ξ−φ(ξ) and it suffices to show that the ∆2 condition is equivalent to the existence of a constantC >0 such that

φ⁰(ξ)·ξ

φ(ξ) ≤C+ 1

which follows from [31, Theorem 3.2].

2.4. Orlicz Spaces Over the Space-Time Cylinder

In this article, we also consider Orlicz classes and spaces over the space-time cylinderQ; the definitions and results introduced earlier are the same with Ω replaced byQ. We emphasize that Lφ(Q;R^d)6=

L_φ(0, T; L_φ(Ω;R^d)), except for the case when φ is equivalent to a power function as proven in [7, Proposition 1.3 on p. 218].

3. Full Discretization

In this section, we describe the numerical approximation of (1.1). We consider an equidistant time grid: for N ∈N, let τ =T /N and t_n =nτ (n= 0,1, . . . , N). In addition to the time discretization, we consider an internal approximation (V_m)_m∈Nof the space

V :={w∈H¹₀(Ω)|∇w∈E_φ(Ω;R^d)}, kwk_V :=k∇wk_2,Ω+k∇wk_φ,Ω

so thatVm ⊆W^1,∞(Ω), which we will construct in the next subsection. With respect to the right- hand side, we consider the following restriction to the time grid: for n = 1,2, . . . , N, let fⁿ =

1 τ

Rtn

tn−1f(·, t) dt. The numerical method we consider now reads as follows: for given u⁰, v⁰ ∈ Vm

andf ∈L¹(0, T; L²(Ω)), find (uⁿ)^N_n=1,(vⁿ)^N_n=1⊂Vm such that forn= 1, . . . , N Z

Ω

vⁿ−vⁿ⁻¹

τ ψ+∇vⁿ· ∇ψ+σ(∇uⁿ)· ∇ψdx = Z

Ω

fⁿψdx for allψ∈Vm, (3.1a) where

uⁿ−uⁿ⁻¹

τ =vⁿ, (3.1b)

that is,uⁿ =u0+τPn

j=1v^j. Note thatσ(∇uⁿ) is in L¹(Ω) becauseuⁿ∈W^1,∞(Ω) andσis continuous.

The scheme (3.1) can also be written as Z

Ω

uⁿ−2uⁿ⁻¹+uⁿ⁻²

τ² ψ+∇uⁿ− ∇uⁿ⁻¹

τ · ∇ψ+σ(∇uⁿ)· ∇ψ= Z

Ω

fⁿψ

for allψ∈Vmforn= 1,2, . . . , N, where u⁻¹:=u⁰−τ v⁰. 3.1. Construction of the Galerkin Scheme

Next, we construct a special Galerkin scheme which provides stability that we will later employ to bound the discrete second time derivative. Letr∈Nbe sufficiently big such that

H^r−1(Ω;R^d),→L^∞(Ω;R^d),→Eφ(Ω;R^d).

Consequently, the space H^r := H^r(Ω)∩H¹₀(Ω) is densely embedded in V (see Lemma A.1 in the Appendix). We can then define (·,·)r as the canonical inner product andk·k_ras the induced norm in the Hilbert spaceH^rand let

T : L²(Ω)→L²(Ω), f 7→u

be the solution operator to the following problem:

Forf ∈L²(Ω) find u∈ H^r such that (u, v)r= (f, v) for all v∈ H^r.

The operator T is well-defined (Lemma of Lax-Milgram), selfadjoint, nonnegative, one-to-one and compact. Similar steps as those in [4, Theorem 6.11 and 9.31] imply the existence of an orthonormal basis (e_m)_m∈Nof L²(Ω) consisting of eigenfunctions of T, i.e.

T em=µmem withµm>0, µm→0 form→ ∞.

Letϕm :=√

µmem,m ∈Nthen (ϕm)_m∈N is an orthonormal basis of H^r. Because of the density of the embedding of H^r in V, the sequence (ϕm)m∈N is a Galerkin basis of V and the spaces Vm :=

span{ϕ1, . . . , ϕm}form a Galerkin scheme with respect toV.

Now, letPm: L²(Ω)→L²(Ω) denote the L²-orthogonal projections ontoVmdefined by P_mv=

m

X

j=1

(v, e_j)e_j =

m

X

j=1

(v, ϕ_j)_rϕ_j, v∈L²(Ω).

In particular we have

(Pmv, vm) = (v, vm) for allvm∈Vm

and, since theej are eigenfunctions of the operatorT, Pm is H^r-orthogonal. Therefore, we have for allv∈ H^r

kPmvk²_r= (Pmv, Pmv)r= (v, Pmv)r≤ kPmvk_rkvk_r and thus,

sup

v∈H^r\{0}

kPmvk_r kvk_r ≤1 for allm∈N.

3.2. Existence of Approximate Solutions

To demonstrate that the numerical scheme (3.1) has a unique solution we use Brouwer’s fixed point theorem.

Theorem 3.1. Let u⁰, v⁰ ∈V_m and f ∈L¹(0, T; L²(Ω)) be given. Then there exists a unique solution (uⁿ)^N_n=1,(vⁿ)^N_n=1 to the numerical scheme (3.1).

The proof of existence of solutions to the numerical scheme is based on the following auxiliary result, which is a direct consequence of Brouwer’s fixed point theorem (see [17]).

Lemma 3.2. For some R >0, let h:B(0, R)→R^m be continuous, whereB(0, R)⊂R^m denotes the closed ball of radiusR with origin0 with respect to some normk·k_Rm onR^m. If

h(v)·v≥0 for allv∈R^m with kvk

R^m =R then there existsv^∗∈B(0, R)such thath(v^∗) = 0.

Proof of theorem3.1. We construct a one-to-one mapping betweenVm= span{ϕ1, . . . , ϕm} and R^m as follows:

w= [w₁, . . . , w_m]∈R^m ↔ V_m3w=

m

X

j=1

w_jϕ_j,

and kwk_Rm := kwk_2,Ω defines a norm. Existence and uniqueness are now shown step by step. Let us assume that uⁿ⁻¹, uⁿ⁻² ∈ Vm are given. We show that there exists uⁿ ∈ Vm corresponding to uⁿ∈R^m being a zero of the mappingh= [h1, . . . , hm] :R^m→R^mdefined by

hj(w) :=

Z

Ω

w−2uⁿ⁻¹+uⁿ⁻²

τ² ϕj+∇w− ∇uⁿ⁻¹

τ · ∇ϕj+σ(∇w)· ∇ϕj−fⁿϕj

dx,

forj = 1,2, . . . , m. The continuity ofh:R^m →R^m is a consequence of the continuity ofσ together with the fact thatVm ⊂W^1,∞(Ω). With the Cauchy-Schwarz inequality and the monotonicity of σ (note thatσ(0) = 0 which follows from the properties ofφ), we obtain

h(w)·w= Z

Ω

w−2uⁿ⁻¹+uⁿ⁻²

τ² w+∇w− ∇uⁿ⁻¹

τ · ∇w+σ(∇w)· ∇w−fⁿw

dx

≥ 1

τ²kwk²_2,Ω− 1 τ²

uⁿ⁻¹

_2,Ω· kwk_2,Ω− 1 τ

vⁿ⁻¹

_2,Ω· kwk_2,Ω +1

τk∇wk²_2,Ω−1 τ

∇uⁿ⁻¹

_2,Ω· k∇wk_2,Ω− kfⁿk_2,Ω· kwk_2,Ω

= 1 τkwk_2,Ω

1

τ kwk_2,Ω−1 τ

uⁿ⁻¹ _2,Ω−

vⁿ⁻¹

_2,Ω−τkfⁿk_2,Ω

+1

τ k∇wk_2,Ω

k∇wk_2,Ω−

∇uⁿ⁻¹ _2,Ω

.

ChoosingR=kwk_2,Ωsufficiently large and incorporating the Poincar´e-Friedrichs inequality allows us to apply Lemma 3.2, providing existence of a zero ofhand thus a solution to (3.1) at leveln.

Let w1, w2 be two solutions of (3.1) at level n. Then, in view of the monotonicity of σ (and σ(0) = 0), we have

1

τ²kw1−w2k²_2,Ω+1

τk∇w1− ∇w2k²_2,Ω

= Z

Ω

w₁−2uⁿ⁻¹+uⁿ⁻²

τ² −w₂−2uⁿ⁻¹+uⁿ⁻² τ²

(w₁−w₂) +

∇w1− ∇uⁿ⁻¹

τ −∇w2− ∇uⁿ⁻¹ τ

·(∇w1− ∇w2)

dx

=− Z

Ω

(σ(∇w₁)−σ(∇w₂))·(∇w₁− ∇w₂) dx≤0

which proves uniqueness.

3.3. A Priori Estimates

The following a priori estimates are the essential prerequisite for the proof of convergence.

Theorem 3.3 (discrete a priori estimate I). The discrete solutions(uⁿ)^N_n=1,(vⁿ)^N_n=1 from theorem 3.1 satisfy the following a priori estimate forn= 1,2, . . . , N:

kvⁿk²_2,Ω+

n

X

j=1

v^j−v^j−1

2 2,Ω+ 2τ

n

X

j=1

∇v^j

2 2,Ω+ 2

Z

Ω

φ(∇uⁿ) dx

≤C

v⁰

2 2,Ω+

Z

Ω

φ(∇u⁰) dx +kfk²_L1(0,T;L²(Ω))

. (3.2) Proof. We test the first equation of (3.1) with ψ = vⁿ and employ the convexity inequality, the Cauchy-Schwarz inequality and the identity

(A−B)·A= 1

2(A²−B²+ (A−B)²), (3.3)

which holds true for allA, B∈Ras well asA, B∈R^d. We find 1

2τ

kvⁿk²_2,Ω− vⁿ⁻¹

2 2,Ω+

vⁿ−vⁿ⁻¹

2 2,Ω

+k∇vⁿk²_2,Ω+1 τ

Z

Ω

(φ(∇uⁿ)−φ(∇uⁿ⁻¹)) dx

≤ kfⁿk_2,Ωkvⁿk_2,Ω.

Summation then implies for alln= 1,2, . . . , N kvⁿk²_2,Ω+

n

X

j=1

v^j−v^j−1

2 2,Ω+ 2τ

n

X

j=1

∇v^j

2 2,Ω+ 2

Z

Ω

φ(∇uⁿ) dx

≤ v⁰

2 2,Ω+ 2

Z

Ω

φ(∇u⁰) dx +2τ

n

X

j=1

f^j _2,Ω

v^j

_2,Ω. (3.4) Takingnsuch thatkvⁿk_2,Ω= maxj=1,2,...,N

v^j

_2,Ω=:X and using τ

N

X

j=1

f^j

_2,Ω≤ kfk_L1(0,T;L²(Ω)),

results in the quadratic inequality X²≤

v⁰

2 2,Ω+ 2

Z

Ω

φ(∇u⁰) dx +2kfk_L1(0,T;L²(Ω))X, implying

X≤ v⁰

_2,Ω+√ 2

Z

Ω

φ(∇u⁰) dx ¹₂

+ 2kfk_L1(0,T;L²(Ω)).

Going back to (3.4) proves the assertion.

Theorem 3.4 (discrete a priori estimate II). The discrete solutions (uⁿ)^N_n=1,(vⁿ)^N_n=1 from Theorem 3.1satisfy the following a priori estimate for n= 1,2, . . . , N:

τ

N

X

n=1

vⁿ−vⁿ⁻¹ τ

(H^r)^∗

≤C

v⁰ _2,Ω+

Z

Ω

φ(∇u⁰) dx +kfk_L1(0,T;L²(Ω))+ max

n=1,...,Nkσ(∇uⁿ)k_φ∗,Ω

. (3.5) Proof. Sincevⁿ andvⁿ⁻¹ are inVm⊂V ⊂L²(Ω) and due to theH^r-orthogonality of the projection Pm, we have

vⁿ−vⁿ⁻¹ τ

(H^r)^∗

= sup

v∈H^r\{0}

1 kvk_r

vⁿ−vⁿ⁻¹ τ , v

r

= sup

v∈H^r\{0}

1 kvk_Hr

kPmvk_Hr

kP_mvk_Hr

vⁿ−vⁿ⁻¹ τ , Pmv

L²(Ω)

.

Since (vⁿ)^N_n=0 satisfies the first equation in equation (3.1) andP_mv∈V_m, we obtain

vⁿ−vⁿ⁻¹ τ

(H^r)^∗

= sup

v∈H^r\{0}

kPmvk_Hr

kvk_Hr

· R

Ω(fⁿ·Pmv− ∇vⁿ· ∇Pmv−σ(∇uⁿ)· ∇Pmv) dx kPmvk_Hr

Employing the Cauchy-Schwarz inequality and the generalized H¨older inequality, we find Z

Ω

(fⁿ·Pmv− ∇vⁿ· ∇Pmv−σ(∇uⁿ)· ∇Pmv) dx

≤C

kfⁿk_2,Ω· kPmvk_2,Ω+k∇vⁿk_2,Ω· k∇Pmvk_2,Ω+kσ(∇uⁿ)k_φ∗,Ω· k∇Pmvk_φ,Ω

≤CkP_mvk_V

kfⁿk_2,Ω+k∇vⁿk_2,Ω+kσ(∇uⁿ)k_φ∗,Ω

. Since the continuity of the embeddingH^r,→V implies _kP ¹

mvk_Hr ≤C_kP¹

mvk_V forv6= 0, together with the stability of the projectionsPmwe obtain

vⁿ−vⁿ⁻¹ τ

(H^r)^∗

≤C

kfⁿk_2,Ω+k∇vⁿk_2,Ω+kσ(∇uⁿ)k_φ∗,Ω

.

Multiplying byτ and summing from n= 1 toN yields τ

N

X

n=1

vⁿ−vⁿ⁻¹ τ

_(Hr)∗

≤C τ

N

X

n=1

kfⁿk_2,Ω+τ

N

X

n=1

k∇vⁿk_2,Ω+ max

n=1,...,Nkσ(∇uⁿ)k_φ∗,Ω

! .

The claim now follows fromτPN

n=1kfⁿk_2,Ω≤ kfk_L1(0,T;L²(Ω)) and the previous a priori estimate in

Theorem 3.2.

4. Existence via Convergence of Approximate Solutions

In the following, let us consider sequences (m_l)_l∈Nand (N_l)_l∈Nof positive integers such thatm_l, N_l→

∞ as l → ∞. The discrete solution to (3.1) corresponding to the discretization parameters ml, Nl

(with τ_l :=T /N_l) shall be denoted by (uⁿ_l)^N_n=0^l ,(vⁿ_l)^N_n=0^l , where u⁰_l ∈ V_ml and v⁰_l ∈V_ml denote the approximate initial values. We do not explicitly denote the dependence oft_n =nτ_lonl.

Regarding the approximation of the initial values, we assume that

u⁰_l →u₀in V and v_l⁰→v₀ in L²(Ω) asl→ ∞. (4.1) From the discrete solution, we construct approximate solutions defined on the whole time interval as follows: letul denote the piecewise constant function such that

ul(·, t) =uⁿ_l ift∈(t_n−1, tn] (n= 1,2, . . . , Nl), ul(·,0) =u¹_l, and letub_l denote the linear spline interpolating (t₀, u⁰_l),(t₁, u¹_l), . . . ,(t_Nl, u^N_l ^l), i.e.

ubl(t) =uⁿ⁻¹_l +uⁿ_l −uⁿ⁻¹_l

τ_l (t−t_n−1)

= tn−t τl

uⁿ⁻¹_l +t−tn−1

τl

uⁿ_l fort∈[t_n−1, tn] (n= 1,2, . . . , Nl).

In an analogous way, we definevlandvbl, as well as the piecewise constant functionfl. The primary result of this paper can be summarized by the following theorem.

Theorem 4.1. Let u0 ∈ V, v0 ∈ L²(Ω) and f ∈ L¹(0, T; L²(Ω)). Further, let σ satisfy the growth condition

φ^∗(σ(A))≤C(1 +φ(A))

for allA∈R^d. Then there exists a weak solution u∈Cw([0, T]; H¹₀(Ω)) with∂tu∈Cw([0, T]; L²(Ω)),

∇u∈Lφ(Q;R^d)andσ(∇u)∈Lφ^∗(Q;R^d) to (1.1)in the sense of distributions, that is, Z

Q

(−∂_tu∂_tw+∇∂_tu· ∇w+σ(∇u)· ∇w) dx dt = Z

Q

f wdx dt for allw∈Cc^∞(Q), withu(·,0) =u0 in V, and∂tu(·,0) =v0 inL²(Ω).

This solution is the limit of a subsequence, denoted byl throughout this paper, of approximate solutions constructed from (3.1) in the following sense: The piecewise constant and piecewise linear temporal interpolationulandublconverge weakly* in L^∞(0, T; H¹₀(Ω)) and strongly inC([0, T]; L²(Ω)), respectively, towardsu. The piecewise constant temporal interpolation vl of the discrete time deriva- tives converges weakly* in the space L^∞(0, T; L²(Ω)) and weakly in L²(0, T; H¹₀(Ω)) towards∂tuand the piecewise linear in time interpolationbv_l converges strongly in L²(Q) towards∂_tu. Moreover,∇ul

converges weakly in L_φ(Q;R^d) towards ∇u and σ(∇ul) converges weakly* in L_φ^∗(Q;R^d) towards σ(∇u).

Lemma 4.2 (Convergence of subsequences I). Letu₀∈V,v₀∈L²(Ω)andf ∈L¹(0, T; L²(Ω)) and let the approximations of the initial values (u⁰_l)and (v⁰_l) satisfy (4.1). Then there exists a subsequence, still denoted by l, and someu∈Cw([0, T]; H¹₀(Ω)) with ∂tu∈L^∞(0, T; L²(Ω))∩L²(0, T; H¹₀(Ω)) and

∇u∈Lφ(Q;R^d), as well as ξ∈H¹₀(Ω) andζ∈L²(Ω)such that, as l→ ∞, (I) ul

−* u∗ inL^∞(0, T; H¹₀(Ω)),

(II) bul−ul→0in L²(0, T; H¹₀(Ω)), (III) bul→uinC([0, T]; L²(Ω)), (IV) vl

−∗

* ∂tuinL^∞(0, T; L²(Ω)), (V) bvl−vl→0 inL²(Q), (VI) vl−* ∂tuinL²(0, T; H¹₀(Ω)), (VII) ∇ul

−∗

*∇uinLφ(Q;R^d),

(VIII) bul(T)−* ξin H¹₀(Ω) andbvl(T)−* ζ inL²(Ω).

Remark 4.3. Under the assumptions of Theorem 4.2, a subsequence of (ul) converges strongly in L^∞(0, T; L²(Ω)) towardsubecause

kul−bulk_L∞(0,T;L²(Ω))≤τlkvlk_L∞(0,T;L²(Ω))→0 asl→ ∞.

Moreover, ifX is an intermediate Banach space between L²(Ω) and H¹₀(Ω) in the sense of Lions and Peetre [25] such that there existsC >0 andθ∈(0,1) with

kwk_X ≤Ck∇wk^θ_2,Ωkwk^1−θ_2,Ω for allw∈H¹₀(Ω),

then (ubl) is a Cauchy sequence and thus converges strongly inC([0, T];X) towardsu. As before, (ul) converges strongly in L^∞(0, T;X) towardsu.

Proof of Theorem 4.2. In view of (4.1), the right hand side in the a priori estimate (3.2) is bounded.

Becauseuⁿ_l =u⁰_l +τ_lPn

j=1v^j_l, we find k∇uⁿ_lk_2,Ω≤

∇u⁰_{l 2,Ω}+τl

n

X

j=1

k∇vⁿ_lk_2,Ω≤

∇u⁰_{l 2,Ω}+C



τl n

X

j=1

∇v^j_l

2 2,Ω





1 2

.

Thus, as a consequence of the discrete a priori estimate (3.2), the sequences (ul)_l∈N,(ubl)_l∈Nare bounded in L^∞(0, T; H¹₀(Ω)) and also (vl)_l∈N,(bvl)_l∈Nare bounded in L^∞(0, T; L²(Ω)). Thus, there are a subse- quence, still denoted byl, and elements u,bu∈L^∞(0, T; H¹₀(Ω)), v,bv∈L^∞(0, T; L²(Ω)) such that

ul

−* u,∗ bul

−*∗ buin L^∞(0, T; H¹₀(Ω)), vl

−* v,∗ bvl

−*∗ bv in L^∞(0, T; L²(Ω)).

In view of (3.2)

kub_l−u_lk²_L2(0,T;H¹₀(Ω))=τ_l³ 3

Nl

X

n=1

k∇vⁿ_lk²_L2(Ω)→0, we find thatubl−ul→0 in L²(0, T; H¹₀(Ω)) and thusub=u. Similarly,

kbvl−vlk²_L2(Q)=τl

3

N_l

X

n=1

v_lⁿ−vⁿ⁻¹_l

2

L²(Ω)→0, and thusbv=v. Because by definition vl=∂tbul, we immediately findv=∂tu.

The sequence (ubl) ⊂ C([0, T]; L²(Ω)) is equicontinuous because (∂bul) = (vl) is bounded in L^∞(0, T; L²(Ω)) and (ubl(t))⊂H¹₀(Ω) is bounded in H¹₀(Ω) and hence relatively compact in L²(Ω) for every t ∈ [0, T]. An application of Arzel`a-Ascoli’s theorem implies strong convergence in the space C([0, T]; L²(Ω)) of a subsequence (again still denoted byl), and the limit can only be the weak*-limitu.

We have seen thatu∈L^∞(0, T; H¹₀(Ω)) with∂_tu∈L^∞(0, T; L²(Ω)) such thatu∈A C([0, T]; L²(Ω)), and in view of [24, Lemma 8.1 on p. 297]u∈Cw([0, T]; H¹₀(Ω)).

Consequently, we have proved the first five statements. The sixth follows directly as kvlk²_L2(0,T;H¹₀(Ω))=τl

N_l

X

n=1

k∇vⁿ_lk²_2,Ω

and the right hand side is bounded due to the a priori estimate (3.2). Hence, we can extract a subsequence (still denoted by l) and ev ∈ L²(0, T; H¹₀(Ω)) such that vl −* ev in L²(0, T; H¹₀(Ω)). As before, we can showev=∂tu.

In view of the discrete a priori estimate (3.2) we observe that Z

Q

φ(∇ul) dx dt =τl N_l

X

n=1

Z

Ω

φ(∇uⁿ_l) dx

is uniformly bounded. However, from the boundedness of the modular boundedness of the Luxemburg norm follows. Therefore, the sequence (∇ul) is bounded in Lφ(Q;R^d), the dual of the separable space E_φ^∗(Q;R^d). Thus, we can extract a subsequence (still denoted byl) such that∇ul

−* χ∗ in L_φ(Q;R^d) for someχ. In view of the sequential lower semicontinuity of the modular in L¹(Q;R^d) as shown in [11, Lemma 2.2], we haveχ∈Lφ(Q;R^d).

Since Cc^∞(Ω;R^d)⊗Cc^∞(0, T)⊆ E_φ^∗(Q;R^d) we find for all functions Φ∈ Cc^∞(Ω;R^d) and Ψ ∈ Cc^∞(0, T) with integration by parts and (I)

Z

Q

χ·ΦΨ dx dt = lim

l→∞

Z

Q

∇ul·ΦΨ dx dt

=−lim

l→∞

Z

Q

ul∇ ·ΦΨ dx dt =− Z

Q

u∇ ·ΦΨ dx dt and thusχ=∇u.

Lastly, with the same argument as in (I), the sequence (ub_l(·, T))_l∈N (with ub_l(·, T) = u^N_l ^l = u_l(·, T)) is bounded in H¹₀(Ω) and the sequence (bv_l(·, T))_l∈N is bounded in L²(Ω). Thus there exist ξ∈H¹₀(Ω) andζ∈L²(Ω) and subsequences (still denoted by l) such thatu(Tb )−* ξandvb_l(T)−* ζin H¹₀(Ω) and L²(Ω), respectively.

Lemma 4.4 (Convergence of subsequences II). Let the assumptions of Lemma 4.2be satisfied and let σfulfill the growth condition

φ^∗(σ(A))≤C(1 +φ(A)) for allA∈R^d.

Additionally, assume that there is a constant C >0 such that the approximations of the initial value v₀ satisfy τ_l

v_l⁰

2

H¹₀(Ω) < C for all l∈N. Then, there exists α∈Lφ^∗(Q;R^d)and a subsequence (still denoted byl) such that

(IX) σ(∇ul)−* α^∗ inLφ^∗(Q;R^d)und (X) bvl→∂tuinL²(Q).

Note that the additional assumption τl

v⁰_l

2

H¹₀(Ω) < C is fulfilled by the projections of v0 onto the spacesVm_l if we couple the time and space discretization parameters appropriately.

Proof. Using the growth condtion, we find Z

Q

φ^∗(σ(∇u_l)) dx dt≤C Z

Q

(1 +φ(∇u_l)) dx dt

and as seen in the previous proof the right-hand side is bounded. Thus, (σ(∇ul)) is bounded in Lφ^∗(Q;R^d), the dual of the separable space Eφ(Q;R^d). Therefore, we can extract a subsequence (still denoted by l) such that (σ(∇ul)) converges weakly* towards someα∈Lφ^∗(Q;R^d). Again, using the weak sequential lower semicontinuity of the modular in L¹(Q;R^d) we findα∈Lφ^∗(Q;R^d).

For the second statement, recall that the sequence (vl) is bounded in L²(0, T; H¹₀(Ω)). A simple calculation then shows

kbvlk²_L2(0,T;H¹₀(Ω))≤C τl

v_l⁰

2

H¹₀(Ω)+τl N_l

X

n=1

kvⁿ_lk²_H1 0(Ω)

!

and because of the assumptionτl

v⁰_l

2

H¹₀(Ω)≤C and the discrete a priori estimate (3.2), we thus find that also (bvl) is bounded in L²(0, T; H¹₀(Ω)).

Considering the time derivative∂tbvlusing the discrete a priori estimate (3.5), we find k∂tbvlk²_L2(0,T;(H^r)^∗)=τl

N_l

X

n=1

v_lⁿ−v_lⁿ⁻¹ τ_l

2

(H^r)^∗

≤C v⁰

_2,Ω+ Z

Ω

φ(∇u⁰) dx +kfk_L1(0,T;L²(Ω))+ max

n=1,...,Nkσ(∇uⁿ)k_φ∗,Ω

! .

As seen before (recall thatk·k_φ∗,Ω≤1 +ρφ^∗,Ω(·)) using the growth condition we find that the right- hand side is bounded. Considering the scale of spaces

H¹₀(Ω),→^c L²(Ω),→(H^r)^∗ (4.2) we have seen that the sequence (bvl) is bounded in the space

Z :={w∈L²(0, T; H¹₀(Ω))|∃∂tw∈L²(0, T; (H^r)^∗)}, equipped with the norm

kwk_Z :=kwk_L2(0,T;H¹₀(Ω))+k∂twk_L2(0,T;(H^r)^∗).

The generalized Lions-Aubin lemma (see [29, Lemma 7.7]) implies thatZ is compactly embedded in L²(0, T; L²(Ω)) = L²(Q). Thus, there exists a subsequence (still denoted byl) that converges strongly.

Because of lemma (4.2) (V), the limit can only be∂tu.

Proof of theorem4.1. Using the piecewise constant and piecewise linear interpolation in time, the numerical scheme (3.1) can be rewritten as

Z

Ω

(∂_tvb_l(·, t)ψ+∇v_l(·, t)· ∇ψ+σ(∇u_l(·, t))· ∇ψ) dx = Z

Ω

f_l(·, t)ψdx, (4.3) for allψ∈Vm_l, which holds almost everywhere as well as in the weak sense on (0, T), such that

Z

Ω

(bvl(·, T)ψΨ(T)−bvl(·,0)ψΨ(0)) dx +

Z

Q

(−vblψΨ⁰+∇vl· ∇ψΨ +σ(∇ul)· ∇ψΨ) dx dt = Z

Q

flψΨ dx dt,

for allψ∈Vm_land Ψ∈C¹([0, T]). Takingψ=Rm_lwfor arbitraryw∈V, whereRm_lis a restriction operator such that

Rm_lw→w in V asl→ ∞for allw∈V (4.4)

(see also [33, pp. 13 ff]), and employing the weak and weak* convergence shown in lemma 4.2 and 4.4, the strong convergence of fl in L¹(0, T; L²(Ω)) towards f (which follows from standard arguments) and the strong convergence ofbvl(·,0) =v⁰_l in L²(Ω) tov0, we obtain the limit equation

Z

Ω

(ζwΨ(T)−v0wΨ(0)) dx + Z

Q

(−∂tuwΨ⁰+∇∂tu· ∇wΨ +α· ∇wΨ) dx dt = Z

Q

f wΨ dx dt (4.5)

for allw∈V and Ψ∈C¹([0, T]). To be precise, we have used that, asl→ ∞, Pm_lw→win L²(Ω),

P_mlwΨ⁰ →wΨ⁰ in L¹(0, T; L²(Ω)), P_mlwΨ→wΨ in L²(0, T; H¹₀(Ω)),

∇Pm_lwΨ→ ∇wΨ in Eφ(Q;R^d), Pm_lwΨ→wΨ in L^∞(0, T; L²(Ω)).

The convergences above follow from (4.4) and the definition of the norm inV. Note also that k∇R_mlwΨ− ∇wΨk_φ,Ω≤max(1, T)kΨk_C([0,T])k∇R_mlw− ∇wk_φ,Ω.

The limit equation (4.5) shows that d

dt Z

Ω

∂tuwdx = Z

Ω

(f w− ∇∂tu· ∇w−α· ∇u) dx for allw∈V, (4.6) in the weak sense on (0, T). The right-hand side in (4.6) is in L¹(0, T) because f ∈L¹(0, T; L²(Ω)), u∈L^∞(0, T; H¹₀(Ω)) andα∈Lφ^∗(Q;R^d)⊆L¹(0, T; Lφ^∗(Ω;R^d)). Since we already know that∂tu∈ L^∞(0, T; L²(Ω)), this shows that the mapping t 7→ R

Ω∂_tu(x, t)w(x) dx is absolutely continuous on [0, T] for every w ∈ V. BecauseV is dense in L²(Ω) and ∂_tu ∈ L^∞(0, T; L²(Ω)) the mapping t 7→

R

Ω∂_tu(x, t)w(x) dx is also continuous on [0, T] for everyw∈L²(Ω) so that∂_tu∈Cw([0, T]; L²(Ω)).

For the last step of the proof, it will be crucial to use the limit equation (4.5) not only for test functions in V ⊗C¹([0, T]), but for a more general class of test functions. We will use the following approximation result almost identical to [10, Lemma 4.3].

Lemma 4.5. Let

w∈W :={w∈W^1,1(0, T; L²(Ω))∩L²(0, T; H¹₀(Ω))|∇w∈Lφ(Q;R^d)}.

Then for anyε >0, there exists a functionw_ε∈V ⊗C¹([0, T]) such that

kwε−wk_W1,1(0,T;L²(Ω))< ε kwε−wk_L2(0,T;H¹₀(Ω))< ε, and for allη∈Lφ^∗(Q;R^d)

Z

Q

η· ∇wε− Z

Q

η· ∇wdx dt

< ε.

For anyε >0, and anyw∈W, there is (recalling also the continuous embedding of the space W^1,1(0, T; L²(Ω)) intoC([0, T]; L²(Ω))) an elementw_ε∈C¹([0, T])⊗V such that

Z

Ω

ζ(wε(·, T)−w(·, T)) dx

+ Z

Ω

v0(wε(·,0)−w(·,0)) dx

+ Z

Q

∂tu∂t(wε−w) dx dt +

Z

Q

∇∂tu· ∇(wε−w) dx dt

+ Z

Q

α· ∇(wε−w) dx dt

+ Z

Q

f(w_ε−w) dx dt

< ε.

Therefore, Z

Ω

(ζw(·, T)−v0w(·,0)) dx + Z

Q

(−∂tu∂tw+∇∂tu· ∇w+α· ∇w) dx dt = Z

Q

f wdx dt, (4.7) for allw∈W.

Identification of initial and final values: Since bul → u in C([0, T]; L²(Ω)) as l → ∞ we have in particularbul(0)→u(0) in L²(Ω). On the other hand,ubl(0) =u⁰_l →u0 inV asl→ ∞thusu(0) =u0

in L²(Ω). Similarly we can identifyu(T) withξ.

In order to identify∂tu(0) and∂tu(T) withv0andζ respectively we test the limit equation (4.6) with Ψ(t) = (T−t)/T. Thus, we find for allw∈V

d dt

Ψ(t)

Z

Ω

∂tu(·, t)wdx

= Ψ⁰(t) Z

Ω

∂tu(·, t)wdx +Ψ(t)d dt

Z

Ω

∂tu(·, t)wdx

= Ψ⁰(t) Z

Ω

∂_tu(·, t)wdx + Ψ(t)

Z

Ω

(f(·, t)w− ∇∂_tu(·, t)· ∇w−α(·, t)· ∇w) dx. Integration over (0, T) and employing (4.5) yields

− Z

Ω

∂_tu(·,0)wdx = Z

Ω

(ζwΨ(T)−v₀wΨ(0)) dx =− Z

Ω

v₀wdx.