text: 4 The modiﬁed Friedmann equations can be used to derive the continuity equation Using this reformulation of the ﬁeld equations, a continuity equation for the exotic ﬂuid similar to the matter content can also be derived, giving ρ˙ + 3H (ρ + p) = 0, (14) ρ˙ + 3H (ρ + p ) = 0. (23) exo exo exo which is essentially the same as GR. By deﬁn- ing an equation of state (EoS) parameter w for the matter component through the relation Another key component in describing the evo- lution of the universe is the deceleration param- eter which is deﬁned to by p = wρ, (15) ≡ −a¨a − H˙ − q(t) = 1, (24) the continuity equation can be solved to give a˙2 H2 − which for f(T) gravity is expressed as ρ = ρ a(t) 3(1+w), (16) 0 − − 3(1 + w) (T + 2Tf f) q(t) = 1 + T . (25) provided that w is constant in time. 2T (1 + f + 2Tf ) T TT By examining the modiﬁed Friedmann equa- tions, one notes that the extra components aris- Evidence shows that the universe is expanding, ing from f(T) gravity can be used to deﬁne an it implies that q (t ) < 0. This criterion is impor- exotic ﬂuid having energy density ρ and pres- 0 exo tant when analysing the models in Secs. V and sure p exo VI. III. PERTURBATIONS OF THE FLAT ≡ − f FLRW METRIC IN f(T) GRAVITY κ2ρ Tf , (17) exo T 2 ≡ − κ2p κ2ρ + 2H˙ (f + 2Tf ) . (18) In this section, we shall consider perturbations exo exo T TT of the homogeneous and isotropic FLRW metric and study their evolution, which ultimately de- which in turn can be used to deﬁne an eﬀective termines whether the cosmological solutions in equation of state parameter w to be f(T) gravity are stable. The perturbations con- exo sidered are of ﬁrst order, and are described by ≡ p − − f + 2Tf w exo = 1 4H˙ T − TT . (19) → → exo ρ f 2Tf H(t) H(t) (1 + δ) , ρ(t) ρ(t) (1 + δ ) , exo T m (26) This can expressed in terms of f(T) only where δ and δ represent isotropic deviation of by using the ﬁeld equations Eqs. (12) m the Hubble parameter and the matter overden- and (13), and then in terms of the EoS sity respectively. Here, H(t) and ρ(t) represent parameter for matter Eq. (15) to give the zero order quantities, and hence satisfy Eqs. − − − (f T 2Tf ) (f + 2Tf ) (12), (13) and (14) (in some references, these are w = 1+(1+w) T T − TT . exo (1 + fT + 2TfTT) (f 2TfT) sometimes denoted as H0(t) and ρ0(t), however (20) such notation is avoided here to easily distinguish from quantities which are evaluated at present In this way, the Friedmann equations can be times). reduced into a more familiar form, The perturbation of the function f and its derivatives are − T = 2κ2 (ρ + ρ ) , (21) exo − 2H˙ = κ2 (ρ + p + ρ + p ) . (22) exo exo δf = f δT, δf = f δT, (27) T T TT ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: modiﬁed

text: Friedmann

text: equations

text: can

text: be

text: used

text: to

text: derive

text: the

text: continuity

text: equation

text: Using

text: this

text: reformulation

text: of

text: the

text: ﬁeld

text: equations,

text: a

text: continuity

text: equation

text: for

text: the

text: exotic

text: ﬂuid

text: similar

text: to

text: the

text: matter

text: content

text: can

text: also

text: be

text: derived,

text: giving

text: ρ˙

text: (ρ

text: p)

text: ρ˙

text: (ρ

text: p

text: exo

text: exo

text: exo

text: which

text: is

text: essentially

text: the

text: same

text: as

text: GR.

text: By

text: deﬁn-

text: ing

text: an

text: equation

text: of

text: state

text: (EoS)

text: parameter

text: w

text: for

text: the

text: matter

text: component

text: through

text: the

text: relation

text: Another

text: key

text: component

text: in

text: describing

text: the

text: evo-

text: lution

text: of

text: the

text: universe

text: is

text: the

text: deceleration

text: param-

text: eter

text: which

text: is

text: deﬁned

text: to

text: by

text: p

text: wρ,

text: −a¨a

text: H˙

text: q(t)

text: the

text: continuity

text: equation

text: can

text: be

text: solved

text: to

text: give

text: which

text: for

text: f(T)

text: gravity

text: is

text: expressed

text: as

text: ρ

text: ρ

text: a(t)

text: w)

text: (T

text: f)

text: q(t)

text: T

text: provided

text: that

text: w

text: is

text: constant

text: in

text: time.

text: f

text: T

text: TT

text: By

text: examining

text: the

text: modiﬁed

text: Friedmann

text: equa-

text: tions,

text: one

text: notes

text: that

text: the

text: extra

text: components

text: aris-

text: Evidence

text: shows

text: that

text: the

text: universe

text: is

text: expanding,

text: ing

text: from

text: f(T)

text: gravity

text: can

text: be

text: used

text: to

text: deﬁne

text: an

text: it

text: implies

text: that

text: q

text: (t

text: This

text: criterion

text: is

text: impor-

text: exotic

text: ﬂuid

text: having

text: energy

text: density

text: ρ

text: and

text: pres-

text: exo

text: tant

text: when

text: analysing

text: the

text: models

text: in

text: Secs.

text: V

text: and

text: sure

text: p

text: exo

text: VI.

text: III.

text: PERTURBATIONS

text: OF

text: THE

text: FLAT

text: f

text: FLRW

text: METRIC

text: IN

text: f(T)

text: GRAVITY

text: κ2ρ

text: Tf

text: exo

text: T

text: κ2ρ

text: (f

text: In

text: this

text: section,

text: we

text: shall

text: consider

text: perturbations

text: exo

text: exo

text: T

text: TT

text: of

text: the

text: homogeneous

text: and

text: isotropic

text: FLRW

text: metric

text: and

text: study

text: their

text: evolution,

text: which

text: ultimately

text: de-

text: which

text: in

text: turn

text: can

text: be

text: used

text: to

text: deﬁne

text: an

text: eﬀective

text: termines

text: whether

text: the

text: cosmological

text: solutions

text: in

text: equation

text: of

text: state

text: parameter

text: w

text: to

text: be

text: f(T)

text: gravity

text: are

text: stable.

text: The

text: perturbations

text: con-

text: exo

text: sidered

text: are

text: of

text: ﬁrst

text: order,

text: and

text: are

text: described

text: by

text: p

text: f

text: w

text: exo

text: T

text: TT

text: exo

text: ρ

text: f

text: H(t)

text: H(t)

text: δ)

text: ρ(t)

text: ρ(t)

text: δ

text: exo

text: T

text: m

text: This

text: can

text: expressed

text: in

text: terms

text: of

text: f(T)

text: only

text: where

text: δ

text: and

text: δ

text: represent

text: isotropic

text: deviation

text: of

text: by

text: using

text: the

text: ﬁeld

text: equations

text: Eqs.

text: m

text: the

text: Hubble

text: parameter

text: and

text: the

text: matter

text: overden-

text: and

text: and

text: then

text: in

text: terms

text: of

text: the

text: EoS

text: sity

text: respectively.

text: Here,

text: H(t)

text: and

text: ρ(t)

text: represent

text: parameter

text: for

text: matter

text: Eq.

text: to

text: give

text: the

text: zero

text: order

text: quantities,

text: and

text: hence

text: satisfy

text: Eqs.

text: (f

text: T

text: (f

text: and

text: (in

text: some

text: references,

text: these

text: are

text: w

text: T

text: T

text: TT

text: exo

text: fT

text: (f

text: sometimes

text: denoted

text: as

text: and

text: however

text: such

text: notation

text: is

text: avoided

text: here

text: to

text: easily

text: distinguish

text: from

text: quantities

text: which

text: are

text: evaluated

text: at

text: present

text: In

text: this

text: way,

text: the

text: Friedmann

text: equations

text: can

text: be

text: times).

text: reduced

text: into

text: a

text: more

text: familiar

text: form,

text: The

text: perturbation

text: of

text: the

text: function

text: f

text: and

text: its

text: derivatives

text: are

text: T

text: (ρ

text: ρ

text: exo

text: κ2

text: (ρ

text: p

text: ρ

text: p

text: exo

text: exo

text: δf

text: f

text: δT,

text: δf

text: f

text: δT,

text: T

text: T

text: TT

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 - − ≈ 1.0 and n = 2, δ(t)/δ (t0) 0.925]. - 6 1.5 - 5 ()DEwt 2.0 4 - 2.5 ()/()δδtt0 3 - 2 3.0 0 10 20 / 30 40 50 1 t Gyr 0 / FIG. 4: The evolution of the exotic ﬂuid’s EoS 0 10 20 30 40 50 parameter w (t) with cosmic time for the model t Gyr −exo − − f(T) = α ( T)n for n = 0, 1 and 2. In ΛCDM FIG. 6: The evolution of the ratio of the Hubble (solid), the EoS parameter is constant and equals to − perturbation parameter δ(t) to its current value δ (t ) 1 (since this describes a co−smological constant). O−n with cosmic time for the model f(T) = α (−T)n fo0r the other hand, the n = 1 (dotted) and n = 2 − − n = 0, 1 and 2. Each model exhibits the same (dashed) models describe a varying EoS parameter, − − behaviour as ΛCDM, in which each model decays to starting at n 1 at t = 0 and reaches 1 during late 0 at late times indicating stability. The diﬀerence times. In other words, such models mimic the cos- only occurs during earlier times where the value of mological constant behaviour at late times. For all δ(t) varies from model to model. cases, the nature of the exotic ﬂuid is phantom. 2.0 0.5 1.8 1.6 0.0 ()0tM ()qt - ()/δδtM 1.4 1.2 0.5 1.0 - 1.0 0.8 / / 0 10 20 30 40 50 0 10 20 30 40 50 t Gyr t Gyr FIG. 7: The evolution of the ratio of the matter per- FIG. 5: The evolution of the deceleration parameter − turbation parameter δ (t) to its current value δ (t ) q(t) with co−smic tim−e for the model f(T) = α ( T)n with cosmic time for tMhe model f(T) = α (−TM)n fo0r for n = 0, 1 and 2. All models mimic the be- − − n = 0, 1 and 2. One again ﬁnds that each model haviour of the ΛCDM (solid) model, and each of evolves in the same way as ΛCDM, in which each of them transition to an accelerating universe at ap- ≈ them approach the same value of 1 at current times proximately the same time (t 8 Gyr). Each model − and decays to a constant value at late times. This starts with q(0) = 0.4 and ends with q (t = t∞) = 1 value is the only major diﬀerence between the mod- as values. What can be noticed however is that ≈ els [for ΛCDM (solid), δ(t)/δ (t ) 0.850 whilst for the rate of q(t) happens much more drastically for − − 0 ≈ − − n = 1 (dotted) and n = 2 (dashed), δ(t)/δ (t ) n = 1 (dotted) and n = 2 (dashed) compared to 0 0.925]. Furthermore, each model is stable and indi- ΛCDM. cate individually that the matter perturbations per- sist at late times. (cid:18) (cid:20) (cid:114) (cid:21)(cid:19) models, and completely decay at late times. In the case for matter perturbations, the models − − T again exhibit a similar behaviour as ΛCDM in B. f(T) = αT 1 exp p stability 0 T that they are stable and decaying with time un- 0 til they reach a limiting value at late times. This (cid:18) (cid:20) (cid:114) (cid:21)(cid:19) means that the matter perturbations persist at In this section, we consider Linder’s exponen- late times in the same way as ΛCDM does with tial gravity model for f(T) which is of the form the only diﬀerence being the limiting values [for − − T ≈ − f(T) = αT 1 exp p , where α and ΛCDM, δ(t)/δ (t0) 0.850 whilst for n = 1 0 T0 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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label: paragraph

font: default

text: and

text: n

text: δ(t)/δ

text: ()DEwt

text: t

text: Gyr

text: FIG.

text: The

label: caption

text: evolution

text: of

text: the

text: exotic

text: ﬂuid’s

text: EoS

text: parameter

text: w

text: (t)

text: with

text: cosmic

text: time

text: for

text: the

text: model

text: t

text: Gyr

text: −exo

text: f(T)

text: α

text: T)n

text: for

text: n

text: and

text: In

text: ΛCDM

text: FIG.

text: The

text: evolution

text: of

text: the

text: ratio

text: of

text: the

text: Hubble

text: (solid),

text: the

text: EoS

text: parameter

text: is

text: constant

text: and

text: equals

text: to

text: perturbation

text: parameter

text: δ(t)

text: to

text: its

text: current

text: value

text: δ

text: (t

text: (since

text: this

text: describes

text: a

text: co−smological

text: constant).

text: O−n

text: with

text: cosmic

text: time

text: for

text: the

text: model

text: f(T)

text: α

text: (−T)n

text: the

text: other

text: hand,

text: the

text: n

text: (dotted)

text: and

text: n

text: n

text: and

text: Each

text: model

text: exhibits

text: the

text: same

text: (dashed)

text: models

text: describe

text: a

text: varying

text: EoS

text: parameter,

text: behaviour

text: as

text: ΛCDM,

text: in

text: which

text: each

text: model

text: decays

text: to

text: starting

text: at

text: n

text: at

text: t

text: and

text: reaches

text: during

text: late

text: at

text: late

text: times

text: indicating

text: stability.

text: The

text: diﬀerence

text: times.

text: In

text: other

text: words,

text: such

text: models

text: mimic

text: the

text: cos-

text: only

text: occurs

text: during

text: earlier

text: times

text: where

text: the

text: value

text: of

text: mological

text: constant

text: behaviour

text: at

text: late

text: times.

text: For

text: all

text: δ(t)

text: varies

text: from

text: model

text: to

text: model.

text: cases,

text: the

text: nature

text: of

text: the

text: exotic

text: ﬂuid

text: is

text: phantom.

text: ()qt

text: ()/δδtM

text: t

text: Gyr

text: t

text: Gyr

text: FIG.

text: The

text: evolution

text: of

text: the

text: ratio

text: of

text: the

text: matter

text: per-

text: FIG.

text: The

text: evolution

text: of

text: the

text: deceleration

text: parameter

text: turbation

text: parameter

text: δ

text: (t)

text: to

text: its

text: current

text: value

text: δ

text: (t

text: q(t)

text: with

text: co−smic

text: tim−e

text: for

text: the

text: model

text: f(T)

text: α

text: T)n

text: with

text: cosmic

text: time

text: for

text: tMhe

text: model

text: f(T)

text: α

text: (−TM)n

text: for

text: n

text: and

text: All

text: models

text: mimic

text: the

text: be-

text: n

text: and

text: One

text: again

text: ﬁnds

text: that

text: each

text: model

text: haviour

text: of

text: the

text: ΛCDM

text: (solid)

text: model,

text: and

text: each

text: of

text: evolves

text: in

text: the

text: same

text: way

text: as

text: ΛCDM,

text: in

text: which

text: each

text: of

text: them

text: transition

text: to

text: an

text: accelerating

text: universe

text: at

text: ap-

text: them

text: approach

text: the

text: same

text: value

text: of

text: at

text: current

text: times

text: proximately

text: the

text: same

text: time

text: (t

text: Gyr).

text: Each

text: model

text: and

text: decays

text: to

text: a

text: constant

text: value

text: at

text: late

text: times.

text: This

text: starts

text: with

text: and

text: ends

text: with

text: q

text: (t

text: t∞)

text: value

text: is

text: the

text: only

text: major

text: diﬀerence

text: between

text: the

text: mod-

text: as

text: values.

text: What

text: can

text: be

text: noticed

text: however

text: is

text: that

text: els

text: [for

text: ΛCDM

text: (solid),

text: δ(t)/δ

text: (t

text: whilst

text: for

text: the

text: rate

text: of

text: q(t)

text: happens

text: much

text: more

text: drastically

text: for

text: n

text: (dotted)

text: and

text: n

text: (dashed),

text: δ(t)/δ

text: (t

text: n

text: (dotted)

text: and

text: n

text: (dashed)

text: compared

text: to

text: Furthermore,

text: each

text: model

text: is

text: stable

text: and

text: indi-

text: ΛCDM.

text: cate

text: individually

text: that

text: the

text: matter

text: perturbations

text: per-

text: sist

text: at

text: late

text: times.

text: models,

text: and

text: completely

text: decay

text: at

text: late

text: times.

text: In

text: the

text: case

text: for

text: matter

text: perturbations,

text: the

text: models

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text: Input: Hs =←(hsi)Ii=1 (cid:46) list of hidden states generated by encoder given input X. Figure 1 shows a typical K- Initialize: s ←0 (cid:48) (cid:46) s: cumulative log-likelihood best beam search algorithm used in the decoder Yˆ ←‘BOS (cid:46) Yˆ : list of generated words of EncDec approach. We deﬁne the (minimal) re- Ht Hs (cid:46) Ht: hidden states to process decoder 1: h ← (s, Yˆ , Ht) (cid:46) triplet of (minimal) info for decoding process quired information h shown in Figure 1 for the j- Q ← Q 2: Qw ← p{}ush( w, h) (cid:46) set initial triplet h to priority queue Qw th decoding process is the following triplet, h = 34:: Recpe←at (cid:46) prepare queue to store complete sentences (sj−1, Yˆj−1, Hjt−1), where sj−1 is th−e cumula- 5: O˜ () (cid:46) prepare empty list tive log-likelihood from step 0 to j 1, Yˆ − j 1 6: Repeat ← Q is a (candidate of) output word sequence gener- 78:: ho˜ ← pcoapl(cLLw()h) (cid:46) pop a candi(cid:46)dasteeehEisqto.r2y ated so far from step 0 to j − 1, that is, Yˆj−1 = ← 190:: U{ nO˜til Qwa}=pp−e∅nd←(O˜, o˜) (cid:46)(cid:46) arepppeeantdulniktiellQihowoids evmecptotyr (syta0te, s. .f.o,ryjc−al1c)ulaantidngHthjt−e1j-itsh tdheecoaldlintgheprhoidcedsesn. 11: {(mˆ}, kˆ−)z Kz←=1C findKBes{t(O˜) } − Then, the function calcLL in Line 8 can be writ- 12: Qh(cid:48)z←Kz=1C makeTrQipl{et(} (m−ˆ , kˆ)z Kz=1C) ten as follows: 111345::: Un(tQilwQ, Qsc=e)l←∅ecSteTpoCpoKm(p(cQ, (cid:48))hz (cid:46)Kz=se1pCa(cid:46)r)atﬁenQish(cid:48) iifntQo Qics eomr Qptyw (cid:0)(cid:0) (cid:1) (cid:0) (cid:1) (cid:1) Output: Qw w c o˜j = v sj−1, M + log Softmax(oj) Figure 1: Algorithm for a K-best beam search de- o = Ωt Hs, Ht− , yˆ − , (2) j j 1 j 1 coding typically used in EncDec approach. · andLeotuXtput=se(qxuie)nIi=ce1s,anredspYect=ive(lyy,j)wJj=he1rebexinapnudt givwenhevreecStoorfatnmdaxΩ(t)(·i)srtehperessoeftnmtsatxhfeuonvcetiroanllfporroa- i cess of a single decoding step. yj are one-hot vectors, which correspond to the × − i-th word in the input and the j-th word in the out- Moreover, O˜ in Line 11 is a (M (K C))- V put. Let t denote the vocabulary (set of words) matrix, where C is the number of complete sen- Q of output. For simpliﬁcation, this paper uses the tences in . The (m, k)-element of O˜ repre- c following four notation rules: sents a likelihood of the m-th word, namely o˜ [m], j Q 1. (xi)Ii=1 is a short notation for representing a that is cal−culated using the k-th candidate in w list of (column) vectors, i.e., (x , . . . , x ) = at the (j 1)-th step. In Line 12, the function 1 I (x )I . makeTriplet constructs a set of triplets based i i=1 on the information of index (mˆ , kˆ). Then, in Line 2. v(a, D) represents a D-dimensional (col- 13, the function selectTopK selects the top-K umn) vector whose elements are all a, i.e., candidates from union of a set of generated triplets (cid:62) { } − v(1, 3) = (1, 1, 1) . at current step h K Cand a set of triplets of z z=Q1 complete sentences in . Finally, the function 3. x[i] represents the i-th element of x, i.e., x = c Q(cid:48) (cid:62) sepComp in Line 13 divides a set of triplets in (0.1, 0.2, 0.3) , then x[2] = 0.2. two distinct sets whether they are complete sen- |V | Q Q Q(cid:48) 4. M = t and, m always denotes the in∈- tences, c, or not, w. If the eleQmentsQin(cid:48) d{ex of ou}tput vocabulary, namely, m aQre all∅complete sentences, namely, c = and 1, . . . , M , and o[m] represents the score of w = , then the algorithm stops according to the the m-th word in Vt, where o ∈ RM. evaluation of Line 15. · Encoder: Let Ωs( ) denote the overall process 3 Word Frequency Estimation of our 2-layer bidirectional LSTM encoder. The This section describes our proposed method, encoder receives input X and returns a list of ﬁnal which roughly consists of two parts: hidden states Hs = (hs)I : i i=1 1. a sub-model that estimates the upper-bound frequencies of the target vocabulary words in Hs = Ωs(X). (1) the output, and 2. architecture for controlling the output words Decoder: We employ a K-best beam-search de- in the decoder using estimations. coder to ﬁnd the (approximated) best output Yˆ ##LTLine## ##LTLine##

text: Input:

text: Hs

text: list

text: of

text: hidden

text: states

text: generated

text: by

text: encoder

text: given

text: input

text: X.

text: Figure

text: shows

text: a

text: typical

text: K-

text: Initialize:

text: s

text: ←0

text: s:

text: cumulative

text: log-likelihood

text: best

text: beam

text: search

text: algorithm

text: used

text: in

text: the

text: decoder

text: Yˆ

text: ←‘BOS

text: Yˆ

text: list

text: of

text: generated

text: words

text: of

text: EncDec

text: approach.

text: We

text: deﬁne

text: the

text: (minimal)

text: re-

text: Ht

text: Hs

text: Ht:

text: hidden

text: states

text: to

text: process

text: decoder

text: h

text: (s,

text: Yˆ

text: Ht)

text: triplet

text: of

text: (minimal)

text: info

text: for

text: decoding

text: process

text: quired

text: information

text: h

text: shown

text: in

text: Figure

text: for

text: the

text: j-

text: Q

text: Q

text: Qw

text: p{}ush(

text: w,

text: h)

text: set

text: initial

text: triplet

text: h

text: to

text: priority

text: queue

text: Qw

text: th

text: decoding

text: process

text: is

text: the

text: following

text: triplet,

text: h

text: Recpe←at

text: prepare

text: queue

text: to

text: store

text: complete

text: sentences

text: where

text: is

text: th−e

text: cumula-

text: O˜

text: prepare

text: empty

text: list

text: tive

text: log-likelihood

text: from

text: step

text: to

text: j

text: Yˆ

text: j

text: Repeat

text: Q

text: is

text: a

text: (candidate

text: of)

text: output

text: word

text: sequence

text: gener-

text: ho˜

text: pcoapl(cLLw()h)

text: pop

text: a

text: ated

text: so

text: far

text: from

text: step

text: to

text: j

text: that

text: is,

text: U{

text: nO˜til

text: Qwa}=pp−e∅nd←(O˜,

text: o˜)

text: arepppeeantdulniktiellQihowoids

text: evmecptotyr

text: s.

text: tdheecoaldlintgheprhoidcedsesn.

text: {(mˆ},

text: kˆ−)z

text: findKBes{t(O˜)

text: Then,

text: the

text: function

text: calcLL

text: in

text: Line

text: can

text: be

text: writ-

text: makeTrQipl{et(}

text: (m−ˆ

text: kˆ)z

text: ten

text: as

text: follows:

text: Un(tQilwQ,

text: Qsc=e)l←∅ecSteTpoCpoKm(p(cQ,

text: iifntQo

text: Qics

text: eomr

text: Qptyw

text: Output:

text: Qw

text: w

text: c

text: o˜j

text: v

text: M

text: log

text: Softmax(oj)

text: Figure

text: Algorithm

text: for

text: a

text: K-best

text: beam

text: search

text: de-

text: o

text: Ωt

text: Hs,

text: Ht−

text: yˆ

text: j

text: j

text: j

text: coding

text: typically

text: used

text: in

text: EncDec

text: approach.

text: i

text: cess

text: of

text: a

text: single

text: decoding

text: step.

text: yj

text: are

text: one-hot

text: vectors,

text: which

text: correspond

text: to

text: the

text: i-th

text: word

text: in

text: the

text: input

text: and

text: the

text: j-th

text: word

text: in

text: the

text: out-

text: Moreover,

text: O˜

text: in

text: Line

text: is

text: a

text: (M

text: (K

text: C))-

text: V

text: put.

text: Let

text: t

text: denote

text: the

text: vocabulary

text: (set

text: of

text: words)

text: matrix,

text: where

text: C

text: is

text: the

text: number

text: of

text: complete

text: sen-

text: Q

text: of

text: output.

text: For

text: simpliﬁcation,

text: this

text: paper

text: uses

text: the

text: tences

text: in

text: The

text: (m,

text: k)-element

text: of

text: O˜

text: repre-

text: c

text: following

text: four

text: notation

text: rules:

text: sents

text: a

text: likelihood

text: of

text: the

text: m-th

text: word,

text: namely

text: o˜

text: [m],

text: j

text: Q

text: is

text: a

text: short

text: notation

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text: a

text: that

text: is

text: cal−culated

text: using

text: the

text: k-th

text: candidate

text: in

text: w

text: list

text: of

text: (column)

text: vectors,

text: i.e.,

text: (x

text: x

text: at

text: the

text: (j

text: step.

text: In

text: Line

text: the

text: function

text: I

text: (x

text: )I

text: makeTriplet

text: constructs

text: a

text: set

text: of

text: triplets

text: based

text: i

text: on

text: the

text: information

text: of

text: index

text: (mˆ

text: kˆ).

text: Then,

text: in

text: Line

text: v(a,

text: D)

text: represents

text: a

text: D-dimensional

text: (col-

text: the

text: function

text: selectTopK

text: selects

text: the

text: top-K

text: umn)

text: vector

text: whose

text: elements

text: are

text: all

text: a,

text: i.e.,

text: candidates

text: from

text: union

text: of

text: a

text: set

text: of

text: generated

text: triplets

text: at

text: current

text: step

text: h

text: K

text: Cand

text: a

text: set

text: of

text: triplets

text: of

text: z

text: complete

text: sentences

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text: Finally,

text: the

text: function

text: x[i]

text: represents

text: the

text: i-th

text: element

text: of

text: x,

text: i.e.,

text: x

text: c

text: sepComp

text: in

text: Line

text: divides

text: a

text: set

text: of

text: triplets

text: in

text: then

text: two

text: distinct

text: sets

text: whether

text: they

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text: complete

text: sen-

text: |V

text: Q

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text: M

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text: and,

text: m

text: always

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text: the

text: in∈-

text: tences,

text: c,

text: or

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text: w.

text: If

text: the

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text: of

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text: vocabulary,

text: namely,

text: m

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text: all∅complete

text: sentences,

text: namely,

text: c

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text: M

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text: o[m]

text: represents

text: the

text: score

text: of

text: w

text: then

text: the

text: algorithm

text: stops

text: according

text: to

text: the

text: the

text: m-th

text: word

text: in

text: Vt,

text: where

text: o

text: RM.

text: evaluation

text: of

text: Line

text: Encoder:

text: Let

text: Ωs(

text: denote

text: the

text: overall

text: process

text: Word

text: Frequency

text: Estimation

text: of

text: our

text: bidirectional

text: LSTM

text: encoder.

text: The

text: This

text: section

text: describes

text: our

text: proposed

text: method,

text: encoder

text: receives

text: input

text: X

text: and

text: returns

text: a

text: list

text: of

text: ﬁnal

text: which

text: roughly

text: consists

text: of

text: two

text: parts:

text: hidden

text: states

text: Hs

text: (hs)I

text: i

text: a

text: sub-model

text: that

text: estimates

text: the

text: upper-bound

text: frequencies

text: of

text: the

text: target

text: vocabulary

text: words

text: in

text: Hs

text: Ωs(X).

text: the

text: output,

text: and

text: architecture

text: for

text: controlling

text: the

text: output

text: words

text: Decoder:

text: We

text: employ

text: a

text: K-best

text: beam-search

text: de-

text: in

text: the

text: decoder

text: using

text: estimations.

text: coder

text: to

text: ﬁnd

text: the

text: (approximated)

text: best

text: output

text: Yˆ

text: ##LTLine##

text: ##LTLine##


text: 12 RICCARDA ROSSI ∈ u(0, x) = u (x), u˙(0, x) = u˙ (x) for a.a. x Ω, (2.23a) 0 0 ∈ e(0, x) = e (x) for a.a. x Ω, (2.23b) 0 ∈ z(0, x) = z (x) for a.a. x Ω, (2.23c) 0 ∈ p(0, x) = p (x) for a.a. x Ω, (2.23d) 0 ∞ ∗ ϑ(0) = ϑ in W1, (Ω) , (2.23e) 0 and with - the kinematic admissibility condition ∈ ∈ (u(t, x), e(t, x), p(t, x)) A(w(t, x)) for a.a. (t, x) Q; (2.24) - the weak formulation (2.8) of the momentum balance (1.3b); - the feasibility and unidirectionality constraints ∈ ≤ × z [0, 1] and z˙ 0 a.e. in Ω (0, T) (2.25) and the subdiﬀerential inclusion for damage evolution (cid:48) (cid:51) −1C(cid:48) ∗ ∂R (z˙) + νA (z˙) + A (z) + W (z) (z)e : e + ϑ in Hs(Ω) a.e. in (0, T); (2.26) 2 s s 2 - the plastic ﬂow rule (cid:51) D C − C E × ∂ H(z, ϑ; p˙) + p˙ ( (z)e˙ + (z)e ϑ (z) ) a.e. in Ω (0, T), (2.27) p˙ D - the strict positivity of ϑ: ∃ ∈ ϑ¯ > 0 for a.a. (t, x) Q : ϑ(t, x) > ϑ¯; (2.28) ∈ ∞ and the weak f(cid:90)ormulation of the heat equation (1.3e) for every test function ϕ W1, (Ω): (cid:104) (cid:90) (cid:105) ∇ ∇ (cid:90) ϑ˙, ϕ (cid:0) ∞ + κ(ϑ) ϑ ϕdx (cid:1) W1, (Ω) Ω (2.29) D − C E | | | | = G + (z)e˙:e˙ ϑ (z) :e˙+R(z˙)+ z˙ 2+ν¯a (z˙, z˙)+H(z, ϑ; p˙)+ p˙ 2 ϕdx + gϕdS . s Ω ∂Ω (cid:55)→ (cid:107) (cid:107) Since the ‘viscous’ contribution z˙ 1 z˙ 2 is diﬀerentiable on Hs(Ω), by the sum rule we have that 2 →L2(Ω) ∗ ∂R (z˙) = ∂R(z˙) + J(z˙), where J : L2(Ω) Hs(Ω) is the embedding operator. Nevertheless, as the spaces 2 ∗ (Hs(Ω), L2(Ω), Hs(Ω) ) form a Hilbert triple, in what follows we will omit the symbol J. Therefore, (2.26) rewrites a(cid:40)s (cid:48) − C(cid:48) ω + z˙ + νA (z˙) + A (z) + W (z) = 1 (z)e : e + ϑ, ∗ ∈ s s 2 in Hs(Ω) a.e. in (0, T) . (2.30) ω ∂R(z˙) We refer to the previously developed calculations, leading to the mechanical and total energy balances (2.18) and (2.19), for the proof of the following result. ##LTLine## ##LTLine## ##LTLine##

text: RICCARDA

text: ROSSI

label: equation

text: x)

text: u

text: (x),

text: x)

text: u˙

text: (x)

text: for

text: a.a.

text: x

text: Ω,

text: x)

text: e

text: (x)

text: for

text: a.a.

text: x

text: Ω,

text: x)

text: z

text: (x)

text: for

text: a.a.

text: x

text: Ω,

text: x)

text: p

text: (x)

text: for

text: a.a.

text: x

text: Ω,

text: ϑ

text: in

text: (Ω)

text: and

text: with

label: list

text: the

text: kinematic

text: admissibility

text: condition

text: (u(t,

text: x),

text: e(t,

text: x),

text: p(t,

text: x))

text: A(w(t,

text: x))

text: for

text: a.a.

text: (t,

text: x)

text: Q;

text: the

text: weak

text: formulation

text: of

text: the

text: momentum

text: balance

text: the

text: feasibility

text: and

text: unidirectionality

text: constraints

text: z

text: and

text: z˙

text: a.e.

text: in

text: Ω

text: T)

text: and

text: the

text: subdiﬀerential

text: inclusion

text: for

text: damage

text: evolution

text: ∂R

text: (z˙)

text: νA

text: (z˙)

text: A

text: (z)

text: W

text: (z)

text: (z)e

text: e

text: ϑ

text: in

text: Hs(Ω)

text: a.e.

text: in

text: T);

text: s

text: s

text: the

text: plastic

text: ﬂow

text: rule

text: D

text: C

text: C

text: E

text: H(z,

text: ϑ;

text: p˙)

text: p˙

text: (z)e˙

text: (z)e

text: ϑ

text: (z)

text: a.e.

text: in

text: Ω

text: T),

text: p˙

text: D

text: the

text: strict

text: positivity

text: of

text: ϑ:

text: ϑ¯

text: for

text: a.a.

text: (t,

text: x)

text: Q

text: ϑ(t,

text: x)

text: ϑ¯;

text: and

text: the

text: weak

text: of

text: the

text: heat

text: equation

text: for

text: every

text: test

text: function

text: ϕ

text: (Ω):

text: ϑ˙,

text: ϕ

text: κ(ϑ)

text: ϑ

text: ϕdx

text: (Ω)

text: Ω

text: D

text: C

text: E

text: G

text: (z)e˙:e˙

text: ϑ

text: (z)

text: :e˙+R(z˙)+

text: z˙

text: (z˙,

text: z˙)+H(z,

text: ϑ;

text: p˙)+

text: p˙

text: ϕdx

text: gϕdS

text: s

text: Ω

text: ∂Ω

text: Since

text: the

text: ‘viscous’

text: contribution

text: z˙

text: z˙

text: is

text: diﬀerentiable

text: on

text: Hs(Ω),

text: by

text: the

text: sum

text: rule

text: we

text: have

text: that

text: ∂R

text: (z˙)

text: ∂R(z˙)

text: J(z˙),

text: where

text: J

text: Hs(Ω)

text: is

text: the

text: embedding

text: operator.

text: Nevertheless,

text: as

text: the

text: spaces

text: (Hs(Ω),

text: Hs(Ω)

text: form

text: a

text: Hilbert

text: triple,

text: in

text: what

text: follows

text: we

text: will

text: omit

text: the

text: symbol

text: J.

text: Therefore,

text: rewrites

text: ω

text: z˙

text: νA

text: (z˙)

text: A

text: (z)

text: W

text: (z)

text: (z)e

text: e

text: ϑ,

text: s

text: s

text: in

text: Hs(Ω)

text: a.e.

text: in

text: T)

text: ω

text: ∂R(z˙)

text: We

text: refer

text: to

text: the

text: previously

text: developed

text: calculations,

text: leading

text: to

text: the

text: mechanical

text: and

text: total

text: energy

text: balances

text: and

text: for

text: the

text: proof

text: of

text: the

text: following

text: result.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A COUPLED VISCOPLASTIC-DAMAGE SYSTEM WITH TEMPERATURE 13 Lemma 2.2. Let (u, e, z, p, ϑ) be a weak energy solution to (the Cauchy problem for) system (1.3, 1.4). Then, ≤ ≤(cid:90) ≤ for every 0 s t T the functions (u, e, z, p, ϑ) comply with the mechanical energy balance (2.18) and with ρ (cid:90)| | (cid:104) (cid:105) u˙(t) 2 dx + ϑ(t), 1 ∞ +Q(e(t), z(t)) + G(z(t)) 2 W1, (Ω) Ω ρ (cid:90) | | (cid:104) (cid:105) (cid:90) (cid:90) (cid:90) (cid:90) = u˙(s) 2 dx + ϑ(s), 1 ∞ +Q(e(s), z(s)) + G(z(s)) 2 W1, (Ω) Ω (2.31) (cid:18)t(cid:90)(cid:104) − (cid:105) (cid:90) t (cid:90) (cid:90) t (cid:19) (cid:90) (cid:90) + L, u˙ w˙ R + Gdxdr + gdS dr H1 (Ω; d) s Dir s Ω s ∂Ω − − t t + ρ u˙(t)w˙(t)dx u˙ w˙(0)dx u˙w¨dxdr + σ : E(w˙)dxdr . 0 (cid:82) Ω Ω s Ω s Ω ∈ ∞ (cid:104) (cid:105) Observe that, since ϑ L (0, T; L1(Ω)), there holds ϑ(t), 1 ∞ = ϑ(t) dx = F(ϑ(t)) for almost all ∈ W1, (Ω) Ω t (0, T) and for t = 0, and in that case (2.31) coincides with the total energy balance (2.19). For the (Cauchy problem associated with the) non-regularized system (1.1, 1.2), we will be able to prove an existence result only for a solution concept containing much less information than that from Def. 2.1. In particular, we will notably weaken the formulations of the heat equation (1.1e), given in terms of an entropy inequality joint with the total energy inequality, and of the damage and plastic ﬂow rules. In order to motivate Deﬁnition 2.3 ahead, we develop some preliminary considerations on the weak formulation of the damage and plastic ﬂow rules, and on the entropy inequality. The latter will be formally obtained from the heat equation (1.1e) assuming the strict positivity of the temperature ϑ, which shall be rigorously proved (cf. Prop. 3.3 ahead). The entropy inequality. It can be formally obtained by multiplying the heat equation (1.1e) by ϕ/ϑ, with ϕ a ⊂ smooth and positive test function. Integrating in space and over a generic interval (s, t) (0, T) leads to the identity (cid:90) (cid:90) (cid:90) (cid:90) (cid:16) (cid:17) t(cid:90) (cid:90) t ∇ ∇ − ϕ∇ ∇ ∂ l(cid:0)og(ϑ)ϕdxdr + κ(ϑ) log(ϑ) ϕ κ(ϑ) log(ϑ) ϑ dxd(cid:1)r t ϑ s Ω s Ω t (cid:90) (cid:90) D − C E | | − ϕ = G + (z)e˙ : e˙ ϑ (z) : e˙ + R(z˙) + z˙ 2 ϑz˙ + H(z, ϑ; p˙) + p˙ : p˙ dxdr (2.32) ϑ s Ω t ϕ + g dxdr ϑ s ∂Ω ≥ The entropy inequality (2.37) ahead is indeed the -estimate in (2.32), with positive test functions, and where the ﬁrst integral on the left-hand side is integrated in time. − C(cid:48) − − − (cid:48) Weak solvability of the damage and plastic ﬂow rules. Setting ξ := ϑ 1 (z)e : e z˙ A (z) W (z), 2 ∈ × s the subdiﬀerential inclusion (1.1c) for damage evolution reformulates as ξ ∂R(z˙) in Ω (0, T). By the 1-homogeneity of R, the latter is in turn equivalent to the system of inequalities ≤ × ∈ −∞ ξζ R(ζ) in Ω (0, T) for all ζ dom(R) = ( , 0], (2.33a) ≥ × ξz˙ R(z˙) in Ω (0, T) . (2.33b) Therefore, the damage ﬂow rule (1.1c) could be formulated in terms of the constraints (2.25), of the integrated ∈ (in space) version of (2.33a) with test functions ζ H−s (Ω), and of the integrated (in space and time) version of (2.33b), which can be interpreted as an energy-dissipation inequality. Namely, (cid:90) (cid:0) (cid:1) - the one-sided variational inequality (cid:48) C(cid:48) − ≥ a (z(t), ζ) + R(ζ) + z˙(t)ζ + W (z(t))ζ + 1 (z(t))e: eζ ϑ(t)ζ dx 0 s 2 (2.34a) Ω ∈ ∈ for all ζ H−s (Ω) for a.a. t (0, T); ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: COUPLED

text: VISCOPLASTIC-DAMAGE

text: SYSTEM

text: WITH

text: TEMPERATURE

text: Lemma

text: Let

text: (u,

text: e,

text: z,

text: p,

text: ϑ)

text: be

text: a

text: weak

text: energy

text: solution

text: to

text: (the

text: Cauchy

text: problem

text: for)

text: system

text: Then,

text: for

text: every

text: s

text: t

text: T

text: the

text: functions

text: (u,

text: e,

text: z,

text: p,

text: ϑ)

text: comply

text: with

text: the

text: mechanical

text: energy

text: balance

text: and

text: with

text: ρ

text: u˙(t)

text: dx

text: ϑ(t),

text: +Q(e(t),

text: z(t))

text: G(z(t))

text: (Ω)

text: Ω

text: ρ

text: u˙(s)

text: dx

text: ϑ(s),

text: +Q(e(s),

text: z(s))

text: G(z(s))

text: (Ω)

text: Ω

text: t

text: t

text: L,

text: u˙

text: w˙

text: R

text: Gdxdr

text: gdS

text: dr

text: (Ω;

text: d)

text: s

text: Dir

text: s

text: Ω

text: s

text: ∂Ω

text: t

text: t

text: ρ

text: u˙(t)w˙(t)dx

text: u˙

text: u˙w¨dxdr

text: σ

text: E(w˙)dxdr

text: Ω

text: Ω

text: s

text: Ω

text: s

text: Ω

text: Observe

text: that,

text: since

text: ϑ

text: L

text: T;

text: there

text: holds

text: ϑ(t),

text: ϑ(t)

text: dx

text: F(ϑ(t))

text: for

text: almost

text: all

text: (Ω)

text: Ω

text: t

text: T)

text: and

text: for

text: t

text: and

text: in

text: that

text: case

text: coincides

text: with

text: the

text: total

text: energy

text: balance

text: For

text: the

text: (Cauchy

text: problem

text: associated

text: with

text: the)

text: non-regularized

text: system

text: we

text: will

text: be

text: able

text: to

text: prove

text: an

text: existence

text: result

text: only

text: for

text: a

text: solution

text: concept

text: containing

text: much

text: less

text: information

text: than

text: that

text: from

text: Def.

text: In

text: particular,

text: we

text: will

text: notably

text: weaken

text: the

text: formulations

text: of

text: the

text: heat

text: equation

text: given

text: in

text: terms

text: of

text: an

text: entropy

text: inequality

text: joint

text: with

text: the

text: total

text: energy

text: inequality,

text: and

text: of

text: the

text: damage

text: and

text: plastic

text: ﬂow

text: rules.

text: In

text: order

text: to

text: motivate

text: Deﬁnition

text: ahead,

text: we

text: develop

text: some

text: preliminary

text: considerations

text: on

text: the

text: weak

text: formulation

text: of

text: the

text: damage

text: and

text: plastic

text: ﬂow

text: rules,

text: and

text: on

text: the

text: entropy

text: inequality.

text: The

text: latter

text: will

text: be

text: formally

text: obtained

text: from

text: the

text: heat

text: equation

text: assuming

text: the

text: strict

text: positivity

text: of

text: the

text: temperature

text: ϑ,

text: which

text: shall

text: be

text: rigorously

text: proved

text: (cf.

text: Prop.

text: ahead).

text: The

text: entropy

text: inequality.

text: It

text: can

text: be

text: formally

text: obtained

text: by

text: multiplying

text: the

text: heat

text: equation

text: by

text: ϕ/ϑ,

text: with

text: ϕ

text: a

text: smooth

text: and

text: positive

text: test

text: function.

text: Integrating

text: in

text: space

text: and

text: over

text: a

text: generic

text: interval

text: (s,

text: t)

text: T)

text: leads

text: to

text: the

text: identity

text: t

text: ϕ∇

text: κ(ϑ)

text: log(ϑ)

text: ϕ

text: κ(ϑ)

text: log(ϑ)

text: ϑ

text: t

text: ϑ

text: s

text: Ω

text: s

text: Ω

text: t

text: D

text: C

text: E

text: ϕ

text: G

text: (z)e˙

text: e˙

text: ϑ

text: (z)

text: e˙

text: R(z˙)

text: z˙

text: ϑz˙

text: H(z,

text: ϑ;

text: p˙)

text: p˙

text: p˙

text: dxdr

text: ϑ

text: s

text: Ω

text: t

text: ϕ

text: g

text: dxdr

text: ϑ

text: s

text: ∂Ω

text: The

text: entropy

text: inequality

text: ahead

text: is

text: indeed

text: the

text: -estimate

text: in

text: with

text: positive

text: test

text: functions,

text: and

text: where

text: the

text: ﬁrst

text: integral

text: on

text: the

text: left-hand

text: side

text: is

text: integrated

text: in

text: time.

text: Weak

text: solvability

text: of

text: the

text: damage

text: and

text: plastic

text: ﬂow

text: rules.

text: Setting

text: ξ

text: ϑ

text: (z)e

text: e

text: z˙

text: A

text: (z)

text: W

text: (z),

text: s

text: the

text: subdiﬀerential

text: inclusion

text: for

text: damage

text: evolution

text: reformulates

text: as

text: ξ

text: ∂R(z˙)

text: in

text: Ω

text: T).

text: By

text: the

text: of

text: R,

text: the

text: latter

text: is

text: in

text: turn

text: equivalent

text: to

text: the

text: system

text: of

text: inequalities

text: ξζ

text: R(ζ)

text: in

text: Ω

text: T)

text: for

text: all

text: ζ

text: dom(R)

text: ξz˙

text: R(z˙)

text: in

text: Ω

text: T)

text: Therefore,

text: the

text: damage

text: ﬂow

text: rule

text: could

text: be

text: formulated

text: in

text: terms

text: of

text: the

text: constraints

text: of

text: the

text: integrated

text: (in

text: space)

text: version

text: of

text: with

text: test

text: functions

text: ζ

text: H−s

text: (Ω),

text: and

text: of

text: the

text: integrated

text: (in

text: space

text: and

text: time)

text: version

text: of

text: which

text: can

text: be

text: interpreted

text: as

text: an

text: energy-dissipation

text: inequality.

text: Namely,

text: the

text: one-sided

text: variational

text: inequality

text: a

text: (z(t),

text: ζ)

text: R(ζ)

text: z˙(t)ζ

text: W

text: (z(t))ζ

text: (z(t))e:

text: eζ

text: ϑ(t)ζ

text: dx

text: s

text: Ω

text: for

text: all

text: ζ

text: H−s

text: (Ω)

text: for

text: a.a.

text: t

text: T);

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A COUPLED VISCOPLASTIC-DAMAGE SYSTEM WITH TEMPERATURE 19 ⊂ M × (cid:107) (cid:107) → M × (e0) Lγ(Ω; d d) such that lim τ1/γ e0 M × = 0 and e0 e in L2(Ω; d d), (3.5a) τ τ sym τ↓0 τ Lγ(Ω; dsymd) τ 0 sym ⊂ M × (cid:107) (cid:107) → M × (p0) Lγ(Ω; d d) such that lim τ1/γ p0 M × = 0 and p0 p in L2(Ω; d d). (3.5b) τ τ D ↓ τ Lγ(Ω; d d) τ 0 D τ 0 D By consistency with the kinematic admissibility condition at time t = 0, we will also approximate the initial ⊂ R datum u with a family (u0) W1,γ(Ω; d) such that 0 τ τ ⊂ R (cid:107) (cid:107) → R (u0) W1,γ(Ω; d) such that lim τ1/γ u0 R = 0 and u0 u in H1(Ω; d). (3.5c) τ τ ↓ τ W1,γ(Ω; d) τ 0 τ 0 The data (u0) may be constructed by a perturbation technique. In connection with the regulariza- τ τ tion of the discrete momentum balance, we will have to approximate the Dirichlet loading w by a family ⊂ ∩ R ∞ R ∩ (w ) W W1,1(0, T; W1,γ(Ω; d)), where we have used the place-holder W := L1(0, T; W1, (Ω; d)) τ τ R W2,1(0, T; H1(Ω; d)) ∩ R H2(0, T; L2(Ω; d)). We will require that (cid:18) (cid:19) → ↓ ∃ ∈ 1 (cid:107) (cid:107) ≤ ∞ w w in W as τ 0, as well as α 0, s.t. sup ταw E(w˙ ) M × C < . (3.6) τ w γ τ>0 τ Lγ(Ω; dsymd) (cid:90) We will then consider the discrete data 1 tkτ wk := w (s)ds . τ τ − τ tkτ 1 For technical reasons related to the proof of Prop. 3.3 (cf. (3.11) ahead), it will be expedient to replace the C argument of the elasticity tensor with its positive part. We will proceed in this way in the thermal expansion terms contributing to the momentum balance and to the heat equation. Since we will ultimately prove that the discrete damage solutions are conﬁned to the admissible interval [0, 1], cf. (4.9) in Prop. 4.3 ahead, the C restriction to the positive part in the argument of will “disappear” in the end. Finally, in the discrete version of the damage ﬂow rule (where we will stay with the notation (2.30)), we will − | | ∈ R resort to the convex-concave decomposition W(z) = β(z) λ2W z 2 from (2.14), with λW > 0 and β C2( +) convex. For shorter notation, in what follows we will use the place-holders X := W1,γ(Ω; Rd) × Lγ(Ω; Md×d) × Hs(Ω) × Lγ(Ω; Md×d) × H1(Ω), Dir sym D B := W1,γ(Ω; Rd) × Lγ(Ω; Md×d) × Lγ(Ω; Md×d) × H1(Ω) Dir sym D to indicate the state spaces for the solutions to system (3.8) below. Problem 3.1. Let γ > 4. Using the notation (3.2) and starting from data − − u0 := u0, u 1 := u τu˙0, e0 := e0, z0 := z0, p0 := p0, ϑ0 := ϑ , (3.7) τ τ τ 0 τ τ τ τ τ τ τ τ 0 − − − − − ∈ ∈ for all k = 1, . . . , K , given (uk 1, ek 1, zk 1, pk 1, ϑk 1) X, ﬁnd zk Hs(Ω) fulﬁlling τ τ τ τ τ τ τ - the discrete damage ﬂow rule (cid:48) − − ωk + D (z) + νA (D (z)) + A (zk) + β (zk) λ zk 1 τ k,τ s k,τ s τ τ W τ − C(cid:48) − − − ∗ ∈ (3.8a) = 1 (zk)ek 1 : ek 1 + ϑk 1 in Hs(Ω) with ωk ∂R(D (z)). 2 τ τ τ τ τ k,τ − − − − − ∈ ∈ ∈ Given (uk 1, ek 1, zk 1, pk 1, ϑk 1) X and zk Hs(Ω), ﬁnd (uk, ek, pk, ϑk) B fulﬁlling τ τ τ τ τ τ τ τ τ τ ∈ - the kinematic admissibility (uk, ek, pk) A(wk) (in the sense of (2.5)); τ τ τ τ ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

text: COUPLED

text: VISCOPLASTIC-DAMAGE

text: SYSTEM

text: WITH

text: TEMPERATURE

text: M

text: M

text: Lγ(Ω;

text: d

text: d)

text: such

text: that

text: lim

text: τ1/γ

text: M

text: and

text: e

text: in

text: d

text: d),

text: τ

text: τ

text: sym

text: τ↓0

text: τ

text: Lγ(Ω;

text: dsymd)

text: τ

text: sym

text: M

text: M

text: Lγ(Ω;

text: d

text: d)

text: such

text: that

text: lim

text: τ1/γ

text: M

text: and

text: p

text: in

text: d

text: d).

text: τ

text: τ

text: D

text: τ

text: Lγ(Ω;

text: d

text: d)

text: τ

text: D

text: τ

text: D

text: By

text: consistency

text: with

text: the

text: kinematic

text: admissibility

text: condition

text: at

text: time

text: t

text: we

text: will

text: also

text: approximate

text: the

text: initial

text: R

text: datum

text: u

text: with

text: a

text: family

text: d)

text: such

text: that

text: τ

text: τ

text: R

text: R

text: d)

text: such

text: that

text: lim

text: τ1/γ

text: R

text: and

text: u

text: in

text: d).

text: τ

text: τ

text: τ

text: d)

text: τ

text: τ

text: The

text: data

text: may

text: be

text: constructed

text: by

text: a

text: perturbation

text: technique.

text: In

text: connection

text: with

text: the

text: regulariza-

text: τ

text: τ

text: tion

text: of

text: the

text: discrete

text: momentum

text: balance,

text: we

text: will

text: have

text: to

text: approximate

text: the

text: Dirichlet

text: loading

text: w

text: by

text: a

text: family

text: R

text: R

text: (w

text: W

text: T;

text: d)),

text: where

text: we

text: have

text: used

text: the

text: place-holder

text: W

text: T;

text: (Ω;

text: d))

text: τ

text: τ

text: R

text: T;

text: d))

text: R

text: T;

text: d)).

text: We

text: will

text: require

text: that

text: w

text: w

text: in

text: W

text: as

text: τ

text: as

text: well

text: as

text: α

text: s.t.

text: sup

text: ταw

text: E(w˙

text: M

text: C

text: τ

text: w

text: γ

text: τ

text: Lγ(Ω;

text: dsymd)

text: We

text: will

text: then

text: consider

text: the

text: discrete

text: data

text: tkτ

text: wk

text: w

text: (s)ds

text: τ

text: τ

text: τ

text: tkτ

text: For

text: technical

text: reasons

text: related

text: to

text: the

text: proof

text: of

text: Prop.

text: (cf.

text: ahead),

text: it

text: will

text: be

text: expedient

text: to

text: replace

text: the

text: C

text: argument

text: of

text: the

text: elasticity

text: tensor

text: with

text: its

text: positive

text: part.

text: We

text: will

text: proceed

text: in

text: this

text: way

text: in

text: the

text: thermal

text: expansion

text: terms

text: contributing

text: to

text: the

text: momentum

text: balance

text: and

text: to

text: the

text: heat

text: equation.

text: Since

text: we

text: will

text: ultimately

text: prove

text: that

text: the

text: discrete

text: damage

text: solutions

text: are

text: conﬁned

text: to

text: the

text: admissible

text: interval

text: cf.

text: in

text: Prop.

text: ahead,

text: the

text: C

text: restriction

text: to

text: the

text: positive

text: part

text: in

text: the

text: argument

text: of

text: will

text: “disappear”

text: in

text: the

text: end.

text: Finally,

text: in

text: the

text: discrete

text: version

text: of

text: the

text: damage

text: ﬂow

text: rule

text: (where

text: we

text: will

text: stay

text: with

text: the

text: notation

text: we

text: will

text: R

text: resort

text: to

text: the

text: convex-concave

text: decomposition

text: W(z)

text: β(z)

text: z

text: from

text: with

text: λW

text: and

text: β

text: convex.

text: For

text: shorter

text: notation,

text: in

text: what

text: follows

text: we

text: will

text: use

text: the

text: place-holders

text: X

text: Rd)

text: Lγ(Ω;

text: Md×d)

text: Hs(Ω)

text: Lγ(Ω;

text: Md×d)

text: Dir

text: sym

text: D

text: B

text: Rd)

text: Lγ(Ω;

text: Md×d)

text: Lγ(Ω;

text: Md×d)

text: Dir

text: sym

text: D

text: to

text: indicate

text: the

text: state

text: spaces

text: for

text: the

text: solutions

text: to

text: system

text: below.

text: Problem

text: Let

text: γ

text: Using

text: the

text: notation

text: and

text: starting

text: from

text: data

text: u

text: u

text: ϑ0

text: ϑ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: for

text: all

text: k

text: K

text: given

text: (uk

text: ek

text: zk

text: pk

text: ϑk

text: X,

text: ﬁnd

text: zk

text: Hs(Ω)

text: fulﬁlling

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: the

text: discrete

text: damage

text: ﬂow

text: rule

text: ωk

text: D

text: (z)

text: νA

text: (D

text: (z))

text: A

text: (zk)

text: β

text: (zk)

text: λ

text: zk

text: τ

text: k,τ

text: s

text: k,τ

text: s

text: τ

text: τ

text: W

text: τ

text: (zk)ek

text: ek

text: ϑk

text: in

text: Hs(Ω)

text: with

text: ωk

text: ∂R(D

text: (z)).

text: τ

text: τ

text: τ

text: τ

text: τ

text: k,τ

text: Given

text: (uk

text: ek

text: zk

text: pk

text: ϑk

text: X

text: and

text: zk

text: Hs(Ω),

text: ﬁnd

text: (uk,

text: ek,

text: pk,

text: ϑk)

text: B

text: fulﬁlling

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: the

text: kinematic

text: admissibility

text: (uk,

text: ek,

text: pk)

text: A(wk)

text: (in

text: the

text: sense

text: of

text: τ

text: τ

text: τ

text: τ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 22 RICCARDA ROSSI and, accordingly, the approximate elliptic operator (cid:90) (cid:90) → ∗ (cid:104) (cid:105) ∇ ∇ − Ak : H1(Ω) H1(Ω) deﬁned by Ak (θ), v := κ (θ) θ vdx gkvdS. (3.15) M M H1(Ω) M τ Ω ∂Ω We are now in the position to introduce the approximate discrete system (3.16). For notational simplicity, we will omit to indicate the dependence of the solution quintuple on the index M. Problem 3.4. Let γ > 4. Starting from the discrete Cauchy data (3.7), for all k = 1, . . . , K , given − − − − − ∈ ∈ τ (uk 1, ek 1, zk 1, pk 1, ϑk 1) X, ﬁnd zk Hs(Ω) fulﬁlling the discrete damage ﬂow rule (3.8a). Given τ− τ− τ− τ− τ− ∈ ∈τ ∈ (uk 1, ek 1, zk 1, pk 1, ϑk 1) X and zk Hs(Ω), ﬁnd (uk, ek, pk, ϑk) B fulﬁlling τ τ τ τ τ τ τ τ τ τ ∈ - the kinematic admissibility (uk, ek, pk) A(wk); τ τ τ τ (cid:90) (cid:90) - the approximate discrete momentum balance ρ D2 (u)vdx + σk : E(v)dx = (cid:104)Lk, v(cid:105) R for all v ∈ W1,γ(Ω; Rd), (3.16a) k,τ M,τ τ H1 (Ω; d) Dir Ω Ω Dir D C | | − − C E with the place-holder σk := (zk)D (e) + (zk)ek + τ ek γ 2ek T (ϑk) ((zk)+) ; M,τ τ k,τ τ τ τ τ M τ τ - the approximate discrete plastic ﬂow rule | | − (cid:51) ∈ − ζk + D (p) + τ pk γ 2pk (σk ) , with ζk ∂ H(zk, ϑk 1; D (p)) a.e. in Ω; (3.16b) τ k,τ τ τ M,τ D τ p˙ τ τ k,τ - the approximate discrete heat equation D − C E D (ϑ) + Ak (ϑk) = Gk + (zk)D (e) : D (e) T (ϑk) ((zk)+) : D (e) k,τ M τ τ τ k,τ k,τ M τ τ k,τ | | − − + R(D (z)) + D (z) 2 + ν¯a (D (z), D (z)) ϑk 1D (z) (3.16c) k,τ k,τ s k,τ k,τ τ k,τ − | | ∗ + H(zk, ϑk 1; D (p)) + D (p) 2 in H1(Ω) . τ τ k,τ k,τ Step 2: Existence of solutions to system (3.16): we have the following result. Lemma 3.5. Under the growth condition (2.κ ), there exists τ¯ > 0 such that for 0 < τ < τ¯ and for every 1 ∈ k = 1, . . . , K there exists a solution (uk, ek, zk, pk, ϑk) X to system (3.16). Furthermore, for any solution τ τ τ τ τ τ (uk, ek, zk, pk, ϑk) the function ϑk complies with the positivity property (3.9). τ τ τ τ τ τ Proof. The positivity property (3.9) (note that the constant providing the lower bound for ϑk is independent τ of the truncation parameter M), follows from the analogue of estimate (3.10), with the same comparison argument. Let us now address the existence of solutions. First of all, we ﬁ(cid:18)n(cid:90)d a solution zk to (3.8a(cid:90)) via the minimum problem (cid:19) τ Min ∈ R(z−zk−1)d(cid:90)x +(cid:0) 1 |z−zk−1|2 dx + ν a (z−zk−1, z−zk−(cid:1)1) + F (z) , z Hs(Ω) τ 2τ τ 2τ s τ τ k Ω Ω (3.17) 1 − − C − − − − with F (z) := a (z, z) + β(z) λ zk 1z+1 (z)ek 1:ek 1 ϑk 1z dx . k 2 s W τ 2 τ τ τ Ω With the direct method in the calculus of variations, it is easy to check that (3.17), whose Euler-Lagrange equation is (3.8a), has a solution zk. τ Let us now brieﬂy address the solvability of the approximate discrete thermoviscoplastic system (3.16), for ∈ { } ﬁxed k 1, . . . , K with zk given. To this end, we reformulate system (3.16) in the form τ τ ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: RICCARDA

text: ROSSI

text: and,

text: accordingly,

text: the

text: approximate

text: elliptic

text: operator

text: Ak

text: deﬁned

text: by

text: Ak

text: (θ),

text: v

text: κ

text: (θ)

text: θ

text: vdx

text: gkvdS.

text: M

text: M

text: M

text: τ

text: Ω

text: ∂Ω

text: We

text: are

text: now

text: in

text: the

text: position

text: to

text: introduce

text: the

text: approximate

text: discrete

text: system

text: For

text: notational

text: simplicity,

text: we

text: will

text: omit

text: to

text: indicate

text: the

text: dependence

text: of

text: the

text: solution

text: quintuple

text: on

text: the

text: index

text: M.

text: Problem

text: Let

text: γ

text: Starting

text: from

text: the

text: discrete

text: Cauchy

text: data

text: for

text: all

text: k

text: K

text: given

text: τ

text: (uk

text: ek

text: zk

text: pk

text: ϑk

text: X,

text: ﬁnd

text: zk

text: Hs(Ω)

text: fulﬁlling

text: the

text: discrete

text: damage

text: ﬂow

text: rule

text: Given

text: τ−

text: τ−

text: τ−

text: τ−

text: τ−

text: ∈τ

text: (uk

text: ek

text: zk

text: pk

text: ϑk

text: X

text: and

text: zk

text: Hs(Ω),

text: ﬁnd

text: (uk,

text: ek,

text: pk,

text: ϑk)

text: B

text: fulﬁlling

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: the

text: kinematic

text: admissibility

text: (uk,

text: ek,

text: pk)

text: A(wk);

text: τ

text: τ

text: τ

text: τ

text: the

text: approximate

text: discrete

text: momentum

text: balance

text: ρ

text: (u)vdx

text: σk

text: E(v)dx

text: R

text: for

text: all

text: v

text: Rd),

text: k,τ

text: M,τ

text: τ

text: (Ω;

text: d)

text: Dir

text: Ω

text: Ω

text: Dir

text: D

text: C

text: C

text: E

text: with

text: the

text: place-holder

text: σk

text: (zk)D

text: (e)

text: (zk)ek

text: τ

text: ek

text: γ

text: T

text: (ϑk)

text: ((zk)+)

text: M,τ

text: τ

text: k,τ

text: τ

text: τ

text: τ

text: τ

text: M

text: τ

text: τ

text: the

text: approximate

text: discrete

text: plastic

text: ﬂow

text: rule

text: ζk

text: D

text: (p)

text: τ

text: pk

text: γ

text: (σk

text: with

text: ζk

text: H(zk,

text: ϑk

text: D

text: (p))

text: a.e.

text: in

text: Ω;

text: τ

text: k,τ

text: τ

text: τ

text: M,τ

text: D

text: τ

text: p˙

text: τ

text: τ

text: k,τ

text: the

text: approximate

text: discrete

text: heat

text: equation

text: D

text: C

text: E

text: D

text: (ϑ)

text: Ak

text: (ϑk)

text: Gk

text: (zk)D

text: (e)

text: D

text: (e)

text: T

text: (ϑk)

text: ((zk)+)

text: D

text: (e)

text: k,τ

text: M

text: τ

text: τ

text: τ

text: k,τ

text: k,τ

text: M

text: τ

text: τ

text: k,τ

text: R(D

text: (z))

text: D

text: (z)

text: ν¯a

text: (D

text: (z),

text: D

text: (z))

text: ϑk

text: (z)

text: k,τ

text: k,τ

text: s

text: k,τ

text: k,τ

text: τ

text: k,τ

text: H(zk,

text: ϑk

text: D

text: (p))

text: D

text: (p)

text: in

text: τ

text: τ

text: k,τ

text: k,τ

text: Step

text: Existence

text: of

text: solutions

text: to

text: system

text: we

text: have

text: the

text: following

text: result.

text: Lemma

text: Under

text: the

text: growth

text: condition

text: there

text: exists

text: τ¯

text: such

text: that

text: for

text: τ

text: τ¯

text: and

text: for

text: every

text: k

text: K

text: there

text: exists

text: a

text: solution

text: (uk,

text: ek,

text: zk,

text: pk,

text: ϑk)

text: X

text: to

text: system

text: Furthermore,

text: for

text: any

text: solution

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: (uk,

text: ek,

text: zk,

text: pk,

text: ϑk)

text: the

text: function

text: ϑk

text: complies

text: with

text: the

text: positivity

text: property

text: τ

text: τ

text: τ

text: τ

text: τ

text: τ

text: Proof.

text: The

text: positivity

text: property

text: (note

text: that

text: the

text: constant

text: providing

text: the

text: lower

text: bound

text: for

text: ϑk

text: is

text: independent

text: τ

text: of

text: the

text: truncation

text: parameter

text: M),

text: follows

text: from

text: the

text: analogue

text: of

text: estimate

text: with

text: the

text: same

text: comparison

text: argument.

text: Let

text: us

text: now

text: address

text: the

text: existence

text: of

text: solutions.

text: First

text: of

text: all,

text: we

text: a

text: solution

text: zk

text: to

text: via

text: the

text: minimum

text: problem

text: τ

text: Min

text: dx

text: ν

text: a

text: F

text: (z)

text: z

text: Hs(Ω)

text: τ

text: 2τ

text: τ

text: 2τ

text: s

text: τ

text: τ

text: k

text: Ω

text: Ω

text: C

text: with

text: F

text: (z)

text: a

text: (z,

text: z)

text: β(z)

text: λ

text: zk

text: (z)ek

text: ϑk

text: dx

text: k

text: s

text: W

text: τ

text: τ

text: τ

text: τ

text: Ω

text: With

text: the

text: direct

text: method

text: in

text: the

text: calculus

text: of

text: variations,

text: it

text: is

text: easy

text: to

text: check

text: that

text: whose

text: Euler-Lagrange

text: equation

text: is

text: has

text: a

text: solution

text: zk.

text: τ

text: Let

text: us

text: now

text: brieﬂy

text: address

text: the

text: solvability

text: of

text: the

text: approximate

text: discrete

text: thermoviscoplastic

text: system

text: for

text: ﬁxed

text: k

text: K

text: with

text: zk

text: given.

text: To

text: this

text: end,

text: we

text: reformulate

text: system

text: in

text: the

text: form

text: τ

text: τ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 RICCARDA ROSSI the Karush-Kuhn-Tucker conditions. The entropic formulation originates from the work by E. Feireisl in ﬂuid mechanics [Fei07] and has been ﬁrst adapted to the context of phase transition systems in [FPR09], and later extended to damage models in [RR15]. It is given by an entropy inequality, formally obtained by dividing the heat equation by ϑ and testing the resulting relation by a suﬃciently regular, positive test function (cf. the calculations at the beginning of Section ), combined with a total energy inequality. The weak formulation of the damage ﬂow rule has been ﬁrst proposed in the context of damage modeling in [HK11, HK13]: the subdif- ferential inclusion for damage is replaced by a one-sided variational inequality, with test functions reﬂecting the sign constraint imposed by the dissipation potential R, joint with a (mechanical) energy-dissipation inequality also incorporating contributions from the momentum balance and the plastic ﬂow rule. Clearly, one of the analytical advantages of the entropy inequality for the heat equation, and of the one-sided inequality for the damage ﬂow rule, is that the troublesome quadratic terms on the right-hand sides of (1.1c) and of (1.1e) feature as multiplied by a negative test function, cf. (2.34a) and (2.37) ahead, respectively. This allows for upper semicontinuity arguments in the limit passage in suitable approximations of such inequalities. Instead, the total and mechanical energy inequalities can be obtained by lower semicontinuity techniques. We will also consider a regularized version of system (1.1, 1.2), where the damage ﬂow rule features the additional term A (z˙), modulated by a positive constant ν, and, accordingly, the term νa (z˙, z˙) (with a s s s the bilinear form associated with A ) occurs on the right-hand side of the heat equation. This leads to the s regularized thermoviscoelastoplastic damage system × E(u) = e + p in Ω (0, T), (1.3a) − × ρu¨ divσ = F in Ω (0, T), (1.3b) D C − C E × σ = (z)e˙ + (z)e (z) ϑ in Ω (0, T), (cid:48) (cid:51) − C(cid:48) × ∂R(z˙) + z˙ + νA (z˙) + A (z) + W (z) 1 (z)e : e + ϑ in Ω (0, T), (1.3c) s s 2 (cid:51) × ∂ H(z, ϑ; p˙) + p˙ σ in Ω (0, T), (1.3d) p˙ D ϑ˙ − div(κ(ϑ)∇ϑ) =G + De˙ : e˙ − ϑC(z)E : e˙ × | | − in Ω (0, T), (1.3e) + R(z˙) + z˙ 2 + ν¯a (z˙, z˙) ϑz˙ + H(z, ϑ; p˙) + p˙ : p˙ s | | with ν¯ = ν/ Ω , supplemented with the boundary conditions × × u = w on Γ (0, T), σn = f on Γ (0, T), (1.4a) Dir Neu × × ∇ × ∂ z = 0 on ∂Ω (0, T), ∂ z˙ = 0 on ∂Ω (0, T), κ(ϑ) ϑn = g on ∂Ω (0, T). (1.4b) n n For this regularized system we will be able to show the existence of an enhanced type of solution. It features a conventional weak formulation of the heat equation with suitable test functions, and the formulation of the ∗ damage ﬂow rule as a subdiﬀerential inclusion in Hs(Ω) . To obtain the latter, a key role is played by the ∈ regularizing term A (z˙), ensuring that z˙ Hs(Ω) a.e. in (0, T). Thanks to this feature, it is admissible to test s the subdiﬀerential inclusion rendering (1.3c) by z˙ itself. This is at the core of the validity of the total energy balance. That is why, also in accordance with the nomenclature from [RR15, Ros16], we shall refer to these enhanced solutions as weak energy solutions. We will prove the existence of entropic solutions, see Theorem 2.5, and of weak energy solutions, see Theorem 2.6, to (the Cauchy problems for) systems (1.1, 1.2) and (1.3, 1.4), respectively, by passing to the limit in a time-discretization scheme carefully devised in such a way as to ensure the validity of discrete versions of the entropy and energy inequalities along suitable interpolants of the discrete solutions. One of our standing assumptions will be a suitable growth of the heat conductivity coeﬃcient κ, namely ∼ κ(ϑ) ϑµ for some µ > 1. (1.5) ##LTLine## ##LTLine## ##LTLine##

text: RICCARDA

text: ROSSI

text: the

text: Karush-Kuhn-Tucker

text: conditions.

text: The

text: entropic

text: formulation

text: originates

text: from

text: the

text: work

text: by

text: E.

text: Feireisl

text: in

text: ﬂuid

text: mechanics

text: and

text: has

text: been

text: ﬁrst

text: adapted

text: to

text: the

text: context

text: of

text: phase

text: transition

text: systems

text: in

text: and

text: later

text: extended

text: to

text: damage

text: models

text: in

text: It

text: is

text: given

text: by

text: an

text: entropy

text: inequality,

text: formally

text: obtained

text: by

text: dividing

text: the

text: heat

text: equation

text: by

text: ϑ

text: and

text: testing

text: the

text: resulting

text: relation

text: by

text: a

text: suﬃciently

text: regular,

text: positive

text: test

text: function

text: (cf.

text: the

text: calculations

text: at

text: the

text: beginning

text: of

text: Section

text: combined

text: with

text: a

text: total

text: energy

text: inequality.

text: The

text: weak

text: formulation

text: of

text: the

text: damage

text: ﬂow

text: rule

text: has

text: been

text: ﬁrst

text: proposed

text: in

text: the

text: context

text: of

text: damage

text: modeling

text: in

text: the

text: subdif-

text: ferential

text: inclusion

text: for

text: damage

text: is

text: replaced

text: by

text: a

text: one-sided

text: variational

text: inequality,

text: with

text: test

text: functions

text: reﬂecting

text: the

text: sign

text: constraint

text: imposed

text: by

text: the

text: dissipation

text: potential

text: R,

text: joint

text: with

text: a

text: (mechanical)

text: energy-dissipation

text: inequality

text: also

text: incorporating

text: contributions

text: from

text: the

text: momentum

text: balance

text: and

text: the

text: plastic

text: ﬂow

text: rule.

text: Clearly,

text: one

text: of

text: the

text: analytical

text: advantages

text: of

text: the

text: entropy

text: inequality

text: for

text: the

text: heat

text: equation,

text: and

text: of

text: the

text: one-sided

text: inequality

text: for

text: the

text: damage

text: ﬂow

text: rule,

text: is

text: that

text: the

text: troublesome

text: quadratic

text: terms

text: on

text: the

text: right-hand

text: sides

text: of

text: and

text: of

text: feature

text: as

text: multiplied

text: by

text: a

text: negative

text: test

text: function,

text: cf.

text: and

text: ahead,

text: respectively.

text: This

text: allows

text: for

text: upper

text: semicontinuity

text: arguments

text: in

text: the

text: limit

text: passage

text: in

text: suitable

text: approximations

text: of

text: such

text: inequalities.

text: Instead,

text: the

text: total

text: and

text: mechanical

text: energy

text: inequalities

text: can

text: be

text: obtained

text: by

text: lower

text: semicontinuity

text: techniques.

text: We

text: will

text: also

text: consider

text: a

text: regularized

text: version

text: of

text: system

text: where

text: the

text: damage

text: ﬂow

text: rule

text: features

text: the

text: additional

text: term

text: A

text: (z˙),

text: modulated

text: by

text: a

text: positive

text: constant

text: ν,

text: and,

text: accordingly,

text: the

text: term

text: νa

text: (z˙,

text: z˙)

text: (with

text: a

text: s

text: s

text: s

text: the

text: bilinear

text: form

text: associated

text: with

text: A

text: occurs

text: on

text: the

text: right-hand

text: side

text: of

text: the

text: heat

text: equation.

text: This

text: leads

text: to

text: the

text: s

text: regularized

text: thermoviscoelastoplastic

text: damage

text: system

text: E(u)

text: e

text: p

text: in

text: Ω

text: T),

text: ρu¨

text: divσ

text: F

text: in

text: Ω

text: T),

text: D

text: C

text: C

text: E

text: σ

text: (z)e˙

text: (z)e

text: (z)

text: ϑ

text: in

text: Ω

text: T),

text: ∂R(z˙)

text: z˙

text: νA

text: (z˙)

text: A

text: (z)

text: W

text: (z)

text: (z)e

text: e

text: ϑ

text: in

text: Ω

text: T),

text: s

text: s

text: H(z,

text: ϑ;

text: p˙)

text: p˙

text: σ

text: in

text: Ω

text: T),

text: p˙

text: D

text: ϑ˙

text: div(κ(ϑ)∇ϑ)

text: =G

text: De˙

text: e˙

text: ϑC(z)E

text: e˙

text: in

text: Ω

text: T),

text: R(z˙)

text: z˙

text: ν¯a

text: (z˙,

text: z˙)

text: ϑz˙

text: H(z,

text: ϑ;

text: p˙)

text: p˙

text: p˙

text: s

text: with

text: ν¯

text: ν/

text: Ω

text: supplemented

text: with

text: the

text: boundary

text: conditions

text: u

text: w

text: on

text: Γ

text: T),

text: σn

text: f

text: on

text: Γ

text: T),

text: Dir

text: Neu

text: z

text: on

text: ∂Ω

text: T),

text: z˙

text: on

text: ∂Ω

text: T),

text: κ(ϑ)

text: ϑn

text: g

text: on

text: ∂Ω

text: T).

text: n

text: n

text: For

text: this

text: regularized

text: system

text: we

text: will

text: be

text: able

text: to

text: show

text: the

text: existence

text: of

text: an

text: enhanced

text: type

text: of

text: solution.

text: It

text: features

text: a

text: conventional

text: weak

text: formulation

text: of

text: the

text: heat

text: equation

text: with

text: suitable

text: test

text: functions,

text: and

text: the

text: formulation

text: of

text: the

text: damage

text: ﬂow

text: rule

text: as

text: a

text: subdiﬀerential

text: inclusion

text: in

text: Hs(Ω)

text: To

text: obtain

text: the

text: latter,

text: a

text: key

text: role

text: is

text: played

text: by

text: the

text: regularizing

text: term

text: A

text: (z˙),

text: ensuring

text: that

text: z˙

text: Hs(Ω)

text: a.e.

text: in

text: T).

text: Thanks

text: to

text: this

text: feature,

text: it

text: is

text: admissible

text: to

text: test

text: s

text: the

text: subdiﬀerential

text: inclusion

text: rendering

text: by

text: z˙

text: itself.

text: This

text: is

text: at

text: the

text: core

text: of

text: the

text: validity

text: of

text: the

text: total

text: energy

text: balance.

text: That

text: is

text: why,

text: also

text: in

text: accordance

text: with

text: the

text: nomenclature

text: from

text: we

text: shall

text: refer

text: to

text: these

text: enhanced

text: solutions

text: as

text: weak

text: energy

text: solutions.

text: We

text: will

text: prove

text: the

text: existence

text: of

text: entropic

text: solutions,

text: see

text: Theorem

text: and

text: of

text: weak

text: energy

text: solutions,

text: see

text: Theorem

text: to

text: (the

text: Cauchy

text: problems

text: for)

text: systems

text: and

text: respectively,

text: by

text: passing

text: to

text: the

text: limit

text: in

text: a

text: time-discretization

text: scheme

text: carefully

text: devised

text: in

text: such

text: a

text: way

text: as

text: to

text: ensure

text: the

text: validity

text: of

text: discrete

text: versions

text: of

text: the

text: entropy

text: and

text: energy

text: inequalities

text: along

text: suitable

text: interpolants

text: of

text: the

text: discrete

text: solutions.

text: One

text: of

text: our

text: standing

text: assumptions

text: will

text: be

text: a

text: suitable

text: growth

text: of

text: the

text: heat

text: conductivity

text: coeﬃcient

text: κ,

text: namely

text: κ(ϑ)

text: ϑµ

text: for

text: some

text: µ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 38 RICCARDA ROSSI (cid:12) (cid:12) (cid:16) (cid:17) (cid:12) (cid:12) → This follows fr(cid:12)om (cid:12)the pointwise convergence 1 1 a.e. in Q, combined with the Dominated Convergence ≤ (cid:16) (cid:17) ϑτk ϑ ∇ ∇ Theorem, since ϑτ1k ϑ1¯ by (2.42). Furthermore, since ϑτ1k = |ϑτϑkτ|k2 , combining (2.42) with estimate (4.10o) we infer that the sequence 1 is bounded in L2(0, T; H1(Ω)). All in all, we have ϑτk k 1 1 (cid:42) in L2(0, T; H1(Ω)). (5.14) ϑ ϑ τk Therefore, we can now take the lim sup →∞ of the fourth integral combining convergences (4.1a) for (G ) , k τk k (4.4) for (ϕ ), and (5.1h), (5.1q), (5.1l), (5.1m), (5.1n), (5.1w), (5.10), (5.13), and again resorting to the Ioﬀe τk theorem. Finally, the limit passage in the ﬁfth integral term ensues from (4.1b) for (h ) , (4.4), and (5.14). ∈ ∞ ∩ τk k We thus conclude the validity (2.37), tested by functions ϕ C0([0, T]; W1, (Ω)) H1(0, T; L6/5(Ω)), on the ∈ ∈ interval (s, t) for almost all t (0, T) and almost all s (0, t). With the very same argument as in the proof of [Ros16, Thm. 1], we establish the summability properties ∇ (2.43) for κ(ϑ) log(ϑ). In view of (2.43), the entropy inequality (2.37) makes sense for all positive test ∪ ∞ functions ϕ in H1(0, T; L6/5(Ω)) L (0, T; W1,d+(cid:15)(Ω)) with (cid:15) > 0. Therefore, with a density argument we conclude it for this larger test space. Step 3: preliminary considerations for the damage and plastic ﬂow rules. Convergences (5.1q)–(5.1s), (5.1x), and (5.11) ensure that the functions (ϑ, e, p, ζ) fulﬁll (cid:51) ζ + p˙ σ a.e. in Q. (5.15) D To conclude the plastic ﬂow rule (2.27), it thus remains to show that ∈ ζ ∂ H(x, z, ϑ; p˙) a.e. in Q, (5.16) p˙ (cid:40) which is equivalent (cf. the considerations at the beginning of Sec. ), to ≤ ∈ M × ζ : η H(x, z, ϑ; η) for all η d d ≥ D a.e. in Q, ζ : p˙ H(x, z, ϑ; p˙) (cid:55)→ by the 1-homogeneity(cid:90)o(cid:90)f the functiona(cid:90)l p˙ H(x, z, ϑ; p˙). In fact, we will prove the (equivalent) relations (cid:90)(cid:90) ≤ (cid:90) T ∈ M × ζ:η dxdt H(z(t), ϑ(t); η(t))dt for all η L2(Q; d d), (5.17a) D Q 0 ≥ T ζ:p˙ dxdt H(z(t), ϑ(t); p˙(t))dt, (5.17b) Q 0 featuring the plastic dissipation potential H from (2.13). With this aim, we will pass to the limit in the inequalitie(cid:90)s (cid:90)satisﬁed at level(cid:90)k using the discrete subdiﬀerential inclusion (4.2c), namely (cid:90)(cid:90) ≤ (cid:90)T ∈ M × ζ :ηdxdt H(z (t), ϑ (t); η(t))dt for all η L2(Q; d d), (5.18a) τk τk τk D Q 0 ≥ T ζ :p˙ dxdt H(z (t), ϑ (t); p˙ (t))dt. (5.18b) τk τk τk τk τk Q 0 In order to take the limit of (5.18a), we use conditions (2.11) on the dissipation metric H. The strong convergences (5.1n) for z and (5.1w) for ϑ , combined with the continuity property (2.11b), ensure that for ∈ τk M × τk → every test function η L2(Q; d d) there holds H(z , ϑ ; η) H(z, ϑ; η) almost everywhere in Q. Using that ≤ | | sym τk τk H(z , ϑ ; η) C η a.e. in Q thanks to (2.12a), we conclude via the Dominated Convergence Theorem that τk τk R ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: RICCARDA

text: ROSSI

text: This

text: follows

text: pointwise

text: convergence

text: a.e.

text: in

text: Q,

text: combined

text: with

text: the

text: Dominated

text: Convergence

text: ϑτk

text: ϑ

text: Theorem,

text: since

text: ϑτ1k

text: ϑ1¯

text: by

text: Furthermore,

text: since

text: ϑτ1k

text: combining

text: with

text: estimate

text: we

text: infer

text: that

text: the

text: sequence

text: is

text: bounded

text: in

text: T;

text: All

text: in

text: all,

text: we

text: have

text: ϑτk

text: k

text: in

text: T;

text: ϑ

text: ϑ

text: τk

text: Therefore,

text: we

text: can

text: now

text: take

text: the

text: lim

text: sup

text: of

text: the

text: fourth

text: integral

text: combining

text: convergences

text: for

text: (G

text: k

text: τk

text: k

text: for

text: (ϕ

text: and

text: and

text: again

text: resorting

text: to

text: the

text: Ioﬀe

text: τk

text: theorem.

text: Finally,

text: the

text: limit

text: passage

text: in

text: the

text: ﬁfth

text: integral

text: term

text: ensues

text: from

text: for

text: (h

text: and

text: τk

text: k

text: We

text: thus

text: conclude

text: the

text: validity

text: tested

text: by

text: functions

text: ϕ

text: T];

text: (Ω))

text: T;

text: on

text: the

text: interval

text: (s,

text: t)

text: for

text: almost

text: all

text: t

text: T)

text: and

text: almost

text: all

text: s

text: t).

text: With

text: the

text: very

text: same

text: argument

text: as

text: in

text: the

text: proof

text: of

text: Thm.

text: we

text: establish

text: the

text: summability

text: properties

text: for

text: κ(ϑ)

text: log(ϑ).

text: In

text: view

text: of

text: the

text: entropy

text: inequality

text: makes

text: sense

text: for

text: all

text: positive

text: test

text: functions

text: ϕ

text: in

text: T;

text: L

text: T;

text: with

text: Therefore,

text: with

text: a

text: density

text: argument

text: we

text: conclude

text: it

text: for

text: this

text: larger

text: test

text: space.

text: Step

text: preliminary

text: considerations

text: for

text: the

text: damage

text: and

text: plastic

text: ﬂow

text: rules.

text: Convergences

text: and

text: ensure

text: that

text: the

text: functions

text: (ϑ,

text: e,

text: p,

text: ζ)

text: fulﬁll

text: ζ

text: p˙

text: σ

text: a.e.

text: in

text: Q.

text: D

text: To

text: conclude

text: the

text: plastic

text: ﬂow

text: rule

text: it

text: thus

text: remains

text: to

text: show

text: that

text: ζ

text: H(x,

text: z,

text: ϑ;

text: p˙)

text: a.e.

text: in

text: Q,

text: p˙

text: which

text: is

text: equivalent

text: (cf.

text: the

text: considerations

text: at

text: the

text: beginning

text: of

text: Sec.

text: to

text: M

text: ζ

text: η

text: H(x,

text: z,

text: ϑ;

text: η)

text: for

text: all

text: η

text: d

text: d

text: D

text: a.e.

text: in

text: Q,

text: ζ

text: p˙

text: H(x,

text: z,

text: ϑ;

text: p˙)

text: by

text: the

text: the

text: p˙

text: H(x,

text: z,

text: ϑ;

text: p˙).

text: In

text: fact,

text: we

text: will

text: prove

text: the

text: (equivalent)

text: relations

text: T

text: M

text: ζ:η

text: dxdt

text: H(z(t),

text: ϑ(t);

text: η(t))dt

text: for

text: all

text: η

text: d

text: d),

text: D

text: Q

text: T

text: ζ:p˙

text: dxdt

text: H(z(t),

text: ϑ(t);

text: p˙(t))dt,

text: Q

text: featuring

text: the

text: plastic

text: dissipation

text: potential

text: H

text: from

text: With

text: this

text: aim,

text: we

text: will

text: pass

text: to

text: the

text: limit

text: in

text: the

text: at

text: using

text: the

text: discrete

text: subdiﬀerential

text: inclusion

text: namely

text: M

text: ζ

text: :ηdxdt

text: H(z

text: (t),

text: ϑ

text: (t);

text: η(t))dt

text: for

text: all

text: η

text: d

text: d),

text: τk

text: τk

text: τk

text: D

text: Q

text: T

text: ζ

text: :p˙

text: dxdt

text: H(z

text: (t),

text: ϑ

text: (t);

text: p˙

text: (t))dt.

text: τk

text: τk

text: τk

text: τk

text: τk

text: Q

text: In

text: order

text: to

text: take

text: the

text: limit

text: of

text: we

text: use

text: conditions

text: on

text: the

text: dissipation

text: metric

text: H.

text: The

text: strong

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