Moring, Kristian; Rainer, Rudolf Stability for systems of porous medium type

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Moring, Kristian; Rainer, Rudolf

Stability for systems of porous medium type

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Journal of Mathematical Analysis and Applications

DOI:

10.1016/j.jmaa.2021.125532 Published: 01/02/2022

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Please cite the original version:

Moring, K., & Rainer, R. (2022). Stability for systems of porous medium type. Journal of Mathematical Analysis and Applications, 506(1), [125532]. https://doi.org/10.1016/j.jmaa.2021.125532

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Contents lists available atScienceDirect

Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

Stability for systems of porous medium type

Kristian Moringa, Rudolf Rainerb,∗

a DepartmentofMathematicsandSystemsAnalysis,AaltoUniversity,P.O.Box11100,FI-00076Aalto, Finland

bFachbereichMathematik,UniversitätSalzburg,HellbrunnerStr.34,5020Salzburg,Austria

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received22July2020 Availableonline30July2021 Submittedby A.Lunardi

Keywords:

Porousmediumtypesystems Stability

Weestablishstability propertiesofweaksolutionsfor systemsofporous medium typewithrespecttotheexponentm.Therebywetreatstabilityforthelocalcase as wellasfor Cauchy-Dirichletproblems.Both degenerateandsingularcasesare covered.

©2021TheAuthors.PublishedbyElsevierInc.Thisisanopenaccessarticle undertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Pointofinterestisthestabilityof weaksolutionstoparabolic systems

tu−divA

x, t, u, D(|u|m−1u)

= 0 (1.1)

inacylindricaldomainwithrespect totheexponentm.Aisavectorfieldwhose structuralpropertiesare detailedfurtherdown.Thisgeneraltypeislabelledsystemoftheporousmediumtype,asitcontainsas its principalprototypetheporousmediumequation

tu−Δ(um) = 0.

Theequation isdividedinto two regimes: If0< m<1 one speaksofthesingular oralso fastdiffusion case,whileform>1 onespeaksofthedegenerateorslowdiffusioncase.Bothcaseswillbetreated,although wewillhavearestrictioninthesingularcase.Inparticular, apositive lowerboundformisrequired.This matchesupwith regularityresultsfor theporousmediumequation,as thesamebound appearse.g.in[9]

and [14, §6.21].

* Correspondingauthor.

E-mailaddresses:kristian.moring@aalto.fi(K. Moring),rudolf.rainer@sbg.ac.at(R. Rainer).

https://doi.org/10.1016/j.jmaa.2021.125532

0022-247X/©2021TheAuthors.PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

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Wewillanswerthequestionwhetherweaksolutionsof(1.1) convergetoasolutionofthelimitproblemas theexponentmvaries.Thisensuresthatthesolutionsoftheequationarestableundersmallperturbationsof theparameterm,whichinapplicationsmaybeknownonlyapproximately.Inthefirstpart,localconvergence will be studied.We assumeweakconvergenceof thesequence ofsolutionsinthis caseinorder toidentify the limit.Inthe second part,weinspect aCauchy-Dirichletproblem,where thesolutionsare expected to attain giveninitialandboundaryvalues.

Fortheparabolicp-LaplaceequationthestabilityquestionhasbeentreatedbyKinnunenandParviainen in[21].Twoingredients wereessentialforthe proof:for one,thelateralboundaryof thecylindermust be sufficientlyregular.Furthermore,toovercomethedifficultythatweaksolutions(totheparabolicp-Laplace equation)fordifferentexponentsareindifferentparabolicSobolevspaces,aglobalhigherintegrabilityresult is essential.Somewhat surprisingly, neither of these were needed to complete theproof when considering an equation of the type(1.1). This could stem from the factthat, incontrast to the parabolic p-Laplace equation, thespacesintheporousmediumsetting arefixed,even whentheexponentsdiffer. Eventhough not needed, the higher integrability can still be applied to obtain better convergence properties for the sequence ofsolutionsand theirgradients.

For the proof of the local result, we proceed as follows: By Caccioppoli type estimates we obtain a uniform bound on thenorms ofthe solutionsinareflexive Banachspace, whichinturn impliesthe weak convergence ofasubsequence.To improvetheconvergenceforthesolutionsfromweak tostrong, weusea dual pairingargumentwhichthenallowsustousethecompactnesspropertiesofparabolicSobolevspaces, morespecificallyTheorem3in[27].Toimprovetheconvergenceforthegradients,in[21] theauthorsshowed thatthey form aCauchysequence inorderto avoidtesting with thelimitfunction itself. Inthiscase, we are ableto showitdirectly.

Inthe globalcase,we applythelocal result.It remainsto extendtheobtainedconvergencesfrom local toglobal,whichwedobyapplyingameasuretheoreticargument:Onecanobservethatstrongconvergence inL1orevenpointwisea.e.convergencetogetherwithboundednessinL2impliesstrongconvergenceinLq for allq <2.WeconducttheargumentindetailinLemma5.5 andthenreuse itseveraltimes throughout theproofs.

Weshall giveabriefrecap oftherecenthistoryintheresearchof stabilityquestions.Lindqviststudied stability questions for the stationary p-Laplace equation in [22], already in 1987. Due to the mentioned difficultiesarising fromvarying Sobolevspaces, thestability problemforparabolic p-Laplacianwassettled onlyafter higherintegrabilitywasproven.First,KinnunenandLewisshowedthelocal higherintegrability in[19],whichwasthenextendeduptotheboundarybyParviainen[26] in2009.ThisallowedKinnunenand Parviainen toprovethestability fortheparabolic p-Laplacian[21] oneyearlater.Lukkariand Parviainen studiedsimilarstabilityquestionsfortheparabolicp-Laplaceinthedegeneratecasein[24].Theyalsotook intoaccountmeasuredataattheinitialboundary.Regardingequationsoftheporousmediumtype,Lukkari inspected nonnegative weaksolutionsto the modelequation in [23]. He used thespecific structure of the modelequation,whichisnotavailableinourgeneralsetting.

Further,in[1] thetheoryofnonlinearsemigroupsisappliedtoobtainastabilityresultforaninitial-value problem forequations of theform tu−Δϕ(u)= 0 withanon-linearity ϕ. Byapplying the “doublingof variables” methodofKruzkov,quantitativestabilityestimatesinthesenseofcontinuousdependenciesand errorestimatesareobtainedin[11,12,18].

Additionally, we mention the following border cases: For stability results for the case m → ∞, where the limitproblemissometimestermedthemesaproblem,wereferto [3,2,10].For m→0,where thelimit problem is tu−Δlogu = 0, we refer to [15,16]. Also worth noting is [17], where the limit m 0 is inspected,soconsideringtheveryfastdiffusionequationwithm<0.

Acknowledgments. K. Moring has been supported by the Magnus Ehrnrooth Foundation. R. Rainer has been supportedbytheFWF-ProjectP31956“DoublyNonlinearEvolutionEquations”.

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2. Preliminaries

2.1. Statementof thelocal result

Weconsiderporousmediumsystemsofthetype

tu−divA(x, t, u, Dum) = 0 in ΩT, (2.1) in which ΩT := Ω×(0,T) is a space-time cylinder. Ω is abounded open subset of Rn and T > 0. We considern≥2 andusetheabbreviationum:=|u|m1u.∂tdenotesthetimederivative,whileD=Dxand div = divx denotethe derivativesand thedivergence withrespect to thespatial variable x.For opensets A,B⊂Rn+1,wewrite AB ifAisacompactsubsetof B.

Theassumptions on thevector fieldA: ΩT ×RN×RN n RN n areas follows. We assume thatAis aCarathéodory function, i.e. it is measurable with respect to (x,t)∈ ΩT for all(u,ξ)∈ RN ×RN n and continuous with respect to (u,ξ) for a.e. (x,t)∈ ΩT. Moreover, we assume thatA satisfies the following structuralconditionswith0< ν≤L<∞:

A(x, t, u, ξ)·ξ≥ν|ξ|2,

|A(x, t, u, ξ)| ≤L|ξ|,

(2.2)

for a.e. (x,t)∈ ΩT and any (u,ξ)∈RN×RN n.We also assumethatthe vector fieldis monotonein the sense thatforsomeμ∈(0,),

A(x, t, u, ξ)A(x, t, v, η)

·−η)≥μ|ξ−η|2 (2.3) holds truefor a.e. (x,t)∈ΩT and for any pairs(u,ξ),(v,η)∈ RN×RN n. Wework withweak solutions, whichwedefinenow.

Definition2.1. AssumethatthevectorfieldA: ΩT×RN×RN nRN nsatisfies(2.2) and(2.3).Weidentify ameasurablemapu: ΩT RN intheclass

um∈L2loc

0, T;Wloc1,2(Ω,RN) ,

with additional assumption u∈Lm+1locT,RN) in casem <1,as aweak solution to theporous medium typesystem(2.1) withexponentmifandonlyiftheidentity

¨

ΩT

−u·∂tϕ+A(x, t, u, Dum)·Dϕ

dxdt= 0 (2.4)

holdstrueforanytesting functionϕ∈C0T,RN).

Theassumptionsonϕcanbe weakened.Itsufficesthat

ϕ∈W1,2(0, T;L2(Ω,RN))∩L2(0, T;W01,2(Ω,RN))

andsuppϕΩT when m≥1.Ifm<1,we furtherdemandtϕ∈L1+mmT,RN) to ensurethefiniteness oftheintegraloftheparabolicpartoftheequation.

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Remark2.2.InSection4.1wewillprovethataweaksolutionuaccordingtoDefinition2.1hasarepresen- tativeinclass C((0,T);Lm+1loc (Ω,RN)).

We denote the critical exponent by mc := (n−2)n+2+, where (n2)+ := max{n−2,0}. Let (mi) be a sequence of real numbers in (mc,∞) such that mi −→ m (mc,∞) as i −→ ∞. Let further ui be a weaksolutiontotheEquation (2.1) withexponentmi.Weassumethatthere existsameasurablefunction u: ΩT RN,suchthatasi→ ∞,

umi ium weakly inL2locT,RN). (2.5) Moreover, ifm∈(mc,1),wemakeanadditionalassumption,namely

umi i+1 is bounded inL1locT,RN). (2.6) Thefollowing isourmain resultinthelocal setting.

Theorem 2.3. Let(mi)i∈N bea sequencein (mc,∞)such that mi −→m∈(mc,∞) asi−→ ∞.Let ui be a weak solution of Equation (2.1) with exponent mi in the sense of Definition 2.1, where the vector field A satisfiesthegrowth andmonotonicity conditions (2.2)and(2.3).Furthermore,assume that theassump- tions(2.5)and(2.6)areinforce.Then,forthefunctionufrom(2.5),wehaveum∈L2loc(0,T;Wloc1,2(Ω,RN)) with

umi i i→∞−→ um inL2loc(0, T;Wloc1,2(Ω,RN)).

Moreover, thelimit functionuisaweaksolution totheEquation(2.1)with exponent m.

2.2. Cauchy-Dirichletproblem

Wefurtherconsiderstability foraCauchy-Dirichletproblemoftheform tu−divA(x, t, u, Dum) = 0 in ΩT,

u=g onparΩT, (2.7)

whereparΩT :=

∂Ω×(0,T)

Ω× {0}

istheparabolicboundaryofΩT.Letm= limi→∞mi(mc,∞) as before.Inthefollowingweusetheshorthandnotation

I(u, g) :=Im(u, g) :=m+11

|u|m+1− |g|m+1

gm(u−g).

Whenconsideringexponentsmi insteadofm, wethenwriteIi(ui,g) forImi(ui,g).

Wedefine aweaksolutionto theCauchy-Dirichletproblem (2.7) asfollows.

Definition 2.4.Assume that the vector field A: ΩT ×RN ×RN n RN n satisfies (2.2) and (2.3). Let g: ΩT RN be intheclass

g∈C0

[0, T];Lm+1(Ω,RN)

with gm∈L2

0, T;W1,2(Ω,RN) .

Weidentifyameasurable mapu: ΩT RN intheclass um∈L2

0, T;W1,2(Ω,RN)

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with additional assumption u∈ Lm+1T,RN) if m< 1,as a weak solution to the porousmedium type system(2.7) withexponentmandinitialandboundaryvaluesg ifandonlyifuisaweaksolutionof(2.1) withexponentminthesenseofDefinition2.1anduattainsinitialandboundaryvaluesg inthesensethat (umgm)(·, t)∈W01,2(Ω), for a.e.t∈(0, T), (2.8) and

1 h ˆh

0

ˆ

Ω

I(u, g) dxdt−→0, (2.9)

ash→0.

Again,theassumptionsonϕinDefinition2.4canbeweakened.Itsufficesthatthetestfunctionsatisfies ϕ∈W1,2(0, T;L2(Ω,RN))∩L2(0, T;W01,2(Ω,RN))

and ϕ(0) = ϕ(T) = 0 when m 1. If m < 1, we further demand tϕ L1+mmT,RN) to ensure the finitenessoftheintegraloftheparabolicpartoftheequation.

Remark2.5.Asinthelocalcase,wewillalsoprovethataglobalweaksolutionuwithinitialandboundary datag accordingto Definition2.4hasarepresentativeinclassC([0,T];Lm+1(Ω,RN)).

Remark2.6.Note thatforthe representative u∈C([0,T];Lm+1(Ω,RN)) thecondition(2.9) is equivalent to

u(·,0) =g(·,0) a.e. in Ω.

Thisisadirectconsequenceoftheestimatesin (3.2).

For theboundarydatumg: ΩT RN we suppose thatforsome m < m, β >2mm˜ and γ > 1+m,we have

⎧⎪

⎪⎨

⎪⎪

gm∈Lβ

0, T;W1,β(Ω,RN) , g∈C0

[0, T], Lγ(Ω,RN) ,

tgm∈Lmγ˜T,RN).

(2.10)

Thereasonforchoosingtheseconditionsistwofold.Firstofall,itensuresthatgcanbechosenasinitial and boundary values for all i N, even though the exponents differ. Secondly, it ensures the uniform boundednessoftherighthandsideoftheenergyestimateinLemma6.2.Wewillshowwhytheseconditions areneededinLemma6.1.

Observe thatwe could make stronger butmore simplifiedassumptions, for examplegm C1T) for somem < m, whichwouldensurethattheconditionsabovearesatisfied.

Further,note thatwith these assumptions,the boundaryproblemfor weaksolutionsmight notbewell defined for small i: The exponent mi, possibly being quite larger than m, could exceed the integrability exponent of g. However,we are only interested inconvergence properties, i.e. the tail of the sequence in question,suchthatthisrestrictionisofnoconcerntous.Wemaythusassumethatmiisalreadysufficiently closetom,ensuringexistenceofweaksolutionsandfinitenessoftheintegralsasinLemma6.1foralli∈N.

WewillextendthelocalresultinTheorem 2.3totheboundary:

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Theorem 2.7. Let(mi)i∈N bea sequencein (mc,∞)such that mi −→m∈(mc,∞) asi−→ ∞.Let ui be a weak solution of Equation (2.7) with exponent mi in the sense of Definition 2.4, where the vector field A satisfies the growth and monotonicity conditions (2.2) and (2.3) and the boundary datum g fulfils the conditions (2.10).

Then thereexists asubsequence, stilldenoted by(umi i),and ameasurablemap u: ΩT RN,such that um∈L2(0,T;W1,2(Ω,RN))with

umi ii→∞−→ um inL2(0, T;W1,2(Ω,RN)). (2.11) Moreover, the limitfunction uisa weaksolution tothe Equation (2.7), attaining theinitialand boundary valuesg in thesense of(2.8)and (2.9).

Remark2.8.InthespecialcaseA(x,t,u,Dum)=Dum thelimitfunctionuinTheorem 2.7isuniqueand theconvergencein (2.11) holdsforthewhole(original)sequence,notonlyonthelevelofsubsequences.

3. Auxiliaryresults

WefirstrecallacompactnessresultbySimon [27]. Letus denote(τhf)(t):=f(t+h) forh>0.

Lemma 3.1. Assume that there is a compact embedding of Banach spaces X B. Let F Lp(0,T;B), where 1≤p≤ ∞.Inaddition, suppose that

F is bounded in L1loc(0, T;X) and

τhf−fLp(0,Th;B)0ash→0, uniformly forf ∈F.

Then F isrelativelycompactinLp(0,T;B).

Furthertherewill betheneedforsomealgebraic inequalities,alsoregardingtheboundarytermI(u,g).

It is often useful to see that it is comparable to um+12 gm+12 . We take the following Lemmas from [9, Lemma3.2,3.3] andfrom[7,Lemma2.3].

Lemma 3.2.Forallα >1thereexistsc=c(α)>0such thatforalla,b∈RN thereholds

|b−a|α≤c|bαaα|. (3.1)

Lemma 3.3.Forallm>0there existsc=c(m)>0suchthatforallu,g∈RN onehas

1

cI(u, g)≤ |um+12 gm+12 |2≤c I(u, g),

1

c|umgm| ≤

|u|+|g|m1

|u−g| ≤c|umgm|.

(3.2)

Further,

I(u, g)≤c|umgm|(1+m)/m form >1,

I(u, g)≤c(|u|+|g|)m−1|u−g|2≤c|umgm||u−g| form >0. (3.3)

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3.1. Sobolev-Gagliardo-Nirenberginequalities

NextwestatetheparabolicSobolevinequalityfrom [13,Prop.I.3.1] andalocalvariantofit.Thefollowing inequality will allow us to gain higher integrability for the functions ui and further, better convergence properties.

Lemma3.4. LetB(xo,)⊂Ω and0< t1< t2< T.If v∈L

t1, t2;Lr(B(xo, ))

∩Lp

t1, t2;W1,p(B(xo, )) forp∈(1,)andr∈[1,),thereexistsa constantc=c(n,p,r)suchthat

t2

ˆ

t1

ˆ

B(xo,)

|v|dxdt

≤c

t2

ˆ

t1

ˆ

B(xo,)

v

p+|Dv|p

dxdt

⎜⎝ sup

t∈(t1,t2)

ˆ

B(xo,)×{t}

|v|rdx

⎟⎠

p n

,

where=pn+rn . Lemma3.5. If

v∈L

0, T;Lr(Ω)

∩Lp

0, T;W01,p(Ω) forp∈(1,)andr∈[1,),thereexistsa constantc=c(n,p,r,Ω)suchthat

¨

ΩT

|v|dxdt

≤c ¨

ΩT

|Dv|pdxdt

sup

t∈(0,T)

ˆ

Ω×{t}

|v|rdx pn

,

where=pn+rn .

3.2. Mollification intime

In order to be able to prove useful estimates for weak solutions of Equation (2.1), we exploit time mollificationofthefollowingtype.

Definition3.6. Forv∈L1T,RN) andh>0,define amollificationintimeby

vh(x, t) := 1 h ˆt

0

es−th v(x, s)ds.

Similarly,definethereversetimemollificationintimeby

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v¯h(x, t) := 1 h

ˆT

t

et−sh v(x, s)ds.

WecollectsomeusefulpropertiesofthemollificationinthefollowingLemma,see [20,Lemma2.9] and [5, AppendixB].Analogousstatementsholdtrueforthereversetimemollification.

Lemma 3.7.Letv andvh beasabove. Thenthefollowingpropertieshold:

(i) Ifv∈LpT,RN) forsome p≥1,then

vhLpT,RN)≤ vLpT,RN), and vh→v inLpT,RN)ash→0.

(ii) Letv∈Lp(0,T;W1,p(Ω,RN))forsome p≥1.Then

vhLp(0,T;W1,p(Ω,RN))≤ vLp(0,T;W1,p(Ω,RN))

and vh→v inLp(0,T;W1,p(Ω,RN))ash→0.

(iii) If v∈Lp(0,T;W01,p(Ω,RN)),thenvh∈Lp(0,T;W01,p(Ω,RN)).

(iv) Ifv∈Lp(0,T;Lp(Ω,RN)),thenvh∈C([0,T];Lp(Ω,RN)).

(v)The weaktimederivative tvh existsin ΩT and isgivenbyformula

tvh= 1

h(v−vh), whereas forthereversemollification wehave

tv¯h= 1

h(v¯h−v).

Remark3.8.Observethatsimilar propertieshold alsoformollificationdefinedas

vh(x, t) :=ehtvo+1 h

ˆt

0

es−th v(x, s)ds

forvo∈L1(Ω,RN).Oneadvantageofthis formulaisthatwecancompute

tvh(x, t) = 1 h ˆt

0

es−th sv(x, s)ds

undersuitableassumptions,see [5,AppendixB,LemmaB.3].FromthistogetherwithLemma3.7onecan deduceconvergencesforthetimederivativeaswell providedthatitexistsinanappropriatespace.Wewill exploitthisintheglobalcase.

4. Continuityintimeandmollifiedformulation

In this section we will provethat weaksolutions haverepresentatives thatare continuous in time, ac- cording toRemarks 2.2and2.5.

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4.1. Continuityin timeforlocal problem

In order to prove the continuity in time of weak solution according to Definition 2.1 as noted in Re- mark2.2,wewillusethefollowingLemma,whichcanbe foundin[29,Lemma2.12],[28,Lemma3.4],[31, Lemma3.8].Weincludetheproofforthecontinuityforcompleteness,where wetaketheapproachof[31].

Observethatinthelocalcasewewillusemollificationsforumdefinedby umh(x, t) := 1

h ˆt

τ1

eshtum(x, s)ds,

for t τ1, in which τ1 >0 is fixed. This is due to the factthat um is onlylocally integrable. Similarly, definethereversetime mollificationintimeby

umh¯(x, t) := 1 h

τ2

ˆ

t

et−sh um(x, s)ds,

fort≤τ2< T.

Lemma4.1. LetV betheset ofallv∈C0

(0,T),L1+mloc (Ω,RN)

suchthat vm∈L2loc

0, T;Wloc1,2(Ω,RN)

and tvm∈L

m+1 m

locT,RN).

Then, foraweaksolution uaccording toDefinition2.1,

¨

ΩT

tζI(u, v)dxdt=

¨

ΩT

ζ∂tvm·(u−v) +A(x, t, u, Dum)·D

ζ(umvm) dxdt

holdstrue forallv∈ V, ζ∈C0T).

Proof. Wewilltesttheweakequation(2.4) foruwithϕ=ζ(vmumh) withsomesmallfixedτ1>0 in themollifier.Wefirstinspect theparabolic partof theequation:

¨

ΩT

u·∂tϕdxdt=

¨

ΩT

tζu·(vmumh) +ζu·∂t(vmumh)dxdt

=

¨

ΩT

tζu·(vmumh) +ζu·∂tvmdxdt

+

¨

ΩT

−ζum1/mh ·∂tumh+ζ(um1/mh −u)·∂tumh)dxdt

¨

ΩT

tζu·(vmumh) +ζu·∂tvmdxdt

+ m

m+ 1

¨

ΩT

tζ|umh|(m+1)/mdxdt

−→h↓0 ¨

ΩT

tζ(u·vm− |u|m+1) +ζu·∂tvm+ m

m+ 1tζ|u|m+1dxdt

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= ˆ

ΩT

ζ∂tvm·(u−v)−∂tζI(u, v)dxdt.

By comparing tothe divergence partofthe equation, weobtain thedirection‘’ ofthe claim.The other directioncanbeshownbytaking thereverse timemollificationinthetestfunctionϕ.

Lemma 4.2. The weak solution u according to Definition 2.1 has a representative that belongs to class C0((0,T);L1+mloc (Ω,RN)).

Proof. LetK Ω andη ∈C0(Ω) with η = 1 onK, |Dη|≤C(K). Letτ (0,12T) andε>0 suchthat τ+ε< 12T. Takeψ∈C([0,]) withψ= 1 on[0,12T],ψ= 0 on[34T,T] and |≤8T−1.Furtherdefine

ξ(t) =ξε,τ(t) :=

⎧⎪

⎪⎨

⎪⎪

0, t < τ,

1

ε(t−τ), t∈[τ, τ+ε], 1, t > τ+ε.

WeapplyLemma4.1withζ=ηψξ,wh=um1/m¯

h ,andτ2(34T,T) inthemollifier.Observethatwh∈ V definedinLemma4.1.Especiallywh∈C0([t1,34T];L1+mloc (Ω,RN)) foranyh>0 andt1>0 canbe seenas follows. Ifm>1 thisisadirectconsequenceof (3.1) andthefactthat um¯h∈C0([t1,34T];L

1+m m

loc (Ω,RN)) together with assumption u Lm+1locT,RN) and Lemma 3.7 (iv). In the case m < 1, we make use of second inequality in (3.2) andHölder’s inequality.Fixing0< t1≤τ and definingE := suppη×(t1,34T), this yields

1 ε

¨

Ω×(τ,τ+ε)

ηI(u, wh)dxdt=

¨

ΩT

η∂tξI(u, wh)dxdt

=

¨

ΩT

ζ∂twmh(u−wh) +A(x, t, u, Dum)·D

ζ(umwmh)

−I(u, wh)ηξ∂tψ dxdt

≤C

¨

E

|Dum|

|Dum−Dwmh|+|Dη||umwmh| + 8

TI(u, wh)

dxdt

≤C

¨

E

|Dum−Dwmh|2+|umwmh|2+I(u, wh) dxdt

byapplyingHölder’sinequalityandusingtheassumption|Dum|∈L2locT).Bytheconvergenceproperties of thetime mollification,thefirstand secondtermontherighthandside vanishash↓0.

For thelast term, we mustdistinguish between two cases.If m≥1, wecanapply estimate I(u,wh) c|umwmh|(1+m)/mby(3.3),whichvanishesash↓0.Inthesingularcasem<1,by(3.3) wecompute

¨

E

I(u, wn)dxdt

≤c

¨

E

|umwmh||u−wh|dxdt

≤c ¨

E

|umwmh|1+mm dxdt

1+mm ¨

E

|u−wh|1+mdxdt 1+m1

.

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Again,since um∈L

1+m m

locT,RN) andu∈L1+mlocT,RN),these integralsvanish as h↓0.Notice thatin thepreviousestimates,therighthandside doesnotdependonτ.Thus,

limh0 sup

τ∈(t1,12T)

ˆ

K×{τ}

I(u, wh)dx= 0. (4.1)

Forthecasem≥1,weusetheinequalities(3.2) and(3.1) andseethat sup

τ(t1,12T)

ˆ

K×{τ}

|wh−u|m+1dx≤C sup

τ(t1,12T)

ˆ

K×{τ}

I(u, wh)dx0.

Inthecasem<1 weuseinequalities (3.2) and (3.3) toobtain ˆ

K×{τ}

|wh−u|m+1dx

≤c ˆ

K×{τ}

(|wh|+|u|)

(1m)(1+m)

2 |w

m+1 2

h um+12 |m+1dx

≤c

⎜⎝ ˆ

K×{τ}

(|wh|+|u|)m+1dx

⎟⎠

1−m

2

⎜⎝ ˆ

K×{τ}

|w

m+1 2

h um+12 |2dx

⎟⎠

m+1 2

≤c

⎜⎝ ˆ

K×{τ}

|wh|m+1dx+ ˆ

K×{τ}

|u|m+1dx

⎟⎠

1−m

2

⎜⎝ ˆ

K×{τ}

I(u, wh)dx

⎟⎠

m+1 2

for any τ (t1,12T). By taking supremum over τ and passing to the limit h 0 the right hand side converges tozeroby(4.1).Observethatthefirsttermof therighthandsidestaysbounded sincewehave thatu∈Lloc(0,T;Lm+1loc (Ω,RN)).ThisistruebyLemma5.1andsinceu∈Lm+1loc (Ω,RN) bydefinition.In addition,from thepropertiesofthemollificationitfollowsthat

um1/m¯h

L(t1,34T;Lm+1(K,RN)) ≤ uL(t1,34T;Lm+1(K,RN)),

whichimplies thatwh isuniformly bounded inL(t1,34T;Lm+1loc (Ω,RN)).Wecanthus come to thesame conclusionasinthecasem≥1.Intotal,wehaveshownthat

wh=um1/m¯h −→u inL(t1,12T;L1+m(K,RN))

as h 0. Observe that K Ω and t1 > 0 were arbitrary. Since wh is continuous map from (t1,t2) to Lm+1loc (Ω,RN), we conclude that u C0((t1,12T],L1+mloc (Ω,RN)) as uniform limit of continuous functions from(t1,t2) toLm+1loc (Ω,RN),afterpossibleredefinitioninasetofmeasurezero.Toobtaintheresultonthe fulltimeinterval,onecaneithermodifycut-offfunctionsξandψsothatτ canbearbitrarilyclosetoT on acompactsubintervalof(0,T),orapplythesameargumentswithusualtimemollificationswh=um1/mh andreversedcut-offfunctions,assuggestedin[31, Lemma3.9].Thiscompletestheproof.

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4.2. Continuityintimein globalcase

Lemma 4.3. The weak solution u according to Definition 2.4 has a representative that belongs to class C0([0,T];L1+m(Ω,RN)).

Proof. Let τ (0,12T), ε >0 such thatτ +ε < 12T. Define ζ =ψξ with ψ,ξ as inLemma 4.2. We test theweakformulation(2.4) foruagainstthetest functionϕ=ζ(umλ¯umhgm¯λ+gmh) fortwo different mollificationparametersλ>0,h>0.Now themollificationsaredefinedas

umh(x, t) :=ehtgm(x,0) + 1 h

ˆt

0

eshtum(x, s)ds,

gmh(x, t) :=ehtgm(x,0) + 1 h

ˆt

0

es−th gm(x, s)ds,

and

umλ¯(x, t) :=et−Tλ gm(x, T) +1 λ

ˆT

t

et−sλ um(x, s)ds,

gmλ¯(x, t) :=et−Tλ gm(x, T) +1 λ

ˆT

t

et−sλ gm(x, s)ds.

Notice thatϕ(·,t)∈W01,2(Ω,RN) fora.e.t∈(0,T).Fortheparabolic partoftheequation wehave

¨

ΩT

u·∂tϕdxdt=

¨

ΩT

tζu·(umλ¯umh) +ζu·∂t(um¯λumh)dxdt

¨

ΩT

tζu·(gm¯λgmh) +ζu·∂t(gmλ¯gmh)dxdt.

The firstlinecanbeestimated as inLemma4.1, whileinthesecondline onecanimmediately passtothe limith↓0.Thuswe obtain

¨

ΩT

tξI(u,um1/mλ¯ )dxdt

¨

ΩT

ζ∂tum¯λ·(uum1/m¯λ ) +ζA(x, t, u, Dum)·D

umumλ¯)dxdt

+

¨

ΩT

tζu·(gmgmλ¯) +ζu·∂t(gmgmλ¯)dxdt

¨

ΩT

ζA(x, t, u, Dum)·D

gmgmλ¯)dxdt

¨

ΩT

tψI(u,um1/m¯λ )dxdt

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=: I + II + III + IV.

Observethatthetermonthelefthandsideisnon-negative.ThefirstintegralItogetherwithIVwillvanish due to the same reasoning as in Lemma4.2. The third integralIII vanishes as λ 0 due to the growth condition(2.2),theassumption|Dum|∈L2T) andLemma3.7(ii).ForII,weestimate

II 1

ε

¨

Ω×(τ,τ+ε)

|u|m+1dxdt 1+m1

1 ε

¨

Ω×(τ,τ+ε)

|gm¯λgm|1+mm dxdt 1+mm

+ 8 T

¨

ΩT

|u|m+1dxdt

m+11 ¨

ΩT

|gmλ¯gm|1+mm 1+mm

+ ¨

ΩT

|u|m+1dxdt

m+11 ¨

ΩT

|∂tgm¯λ−∂tgm|m+1m 1+mm

Afterpassingtothelimitε0 the firsttermequals ˆ

Ω×{τ}

|u|m+1dx

1+m1 ˆ

Ω×{τ}

|gm¯λgm|1+mm dx 1+mm

for a.e. τ (0,12T). ByCaccioppoli inequalityin Lemma 6.2 (see also Remark6.3) the assumption u∈ L1+mT) andassumptionsforgthefirstintegralisuniformlyboundedinτ,andthesecondintegralvanishes asλ↓0 bypropertiesofmollification.Usingu∈L1+mT,RN),aswellasgm,∂tgm∈L(1+m)/mT,RN), itfollowsthatIIvanishesasλ↓0.Fortheremainingterms,wecanusethesameargumentsasinLemma4.2 toobtain

limλ↓0 sup

τ(0,12T)

ˆ

Ω

|u−um1/mλ¯ |m+1dx= 0.

ThisprovesthatuisauniformlimitoffunctionsinC0([0,12T];L1+m(Ω,RN)),whichimpliestheexistence of continuous representative u∈C0([0,12T];Lm+1(Ω,RN)). Forthe interval [12T,T], we mayuse reversed mollificationsandcut-off functionsasmentioned attheend oftheproof ofLemma4.2.Finally,we obtain thatu∈C0([0,T];L1+m(Ω,RN)).

4.3. Mollifiedformulation

Inordertoproveusefulestimatesweusethefollowingmollifiedformulationof (2.4),whichcanbederived similarlyas in [4].Weincludetheproofforexpositorypurposes.Inthelocalcaseweusemollifications

uh(x, t) := 1 h ˆt

τ1

eshtu(x, s)ds,

and

ϕh¯(x, t) := 1 h ˆT

t

et−sh ϕ(x, s)ds,

forsolutionuand testfunctionϕ.

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