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Grunert,HoldenResMathSci(2022)9:19
/s40687-022-00314-6
RESEARCH
Uniquenessofconservativesolutionsfor
theHunter–Saxtonequation
KatrinGrunert∗andHelgeHolden
*Correspondence:
katrin.******@;Abstract
x∞
–Saxtonequationut+uux=4−∞dμ(t,z)−xdμ(t,z)
DepartmentofMathematicalandμt+(uμ)x=0hasaunique,global,weak,andconservativesolution(u,μ)ofthe
Sciences,NTNUNorwegianCauchyproblemontheline.
UniversityofScienceand
Technology,No-7491,Keywords:Hunter–Saxtonequation,Uniqueness,Conservativesolutions
Trondheim,Norway
WeacknowledgesupportbytheMathematicsSubjectClassification:Primary:35A02,35L45;Secondary:35B60
grantsWavesandNonlinear
Phenomena(WaNP)andWave
PhenomenaandStability—a
ShockingCombination(WaPheS)1Introduction
fromtheResearchCouncilofTheHunter–Saxton(HS)equation[14]reads
Norway
x∞
1
ut+uux=dμ(t,z)−dμ(t,z),
4−∞x
μt+(uμ)x=0.
HereuisanH1(R)functionforeachtimet,andμ(t)isanonnegativeRadonmeasure.
Derivedinthecontextofmodelingliquidcrystals,theHSequationhasturnedoutto
,forexample,ageometricinterpretation
[17–21],convergentnumericalmethods[9,12],andastochasticversion[11],inaddition
tonumerousextensionsandgeneralizations[24,25],
comprehensivestudyappearedin[15,16].WhiletheHSequationwasoriginallyderived
ondifferentialform
12
(ut+uux)x=ux,
2
where,inthecaseofsmoothfunctions,μequaltou2willautomaticallysatisfythesecond
x
equation,
integratethisequation,say
x
12
ut+uux=ux(t,y)dy,
20
forwhichtheuniquenessofconservativesolutionsonthehalf-linehasbeenestablishedin
[5],,itisessentialtointroduceameasure
μ(t)onthelinesuchthatforalmostalltimesdμ=dμac=
x
dμ=
x
123©TheAuthor(s),whichpermitsuse,
sharing,adaptation,distributionandreproductioninanymediumorformat,aslongasyougiveappropriatecredittotheoriginal
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thirdpartymaterialinthisarticleareincludedinthearticle’sCreativeCommonslicence,unlessindicatedotherwiseinacreditline
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19Page2of54Grunert,HoldenResMathSci(2022)9:19
toanalyzethissituationindetailandrestoreuniquenessbycarefullyselectingparticular
uniquenessresult():
Foranyinitialdata(u0,μ0)∈DtheHunter–Saxtonequationhasauniqueglobalconser-
vativeweaksolution(u,μ)∈.
Inthecaseoftheso-calleddissipativesolutions,whereenergyisremovedexactlyatthe
timeswhenthemeasureμceasestobeabsolutelycontinuous,theuniquenessquestion
hasbeenaddressedin[7]byshowingtheuniquenessofthecharacteristics.
Theproblemathandcanbeillustratedbythefollowingexplicitexample[6].Consider
thetrivialcaseu0=0,whichclearlyhasu(t,x)=,ascanbe
easilyverified,also
α2xα
u(t,x)=−tI(−∞,−α8t2)(x)+I(−α8t2,α8t2)(x)+tI(α8t2,∞)(x)()
4t4
isasolutionforanyα≥0,withμ(0)=αδ0anddμ(t)=4t−2I22(x)dxfor
(−αt/8,αt/8)
t=(characteristic),theinitialvalue
problemisnotwell-posedwithoutfurtherconstraints.
Furthermore,itturnsoutthatthesolutionuoftheHSequationmaydevelopsingularities
infinitetimeinthefollowingsense:Unlesstheinitialdataaremonotonouslyincreasing,
wefind
inf(u)→−∞ast↑t∗=2/sup(−u ).()
x0
Pastwavebreakingthereareatleasttwodifferentclassesofsolutions,denotedconser-
vative(energyisconserved)anddissipative(whereenergyisremovedlocally)solutions,
respectively,andthisdichotomyisthesourceoftheinterestingbehaviorofsolutionsof
theequation(butseealso[8,10]).Wewillinthispaperconsidertheso-calledconservative
casewheretheassociatedenergyispreserved.
ThenaturalapproachtosolvetheHSequationisbytheuseofcharacteristics,.,to
solvetheequation
yˇt(t,ξ)=u(t,yˇ(t,ξ)),yˇ(0,ξ)=yˇ0(ξ).()
However,inthiscasethefunctionu=u(t,x)willingeneralonlybeHölderandnot
,wecannotexpectuniqueness
,itispreciselyinthecasewhereuniquenessfails
[3,5,7,26–28].Wewillreformulatethe
HSequationinnewvariables,theaimbeingtoidentifyvariableswherethesingularities
disappear.
RewritingtheHSequation,usingcharacteristics,yieldsalinearsystemofdifferential
equations[4],
yˇt(t,ξ)=Uˇ(t,ξ),
ˇ1ˇ1
Ut(t,ξ)=(H(t,ξ)−C),
22
Hˇt(t,ξ)=0,():.
Grunert,HoldenResMathSci(2022)9:19Page3of5419
yˇ(t,ξ)2
whereC=μ(0,R)=μ(t,R).HereUˇ(t,ξ)=u(t,yˇ(t,ξ))andHˇ(t,ξ)=−∞ux(t,x)dx.
Thissystemdescribesweak,conservativesolutionsandcanbeintegratedtoyield
t1
yˇ(t,ξ)=yˇ(0,ξ)+tUˇ(0,ξ)+(Hˇ(0,ξ)−C),
42
ˇˇtˇ1
U(t,ξ)=U(0,ξ)+(H(0,ξ)−C),
22
Hˇ(t,ξ)=Hˇ(0,ξ).
Herewemayrecover(u,μ)fromu(t,x)=Uˇ(t,ξ)forsomeξsuchthatx=yˇ(t,ξ)and
μ=yˇ#(Hˇξ),ithasbeenshownin[4]thatgivenanyinitialdata(u0,μ0)∈D
thereexistsatleastoneconservativesolutionandthissolutionsatisfies().Ontheother
hand,
questioncanalsoberephrasedas:Doallconservativesolutionssatisfy()?
In[6],weintroducedanewsetofcoordinates,whichallowedus,incontrastto[4],to
constructaLipschitzmetricd,
systemofdifferentialequations,whichhasbeenderivedusingpseudo-inversesandthe
system(),issurprisinglysimple,butforcedustoimposeanadditionalfirstmoment
1
condition,R(1+|x|)dμ0<∞,,dsatisfies
12
d((u1,μ1)(t),(u2,μ2)(t))≤1+t+td((u1,μ1)(0),(u2,μ2)(0))
8
foranytwoweak,conservativesolutions(ui,μi)∈D,whichsatisfytheadditionalcon-
ditionR(1+|x|)dμi,0<∞.(t,η)=
sup{x|μ(t,(−∞,x))<η}andU(t,η)=u(t,χ(t,η))andintroduceˆχ(t,η)=χ(t,Cη)and
Uˆ(t,η)=U(t,Cη)whereC=μ(t,R).Thenwedefine
d((u,μ)(t),(u,μ)(t))=
Uˆ(t)−Uˆ(t)
+
χˆ(t)−χˆ(t)
+|C−C|.
1122
12
∞12L112
L
However,acloserlookrevealsthatoneexplicitlyassociatestoanyinitialdatatheweak
conservativesolutioncomputedusing().Thus,thequestionifallweakconservative
solutionssatisfy()isneveraddressed.
Furthermore,tostudystabilityquestionsforconservativesolutionsthecoordinatesfrom
[6]seemtobefavorable,
fromthefactthatforeacht∈R,thefunctionF(t,x)=μ(t,(−∞,x)),whereμdenotes
apositive,finiteRadonmeasure,
means,inparticular,thatitsspatialinverseχ(t,η)
increasingfunctionswithpossiblejumpscanleadtothesameproblemsasforconservation
?DotheysatisfysomekindofRankine–
Hugoniotconditionordotheybehavemorelikerarefactionwaves?In[6]thisissue
hasbeenresolvedbyusingthesystem()toshowthatanyjumppreservesposition
,theassociatedsystemfor(χ(t,η),U(t,η))cannotbetreatedusingthe
classicalODEtheory,,
thesenewvariableswouldnotsimplifythestudyofuniquenessquestions.
1Condition()in[6],:.
19Page4of54Grunert,HoldenResMathSci(2022)9:19
Givenaconservativesolution(u,μ),definethequantities
y(t,ξ)=sup{x|x+μ(t,(−∞,x))<ξ},
U(t,ξ)=u(t,y(t,ξ)),
H˜(t,ξ)=ξ−y(t,ξ).
Thenonecanderive,,thatthesequantitiessatisfy
yt(t,ξ)+Uyξ(t,ξ)=U(t,ξ),
H˜t(t,ξ)+UH˜ξ(t,ξ)=0,
1˜1
Ut(t,ξ)+UUξ(t,ξ)=H(t,ξ)−C.
22
Incontrasttou(t,x),thefunctionU(t,ξ)isLipschitzcontinuous,andhence,theabove
systemcanbesolveduniquelyusingthemethodofcharacteristics,whichissufficientto
,itcanbeshownthatby
applyingthemethodofcharacteristicstheabovesystemturnsinto(),.
Althoughtheuniquenessquestionissuccessfullyaddressed,theabovesystemhasone
maindrawback:Thedefinitionofthefunctiony(t,ξ)
hand,theabovesystemcanbeusedtofindotherequivalentformulationsoftheHunter–
Saxtonequation,whichmightbeadvantageousforaddressing,.,stabilityquestions.
Asanillustration,wehereintroduceanovelsetofcoordinates,whichcanbestudied
onitsown,withoutrelyingonspecialpropertiesofsolutionsto()andwhichavoids
ideaistointroduceanauxiliarymeasureν,suchthatG(t,x)=ν(t,(−∞,x))isstrictly
increasingforeacht∈,definetheauxiliaryfunction(thepowerntobe
fixedlater)
1
p(t,x)=2ndμ(t,y),
R(1+(x−y))
whichwillbeasmoothfunctionforallRadonmeasuresμandlet
χ(t,η)=sup{x|ν(t,(−∞,x))<η}
x
=sup{x|p(t,y)dy+μ(t,(−∞,x))<η},
−∞
U(t,η)=u(t,χ(t,η)),
P(t,η)=p(t,χ(t,η)).
Provided(u,μ)isaweak,conservativesolutionoftheHSequation,whichsatisfies
anadditionalmomentcondition,see(),weshow,,thatthetriplet
(χ(t,η),U(t,η),P(t,η))satisfies
χt+hχη=U,()
η
11
Ut+hUη=η−Pχη(t,η˜)d˜η−C,()
204
Pt+hPη=R():.
Grunert,HoldenResMathSci(2022)9:19Page5of5419
where
B+C
h(t,η)=UP(t,η)−U(t,η˜)K(χ(t,η)−χ(t,η˜))1−Pχη(t,η˜)d˜η,()
0
B+C
R(t,η)=−U(t,η˜)K(χ(t,η)−χ(t,η˜))1−Pχη(t,η˜)d˜η
0
B+C
+U(t,η)K(χ(t,η)−χ(t,η˜))1−Pχη(t,η˜)d˜η.()
0
Inparticular,h(t,η),so
thattheabovesystemhasauniquesolutionandcanbesolvedbyapplyingthemethodof
satisfyanadditionalmomentcondition,.
2Background
Inthissection,weintroducetheconceptofweakconservativesolutionsfortheHunter–
,weshowthatthereindeedexistsatleastoneweakconserva-
∞todenotesmoothfunctionswith
c
compactsupportandC∞todenotesmoothfunctionsthatvanishatinfinity.
0
Asastartingpoint,weintroducethespacesweworkin.
E={f∈L∞(R)|f ∈L2(R)}()
equippedwiththenorm
f
=
f
∞+
f
2.
ELL
Furthermore,let
H1(R)=H1(R)×RandH1(R)=H1(R)×R2.
12
WriteRasR=(−∞,1)∪(−1,∞)andconsiderthecorrespondingpartitionofunityψ+
andψ−,.,ψ+andψ−belongtoC∞(R),ψ−+ψ+≡1,0≤ψ±≤1,supp(ψ−)⊂
(−∞,1),andsupp(ψ+)⊂(−1,∞).Furthermore,introducethelinearmappingR1from
H1(R)toEdefinedas
1
(f,a¯)→f=f¯+aψ+,
andthelinearmappingR2fromH1(R)toEdefinedas
2
(f,a,b¯)→f=f¯+aψ++bψ−.
ThemappingsR1andR2arelinear,continuous,
andE2,theimagesofH1(R)andH1(R)byR1andR2,respectively,.,
12
E1=R1(H1(R))andE2=R2(H1(R)).()
12
Thecorrespondingnormsaregivenby:
2
21/2
f
=
f¯+aψ+
=
f¯
2+
f¯
2+a2
E1E1LL
and
2
21/2
f
=
f¯+aψ++bψ−
=
f¯
+
f¯
+a2+b2.
E2E2L2L2:.
19Page6of54Grunert,HoldenResMathSci(2022)9:19
NotethatthemappingsR1andR2arealsowell-definedforall(f,a¯)∈L2(R)=L2(R)×R
1
and(f,a,b¯)∈L2(R)=L2(R)×,let
2
E0=R1(L2(R))andE0=R1(L2(R))
1122
equippedwiththenorms
¯+
¯
221/2
f
0=
f+aψ
0=
f
2+a
E1E1L
and
+−
2221/2
f
0=
f¯+aψ+bψ
0=
f¯
2+a+b,
E2E2L
respectively.
Withthesespacesinmind,wecandefinenexttheadmissiblesetofinitialdata.
(Euleriancoordinates)ThespaceDconsistsofallpairs(u,μ)suchthat
•u∈E2,
•μ∈M+(R),