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P.L. Sachdev Ch. Srinivasa Rao
Large Time Asymptotics
for Solutions of Nonlinear
Partial Differential Equations
123
P.L.Sachdev Ch.SrinivasaRao
(Deceased) DepartmentofMathematics
IndianInstituteofTechnologyMadras
SardarPatelRoad
Chennai-600036
India
[email protected]
[email protected]
ISSN1439-7382
ISBN978-0-387-87808-9 e-ISBN978-0-387-87809-6
DOI10.1007/978-0-387-87809-6
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2009937490
MathematicsSubjectClassification(2000):35Kxx,35Lxx,35Qxx,76xx
(cid:2)c SpringerScience+BusinessMedia,LLC2010
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Preface
The present authors have been engaged in the study – both analytical and numer-
ical – of large time asymptotic behaviour of nonlinear partial differential equa-
tions (PDEs) for many years. They had considerable interaction on this subject
with Professor B. Enflo, KTH, Stockholm and Professor R. E. Grundy, University
of St.Andrews, St. Andrews. Their contribution is gratefully acknowledged. The
presentventurewasembarkeduponwhilethefirstauthor(PLS)wasSeniorScien-
tist(IndianNationalScienceAcademy) attheDepartmentofMathematics,Indian
InstituteofScience,Bangalore,andwascompletedwhenhemovedtotheDepart-
mentofMathematics,UniversityofDelhi,SouthCampus.Hewishestothankboth
theIIScandtheUniversityofDelhiforthefacilitiesandtoINSAforfinancialsup-
port. The second author (ChSR) is grateful to the Indian Institute of Technology
Madras, Chennai for the financial support under Golden Jubilee book writing and
new faculty schemes as he worked on this monograph. The authors also acknowl-
edgetheeffortsofProfessorK.T.Joseph,TataInstituteofFundamentalResearch,
Bombay and Mr. Manoj Kumar Yadav, IIT Madras, who carefully went through
somepartsofthisbook.
NewDelhi,Chennai P.L.Sachdev
August2008 Ch.SrinivasaRao
v
Contents
1 Introduction .................................................. 1
References..................................................... 6
2 LargeTimeAsymptoticsforSolutionsofNonlinearFirst-Order
PartialDifferentialEquations ................................... 9
2.1 Introduction ............................................... 9
2.2 First-order nonlinear partial differential equations – Some
examples.................................................. 10
2.3 Decay estimates for solutions of nonlinear first-order partial
differentialequations ....................................... 21
2.4 Conclusions ............................................... 31
References..................................................... 32
3 LargeTimeAsymptoticAnalysisofSomeNonlinearParabolic
Equations–SomeConstructiveApproaches....................... 33
3.1 Introduction ............................................... 33
3.2 Travelling waves as asymptotics of solutions of initial value
problems–TheBurgersequation ............................. 34
3.3 Profileatinfinity–InitialboundaryvalueproblemforBurgers
equation .................................................. 38
3.4 Travellingwavesdescribingflowinaporousmedium ............ 47
3.5 Evolution of a stable profile describing cross-field diffusion
intoroidalmultipleplasma................................... 56
3.6 Asymptoticsolutionsdescribingfastnonlineardiffusion.......... 63
3.7 Largetimeasymptoticbehaviourofperiodicsolutionsofsome
generalisedBurgersequations ................................ 74
3.8 AsymptoticbehaviourofsomegeneralisedBurgersequationsvia
balancingargument......................................... 82
3.9 EvolutionoftravellingwavesingeneralisedFisher’sequationsvia
matchedasymptoticexpansions...............................106
vii
viii Contents
3.10 Periodictravellingwavesolutionsinreaction–diffusionsystems ...115
3.11 Conclusions ...............................................122
References.....................................................124
4 Self-similar Solutions as Large Time Asymptotics for Some
NonlinearParabolicEquations ..................................129
4.1 Introduction ...............................................129
4.2 Self-similarsolutionsaslargetimeasymptoticsforlinearheat
equation ..................................................132
4.3 Self-similar solutions as intermediate asymptotics for
filtration–absorptionmodel ..................................136
4.4 Analysisofaclassofsimilaritysolutionsoftheporousmedia
equation ..................................................145
4.5 Similaritysolutionsofnonlinearheatconductionproblemsas
largetimeasymptotics ......................................154
4.6 Large time asymptotics for solutions of the porous media
equation ..................................................161
4.7 Largetimebehaviourofsolutionsofadissipativesemilinearheat
equation ..................................................170
4.8 Largetimeasymptoticsforthesolutionsofaveryfastdiffusion
equation ..................................................176
4.9 Conclusions ...............................................184
References.....................................................186
5 AsymptoticsinFluidMechanics .................................189
5.1 Introduction ...............................................189
5.2 Strongexplosioninapowerlawdensitymedium–Self-similar
solutionsoffirstandsecondkind .............................190
5.3 Self-similarsolutionsforcollapsingcavities ....................199
5.4 Largetimebehaviourofsolutionsofcompressibleflowequations
withdamping..............................................203
5.5 Largetimebehaviourofsolutionsofunsteadyboundarylayer
equationsforanincompressiblefluid ..........................211
5.6 AsymptoticbehaviourofvelocityprofilesinPrandtlboundary
layertheory ...............................................219
5.7 Conclusions ...............................................230
References.....................................................230
Index .............................................................233
Chapter 1
Introduction
Nonlinear partial differential equations (PDEs) do not, in general, admit exact
solutions; these solutions are even more rare when initial/boundary conditions
are imposed. There are exceptional circumstances when the PDEs enjoy certain
symmetries: they are invariant to a class of finite or infinitesimal transformations
(Sachdev 2000). When this is the case, the PDEs are exactly reducible to ordi-
nary differential equations (ODEs) if they are functions of two independent vari-
ables; the ODEs may occasionally be integrated in a closed form. Alternatively,
one may study their qualitative properties and obtain the actual solutions numeri-
callywithreferencetoappropriateinitial/boundaryconditions.Thesesolutionsare
calledself-similarandbelongtooneofthetwoclasses,firstkindandsecondkind
(Zel’dovich 1956, Zel’dovich and Raizer 1967, Barenblatt and Zel’dovich 1972,
Sachdev2000),andsolvesomedegenerateproblemsforwhich‘all,oratleastsome,
of the constant parameters in the initial and boundary conditions of the problem,
havingthedimensionalityofindependentvariables,tendtozeroorinfinity.’These
solutionsdescribethosepropertiesofthephenomenathatdonotdependonthede-
tails of the initial and boundary conditions; they do involve some nondimensional
parameterswhich,insomeintegralsense,representthememoryofinitial/boundary
conditions. Exceptionally, there may not be any nondimensional parameter of the
problem in the asymptotic solution (Barenblatt and Zel’dovich 1972). These spe-
cialsolutionsdonotdescribeequilibriumstates;theydescribeintermediatestages
when the process of evolution of the solution is continuing and yet the details of
initial/boundaryconditionshavealreadydisappeared.Thesesolutionssatisfysome
singular,deltafunctionslike,initialconditions.
An interesting denomination of this class of solutions is ‘the profile at infinity’
dueoriginallytoPhilip(1957,1974).Thesesolutionsgive,forlargetime,aremark-
ably accurate behaviour of an entire family of solutions of initial/boundary value
problems.
Thenotion,intermediateasymptotics,wasfirstintroducedformallybyBarenblatt
andZel’dovich(1971,1972).However,theappearanceofintermediateasymptotics
may be seen implicitly in the work of Kolmogorov et al. (1937); this work deals
with the propagation, in a certain spatial region, of a gene whose carriers have an
P.L.Sachdev,Ch.SrinivasaRao,LargeTimeAsymptoticsforSolutionsofNonlinear 1
PartialDifferentialEquations,SpringerMonographsinMathematics,
DOI10.1007/978-0-387-87809-6 1,(cid:2)c SpringerScience+BusinessMedia,LLC2010
2 1 Introduction
advantageinthestruggleforexistence.Theyconsideredtheequation
v −kv =F(v), (1.1.1)
t xx
where the function F(v), 0≤v≤1, is sufficiently smooth and has the properties
that
F(0)=F(1)=0, F(v)>0 (0<v<1),
F(cid:3)(0)=α>0, F(cid:3)(v)<α (0<v<1); (1.1.2)
here, k is a constant. Kolmogorov et al. (1937) proved that (1.1.1) has uniformly
propagatingwavesor‘stationarysolutions’oftheform
v=γ(x−λt+c), (1.1.3)
where the constant c is arbitrary, satisfying the boundary con√ditions γ(−∞)=0,
γ(∞)=1foranyspeedofpropagationλsuchthatλ≥λ =2 kα.Itwasfurther
0
shown that, among all the solutions (1.1.3), only the solution with the minimum
speedλ=λ isstableinthefollowingsense.ThesolutionoftheCauchyproblem
0
satisfyingtheinitialcondition
v(x,0)≡0, x≤a,
0<v(x,0)<1, a<x<b,
v(x,0)≡1, x≥b,
whereaandbarearbitrarynumbersandv (x,0)≥0,tendsast→∞,toasolution
x
ofthetype(1.1.3)withλ=λ.The‘stationary’ortravellingwavesolutionofthe
0
form (1.1.3) is possible with a continuous spectrum of speeds of propagation λ;
onlythesolutioncorrespondingtotheextreme(minimum)valueλ isasymptoticto
0
anonstationarysolutionoftheCauchyproblemast→∞.Theconstantcin(1.1.3)
remainsundeterminedifonlytravellingwavesareconsidered.Itmaybefoundby
numericallytracingtheevolutionofthenonstationarysolutionoftheCauchyprob-
lem until the travelling waveform emerges. The intermediate asymptotic character
of the above solution for λ=λ was demonstrated much later by Kanel’ (1962).
0
WemaypointoutthattheconditionF(cid:3)(v)<αisnotrequiredfortheexistenceofa
travellingwavesolutionof(1.1.1)withthespeedofpropagationλ=λ.Theabove
0
investigationbyKolmogorovetal.(1937)wasfollowedbytheworkofZel’dovich
andFrank-Kamenetskii(1938)whotreateduniformlypropagatingflamesinthethe-
oryofcombustion.AnexcellentdescriptionoftheSovietcontributiontothistopic
maybefoundinthereviewarticlebyBarenblattandZel’dovich(1972).
Therearetwodistincttypesofself-similarsolutions,eachdependingonasim-
ilarity variable, say, ξ=rt−α/A, which is a combination of the independent vari-
ables r and t. Here, A is a dimensional parameter. From a physical point of view,
such solutions exist when the length and time scales cannot be constructed from
theparametersoftheproblemsothatrandtdonotoccurseparatelyinthesolution.
Theunknownfunctionscandependonlyonadimensionlesscombinationofr,t,and
1 Introduction 3
a dimensional parameter A with the dimensions [A]=LT−α, where αis real. We
illustratethismatterwithreferencetoone-dimensionalequationsofgasdynamics,
∂lnρ ∂lnρ ∂u u
+u + +(ν−1) =0, (1.1.4)
∂t ∂r ∂r r
∂u ∂u 1 ∂p
+u + =0, (1.1.5)
∂t ∂r ρ∂r
∂ ∂
lnpρ−γ+u lnpρ−γ=0. (1.1.6)
∂t ∂r
Here, p,ρ,anduarepressure,density,andparticlevelocityatanypointrandtime
t.Theparameterνequals1,2,3fortheplane,cylindrical,andsphericalsymmetries,
respectively.Here,thesolutionwouldassumetheform
a
p= P(ξ), (1.1.7)
rk+1ts+2
a
ρ= G(ξ), (1.1.8)
rk+3ts
r
u= V(ξ), (1.1.9)
t
whereξ=rt−α/Aandaisaparameterwhichappearsinthegivenproblem.Itcon-
tainstheunitofmassandhasthedimensions[a]=MLkTs,wherek andsarereal
numbers. If one substitutes the form (1.1.7)–(1.1.9) into (1.1.4)–(1.1.6), the latter
reduce to a system of nonlinear ODEs and the motion is said to be self-similar. If
theexponentαappearinginthesimilarityvariableξandtheexponentsofr andt
arefullydeterminedfromdimensionalconsiderationsand/ortheconservationlaws
(1.1.4)–(1.1.6),thesimilaritysolutionissaidtobelongtothefirstkind.Thisclass
ofsolutionsmaybeexemplifiedbythesolutiondescribingastrongexplosionintoa
uniformmediumwithdensityρ.Herewehavetwophysicalparameters,theundis-
0
turbeddensity,ρ ∼ML−3,andtheenergyofexplosion,E ∼ML2T−2,appearing
0 0
in the problem. The energy of explosion is assumed to be constant and equals the
energy of the moving gas. This constitutes an integral of the motion. For a strong
explosion,thepressure p andthesoundspeedc intheundisturbedmotionareas-
0 0
sumedtobezeroanddonotentertheproblemasparameters.Theconstantsρ andE
0
helpconstructtheparameterAinthesimilarityvariable:A=(E/ρ)1/5∼LT−2/5.
0
Therefore,wehavethesimilarityvariableξ=r/(E/ρ)1/5t2/5.Theexponentαis
0
thusexplicitlyfoundtobe2/5.Inthiscase,the(reduced)systemofODEsfromthe
system(1.1.4)–(1.1.6)issolvedanalyticallyornumericallysubjecttotheRankine–
Hugoniotconditionsattheshockandzeroparticlevelocityatthecentreofthespher-
icalexplosion(seeTaylor1950,Sedov1946,Sachdev2004).
Intheself-similarsolutionsofthesecondkind,thesubstitutionofthesimilarity
form(1.1.7)–(1.1.9)intothesystem(1.1.4)–(1.1.6)reducesthelattertoasystemof
nonlinearODEsbuttheexponentαisnotdeterminedfromdimensionalconsider-
ations or conservation laws. This exponent is found from the requirement that the
integralcurvepassesthroughanappropriatesingularpoint,usuallyasaddlepoint,
4 1 Introduction
so that the boundary conditions, say, at the shock and the point at infinity behind
theshockaresimultaneouslysatisfied.Anexampleofself-similarsolutionsofthe
secondkindisaconvergingshockfromanimplosion.Ithasbeendiscussedinsome
detailbyZel’dovichandRaizer(1967)andSachdev(2004).
Wehavealreadydiscussedthetravellingwavesolutionsfortheequation(1.1.1)
which constitute self-similar solutions of the second kind. Here, in general, the
wavespeed λis not found by solving a boundary value problem for the ODE re-
sultingfromsubstituting(1.1.3)into(1.1.1)andsatisfyingtheboundaryconditions
γ(−∞)=0, γ(∞)=1. The wavespeed λpossesses an infinite number of possible
values,theminimumofwhichgivesauniquestablesolution.Somethinganalogous
tothishappensfortheconvergingshockproblem(Sachdev2004).
The nonlinear PDEs with two (or more) independent variables may admit self-
similar or travelling wave solutions which are governed by nonlinear ODEs. This
comesaboutwhenthePDEsenjoycertaininvariancepropertiessubjecttofiniteor
infinitesimaltransformationsaswehaveremarkedearlier(BlumanandKumei1989,
Sachdev 2000). Using group-theoretic methods or the so-called direct method of
Clarkson and Kruskal (1989), it becomes possible to exactly reduce the nonlinear
PDEs with two independent variables to nonlinear ODEs and hence analyse the
existence and uniqueness properties of the latter subject to appropriate boundary
and/orinitialconditions.IthelpstounderstandwhentheseODEspossesssolutions
andknowtheirbehaviourforlargedistances.Theirsingularbehaviour,ifany,may
alsobediscovered.Thisstudyisoftenmotivatedbyacanonicalequationofacertain
class for which one may have access to all the relevant information. To exemplify
this, one may refer to the Burgers equation which admits exact linearisation by
theso-calledCole–Hopftransformationandhenceyieldsconsiderableinformation
regardingitssolution.Thismotivatestheanalysisforthelargeclassofgeneralised
BurgersequationswhichdonotenjoythelinearisingfelicityoftheCole–Hopftype
transformationandmustthereforebetreateddirectly(Sachdev1987,2000).
When a given PDE (or a system) does not possess sufficient invariance proper-
ties and therefore cannot be reduced to an ODE (or a system of ODEs), one must
turntoothermethodstofindanalyticsolutions.Oneintuitiveapproachistomimic
thesolutionofasimplerequation;forexample,forgeneralisedBurgersequations,
onecouldsimulatethebehaviourofthesolutionsoftheBurgersequationandobtain
largetimesolutionswhichembedinthemlimitingformssuchasinviscidand/orlin-
earviscousones.ThisbecomespossibleforaclassofgeneralisedBurgersequations
withrespecttoN-waveorperiodicinitialconditionsandinvolvessolutionofanin-
finite system of coupled nonlinear ODEs (see Sachdev and Joseph 1994, Sachdev
etal.1996,2005).
A more general approach to finding asymptotic solutions, usually not leading
to a closed form, is called the balancing argument. It was originally developed by
Grundy and his collaborators (Grundy 1988, Grundy et al. 1994a, 1994b) and has
sincebeenexploitedextensivelybySachdevandhiscoworkers(Sachdevetal.1994,
1999,SachdevandSrinivasaRao2000).Here,itishelpfultointroduceinthegiven
PDE a transformation such as u=tαU(η,t), where η=xtβ is the relevant sim-
ilarity variable; αand β are constants found later. The given PDE in u=u(x,t)
