(守恒律) Hyperbolic Conservation Laws - An Illustrated Tutorial.pdf

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1、HyperbolicConservationLawsAnIllustratedTutorialAlbertoBressanDepartmentofMathematics,PennStateUniversity,UniversityPark,Pa.16802,USA.bressan@math.psu.eduDecember5,2009AbstractThesenotesprovideanintroductiontothetheoryofhyperbolicsystemsofconservationlawsinone

2、spacedimension.Thevariouschapterscoverthefollowingtopics:1.Meaningofaconservationequationanddefinitionofweaksolutions.2.Hyperbolicsystems.Ex-plicitsolutionsinthelinear,constantcoefficientscase.Nonlineareffects:lossofregularityandwaveinteractions.3.Shockwaves:Rank

3、ine-Hugoniotequationsandadmissibilityconditions.4.Genuinelynonlinearandlinearlydegeneratecharacteristicfields.Centeredrarefactionwaves.ThegeneralsolutionoftheRiemannproblem.Waveinteractiones-timates.5.WeaksolutionstotheCauchyproblem,withinitialdatahavingsmallt

4、otalvariation.Approximationsgeneratedbythefront-trackingmethodandbytheGlimm1scheme.6.Continuousdependenceofsolutionsw.r.t.theinitialdata,intheLdis-tance.7.Characterizationofsolutionswhicharelimitsoffronttrackingapproximations.Uniquenessofentropy-admissiblewea

5、ksolutions.8.Vanishingviscosityapproximations.9.Extensionsandopenproblems.ThesurveyisconcludedwithanAppendix,reviewingsomebasicanalyticaltoolsusedintheprevioussections.Throughouttheexposition,technicaldetailsaremostlyleftout.Themaingoalofthesenotesistoconveyb

6、asicideas,alsowiththeaidofalargenumberoffigures.1ConservationLaws1.1ThescalarconservationlawAscalarconservationlawinonespacedimensionisafirstorderpartialdifferentialequationoftheformut+f(u)x=0.(1.1)Hereu=u(t,x)iscalledtheconservedquantity,whilefistheflux.Thevaria

7、bletdenotestime,whilexistheone-dimensionalspacevariable.Equationsofthistypeoftendescribetransportphenomena.Integrating(1.1)overagiven1interval[a,b]oneobtainsZZZdbbbu(t,x)dx=ut(t,x)dx=−f(u(t,x))xdxdtaaa=f(u(t,a))−f(u(t,b))=[inflowata]−[outflowatb].Inotherwords,t

8、hequantityuisneithercreatednordestroyed:thetotalamountofucontainedinsideanygiveninterval[a,b]canchangeonlyduetotheflowofuacrossboundarypoints(fig.1).uabξxFigure1:Flowacrosstwopoints.Usingth