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Perspectives in nonlinear diffusion: between analysis, physics and geometry

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Perspectives in nonlinear diffusion: between analysis, physics and geometry
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We review some topics in the mathematical theory of nonlinear diffusion. Attention is focused on the porous medium equation and the fast diffusion equation, including logarithmic diffusion. Special features are the existence of free boundaries, the limited regularity of the solutions and the peculiar asymptotic laws for porous medium flows, while for fast diffusions we find the phenomena of finite-time extinction, delayed regularization, nonuniqueness and instantaneous extinction. Logarithmic diffusion with its strong geometrical flavor is also discussed. Connections with functional analysis, semigroup theory, physics of continuous media, probability and differential geometry are underlined.
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Mathematical analysisGoodness of fitEvent horizonMathematicianSheaf (mathematics)Lattice (order)GeometryExpressionTheoryPresentation of a groupFunction (mathematics)EquationElementary arithmeticBasis (linear algebra)Energy levelInfinityMathematicsHypothesisHelmholtz decompositionMomentumSpectrum (functional analysis)PropagatorDynamical systemPartial differential equationDifferential geometryPartial differential equationDifferential equationMultiplication signGroup representationPosition operatorMereologyConvolutionUniverse (mathematics)AreaMathematical physicsGenerating set of a groupObservational studyNumerical analysisSpacetimeDifferent (Kate Ryan album)Direction (geometry)Partial derivativeStochastic kernel estimationFlow separationContinuous functionFluidMathematical modelProcess (computing)Körper <Algebra>Matching (graph theory)State of matterRight angleHeat transferAssociative propertyRegular graphDivision (mathematics)AnalogyPhysical systemLogicExplosionArithmetic progressionSeries (mathematics)Military baseIdentical particlesStatistical hypothesis testingDressing (medical)Greatest elementPoint (geometry)Lecture/ConferenceMeeting/Interview
Group actionContinuous functionFood energyDifferent (Kate Ryan album)Condition numberMathematicsPhysicalismMusical ensembleMathematicianAreaTerm (mathematics)Monster groupModel theoryNichtlineares GleichungssystemCuboidStochastic calculusDifferential equationSlide ruleBrownian motionTheoryOperator (mathematics)Interface (chemistry)CoefficientMatrix (mathematics)Parabolische DifferentialgleichungMaxima and minimaFirst-order logicForcing (mathematics)SpacetimeAlpha (investment)DiffusionsgleichungFourier seriesFunction (mathematics)LinearizationPoint (geometry)Complex (psychology)Derivation (linguistics)EquationAnalytic setExtension (kinesiology)Combinatory logicGenerating set of a groupSemigroupGeometryExergieFree groupConformal mapBoundary value problemPartial differential equationEnergy levelExistenceKorrekt gestelltes ProblemMathematical analysisProof theoryPhase transitionPhysical lawEvolutePartial differential equationComplex numberWater vaporDuality (mathematics)DivisorMereologyProcess (computing)TheoremUniqueness quantificationPotenzialtheorieChaos (cosmogony)Set theoryDecision theoryStructural loadDiscounts and allowancesPosition operatorComputer programmingConcentricMilitary baseVotingObservational studyDivision (mathematics)Algebraic structureMoment (mathematics)ExplosionState of matterStaff (military)Principal idealLecture/Conference
Moment (mathematics)Tensor2 (number)MereologyLaplace-OperatorBlock (periodic table)Presentation of a groupAreaTheoryMathematicsPartial differential equationKörper <Algebra>Glatte FunktionPower (physics)EquationPopulation densityPoint (geometry)Complete metric spaceMass flow rateNichtlineares GleichungssystemFluidModel theoryTerm (mathematics)TheoremContinuous functionPhysical lawEllipseParabolische DifferentialgleichungGradientPressureMultiplication signVariable (mathematics)PhysicalismMathematical singularityDivergenceOpen setNatural numberMusical ensembleOperator (mathematics)Boundary value problemAverageFinitismusState of matterSquare numberGamma functionFree groupDiffusionsgleichungWater vaporArithmetic meanVarianceApproximationMathematicianScaling (geometry)Differential operatorMomentumDrag (physics)Arithmetic progressionVotingValidity (statistics)GravitationLimit (category theory)Volume (thermodynamics)Observational studyFamilyMatching (graph theory)Euklidischer RingFigurate numberRight angleGame theoryDifferential equationHydraulic motorTheory of relativityForcing (mathematics)Grothendieck topology
Mathematical analysisProcess (computing)Fraction (mathematics)Operator (mathematics)Laplace-OperatorScaling (geometry)Power (physics)Square numberPhysical lawVariable (mathematics)MathematicsTransformation (genetics)Arithmetic meanBoundary value problemTheoremModel theoryINTEGRALExponential functionInfinityMusical ensembleInterface (chemistry)SurfaceMonster groupEvoluteSinc functionAsymptotic analysisDistanceEstimatorNichtlineares GleichungssystemVarianceKinematicsLimit (category theory)Term (mathematics)Stochastic kernel estimationFunktionenalgebraDifferential equationGroup representationDirected graphPhysicalismSpacetimeGroup theoryTheorySimilarity (geometry)Function (mathematics)Brownian motionParabolische DifferentialgleichungAlgebraic functionEquationPhysical systemAlpha (investment)Beta functionDerivation (linguistics)ParabolaThermal radiationObject (grammar)Multiplication signPerturbation theoryChemical equationWave equationPropagatorMomentumFood energyEnergy levelMereologyEqualiser (mathematics)Mathematical singularityHyperbolischer RaumStudent's t-testNatural numberMathematicianFigurate numberMany-sorted logicReduction of orderSurface of revolutionMortality rateState of matterUniqueness quantificationWater vaporAreaGraph (mathematics)Arithmetic progressionGame theorySeries (mathematics)Numerical analysisUniverse (mathematics)Lecture/Conference
PhysicalismDerivation (linguistics)MathematicianPhysical lawNumerical analysisRight angleMathematical analysisMereologyAlgebraic functionParabolaExponential functionSpacetimeNoise (electronics)Function (mathematics)Similarity (geometry)Statistical mechanicsMultiplication signGroup representationCentralizer and normalizerLimit (category theory)CuboidTerm (mathematics)Classical physicsSign (mathematics)Distribution (mathematics)Profil (magazine)Process (computing)OvalGraph (mathematics)Water vaporCumulative distribution functionState of matterSocial classVariable (mathematics)
Duality (mathematics)Right angleTransformation (genetics)Differential equationCumulative distribution functionProduct (business)Decision theoryPower (physics)Multiplication signDerivation (linguistics)EvoluteFunction (mathematics)SemigroupCalculusFamilyNichtlineares GleichungssystemTheoryElliptic curve
Parabolische DifferentialgleichungMultiplication signEquationFluidTheoryEntropyEstimatorGame theoryGenderProcess (computing)Lecture/Conference
Slide ruleNormal (geometry)Multiplication signProcess (computing)Food energyProof theory1 (number)Compact spaceEstimatorApproximationSpacetimeMass flow ratePhysicalismExistenceGoodness of fitDerivation (linguistics)DissipationConservation lawGradientWater vaporImage resolutionHand fanLecture/ConferenceMeeting/Interview
Kontraktion <Mathematik>Nichtlineares GleichungssystemDerivation (linguistics)Regular graphLogarithmMultiplication signFunktionalanalysisDifferent (Kate Ryan album)Physical lawTheorySpacetimeStandard deviationSlide rulePoint (geometry)EstimatorMetastabilitätPressureFunktionenalgebraReflexive spaceFunction (mathematics)Cumulative distribution functionContinuous functionMass flow rateCurveInfinityInterface (chemistry)Dimensional analysisAlpha (investment)2 (number)ThermodynamicsResultantDynamical systemDissipationScaling (geometry)FluidGlatte FunktionHarmonic analysisGeometryState of matterTheory of relativityInequality (mathematics)Forcing (mathematics)MereologyFree groupFinitismusFlow separationBlock (periodic table)CurvatureHypothesisTime domainSign (mathematics)Compact spaceTerm (mathematics)Category of beingAxiom of choiceCentral limit theoremPartial differential equationPhysical systemAmenable groupLine (geometry)Distortion (mathematics)Water vaporEvent horizonComputer programmingGroup actionNatural numberPerimeterStatistical hypothesis testingFigurate numberCycle (graph theory)Covering spaceCollisionGraph (mathematics)Grothendieck topology4 (number)Stability theoryRule of inferenceLecture/Conference
Different (Kate Ryan album)RenormalizationAlpha (investment)Exponential functionScaling (geometry)InfinityEntropyNormal (geometry)Goodness of fitMortality rateCalculationL1-NormMass flow rateCentral limit theoremDimensional analysisFunction (mathematics)Cumulative distribution function1 (number)GeometryTerm (mathematics)Hessian matrixPhysical quantityDerivation (linguistics)Algebraic functionDissipationMoment (mathematics)VarianceDiagonal matrixPower (physics)Multiplication signBoltzmann constantDiffusionsgleichungNichtlineares GleichungssystemDirection (geometry)Profil (magazine)ParabolaPressureMatrix (mathematics)Group action2 (number)ResultantThermodynamicsFinitismusTheoremSet theorySocial classRight angleDistanceRule of inferenceTime zoneSpacetimeSinc functionGreatest elementCharge carrierAreaGraph (mathematics)Meeting/Interview
Price indexMereologyGravitationMultiplication signResultantCoefficient of determinationMathematicsObservational studyCalculationMonster groupGlatte FunktionDiffusionsgleichungEstimatorLimit (category theory)Archaeological field surveySpacetimeNichtlineares GleichungssystemLecture/Conference
KnotProcess (computing)Set theoryRule of inferenceObservational studyTransportation theory (mathematics)Musical ensembleMathematical optimizationKontraktion <Mathematik>Central limit theoremDifferent (Kate Ryan album)Normal (geometry)Division (mathematics)Maß <Mathematik>VotingGlatte FunktionHyperbolischer RaumEquationLecture/Conference
Nichtlineares GleichungssystemBounded variationGradientOperator (mathematics)Limit (category theory)Total S.A.Multiplication signEvoluteGroup representationSurfaceMaß <Mathematik>Coefficient of determinationCartesian coordinate systemDimensional analysisGeometryCategory of beingLecture/Conference
Connected spaceMultiplication signMaß <Mathematik>MassInfinityGeometryTheoremBoundary value problemNichtlineares GleichungssystemMass flow rateCentral limit theoremPartial differential equationFunction (mathematics)PhysicalismWater vaporMathematicsLecture/Conference
Transcript: Englisch(auto-generated)