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.
Keywords nonlinear diffusion degenerate parabolic equations flows in porous media
Statistical hypothesis testing Point (geometry) Greatest element Observational study State of matter Multiplication sign Heat transfer Mereology Regular graph Goodness of fit Mathematics Different (Kate Ryan album) Energy level Differential equation Matching (graph theory) Graph (mathematics) Generating set of a group Expression Lattice (order) Numerical analysis Universe (mathematics) Right angle Mathematician Arithmetic progression Identical particles
Geometry Process (computing) Observational study State of matter Sheaf (mathematics) Division (mathematics) Basis <Mathematik> Funktionalanalysis Event horizon Dressing (medical) Theory Explosion Computer animation Analogy Physical system
Principal ideal Group action State of matter Decision theory Multiplication sign Chaos (cosmogony) Water vapor Mereology Food energy Grothendieck topology Explosion Group representation Mathematics Different (Kate Ryan album) Cuboid Körper <Algebra> Series (mathematics) Position operator Process (computing) Theory of relativity Generating set of a group Concentric Structural load Moment (mathematics) Physicalism Staff (military) Funktionalanalysis Flow separation Discounts and allowances Phase transition Equation Figurate number Mathematician Arithmetic progression Spacetime Computer programming Existence Momentum Observational study Natural number Operator (mathematics) Nichtlineares Gleichungssystem Set theory Condition number Validity (statistics) Military base Forcing (mathematics) Model theory Algebraic structure Division (mathematics) Volume (thermodynamics) Numerical analysis Approximation Voting Logic Combinatory logic Hydraulic motor Gravitation Game theory Spectrum (functional analysis)
Logical constant Group action Momentum Drag (physics) Observational study State of matter Water vapor Student's t-test Mathematics Many-sorted logic Natural number Operator (mathematics) Interface (chemistry) Reduction of order Körper <Algebra> Nichtlineares Gleichungssystem Series (mathematics) Euklidischer Ring Area Matching (graph theory) Graph (mathematics) Theory of relativity Chemical equation Model theory Moment (mathematics) Physical law Physicalism Mortality rate Limit (category theory) Evolute Surface of revolution Numerical analysis Wave Voting Computer animation Universe (mathematics) Right angle Object (grammar) Game theory Figurate number Mathematician Arithmetic progression Family Spacetime
Computer programming Group action State of matter Transformation (genetics) Image resolution Multiplication sign Decision theory Water vapor Dimensional analysis Product (business) Group representation Meeting/Interview Different (Kate Ryan album) Oval Algebra Social class Physical system Dissipation Noise (electronics) Distribution (mathematics) Process (computing) Graph (mathematics) Gender Reflexive space Line (geometry) Variable (mathematics) Hand fan Duality (mathematics) Right angle Game theory Figurate number Spacetime Distortion (mathematics)
Statistical hypothesis testing Covering space Point (geometry) Graph (mathematics) Theory of relativity State of matter Forcing (mathematics) Multiplication sign Water vapor Funktionalanalysis Mereology Rule of inference Event horizon Natural number Interface (chemistry) Cycle (graph theory) Figurate number Collision Resultant Stability theory
Group action Greatest element Transportation theory (mathematics) Observational study Multiplication sign Water vapor Distance Mereology Rule of inference Grothendieck topology 4 (number) Group representation Goodness of fit Coefficient of determination Mathematics Set theory Amenable group Perimeter Social class Dissipation Area Time zone Graph (mathematics) Process (computing) Block (periodic table) Forcing (mathematics) Surface Moment (mathematics) Exponentiation Computability Division (mathematics) Knot Funktionalanalysis Mortality rate Price index Evolute Voting Physical quantity Charge carrier Normal (geometry) Gravitation Right angle Pressure Resultant Sinc function Spacetime Maß <Mathematik>
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