THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION

By Felix Otto. 2001

1,798 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/0f0a732933e68ac5e6a977327ffd0410ae08bec6

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Kac’s program in kinetic theory (2011, neutral): “[41, 61].”
Equilibria of homogeneous functionals in the fair-competition regime (2016, methodology): “(1.4) This illuminating statement has been clarified in the seminal paper by Otto [42] for the porous medium equation, and generalised to a large family of equations subsequently in [21,1,22], we refer to [55,1] for a comprehensive presentation of this theory of gradient flows in Wasserstein metric ”
Heat Flow on Alexandrov Spaces (2010, background): “This intuition, with the further studies of Otto [31], has been one of the fundamental ingredients that drove the research in the field of gradient flows in relation with optimal transportation problems in the past decade (see, e.g., [1, 42]).”
Gradient flow structures for discrete porous medium equations (2012, background): “11 29 v1 [ m at h. FA ] 5 D ec 2 01 2 n discovered by Otto.”
The derivation of swarming models: Mean-field limit and Wasserstein distances (2013, methodology): “The idea is similar to the evolution of the euclidean Wassertein distance in [30, 31, 70].”
On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion (2013, neutral): “This class was first identified in the seminal work by Jordan, Kinderlehrer, and Otto [22, 41] and later shown to encompass a wide range of known and unknown partial differential equations [4, 27, 33, 38, 42].”
WEIGHTED ENERGY-DISSIPATION FUNCTIONALS FOR GRADIENT FLOWS (2011, methodology): “More recently, following the pioneering work by Otto [32], an even larger class of PDE problems have been translated into gradient flows by resorting to probability spaces endowed with the Wasserstein metric.”
Uniqueness of the solution to the Vlasov-Poisson system with bounded density (2005, background): “The path {ρθ, θ ∈ [1, 2]} is the geodesic linking ρ1 to ρ2, with respect to the Wasserstein metric (see [9], and also [7] where this notion was introduced).”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “This idea has first been underlined in [199, 246].”
Long-Time Asymptotics of Kinetic Models of Granular Flows (2004, neutral): “76P05, 82C40.”
Refining Deep Generative Models via Discriminator Gradient Flow (2020, methodology): “For example, they have been applied to the study of the porous-medium equation (Otto, 2001), crowd modeling (Maury et al., 2010; 2011), and mean-field games (Almulla et al., 2017).”