Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory

By Martial Agueh. Advances in Differential Equations, 2002

154 citations recorded by the source corpus.

We obtain solutions of the nonlinear degenerate parabolic equation ∂ρ ∂t = div { ρ∇c [ ∇ ( F ′(ρ) + V )] } as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires fewer uniform convexity assumptions than those imposed by Alt and Luckhaus in their pioneering work [4]. In fact, their assumptions may fail in our equation. This class of equations includes the Fokker-Planck equation, the porous-medium equation, the fast diffusion equation and the parabolic p-Laplacian equation.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f13d348351d66ffcef974753a8f57428b68326e0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A gradient flow approach to a thin film approximation of the Muskat problem (2011, background): “Indeed, since the pioneering works [12] on the linear Fokker-Planck equation and [15, 16] on the porous medium equation, several equations have been interpreted as gradient flows with respect to some Wasserstein metrics, including doubly degenerate parabolic equations [1], a model for type-II semico”
Contractive Probability Metrics and Asymptotic Behavior of Dissipative Kinetic Equations (2007, background): “These metrics has been considered in the PDE’s analytic community quite recently in connection to gradient flows and steepest descent schemes of linear and nonlinear diffusions [74, 87, 1, 47, 3] and homogeneous kinetic models [43, 3, 44] as well as for describing their asymptotic behavior.”
Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations (2011, methodology): “The approach of [2] in proving the existence of a curve of maximal slope for a functional on P2 is based on a variational version of the implicit Euler scheme, sometimes referred to as the Jordan–Kinderlehrer– Otto (JKO) scheme or minimizing movement scheme [25, 1, 2].”
A new optimal transport distance on the space of finite Radon measures (2015, methodology): “This is similar to the dynamic formulation of the Wasserstein distances Wp of order p [3, 43], and the gradient flows with suitable driving entropies can involve the p-Laplacian operator as in [1].”
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, methodology): “63”
A new transportation distance between non-negative measures, with applications to gradients flows with Dirichlet boundary conditions (2010, unknown): “• Although our strategy is not new (for instance, it has been used in the context of the classical transportation problem in [9, 1] to prove a maximum principle), the fact of having no mass constraints makes it more e ffici nt, and the properties of minimizers that we are able to deduce are in some ”
Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media (2006, methodology): “The application of displacement convexity to rates of convergence in nonlinear evolution equations begun by Otto [43], was recently explored for more general costs associated with different nonlinearities by Agueh [1, 4] and Agueh, Ghoussoub & Kang [2, 3].”
Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces (2009, background): “Among many papers dealing with similar problems, particularly significant are [1,8,9,26,27].”
Gradient flows with metric and differentiable structures, and applications to the Wasserstein space (2004, result): “This leads in many situations to the existence and the uniqueness of gradient flows (see also [9] for related results, mostly concerning uniqueness and rate of convergence as tKQ), to the convergence of the implicit time discretization scheme (see also [15, 8, 1]) and, under a slightly stronger conv”
Gradient Flows: In Metric Spaces and in the Space of Probability Measures (2005, background): “The second Section of Chapter 7 deals with the characterization of constant speed geodesics in Pp(X) (here X is an Hilbert space), parametrized on the unit interval [0, 1].”
Free boundaries in optimal transport and Monge-Ampere obstacle problems (2010, background): “Agueh [2], Ambrosio, Gigli and Savare [8], Cullen and Gangbo [29], Gianazza, Savare and Toscani [45], Savin [71]; in shape optimization since the work of Bouchitte, Buttazzo and Seppecher [11], see also e.”
Chapter 1 – Gradient Flows of Probability Measures (2007, background): “Finally, if γ is an invariant measure, then St[1] = 1; the order preserving property shows that ‖St[ρ]‖L∞(γ) ≤ ‖ρ‖L∞(γ).”