A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

DOI10.1080/03605300903296256zbMath1187.35131arXiv0901.0540OpenAlexW2032254325MaRDI QIDQ3653278

Giuseppe Savaré, Daniel Matthes, Robert J. McCann

Publication date: 22 December 2009

Published in: Communications in Partial Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0901.0540

zbMATH Keywords

nonnegative solutionsentropy methodasymptotic profileWasserstein distanceBarenblatt profilesDerrida-Lebowitz-Speer-Spohn

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Thin fluid films (76A20) Degenerate parabolic equations (35K65) Methods involving semicontinuity and convergence; relaxation (49J45) Initial value problems for higher-order parabolic equations (35K30) Quasilinear parabolic equations (35K59) Fokker-Planck equations (35Q84) Pattern formations in context of PDEs (35B36)

Related Items (96)

A Keller-Segel type system in higher dimensions ⋮ Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type ⋮ A fitness-driven cross-diffusion system from population dynamics as a gradient flow ⋮ Convergence to Equilibrium for a Thin-Film Equation on a Cylindrical Surface ⋮ Nonlinear mobility continuity equations and generalized displacement convexity ⋮ An extension of the Derrida–Lebowitz–Speer–Spohn equation ⋮ High-frequency limit of non-autonomous gradient flows of functionals with time-periodic forcing ⋮ On nonlinear cross-diffusion systems: an optimal transport approach ⋮ Existence of weak solutions to a class of fourth order partial differential equations with Wasserstein gradient structure ⋮ Incompressible immiscible multiphase flows in porous media: a variational approach ⋮ Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation ⋮ Gradient flow structure of a multidimensional nonlinear sixth-order quantum-diffusion equation ⋮ Convergence of a One-Dimensional Cahn--Hilliard Equation with Degenerate Mobility ⋮ Degenerate Cahn-Hilliard equation: from nonlocal to local ⋮ A two-phase two-fluxes degenerate Cahn-Hilliard model as constrained Wasserstein gradient flow ⋮ On cross-diffusion systems for two populations subject to a common congestion effect ⋮ Gradient flows of modified Wasserstein distances and porous medium equations with nonlocal pressure ⋮ Entropic Regularization of NonGradient Systems ⋮ A positivity‐preserving and energy stable scheme for a quantum diffusion equation ⋮ Existence of solutions for a nonlinear system of parabolic equations with gradient flow structure ⋮ Harmonic mappings valued in the Wasserstein space ⋮ Time-convexity of the entropy in the multiphasic formulation of the incompressible Euler equation ⋮ Exponential Convergence to Equilibrium for Coupled Systems of Nonlinear Degenerate Drift Diffusion Equations ⋮ An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems ⋮ On the total variation Wasserstein gradient flow and the TV-JKO scheme ⋮ A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling ⋮ Fractional higher order thin film equation with linear mobility: gradient flow approach ⋮ Porous medium equation and cross-diffusion systems as limit of nonlocal interaction ⋮ On the well-posedness via the JKO approach and a study of blow-up of solutions for a multispecies Keller-Segel chemotaxis system with no mass conservation ⋮ A Wasserstein gradient flow approach to Poisson−Nernst−Planck equations ⋮ Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction ⋮ Invariant manifolds for the thin film equation ⋮ A gradient flow approach to the porous medium equation with fractional pressure ⋮ JKO estimates in linear and non-linear Fokker-Planck equations, and Keller-Segel: \(L^p\) and Sobolev bounds ⋮ Cahn-Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics ⋮ A gradient flow approach to a thin film approximation of the Muskat problem ⋮ A variational finite volume scheme for Wasserstein gradient flows ⋮ From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage ⋮ A multidimensional nonlinear sixth-order quantum diffusion equation ⋮ Weak solutions to the Muskat problem with surface tension via optimal transport ⋮ On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion ⋮ Lecture notes on variational mean field games ⋮ A blob method for diffusion ⋮ Localization, smoothness, and convergence to equilibrium for a thin film equation ⋮ Variational methods for evolution. Abstracts from the workshop held November 12--18, 2017 ⋮ A thin film approximation of the Muskat problem with gravity and capillary forces ⋮ The stochastic thin-film equation: existence of nonnegative martingale solutions ⋮ Optimal density evolution with congestion: L^∞ bounds via flow interchange techniques and applications to variational Mean Field Games ⋮ A BDF2-approach for the non-linear Fokker-Planck equation ⋮ Stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem ⋮ The Waiting Time Phenomenon in Spatially Discretized Porous Medium and Thin Film Equations ⋮ Weighted ultrafast diffusion equations: from well-posedness to long-time behaviour ⋮ A note on the variational analysis of the parabolic-parabolic Keller-Segel system in one spatial dimension ⋮ Lagrangian schemes for Wasserstein gradient flows ⋮ Long-time behavior of a fully discrete Lagrangian scheme for a family of fourth order equations ⋮ On a fractional thin film equation ⋮ Hardy-Littlewood-Sobolev inequalities via fast diffusion flows ⋮ The thin-film equation close to self-similarity ⋮ Nonlinear degenerate cross-diffusion systems with nonlocal interaction ⋮ Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations ⋮ \{Euclidean, metric, and Wasserstein\} gradient flows: an overview ⋮ Global existence for a thin film equation with subcritical mass ⋮ The gradient flow of a generalized Fisher information functional with respect to modified Wasserstein distances ⋮ Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions ⋮ A convergent Lagrangian discretization for a nonlinear fourth-order equation ⋮ The variational formulation of the fully parabolic Keller-Segel system with degenerate diffusion ⋮ NUMERICAL GRADIENT FLOW DISCRETIZATION OF VISCOUS THIN FILMS ON CURVED GEOMETRIES ⋮ A JKO Splitting Scheme for Kantorovich--Fisher--Rao Gradient Flows ⋮ On Patlak-Keller-Segel system for several populations: a gradient flow approach ⋮ The Navier-slip thin-film equation for 3D fluid films: existence and uniqueness ⋮ Gradient flows and evolution variational inequalities in metric spaces. I: structural properties ⋮ Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions ⋮ A family of fourth-order \(q\)-logarithmic equations ⋮ Transport distances and geodesic convexity for systems of degenerate diffusion equations ⋮ A dynamical systems approach for the contact-line singularity in thin-film flows ⋮ Small noise limit and convexity for generalized incompressible flows, Schrödinger problems, and optimal transport ⋮ Unconditionally positivity preserving and energy dissipative schemes for Poisson-Nernst-Planck equations ⋮ Self-Similarity in a Thin Film Muskat Problem ⋮ A variational formulation of the BDF2 method for metric gradient flows ⋮ An optimal transport approach for the kinetic Bohmian equation ⋮ Multiple patterns formation for an aggregation/diffusion predator-prey system ⋮ Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model ⋮ Gradient structures and geodesic convexity for reaction–diffusion systems ⋮ The Parabolic-Parabolic Keller-Segel System with Critical Diffusion as a Gradient Flow in ℝ^d,d ≥ 3 ⋮ Simulation of multiphase porous media flows with minimising movement and finite volume schemes ⋮ Large time behavior of a two phase extension of the porous medium equation ⋮ Well-posedness of evolution equations with time-dependent nonlinear mobility: a modified minimizing movement scheme ⋮ Well-Posedness and Self-Similar Asymptotics for a Thin-Film Equation ⋮ The Spectrum of a Family of Fourth-Order Nonlinear Diffusions Near the Global Attractor ⋮ VARIATIONAL FORMULATION OF THE FOKKER–PLANCK EQUATION WITH DECAY: A PARTICLE APPROACH ⋮ Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients ⋮ Construction of a two-phase flow with singular energy by gradient flow methods ⋮ On the critical mass Patlak-Keller-Segel system for multi-species populations: global existence and infinte time aggregation ⋮ The $L^1$-contraction principle in optimal transport ⋮ The Hele–Shaw flow as the sharp interface limit of the Cahn–Hilliard equation with disparate mobilities ⋮ Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Contractions in the 2-Wasserstein length space and thermalization of granular media
• Higher order nonlinear degenerate parabolic equations
• Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, and nonradial focusing geometries
• The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation
• Long-Time asymptotics for strong solutions of the thin film equation
• A convexity principle for interacting gases
• Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions.
• Fast diffusion to self-similarity: Complete spectrum, long-time asymptotics, and numerology
• Convex functionals of probability measures and nonlinear diffusions on manifolds
• Asymptotic equipartition and long time behavior of solutions of a thin-film equation
• Global Nonnegative Solutions of a Nonlinear Fourth-Order Parabolic Equation for Quantum Systems
• THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION
• Transport inequalities, gradient estimates, entropy and Ricci curvature
• An algorithmic construction of entropies in higher-order nonlinear PDEs
• The Derrida–Lebowitz–Speer–Spohn Equation: Existence, NonUniqueness, and Decay Rates of the Solutions
• Dynamics of an anchored Toom interface
• Thin Films with High Surface Tension
• Lubrication approximation with prescribed nonzero contact anggle
• Existence and positivity of solutions of a fourth‐order nonlinear PDE describing interface fluctuations
• The Variational Formulation of the Fokker--Planck Equation
• On a Fourth-Order Degenerate Parabolic Equation: Global Entropy Estimates, Existence, and Qualitative Behavior of Solutions
• Fluctuations of a stationary nonequilibrium interface
• Droplet Spreading Under Weak Slippage—Existence for the Cauchy Problem
• Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities
• Variational formulation for the lubrication approximation of the Hele-Shaw flow

This page was built for publication: A Family of Nonlinear Fourth Order Equations of Gradient Flow Type