ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS

By Anton Arnold, Peter A. Markowich, Giuseppe Toscani, Andreas Unterreiter. 2001

464 citations recorded by the source corpus.

Subfields: Applied Mathematics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2439af7792127974db4fb1abcee0e13521176be7

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Citations in the corpus

Existence, Uniqueness and Lipschitz Dependence for Patlak-Keller-Segel and Navier-Stokes in R2 with Measure-valued Initial Data (2012, background): “Indeed, recall that ‖ f ‖∞ ‖ f ‖ 2 ‖∇ f ‖ 4 ‖ f ‖X , and since m > 2, for p ∈ [1, 2), ‖ f ‖p ‖ f ‖L2(m) ‖ f ‖X .”
Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$R2 with Measure-Valued Initial Data (2012, background): “Then for some ν ∈ (0, 1/2) (which depends on α), all q ∈ [1, 2], all β sufficiently small (in absolute value), and all f ∈ Lq(m), ‖(Tα+β(τ)− Tα(τ))∇f‖L2(m) .”
Phi-entropy inequalities for diffusion semigroups (2008, methodology): “Remark 3 A general study of admissible functions is performed in [AMTU01], [Cha04, Cha06] and [EVO09]: for instance a C(4) strictly convex function Φ on an interval I of R is admissible if and only if Φ((4))(x) Φ(x) ≥ 2 Φ(x) for all x, and if and only if the map (x, y) 7→ Φ(x) y(2) is convex on I × ”
Factorization for non-symmetric operators and exponential H-theorem (2010, background): “The latter is satisfied for instance under the so-called Bakry-Emery condition Hess (φ) ≥ ν Id (see [4, 28, 3]).”
Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities (2001, methodology): “This method has been applied successfully in the linear case (general linear Fokker±Planck equations see [3], [16]) and in the porous medium/ fast diffusion cases [17], [28], [56].”
Long-Time Asymptotics of Kinetic Models of Granular Flows (2004, background): “The previour analysis helps to recognize that, at least for γ < 0, the stationary solutions to equation (24) do not play a general role of attraction, like it happens for a large class of nonlinear Fokker-Planck like equations studied in [1, 8].”
Cercignani's Conjecture is Sometimes True and Always Almost True (2003, background): “This question is very representative of recent trends in partial dieren tial equations: in many cases of interest, an entropy principle has been identied, and a detailed study of the entropy production principle provides a sharp insight into the problem of asymptotic behavior of the equation under c”
Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality (2000, methodology): “Let [0; 1 ] 3 t 7! „t be a curve on P. According to the above definitions, its tangent vector field [0;1] 3 t 7! 't is given by the identity (holding in weak sense)”
Nonlinear diffusion equations with variable coefficients as gradient flows in Wasserstein spaces (2009, unknown): “This condition is exactly the Bakry-Emery condition for logarithmic Sobolev inequalities (see condition (A1) in [6]), precisely 1 2 ΔaI − 1 2 〈∇a,∇V 〉I −H(a) + aH(V ) + 1 2 ∇a⊗∇V + 1 2 ∇V ⊗∇a− αI ≥ 0.”
Sharp logarithmic Sobolev inequalities on gradient solitons and applications (2008, methodology): “However, as it was discovered in the case of Rn, and in the case of a manifold in [1,2] for linear diffusions or in [7,16,32] for nonlinear diffusions, this relation is really through functional inequalities, see also [33].”
Stability and Asymptotic Analysis of a Fluid-Particle Interaction Model (2006, unknown): “By using the Sobolev inequality Arnold et al., 2001, Corollary 2.18 we obtain for some $ > 0 ∫ 3 ∣∣f − M ∣∣d ≤ ( ∫ 3 ∣∣f − M ∣∣2 1M d )1/2 ≤ ( ∫ 3 ∣∣∣∣ f M − 1 ∣∣∣∣2M d )1/2 ≤ $ ( ∫ 3 ∣∣∣∣ ( f M )∣∣∣∣2M d )1/2 ≤ $ ( ∫ 3 ∣∣∣∣ ( f M )∣∣∣∣2M d )1/2 The integral of the right-hand side over 0 T × " is…”
A Kinetic Flocking Model with Diffusion (2010, background): “Firstly, it is well-known from [1] that LFP enjoys some dissipative properties, stated in the following Proposition 2.”
Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions☆ (2002, unknown): “The entropy-entropy dissipation method has been used for the heat equation in [42,43,1] and generalized to nonlinear diffusions in [17,12] (also see [36] by F.”
Chapter 1 – Gradient Flows of Probability Measures (2007, background): “4a,b,c) [74, 13, 25, 29, 2, 83, 42], or to prove uniqueness under weak assumptions on the data.”