Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities

By José Antonio Carrillo, A. Jungel, Peter A. Markowich, Giuseppe Toscani, Andreas Unterreiter. 2001

374 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/ca7515d337cabd9402ac35b40567883be5a17fa4

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Citations in the corpus

Local and global well-posedness for aggregation equations and Patlak–Keller–Segel models with degenerate diffusion (2010, background): “(11) See [43].”
Asymptotic equipartition and long time behavior of solutions of a thin-film equation☆ (2007, methodology): “Differential Equations 241 (2007) 279–292 By using the explicit formula for the function v(∞) and proceeding as in the analysis of second-order degenerate diffusion in [5] we obtain”
Fine Asymptotics for Fast Diffusion Equations (2003, methodology): “There are some technical issues concerning the convergence of the dissipation of entropy Dð Þ (see Otto, 2001; Carrillo et al., 2001), which takes place only in a weak sense and thus, one can only justify d ds Eð j 1Þ Dð Þ in this way.”
Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates (2003, methodology): “With this result at hand, one can easily work out, for instance, the Bakry-Emery strategy for generalized logarithmic Sobolev inequalities in the same way as in Carrillo, Jüngel, Markowich, Toscani and Unterreiter [9] (in this reference, the authors always perform the Bakry-Emery strategy on very s”
Global existence and convergence to steady states in a chemorepulsion system (2008, background): “, [5] and the references therein) to obtain 1 2u ‖u− u‖(2)1 ≤ G(u(t), v(t)) ≤ G(u0, v0)e, and hence the exponential convergence in L(1)(Ω).”
Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation (2002, methodology): “A Csiszár–Kullback type-inequality [20,14] shows that the L1 deviation of v with respect tov∞ is bounded byH(v(t)|v∞) and thus, one proves the exponential convergence to the equilibriumv∞ in L1.”
Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold (2008, neutral): “also [7], [9], [12], [14], [16], [20], [22], [24], [26].”
Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the p-Laplacian (2002, methodology): “Recently, an approach based on an entropy-entropy production method [1,3,11,2] has been developed, but it has apparently not been sufficient to catch the int rmediate asymptotics , i.”
Weak–Strong Uniqueness Property for the Full Navier–Stokes–Fourier System (2011, neutral): “[4], Dafermos [5], SaintRaymond [23]), represented in the present context by the quantity E(̺, θ | ̺̃, θ̃) = Hθ̃(̺, θ)− ∂̺Hθ̃( ̺̃, θ̃)(̺− ̺̃)−Hθ̃( ̺̃, θ̃), (1.”
A Family of Nonlinear Fourth Order Equations of Gradient Flow Type (2009, result): “H; . The following result can be found in [ 11, Theorem 30 ].”
The boundedness-by-entropy method for cross-diffusion systems (2015, background): “vergence in terms of the relative entropy H∗. By the Csisz´ar-Kullback inequality, we conclude the exponential decay in the L1 norm, ku(t) −u∞kL1(Ω) ≤ Cexp(−λt/2) for t>0. For details, we refer to [11]. The main task is to derive the entropy-dissipation relation (57). According to Hypothesis H2’,”