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1. Refined asymptotics for the Cauchy problem for the fast $p$-Laplace evolution equation

Author

Bonforte, Matteo, Chlebicka, Iwona, and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs
Abstract

Our focus is on the fast diffusion equation $\partial_t u=\Delta_p u$ with $p<2$ in the whole Euclidean space of dimension $N\geq 2$. The properties of the solutions to the $p$-Laplace Cauchy problem change in several special values of the parameter $p$. In the range of $p$ when mass is conserved, non-negative and integrable solutions behave like the Barenblatt (or fundamental) solutions for large times. By making use of the entropy method, we establish the polynomial rates of the convergence in the uniform relative error for a natural class of initial data. The convergence was established in the literature for $p$ close to $2$, but no rates were available. In particular, we allow for the values of $p$, for which the entropy is not displacement convex, as we do not apply the optimal transportation tools. We approach the issue of long-term asymptotics of the gradients of solutions. In fact, in the case of the radial initial datum, we provide also polynomial rates of the uniform convergence in the relative error of radial derivatives of solutions for $\frac{2N}{N+2}

Published
2024

2. Stability for the logarithmic Sobolev inequality

Author

Brigati, Giovanni, Dolbeault, Jean, and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs, Mathematics - Functional Analysis, 39B62, 47J20, 49J40, 35A23, 35K85
Abstract

This paper is devoted to stability results for the Gaussian logarithmic Sobolev inequality, with explicit stability constants.

Published
2023

3. On Gaussian interpolation inequalities

Author

Brigati, Giovanni, Dolbeault, Jean, and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs, Mathematics - Functional Analysis, 26D10, 46T12, 46E35, 39B62, 43A90, 49J40, 58E35
Abstract

This paper is devoted to Gaussian interpolation inequalities with endpoint cases corresponding to the Gaussian Poincar\'e and the logarithmic Sobolev inequalities, seen as limits in large dimensions of Gagliardo-Nirenberg-Sobolev inequalities on spheres. Entropy methods are investigated using not only heat flow techniques but also nonlinear diffusion equations as on spheres. A new stability result is established for the Gaussian measure, which is directly inspired by recent results for spheres.

Published
2023

4. Logarithmic Sobolev and interpolation inequalities on the sphere: constructive stability results

Author

Brigati, Giovanni, Dolbeault, Jean, and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs, Mathematics - Functional Analysis, 26D10, 58J99, 39B62, 43A90, 49J40, 46E35
Abstract

We consider Gagliardo-Nirenberg inequalities on the sphere which interpolate between the Poincar\'e inequality and the Sobolev inequality, and include the logarithmic Sobolev inequality as a special case. We establish explicit stability results in the subcritical regime using spectral decomposition techniques, and entropy and carr\'e du champ methods applied to nonlinear diffusion flows.

Published
2022

6. Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs, 26D10, 46E35, 35K55, 35B40, 49K20
Abstract

We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy - entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case.

Published
2022

7. Functional inequalities and applications to doubly nonlinear diffusion equations

Author

Chlebicka, Iwona and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs, Mathematics - Functional Analysis, 26D10 35K55 35B40 49K35
Abstract

We study weighted inequalities of Hardy and Hardy-Poincar\'e type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile., Comment: 20 pages, no figures

Published
2021

8. The Cauchy problem for the fast $p-$Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

Author

Bonforte, Matteo, Simonov, Nikita, and Stan, Diana

Subjects
Mathematics - Analysis of PDEs, 35K55, 35K92, 35B45, 35B40, 35K15, 35K67, 35C06
Abstract

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=\Delta_p u$ on the whole Euclidean space, in the so-called "good fast diffusion range" $\tfrac{2N}{N+1}

Published
2021

9. Stability in Gagliardo-Nirenberg inequalities - Supplementary material

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs, 26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20
Abstract

This document comes as supplementary material of the paper Stability in Gagliardo-Nirenberg inequalities by the same authors. It is intended to state a number of classical or elementary statements concerning constants and inequalities for which we are not aware of existing published material or expressions detailed enough for our purpose. We claim no originality on the theoretical results and rely on standard methods in most cases, except that we keep track of the constants and provide constructive estimates.

Published
2020

10. Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs, 26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20
Abstract

The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods., Comment: This manuscript merges 2007.03674: Stability in Gagliardo-Nirenberg inequalities and 2007.03419: Stability in Gagliardo-Nirenberg inequalities. Supplementary material with several additions including: Chapter 1 (variational methods) and Chapter 6 (Sobolev's inequality)

Published
2020

11. Fine properties of solutions to the Cauchy problem for a Fast Diffusion Equation with Caffarelli-Kohn-Nirenberg weights

Author

Bonforte, Matteo and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs, 35B40, 35B45, 35K55, 35K67, 35K65
Abstract

We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the Fast Diffusion Equation with weights (WFDE) $u_t=|x|^\gamma\mathrm{div}\left(|x|^{-\beta}\nabla u^m\right)$ posed on $(0,+\infty)\times\mathbb{R}^d$, with $d\ge 3$, in the so-called good fast diffusion range $m_c

Published
2020

12. Quantitative a Priori Estimates for Fast Diffusion Equations with Caffarelli-Kohn-Nirenberg weights. Harnack inequalities and H\'older continuity

Author

Bonforte, Matteo and Simonov, Nikita

Subjects
Mathematics - Analysis of PDEs, 35B45, 35B65, 35K55, 35K67, 35K65
Abstract

We study a priori estimates for a class of non-negative local weak solution to the weighted fast diffusion equation $u_t = |x|^{\gamma} \nabla\cdot (|x|^{-\beta} \nabla u^m)$, with $0 < m <1$ posed on cylinders of $(0,T)\times{\mathbb R}^N$. The weights $|x|^{\gamma}$ and $|x|^{-\beta}$, with $\gamma < N$ and $\gamma -2 < \beta \leq \gamma(N-2)/N$ can be both degenerate and singular and need not belong to the class $\mathcal{A}_2$, a typical assumption for this kind of problems. This range of parameters is optimal for the validity of a class of Caffarelli-Kohn-Nirenberg inequalities, which play the role of the standard Sobolev inequalities in this more complicated weighted setting. The weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios: for instance, the scaling properties of the equation change when considering the problem around the origin or far from it. We therefore prove quantitative - with computable constants - upper and lower estimates for local weak solutions, focussing our attention where a change of geometry appears. Such estimates fairly combine into forms of Harnack inequalities of forward, backward and elliptic type. As a consequence, we obtain H\"older continuity of the solutions, with a quantitative (even if non-optimal) exponent. Our results apply to a quite large variety of solutions and problems. The proof of the positivity estimates requires a new method and represents the main technical novelty of this paper. Our techniques are flexible and can be adapted to more general settings, for instance to a wider class of weights or to similar problems posed on Riemannian manifolds, possibly with unbounded curvature. In the linear case, $m=1$, we also prove quantitative estimates, recovering known results in some cases and extending such results to a wider class of weights., Comment: 53 pages, 2 figures. This is a much improved version of the older manuscript, thanks to the suggestions of the anonymous referee

Published
2018

18. Functional inequalities and applications to doubly nonlinear diffusion equations.

Author

Chlebicka, Iwona and Simonov, Nikita

Subjects
*BURGERS' equation, *HEAT equation
Abstract

We study weighted inequalities of Hardy and Hardy–Poincaré type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile. [ABSTRACT FROM AUTHOR]

Published
2024
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23. Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations.

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, and Simonov, Nikita

Subjects
HEAT equation, EVOLUTION equations, INTERPOLATION, FAMILY stability, REACTION-diffusion equations, ENTROPY
Abstract

We provide a scheme of a recent stability result for a family of Gagliardo-Nirenberg-Sobolev (GNS) inequalities, which is equivalent to an improved entropy – entropy production inequality associated with an appropriate fast diffusion equation (FDE) written in self-similar variables. This result can be rephrased as an improved decay rate of the entropy of the solution of (FDE) for well prepared initial data. There is a family of Caffarelli-Kohn-Nirenberg (CKN) inequalities which has a very similar structure. When the exponents are in a range for which the optimal functions for (CKN) are radially symmetric, we investigate how the methods for (GNS) can be extended to (CKN). In particular, we prove that the solutions of the evolution equation associated to (CKN) also satisfy an improved decay rate of the entropy, after an explicit delay. However, the improved rate is obtained without assuming that initial data are well prepared, which is a major difference with the (GNS) case. [ABSTRACT FROM AUTHOR]

Published
2023
Full Text
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24. Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, Simonov, Nikita, Instituto de Ciencias Matemàticas [Madrid] (ICMAT), Universidad Carlos III de Madrid [Madrid] (UC3M)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Autónoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Project MTM2017-85757-P (Ministry of Science and Innovation, Spain) and the Spanish Ministry of Science and Innovation, through the 'Severo Ochoa Programme for Centres of Excellence in R&D' (CEX2019-000904-S)E.U. H2020 MSCA programme, grant agreement 777822Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR)Inria Mokaplan teamDIM Math-Innov of the Region Île-de-France, ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), and European Project: 777822,GHAIA(2017)

Subjects
intermediate asymptotics, Mathematics::Analysis of PDEs, fast diffusion equation, rates of convergence, stability, Harnack Principle, Hardy-Poincaré inequalities, spectral gap, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], self-similar Barenblatt solutions, asymptotic behavior, 2020 Mathematics Subject Classification.26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20, Gagliardo-Nirenberg inequality, entropy methods
Abstract

Ce document contient hal-02887010, v1: Stability in Gagliardo-Nirenberg inequalities and hal-02887013, v1: Stability in Gagliardo-Nirenberg inequalities. Supplementary material, avec plusieurs additions dont: Chapitre 1 (variational methods) et Chapitre 6 (Sobolev's inequality).; International audience; The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods.

Published
2022

26. Different Behavior of 2-Substituted 3-Nitro-2 H -chromenes in the Reaction with Stabilized Azomethine Ylides Generated from α-Iminoesters.

Author

Kochnev, Ivan A., Barkov, Alexey Y., Simonov, Nikita S., Ulitko, Maria V., Zimnitskiy, Nikolay S., Korotaev, Vladislav Y., and Sosnovskikh, Vyacheslav Y.

Subjects
MANNICH reaction, YLIDES, ISOMERS, IMINES
Abstract

The AgOAc-catalysed reaction of 3-nitro-2-phenyl-2H-chromenes with stabilized azomethine ylides generated from the imines based on methyl glycinate and arylaldehydes leads to a mixture of endo and endo' isomers of the corresponding chromeno[3,4-c]pyrrolidines in a ratio of 2.0–2.3:1 in 85–93% total yields as a result of a Michael addition/Mannich reaction sequence. In a similar reaction involving 2-trifluoromethyl-3-nitro-2H-chromenes, only endo chromeno[3,4-c]pyrrolidines are formed in 85–94% yields. 3-Nitro-2-(trichloromethyl)-2H-chromenes under the same conditions react with these azomethine ylides to give the corresponding Michael adducts as individual anti-isomers with the cis,trans-configuration of the chromane ring in 40–67% yields. Some 4-CF3-substituted chromano[3,4-c]pyrrolidines exhibited high cytotoxic activity against HeLa human cervical carcinoma cells. [ABSTRACT FROM AUTHOR]

Published
2022
Full Text
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27. Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, Simonov, Nikita, Instituto de Ciencias Matemàticas [Madrid] (ICMAT), Universidad Autónoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Carlos III de Madrid [Madrid] (UC3M), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), SAMM - Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Project MTM2017-85757-P (Ministry of Science and Innovation, Spain) and the Spanish Ministry of Science and Innovation, through the 'Severo Ochoa Programme for Centres of Excellence in R&D' (CEX2019-000904-S)E.U. H2020 MSCA programme, grant agreement 777822Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR)Inria Mokaplan teamDIM Math-Innov of the Region Île-de-France, ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), European Project: 777822,GHAIA(2017), Universidad Autonoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Carlos III de Madrid [Madrid] (UC3M), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)

Subjects
intermediate asymptotics, Mathematics::Analysis of PDEs, fast diffusion equation, rates of convergence, stability, Harnack Principle, 26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20, Mathematics - Analysis of PDEs, Hardy-Poincaré inequalities, spectral gap, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], self-similar Barenblatt solutions, asymptotic behavior, 2020 Mathematics Subject Classification.26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20, Gagliardo-Nirenberg inequality, Analysis of PDEs (math.AP), entropy methods
Abstract

The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods., Comment: This manuscript merges 2007.03674: Stability in Gagliardo-Nirenberg inequalities and 2007.03419: Stability in Gagliardo-Nirenberg inequalities. Supplementary material with several additions including: Chapter 1 (variational methods) and Chapter 6 (Sobolev's inequality)

Published
2021

28. Explicit constants in Harnack inequalities and regularity estimates, with an application to the fast diffusion equation

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, Simonov, Nikita, Universidad Autonoma de Madrid (UAM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris sciences et lettres (PSL), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), Universidad Autónoma de Madrid (UAM), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Inria de Paris

Subjects
Entropy methods, Rates of convergence, Self-similar Barenblatt solutions, Moser's Harnack inequality, MathematicsofComputing_NUMERICALANALYSIS, Harnack Principle, Asymptotic behavior, Fast diffusion equation, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20, Intermediate asymptotics, Stability, Gagliardo-Nirenberg inequality
Abstract

This paper is devoted to the computation of various explicit constants in functional inequalities and regularity estimates for solutions of parabolic equations, which are not available from the literature. We provide new expressions and simplified proofs of the Harnack inequality and the corresponding Hölder continuity of the solution of a linear parabolic equation. We apply these results to the computation of a constructive estimate of a threshold time for the uniform convergence in relative error of the solution of the fast diffusion equation.

Published
2020

29. Stability in Gagliardo-Nirenberg inequalities

Author

Bonforte, Matteo, Dolbeault, Jean, Nazaret, Bruno, Simonov, Nikita, Universidad Autonoma de Madrid (UAM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Statistique, Analyse et Modélisation Multidisciplinaire (SAmos-Marin Mersenne) (SAMM), Université Paris 1 Panthéon-Sorbonne (UP1), Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales (MOKAPLAN), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), This work has been partially supported by the Project EFI (ANR-17-CE40-0030) of the French National Research Agency (ANR). M.B. and N.S. were partially supported by Projects MTM2017-85757-P (Spain) and by the E.U. H2020 MSCA programme, grant agreement 777822. N.S. was partially funded by the FPI-grant BES-2015-072962, associated to the project MTM2014-52240-P (Spain)., and ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017)

Subjects
Fast diffusion equation, Self-similar Barenblatt so-lutions, Entropy methods, Rates of convergence, Spectral gap, Hardy-Poincaré inequalities, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Intermediate asymptotics, 2020 Mathematics Subject Classification.26D10, 46E35, 35K55, 49J40, 35B40, 49K20, 49K30, 35J20, Stability, Harnack Principle, Gagliardo-Nirenberg inequality, Asymptotic behavior
Abstract

The purpose of this paper is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg inequalities. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. In both cases, the tail behaviour plays a key role. The regularity properties of the parabolic flow allow us to connect an improved entropy-entropy production inequality during the initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymp-totic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms the distance to the manifold of optimal functions.

Published
2020

30. Fast Diffusion Equations with Caffarelle-Kohn-Nirenberg Weights: Regularity and Asymptotics

Author

Simonov, Nikita, Bonforte, Matteo, UAM. Departamento de Matemáticas, and Bonforte, Matteo (dir.)

Subjects
Matemáticas, Ecuaciones de difusión - Propiedades - Tesis doctorales
Abstract

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 25-02-2020

Published
2020

31. Перетренированность в спорте: симптомы, причины и вред

Author

Simonov Nikita Aleksandrovich and Kaznacheev Vladimir Vasilevich

Subjects
профилактика, спорт, лечение, перетренированность
Abstract

В статье рассматриваются актуальные проблемы темы перетренированности, в частности её симптомы, причины возникновения, а также способы лечения и предотвращения.

Published
2019

33. Перетренированность в спорте: симптомы, причины и вред

Author

Симонов Никита Александрович, ФГБОУ ВО «Донской государственный технический университет», Simonov Nikita Aleksandrovich, FSBEI of HE "Don State Technical University", Казначеев Владимир Васильевич, Kaznacheev Vladimir Vasilevich, Симонов Никита Александрович, ФГБОУ ВО «Донской государственный технический университет», Simonov Nikita Aleksandrovich, FSBEI of HE "Don State Technical University", Казначеев Владимир Васильевич, and Kaznacheev Vladimir Vasilevich

Abstract

В статье рассматриваются актуальные проблемы темы перетренированности, в частности её симптомы, причины возникновения, а также способы лечения и предотвращения.

Published
2019

35. The Cauchy problem for the fast p−Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

Author

Matteo Bonforte, Nikita Simonov, Diana Stan, Simonov, Nikita, Universidad Autonoma de Madrid (UAM), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and Universidad de Cantabria [Santander]

Subjects
Gradient decay estimates, Parabolic Harnack inequalities, Applied Mathematics, General Mathematics, 35B40, Mathematics::Analysis of PDEs, p-Laplacian Equation, 35C06, 35K55, 35K92, 35B45, 35B40, 35K15, 35K67, 35C06, 35B45, 35K92, Mathematics - Analysis of PDEs, Asymptotic behaviour Mathematics Subject Classification. 35K55, 35K15, FOS: Mathematics, 35K67, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Fast Diffusion, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Tail behaviour, Global Harnack principle, Analysis of PDEs (math.AP)
Abstract

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=\Delta_p u$ on the whole Euclidean space, in the so-called "good fast diffusion range" $\tfrac{2N}{N+1}, Comment: 47 pages, 1 figure

Published
2021

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