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

Matteo Bonforte, Jean Dolbeault, Bruno Nazaret, Nikita Simonov

TL;DR

This work provides a constructive stability theory for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities by linking deficit to distance from the optimizer manifold through entropy methods. The authors couple a nonlinear flow–the fast diffusion equation–with refined entropy-entropy production inequalities, a global Harnack principle, and spectral analysis of linearized problems to achieve explicit stability constants, even in the critical Sobolev case. They formulate a generalized deficit using relative entropy and nonlinear Fisher information, obtain uniform-in-time and asymptotic improves in decay rates, and prove stability against perturbations measured in strong norms via the entropy framework. The approach hinges on self-similar variables, Barenblatt profiles, a nonlinear carré du champ method, and a careful control of moments, yielding a quantitative bridge between variational stability and dynamical convergence to equilibrium with explicit constants. This framework advances sharp stability results for GNS inequalities and enriches the toolbox for quantitative functional-analytic estimates with potential applications in PDEs and geometric analysis.

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.

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

TL;DR

This work provides a constructive stability theory for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities by linking deficit to distance from the optimizer manifold through entropy methods. The authors couple a nonlinear flow–the fast diffusion equation–with refined entropy-entropy production inequalities, a global Harnack principle, and spectral analysis of linearized problems to achieve explicit stability constants, even in the critical Sobolev case. They formulate a generalized deficit using relative entropy and nonlinear Fisher information, obtain uniform-in-time and asymptotic improves in decay rates, and prove stability against perturbations measured in strong norms via the entropy framework. The approach hinges on self-similar variables, Barenblatt profiles, a nonlinear carré du champ method, and a careful control of moments, yielding a quantitative bridge between variational stability and dynamical convergence to equilibrium with explicit constants. This framework advances sharp stability results for GNS inequalities and enriches the toolbox for quantitative functional-analytic estimates with potential applications in PDEs and geometric analysis.

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.

Paper Structure

This paper contains 132 sections, 86 theorems, 910 equations, 2 figures, 1 table.

Table of Contents

  1. Gagliardo-Nirenberg-Sobolev inequalities by variational methods
  2. Gagliardo-Nirenberg-Sobolev inequalities
  3. A one-parameter family of inequalities
  4. Optimality
  5. Related inequalities
  6. An elementary proof of the inequalities
  7. Sobolev's inequality in a nutshell
  8. Schwarz symmetrization
  9. Equivalent formulations
  10. Existence and uniqueness of an optimal function
  11. Existence of an optimal function in the subcritical range
  12. Non-scale invariant Gagliardo-Nirenberg-Sobolev inequalities
  13. Restriction to nonnegative functions
  14. A variational problem
  15. Concentration-compactness and the subcritical inequalities
  16. ...and 117 more sections

Key Result

Theorem A

Assume that $d\ge2$, $m\in[m_1,1)$. If $v$ solves FDr-Intro for some initial datum $v_0\ge0$ such that $\int_{{\mathbb R}^d}{v_0}\,\,{\rm d}x=\int_{{\mathbb R}^d}{\mathcal{B}}\,\,{\rm d}x$, and then there exists an explicit $\varepsilon_\star$ such that, for any $\varepsilon\in(0,\varepsilon_\star)$,

Figures (2)

  • Figure 1: In $(X,Y)$ coordinates, arrows represent the vector field which corresponds to \ref{['XY']}, i.e., $(\hbox{\sc a}\,Y-4\,X,-\,\hbox{\sc b}\,Y)$. The line $\hbox{\sc a}\,Y-4\,X=0$ determines the sign of $X'$. The ellipse defined by \ref{['Ellipse']} is also represented. White areas are the regions such that \ref{['Constraints:KS']} holds: the upper limit is $Y=\psi(X)$. Here we have $d=3$, $m=m_1=2/3$, $\hbox{\sc a}=2\,d\,(1-m)/m=3$ and $\hbox{\sc b}=2\,d\,(m-m_c)=2\,\alpha$ and scales are in units of $\mathcal{M}$. The qualitative properties are independent of the dimension $d\ge3$ and of the exponent $m\in[m_1,1)$.
  • Figure 2: Regions A, B, C appear (from left to right) in green with $d=3$, $m=m_1=2/3$, $\hbox{\sc a}=\frac{2\,d}{m}\,(1-m)$ and $\hbox{\sc b}=2\,d\,(m-m_c)$.

Theorems & Definitions (153)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • Theorem 1.5
  • ...and 143 more