Imagen: Álvaro Minguito/ICMAT

Functional inequalities are a central tool in mathematical analysis and in the study of partial differential equations. Among them, Gagliardo–Nirenberg inequalities occupy a prominent place: they allow us to relate different norms of a function (for example, how much it ‘oscillates’ and how much it ‘grows’) and are fundamental in problems ranging from diffusion (including non-linear diffusion) and fluid mechanics to mathematical physics, biology, engineering and, more recently, machine learning.

In a recent paper, researchers Matteo Bonforte (ICMAT-UAM), Jean Dolbeault (University of Paris-Dauphine-CNRS, France), Bruno Nazaret (University of Paris 1 Panthéon-Sorbonne, France) and Nikita Simonov (Sorbonne University, France) establish a stability result for a broad class of Gagliardo–Nirenberg-Sobolev inequalities, in the subcritical and critical regimes.

The breakthrough not only confirms when the best possible case is achieved in these inequalities, but also measures precisely and constructively – that is, all the constants involved are calculated explicitly – what happens when we are ‘close’ to that optimum.

Many mathematical inequalities have functions that reach the optimal value of the constants in the inequality, i.e., those that satisfy the equality, which are called extremal functions. In the case of Sobolev inequalities (in the critical case), these functions were found in the 1970s, independently, by Thierry Aubin, Giorgio Talenti and Eugene Rodemich. In the case of Gagliardo-Nirenberg inequalities (in the subcritical case, which is more delicate), they were discovered by Manuel Del Pino and Jean Dolbeault in 2002. However, in real applications, it is common to work with functions that are not exactly optimal, but almost optimal (as Plato said, the difference between the ‘world of ideas’ and the ‘real world’).

Now, if a function ‘almost reaches’ equality in a Gagliardo–Nirenberg-Sobolev inequality, is it necessarily close to the optimal functions? A stability result answers this question in the affirmative and, moreover, quantifies this closeness so that the result is useful in applications and allows quantitative information to be extracted about this ‘almost optimal’ function, which is particularly interesting in applications, as it allows accurate predictions to be made.

In this new work, the deviation from the optimal case is measured by means of a deficit, which controls in strong norms the distance to the variety (or family) of extremal functions. In other words, not only is it known that the function is close, but it is also possible to measure precisely how close it is and to state that a ‘quasi-optimal’ function is always close to the extremal functions, which are explicit in this case.

The approach developed in the article connects the stability of a static inequality with the dynamic behaviour of a partial differential equation, specifically, the rapid diffusion equation. Thus, demonstrating stability in the inequality is equivalent to proving that the dynamic flow converges more rapidly towards its steady state (or equilibrium), using an entropy method.

One of the key points is to understand the ‘entropic nature’ of these inequalities: in fact, they can be rewritten as inequalities between entropy and entropy production, associated with the fast diffusion equation. This also reveals a ‘hidden’ feature in these inequalities: their deep connection to the reality of certain physical phenomena.

In addition, the authors propose an interaction between nonlinear analysis, spectral theory, and partial differential equations, which allows the different time scales of the problem (from the initial behaviour to the asymptotic regime) to be connected in a unified framework. The underlying idea is that the non-linear rapid diffusion flow transports a certain function, which represents an initial temperature distribution, for example, in its state of dynamic equilibrium (the non-linear analogue of the Gaussian), which is precisely one of the extremal functions of the Gagliardo-Nirenberg-Sobolev inequalities ‘with the same mass’, which is conserved throughout the evolution. One can thus understand the deep relationship between convergence to equilibrium in nonlinear diffusions and proximity to extremals in functional inequalities. These are two equivalent, but not obvious, ways of interpreting the stability described above. This profound idea, which connects two fields that are a priori very distant, has been introduced in this work for the first time and has the potential to be used for the study of other stability problems, both static and dynamic.

Reference

Bonforte, Matteo; Dolbeault, Jean; Nazaret, Bruno; Simonov, Nikita. Stability in Gagliardo–Nirenberg–Sobolev Inequalities Flows, Regularity, and the Entropy Method. Memoirs of the American Mathematical Society, vol. 308, no. 1554, 2025. American Mathematical Society. Publicado electrónicamente el 8 de abril de 2025. ISBN 978-1-4704-7181-1 (print), 978-1-4704-8141-4 (online). https://doi.org/10.1090/memo/1554