Juan Luis Vázquez

We examine the validity of the principle of mass conservation for solutions of some typical equations in the theory of nonlinear diffusion, including equations in standard differential form and also their fractional counterparts. In Part I, consisting of the first nine sections, we use as main examples the heat equation, the porous medium equation and the p -Laplacian equation. Though these equations have the form of conservation laws, it happens that in some ranges of exponents, the solutions posed in the whole Euclidean space actually lose mass over time. From the start, we pay attention to the close connection between the validity of mass conservation and the existence of finite-mass self-similar solutions, as well as their role in the asymptotic behaviour of more general classes of solutions. Describing the extent of this connection is the common thread throughout the paper. When mass conservation does not hold, we are led to examine the situation when it is replaced by its extreme alternative, extinction in finite time, a very surprising fact. The equations offer difficult borderline cases in the presence of some critical parameters that we identify. New results are proved to complete the picture; we establish mass conservation or its failure in those pending cases. We also explain the occasional disappearance of the fundamental solutions in a very graphical way, as “escape to infinity”. In this part of the paper, we aim to provide a comprehensive overview of the types of behaviour to be expected in these basic models. Additionally, Section 10 on the doubly nonlinear equation complements this foundational block. The next sections extend the detailed study to other models that occupy a relevant niche in the current literature. Thus, the three sections of Part II are devoted to the discussion of mass conservation for some fractional nonlinear diffusion equations, where the situation is surveyed and a number of new theorems are proved. This is followed by the analogous study of scalar conservation laws in Part III, with surprising similarities. We conclude with a long review of related equations and topics in Part IV. The survey contains an extensive bibliography. Summing up, the paper aims at surveying important topics in nonlinear diffusion related to mass conservation. At the same time, we solve a number of open problems on key issues and point out interesting connections and new directions.

Algorithms rule our world, shaping everything from social media feeds to stock markets. This chapter uncovers the power and peril of data-driven decision-making. We explore how algorithms can manipulate behaviors and entrench inequalities, acting as invisible forces that govern modern life. In the hands of corporations and governments, these “weapons of math destruction” can influence everything from elections to warfare, raising urgent questions about control and accountability in the algorithmic age.

The Argentine mathematician is a leading figure in the development of partial differential equations in the last 50 years, has been awarded the 2023 Abel Prize “for his seminal contributions to regularity theory for nonlinear partial differential equations including free-boundary problems and the Monge–Ampère equation.” His contributions are not limited to these fields, they include problems in fluid mechanics, optimal transportation, non-local operators, homogenization, and other topics with important connections with other sciences. In a series of landmark contributions Luis used brilliant geometric insights together with ingenious analytical tools to study areas of nonlinear mathematics that were barely explored some decades ago and are now in full bloom. This text combines a description of Luis’s main contributions with a glimpse into his life and personality.

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann–Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense.

We construct the Very Singular Solution (VSS) for the Anisotropic Fast Diffusion Equation (AFDE) in the suitable good exponent range of fast diffusion. VSS is a solution that, starting from an infinite mass located at one point as initial datum, evolves according to the corresponding equation as an admissible solution away from the singularity. It is expected to represent important properties of the fundamental solutions when the initial mass is very big. We work in the whole space. In this setting we show that the diffusion process distributes mass from the the initial infinite singularity along the different space directions: up to constant factors, there is a simple partition formula for the anisotropic mass expansion, given approximately as the minimum of separate 1-D VSS solutions. This striking fact is a consequence of the improved scaling properties of the special solution, and it has strong consequences. If we consider the family of Fundamental Solutions for different masses, we prove that they share the same universal tail behaviour as the VSS. Actually, their tail is asymptotically convergent to the unique VSS tail, so there is a VSS partition formula for their profile at spatial infinity. With the help of this analysis we study the behaviour of the class of nonnegative finite-mass solutions of the Anisotropic FDE, and prove the Global Harnack Principle (GHP) and the Asymptotic Convergence in Relative Error (ACRE) under a natural assumption on the decay of the initial tail.