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A survey on mass conservation, self-similarity and related topics in nonlinear diffusion

Juan Luis Vázquez

TL;DR

This survey investigates when mass is conserved in nonlinear diffusion and related equations, including fractional variants, by linking mass conservation to the existence of self-similar solutions and long-time asymptotics. It develops a unifying framework around the Heat Equation, the Porous Medium Equation, and the p-Laplacian, detailing how Barenblatt and Gaussian self-similar profiles govern mass persistence or extinction across slow, good fast, and very fast diffusion regimes, and extending these ideas to fractional diffusion and scalar conservation laws. The work provides new results on relative mass conservation, critical-exponent behavior, and measure-valued limits, while also exposing regimes where mass can vanish in finite or infinite time and where contraction in $L^p$ spaces fails. Overall, it clarifies the systematic connections among MC, self-similarity, and asymptotics for a broad class of nonlinear diffusion models and their fractional/nonlocal variants, offering rigorous insights and practical implications for modeling diffusion phenomena with mass transfer or loss.

Abstract

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 1, consisting of the first 9 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 lose mass in 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 manuscript. 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 next sections extend the detailed study to other models that occupy a relevant role in the current literature. Thus, the 3 sections of Part 2 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 3, with surprising similarities. We conclude with a long review of related equations and topics in Part 4. The survey contains an extensive bibliography.

A survey on mass conservation, self-similarity and related topics in nonlinear diffusion

TL;DR

This survey investigates when mass is conserved in nonlinear diffusion and related equations, including fractional variants, by linking mass conservation to the existence of self-similar solutions and long-time asymptotics. It develops a unifying framework around the Heat Equation, the Porous Medium Equation, and the p-Laplacian, detailing how Barenblatt and Gaussian self-similar profiles govern mass persistence or extinction across slow, good fast, and very fast diffusion regimes, and extending these ideas to fractional diffusion and scalar conservation laws. The work provides new results on relative mass conservation, critical-exponent behavior, and measure-valued limits, while also exposing regimes where mass can vanish in finite or infinite time and where contraction in spaces fails. Overall, it clarifies the systematic connections among MC, self-similarity, and asymptotics for a broad class of nonlinear diffusion models and their fractional/nonlocal variants, offering rigorous insights and practical implications for modeling diffusion phenomena with mass transfer or loss.

Abstract

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 1, consisting of the first 9 sections, we use as main examples the heat equation, the porous medium equation and the -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 lose mass in 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 manuscript. 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 next sections extend the detailed study to other models that occupy a relevant role in the current literature. Thus, the 3 sections of Part 2 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 3, with surprising similarities. We conclude with a long review of related equations and topics in Part 4. The survey contains an extensive bibliography.
Paper Structure (79 sections, 47 theorems, 512 equations, 9 figures, 1 table)

This paper contains 79 sections, 47 theorems, 512 equations, 9 figures, 1 table.

Table of Contents

  1. Conservation of mass in Diffusion
  2. General concepts, notations and comments
  3. Models of nonlinear diffusion (NLD)
  4. Outline and main results in standard nonlinear diffusion
  5. First study of mass conservation and self-similarity
  6. Mass conservation and the Heat Equation
  7. Mass conservation and its connections for the PME
  8. Slow diffusion range of the PME
  9. The limit case $m\to \infty$ of the PME. A break of the theory
  10. Fast Diffusion. The critical exponent
  11. The very singular solution, VSS
  12. Mass conservation and self-similarity for the PLE
  13. The very singular solution of the Fast PLE
  14. Limit $p\to \infty$ of the profiles. Sand piles
  15. Extending the MC and the contraction laws
  16. ...and 64 more sections

Key Result

Proposition 3.1

For every nonnegative solution $u(x,t)$ of the PME with initial datum $u_0\in L^1({\mathbb{R}^N})$ and every $t>0$ we have where $M=\int_{{\mathbb{R}^N}} u_0(x)\,dx$, and $\alpha$ and $\beta$ are the similarity exponents found in for.simexp.pme. Equality holds at a certain time $t=t_1$ if $u(x,t_1)$ is the Barenblatt solution with mass $M$ or a space translation thereof.

Figures (9)

  • Figure 1: Gaussian function evolution by the HE
  • Figure 2: Evolving Barenblatt Solution represented at different times, starting from a Dirac delta at $t=0$. The free boundary $x(t)$ lies at a finite distance, moving away with time as indicated by the obtained formulas
  • Figure 3: Barenblatt solutions for large $m$. Mesa formation
  • Figure 4: Fast diffusion. Barenblatt profiles
  • Figure 5: Formation of a conic sandpile in the limit $p\to\infty$
  • ...and 4 more figures

Theorems & Definitions (48)

  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 5.1
  • Lemma 5.2
  • Corollary 5.3
  • Proposition 5.1
  • ...and 38 more