## Revista Matemática Iberoamericana

- Rev. Mat. Iberoamericana
- Volume 27, Number 3 (2011), 803-839.

### Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski's coagulation equation

José A. Cañizo and Stéphane Mischler

#### Abstract

We consider Smoluchowski's equation with a homogeneous kernel of the form $a(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha$ with $-1 < \alpha \leq \beta < 1$ and $\lambda := \alpha + \beta \in (-1,1)$. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at $y = 0$ in the case $\alpha < 0$. We also give some partial uniqueness results for self-similar profiles: in the case $\alpha = 0$ we prove that two profiles with the same mass and moment of order $\lambda$ are necessarily equal, while in the case $\alpha < 0$ we prove that two profiles with the same moments of order $\alpha$ and $\beta$, and which are asymptotic at $y = 0$, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.

#### Article information

**Source**

Rev. Mat. Iberoamericana, Volume 27, Number 3 (2011), 803-839.

**Dates**

First available in Project Euclid: 9 August 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.rmi/1312906778

**Mathematical Reviews number (MathSciNet)**

MR2895334

**Zentralblatt MATH identifier**

1242.82031

**Subjects**

Primary: 82C21: Dynamic continuum models (systems of particles, etc.) 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 82C05: Classical dynamic and nonequilibrium statistical mechanics (general)

**Keywords**

coagulation self-similarity regularity uniqueness asymptotic behavior

#### Citation

Cañizo, José A.; Mischler, Stéphane. Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski's coagulation equation. Rev. Mat. Iberoamericana 27 (2011), no. 3, 803--839. https://projecteuclid.org/euclid.rmi/1312906778