# One day meeting in Nonlinear PDEs

#### 4th May 2017

###### Organisers: Charlie Elliott, Jose Rodrigo, Juan Luis Vázquez (UAM-Madrid)*

This one day meeting integrates in the Warwick-EPSRC Symposium in PDEs and Applications. It celebrates the visit of three distinguished mathematicians to Warwick, Yoshikazu Giga (Tokyo), Juan Luis Vázquez (currently a Warwick IAS Fellow), and Juan José López Velázquez (Bonn). More details about the specific topics can be found below.

###### Programme

All lectures will take place in B3.03.

12:00-13:00 | Juan Luis Vázquez | Linear and nonlinear diffusion with nonlocal fractional operators |

13:00-14:30 | Lunch | |

14:30-15:30 | Juan José López Velázquez | Long time asymptotics of homoenergetic solutions for the Boltzmann equation |

15:30-16:00 | Tea | |

16:00-17:00 | Yoshikazu Giga | Approximation by Cahn-Hoffman facets and the crystalline mean curvature flow |

17:00 | Wine Reception |

###### Abstracts

**Juan Luis Vázquez** - *Linear and nonlinear diffusion with nonlocal fractional operators*

This is a report on some of the progress made by the author and collaborators on the topic of linear and nonlinear diffusion equations involving long distance interactions in the form of fractional Laplacian operators. The heat equation is the well-established paradigm for heat transportation and diffusion processes. The probabilistic model behind it is the Brownian motion. Diffusion equations involving non-local effects, hence non-Brownian processes, have been studied for a number of years but only recently they have become the object of intense work in mathematical analysis. Results cover well-posedness, regularity, free boundaries, asymptotics, extinction, and others. Differences with standard diffusion have been specially examined.

**Juan José López Velázquez** - *Long time asymptotics of homoenergetic solutions for the Boltzmann equation*

Homoenergetic solutions are a particular class of solutions of the Boltzmann equation which were introduced in the 1950's by Galkin and Truesdell. They are useful to describe the dynamics of Boltzmann gases under shear, expansion or compression in nonequilibrium situations. Homoenergetic solutions are much simpler than the general solutions of the Boltzmann equation. Their well posedness theory, which has many similarities with the theory of homogeneous solutions of the Boltzmann equation was studied by Cercignani in the 1980s. However, the corresponding long time asymptotics theory differs much of the analogous theory for homogeneous solutions. Actually, in the case of homoenergetic solutions, the long time asymptotics cannnot always be described using Maxwellian distributions. For several collision kernels the long time behaviour of homoenergetic solutions is given by particle distributions which do not satisfy the detailed balance condition. In this talk I will describe different possible long time asymptotics of homoenergetic solutions of the Boltzmann equation as well as some open problems in this direction. (Joint work with R.D. James, S. Müller and A. Nota).

**Yoshikazu Giga** - *Approximation by Cahn-Hoffman facets and the crystalline mean curvature flow*

We are interested in approximation of a general compact set in an Euclidean space by nicer sets. In fact, we show that every compact set can be monotonically approximated by a set admitting a certain vector field called the Cahn-Hoffman vector field. Such a set is called a Cahn-Hoffman facet. If the divergence of the minimal Cahn-Hoffman vector field is constant such a set is often called a Cheeger set, which has been widely studied by B. Kawohl, V. Caselles and others.

More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner.

It turns out that this approximation is useful to construct suitable test functions necessary to establish comparison principle for level-set crystalline mean curvature flow equations. As a consequence, we obtained the well-posedness in arbitrary dimension. For a total variation flow of non-divergence type such a comparison result has been established by a joint work with M.-H. Giga and N. Pozar (2014). This lecture is based on my joint work with Norbert Pozar of Kanazawa University.

*IAS Visiting Fellow

Funded by:

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Mathematics Research Centre

Mathematical Interdisciplinary Research at Warwick (MIR@W)

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