Programme

The conference will be split into two sessions per day in order to accomodate participants from various parts of the world. The morning session (9:30 -- 12:00) will be focused on evolution of interfaces and related problems. The afternoon session (17:00 -- 19:00) will be dedicated to fractional differential equations. 

Each talk is 20 up to 30 minutes long followed by a discussion. Additional breakout rooms will be created for possible private discussions. 

 

Schedule

	October 1	October 2
09:30 – 10:00 CEST / 16:30 – 17:00 JST	Y. Giga	S. Moll
10:20 – 10:50 CEST / 17:20 – 17:50 JST	M. Novaga	P.B. Mucha
11:10 – 11:40 CEST / 18:10 – 18:40 JST	M. Latorre	L. Giacomelli
12:00 – 17:00 CEST	Break	Break
17:00 – 17:30 CEST / 10:00 CDT / 12:00 ART	R. Zacher	S. Roscani
17:50 – 18:20 CEST / 10:50 CDT / 12:50 ART	K. Ryszewska	A. Kubica

 

	October 3
10:30 – 11:00 CEST / 17:30 – 18:00 JST	K. Shirakawa
11:20 – 11:50 CEST / 18:10 – 18:40 JST	W. Górny
12:00 – 12:30 CEST / 19:00 – 19:30 JST / 07:00 – 07:30 ART	M. Yamamoto
13:00 – 17:00 CEST	Break
17:00 – 17:30 CEST / 10:00 CDT / 12:00 ART	V. Voller
17:50 – 18:20 CEST / 10:50 CDT / 12:50 ART	S. Margenov

 

Guide to the time

ART – Argentina Time

CDT – Central Daylight Time / Central Daylight Saving Time

JST – Japan Standard Time

CEST – Central European Summer Time

CEST = JST - 7 = ART + 5 = CDT + 7

 

List of talks

Lorenzo Giacomelli (Sapienza Università di Roma), The 1-harmonic flow 

The 1-harmonic flow is the formal gradient flow -with respect to the L²-distance- of the total variation of a manifold-valued unknown function. The problem originates from image processing and has an intrinsic analytical interest as prototype of constrained and vector-valued evolution equations whose natural energy space is BV. After introducing the problem, I will discuss some tentative steps into this rather uncharted territory, which I took together with Agnese Di Castro, Michal Lasica, José Mazòn, and Salvador Moll: local/global-in-time well-posedness and blow-up of smooth (Lipschitz) solutions, as well as global existence (occasionally, uniqueness) of BV solutions in some special but instructive cases, including the hyper-octant of the N-sphere as target manifold or a generic target manifold when the domain is one-dimensional. I will conclude by highlighting some open questions.

Yoshikazu Giga (The University of Tokyo), A finer singular limit of a single-well Modica-Mortola functional and its applications 

We characterize the Gamma limit of a single-well Modica-Mortola functional under graph convergence in one-dimensional space which is finer than L¹ convergence.  As an application, we give an explicit representation of a singular limit of the Kobayashi-Waren-Carter energy, which is popular in materials science. We also establish compactness under the graph convergence. For these purposes we change an independent variable of a function by introducing arc-length parameter of its graph, which we call an unfolding of a function.  This is a joint work of  Jun Okamoto (University of Tokyo) and Masaaki Uesaka (University of Tokyo, Arithmer Inc.). 

Wojciech Górny (University of Warsaw), Optimal transport methods in the least gradient problem 

The least gradient problem is the Dirichlet problem for the 1-Laplacian operator. On convex domains in two dimensions, it is known to be equivalent to the optimal transport problem with source and target measures located on the boundary of the domain. Here, we present how the methods developed for optimal transport can be used in the least gradient problem. A standard assumption in both problems is convexity of the domain, but some of the methods can also be used in atypical situations. We will focus on the case when the domain is an annulus. 

Marta Latorre Balado (Universidad Rey Juan Carlos), An evolution problem involving the 1-Laplacian operator 

In this talk we present an existence and uniqueness result for an elliptic equation involving the 1‑Laplacian operator and a dynamical boundary condition. Using nonlinear semigroups theory we show the existence of a mild solution and we prove that this solution is, in fact, a strong solution. This is a joint work with S. Segura de León. 

Adam Kubica (Warsaw University of Technology), A self-similar solution to time-fractional Stefan problem. 

We derive the fractional version of one-phase one-dimensional Stefan model, where we assume that the diffusive flux is given by the time-fractional Riemann-Liouville derivative, i.e. we impose the memory effect in the examined model. Furthermore, we find a self-similar solution to this problem. It is a joint work with Katarzyna Ryszewska.

Svetozar Margenov (Bulgarian Academy of Scienes), BURA methods for spectral space fractional diffusion problems 

Abstract

Salvador Moll (Universitat de València), An augmented Lagrangian model for signal segmentation 

In this talk, I will present some results of a recent work with V. Pallardó, in which we study a very simple model for the two–phase signal segmentation problem. This is an augmented Lagrangian variational model based on Chan–Vese’s original one. By using energy methods and PDE methods, we show, in the one dimensional case, that the set of minimizers to the functional contains only binary functions and it coincides with the set of minimizers to Chan–Vese’s one. This fact allows us to obtain two important features of the minimizers. First of all, for piecewise constant signals, the jump set of the minimizer is contained into the jump set of the given signal. Secondly, all of the jump points of the minimizer belong to the same level set of the signal. This last property permits to obtain a trivial algorithm for computing the minimizers.

Piotr Bogusław Mucha (University of Warsaw), DaPrato and Grisvard on a free surface

I would like to present a technique for free boundary problems based on an application of the classical DaPrato and Grisvard theorem concerning maximal regularity estimates for abstract semigroups. Thanks to properties of real interpolation we are able to obtain estimates in L¹(0,∞;Ḃ^s_p,1) Besov-type space and define global Lagrangian coordinates. 

The talk is based on join results with R Danchin, M Hieber, Y Shibata and P Tolksdorf. 

A version of the talk can be seen here: https://bibli.cirm-math.fr/Record.htm?idlist=2&record=19286541124910047239
 

Matteo Novaga (Università di Pisa), Fattening for nonlocal mean curvature flows

I will discuss the fattening phenomenon for the nonlocal mean curvature flow. In particular, I will show a general result about uniqueness of the evolutions starting from starshaped sets or from sets with positive nonlocal mean curvature, and I will discuss the fattening in two dimensions for the evolution starting from the standard cross, showing that the phenomenon is very sensitive to the strength of the nonlocal interaction. 

Sabrina Roscani (Universidad Austral de Rosario), About Explicit Solutions to Different One-dimensional Fractional Stefan-Like Problems 

We present different Stefan-like problems governed by fractional diffusion equations, involving Caputo and Riemann-Liouville derivatives in time or spacial variable, of order α between 0 and 1. We compared the mathematical and the physical approaches and, in each case we present self-similar explicit solutions. Some of these solutions are expressed in terms of special functions and we make a comparison between them as well as with the classical solutions realted to each problem, when the order of differentiation is iqual to 1 and we recover different classical one or two phases Stefan problems governed by heat equations. 

Katarzyna Ryszewska (Warsaw University of Technology), A space-fractional Stefan problem 

In this talk we will consider a non-local in space, one-phase one-dimensional Stefan problem, where the diffusive flux takes the form of the fractional Caputo derivative. The motivation for studying such a problem originates from modelling the diffusion and mass transport in heterogeneous media. A typical example of such phenomenon is a sub-surface water motion. During the talk we will present the result concerning existence of regular solution to this problem. The proof relies mainly on analytic evolution operator theory and fixed point argument.

Ken Shirakawa (Chiba University), Optimal control problems of grain boundary motions with temperature constraints 

This work is based on recent jointwork with H. Antil (George Mason Univ., Japan), S. Kubota (Chiba Univ., Japan), and N. Yamazaki (Kanagawa Univ., Japan). In this talk, we consider a class of optimal control problems, which are governed by state systems based on the phase-field model of grain boundary motion (cf. [Kobayashi et al.; Phys. D, 140 (2000), 141–150]). Each state system is formulated as a coupled system of: Allen–Cahn type equation; and quasilinear type diffusion equation. On this basis, the optimal control problem is prescribed as a minimization problem of a cost functional with respect to the temperature-control and the control of crystalline orientation. Additionally, in our optimal control problems, the temperature-controls are supposed to satisfy the so-called box-constraints, i.e. the range-constraints in L^∞-topology. The focus of this talk is on the theoretical analysis for the class of our optimal control problems. With this view in mind, the mathematical results concerned with: the solvability of optimal control problems; and the continuous-dependence of optimal controls with the first necessary optimality conditions; will be presented as the main theorems of this talk.

Vaughan Voller (University of Minnesota), A physically consistent fractional derivative model of an anomalous Stefan problem

Through using appropriate statistical mechanics treatments the connection between fractional derivative operators and anomalous transport signals is well understood. From both  a physical and mathematical point of view, however, the task of writing down, for a particular anomalous transport system, a consistent fractional derivative based model is still difficult. In such an endeavor we always need to answer the questions;

Can we characterize the parameters that describe the observed anomalous transport in terms of features (geometry etc. ) within the system of interest?
and

Can we associate these  parameters with a particular fractional derivative based transport model.  

To demonstrate how to answer these questions, we consider a problem of saturated moisture infiltration into a  porous tube. Initially the tube contains dry air, the moisture is transported down the tube by maintaining a fixed pressure head at the entrance, and at, and beyond, the sharp wetting front, a head of zero is assumed.  This process can be considered to be a limit case Stefan melting problem with a vanishing specific heat. When the hydraulic conductivity  K in the tube is constant, the head profile in the saturated moist section of the tube is linear and the moisture front moves as the square-root of time; the characteristic time exponent associated with a diffusion process.  When, however, the hydraulic conductivity is arranged as a fractal Cantor set with two contrasting conductivities, a unit conductivity K=1 in the elements of the set and a very large value  K >>1 in the spaces between the elements, so called anomalous behavior is observed. In particular, a direct simulation of the physical model, shows that, the time exponent of the moisture front movement is super-diffusive (i.e., a time exponent larger than the square root), and that the pressure head profile in the saturated region has a power-law form. Fortunately, through an algebraic analysis of the direct simulation, it straight forward to link the exponents in the moisture front movement and head profile directly to the fractal dimension of the Cantor set; answering our first question on identifying the parametrization of the anomalous system.

In addition to the direct simulation, we also propose a fractional derivative based diffusion model for the tube infiltration with a Cantor set conductivity. We show that  a particular solution of the governing equations recovers, both the super-diffusive moisture front movement and the power-law pressure profile observed in  the direct simulation;  thus, by arriving at a suitable fractional derivative based model, answering the second
of our questions.     

Masahiro Yamamoto (The University of Tokyo), Unique existence of solutions for some time fractional partial differential equations and some inverse problems: some recent results 

I will talk on the well-posedness and qualitative properties for generalized time fractional diffusion equations and a fractional transport equation. I plan to touch recent results for inverse problems.

Rico Zacher (Universität Ulm), Quasilinear degenerate subdiffusion problems 

I will prove existence of a bounded weak solution to a degenerate quasilinear subdiffusion problem with bounded measurable coefficients that may explicitly depend on time. The kernel in the involved integro-differential operator with respect to time belongs to the large class of PC kernels. In particular, the case of a fractional time derivative of order less than 1 is included. A key ingredient in the proof is a new compactness criterion of Aubin-Lions type which involves function spaces defined in terms of the integro-differential operator in time. Boundedness of the solution is obtained by the De Giorgi iteration technique. Sufficiently regular solutions are shown to be unique by means of an L¹-contraction estimate. This is joint work with Petra Wittbold (Essen) and Patryk Wolejko (Ulm).