Recent Trends in Differential Equations: Analysis and Discretisation Methods November 15-17, 2010 at Bielefeld University Organised by Etienne Emmrich (Bielefeld), Moritz Kaßmann (Bielefeld), and Petra Wittbold (Essen) Contact e-mail: [email protected] fax: +49.521.106-6498 http://www.math.uni-bielefeld.de/recent2010/ Etienne Emmrich and Moritz Kaßmann Universitat¨ Bielefeld Fakultat¨ fur¨ Mathematik Postfach 100131 33501 Bielefeld, Germany e-mail: [email protected] [email protected] http: www.math.uni-bielefeld.de/˜emmrich www.math.uni-bielefeld.de/˜kassmann Petra Wittbold Universitat¨ Duisburg-Essen, Campus Essen Fakultat¨ fur¨ Mathematik 45117 Essen, Germany e-mail: [email protected] http: www.uni-due.de/˜mat201/wittbold/wittbold.html This workshop is supported by Collaborative Research Center 701 at Biele- feld University, which is funded by DFG (German Research Foundation). The workshop focusses on mathematical approaches to nonlinear and nonlocal phenomena by partial differential and integro-differential equa- tions. The aim is to discuss the use of various methods and concepts such as fractional derivatives, truncation techniques, appropriate function spaces, and stochastic perturbations. We wish you all an inspiring workshop and a pleasant stay in Bielefeld. Etienne Emmrich Moritz Kaßmann Petra Wittbold Contents 1 Useful Information 1 2 Program 6 3 Abstracts 8 4 List of Participants 28 1 Useful Information The university building The university is located in one large building. There is one main hall in the middle connecting all parts of the building. These are labelled by letters. The main hall is on level 0 and it houses shops (books, stationaries, gro- cery), a post office, and several restaurants and coffee shops. All rooms in the university are labelled like V3-106. This means the room is in part V, 3rd floor and has number 106. For a plan of the building, see the last page. Restaurants at the university Westend, opening hours: 11am – 4pm Located next to the swimming pool at one end of the main hall. Serves full meals but also cake and salad. Self-service. Mensa, opening hours: 11:30am – 2pm You must buy a receipt at the service point in front of the mensa. Four menus to choose every day. Salad bar on the east end, here you can also pay in cash. Self-service. Univarza–Restaurant, opening hours: 10am – 12pm Serves full meals, but also pizza, salad, or just coffee or tea. Besides the restaurant, there is also a snack bar. Cafeteria, opening hours: 8am – 6pm Between the main entrance and the Mensa. Serves full meals, sandwiches and light meals as well as cakes, salad, or just coffee or tea. There are also coffee shops on the first floor of the main hall. 1 Computers and internet There is a document in your conference folder with an accountname and password. Please use it to log-in in U5-139 or any other computer in the department of mathematics. There is also a WLAN-account, which you can use to connect your own computer to the internet. Library The university’s library is situated on the first floor, around the whole build- ing. Math books and journals are in part V1. Entrances to the library are from the first floor of the main hall. To go to the mathematics part use the entrances in part U or V. Opening hours: Entrance U1: Sun 9am – 10pm, Mon – Wed 8am – 1 am. Entrance V1: Mon – Wed 9am – 4pm. Public transport. How to find the university The tram Stadtbahn Linie 4 connects the university to the city center. Tickets for one or four trips can be obtained from the machines at each station. You need tickets “Preisstufe 1”, single ticket 2.10 Euro, four tick- ets 6.80 Euro. Make sure to validate your ticket at the stamp machines inside the tram when boarding. From the hotel: cross the street and take the tram Stadtbahn Linie 4 to “Lohmannshof” and get off at “Universitat”.¨ 2 From the main station: take the exit in direction to the city and cross the street to get to the underground. Take the tram Stadtbahn Linie 4 to “Lohmannshof” and get off at “Universitat”.¨ Get together There will be a “Get together” in “Wernings Weinstube” on Sunday evening starting at 6pm. We will be there until 22:30pm. From the hotel: Cross the street into the pedestrian area “Rathausstraße”. “Wernings Weinstube” will appear on the right-hand side at “Alter Markt”. Alter Markt 1 33602 Bielefeld http://www.wernings-weinstube.de Conference dinner You are invited to attend the conference dinner on Monday evening at 7pm in the restaurant “Bernstein”. We kindly ask for a contribution of 35 Euro (20 Euro for PhD students, no fee for students). From the university: take the tram Stadtbahn Linie 4 to the city center and get off at “Jahnplatz”. Turn into the pedestrian area between “Sport Scheck” and “Bertelsmann”. Turn left directly after “Sport Scheck” and get into the entrance with the elevator. From the hotel: Cross the street into the pedestrian area “Rathausstraße”. Turn right at “Alter Markt” and follow the road. The elevator, which is the 3 entrance of “Bernstein” will appear on the right-hand side. Niederwall 2 33602 Bielefeld http://www.bernstein-live.de Reimbursement Reasonable travel expenses will be covered for invited participants. For the reimbursement, we need a copy of your travel documents and your international banc account. Please, have both with you then. For German residents, we probably also need your tax number (due to regulations by the university). The hotel will be paid directly by us. How to find “Arcadia Hotel Bielefeld” (previously “Tulip Inn”) From the main station: take the exit in direction to the city and cross the street to get to the underground. Take any of the trams Stadtbahn Linie 1 to “Senne”, Stadtbahn Linie 2 to “Sieker”, Stadtbahn Linie 3 to “Stieghorst” or Stadtbahn Linie 4 to “Rathaus” and get off at “Rathaus”. Turn left (in the direction of train) and cross the street. From the university: take the tram Stadtbahn Linie 4 to the city center and get off at “Rathaus”. Turn left (in the direction of train) and cross the street. Niederwall 31-35 33602 Bielefeld http://www.arcadia-hotellerie.com/tulip inn bielefeld 4 5 2 Program Monday and Wednesday the workshop takes place in the main building of the university in room V2-105. Tuesday it takes place in room V3-201. November 14th, 2010 (Sunday) -18:00 Arrival 18:00-22:30 Get together in the restaurant Wernings Weinstube, which is at Alter Markt. For those arriving early on Sun- day, we recommend to visit the Kunsthalle Bielefeld. November 15th, 2010 (Monday) 09:15-09:45 Registration 09:45-10:00 Opening 10:00-10:40 Imbert 10:40-11:20 Gwiazda 11:20-12:00 Break (V3-106) 12:00-12:40 Biler 12:40-14:00 Lunch 14:00-14:40 Ru˚ziˇ ckaˇ 14:40-15:20 Boyaval 15:20-16:00 Break (V3-106) 16:00-16:40 Wroblewska´ 16:40-17:20 Hieber 19:00 Conference Dinner 6 November 16th, 2010 (Tuesday) 10:00-10:40 Karch 10:40-11:20 Alexandre 11:20-12:00 Break (V3-106) 12:00-12:40 Coville 12:40-14:00 Lunch 14:00-14:40 Ostermann 14:40-15:20 Karper 15:20-16:00 Break (V3-106) 16:00-16:40 Vovelle 16:40-17:20 Hausenblas 17:20-18:00 Prohl November 17th, 2010 (Wednesday) 10:00-10:40 Vallet 10:40-11:20 Azerad 11:20-12:00 Break (V3-106) 12:00-12:40 Murat 12:40-14:00 Lunch 14:00-14:40 Zacher 14:40-15:20 Schwab 15:20-16:00 Break (V3-106) 7 3 Abstracts (ordered alphabetically) Radjesvarane Alexandre Ecole Navale/ENSAM, France Boltzmann equation: some qualitative properties We report on some recent works on Boltzmann equation, with non cut off cross sections, and in particular existence, regularization properties ... Some of these works were done in collaboration with Yoshinori Morimoto (Kyoto), Seiji Ukai (Kyoto), Chao-Jiang Xu (Wuhan and Rouen) and Tong Yang (Hong Kong). 8 Pascal Azerad Universite´ de Montpellier II, France About a nonlocal model for morphodynamics P. Azerad and A. Bouharguane We will introduce a PDE describing the morphodynamics of sand dunes sheared by a fluid flow. This model involves a non local term which can be seen as a fractional differential operator. The equation is able to describe both erosion and accretion phenomena. We will present mathematical results about the well posedness, violation of maximum principle, existence and instability of travelling waves. We will also give some insight about the numerical discretization of the non local term. 9 Piotr Biler Uniwersytet Wrocławski, Poland Barenblatt profiles for a nonlocal porous medium equation Piotr Biler (Wrocław), Cyril Imbert (Paris IX), Grzegorz Karch (Wrocław), Regis´ Monneau (Paris-Est) We study a generalization of the porous medium equation involving nonlo- cal terms. More precisely, our equation is α−1 m−1 @tu − r · ur juj = 0; d with α 2 (0; 2), m > 1, and a nonlocal gradient operator in R defined by rβu = F −1(iξjξjβ−1Fu). Explicit self-similar solutions with compact support generalizing the Baren- blatt solutions are constructed. They are of the form 1 α m−1 − d 2 2 − 2 2 u(t; x) = kt d+α R − jxj t d+α + with R > 0 and suitable k > 0. We also present an argument to get the Lp decay of weak solutions of the Cauchy problem constructed by Caffarelli and Vazquez´ in the model case m = 2.