## Forward-backward parabolic equations

Summer Term 2015
Selected Topics in PDE and Mathematical Models (V5B4)
Dr. M. Helmers

Lectures Friday 12–14 in room SR0.006

### Synopsis

The nonlinear diffusion equation $\partial_t u = \Delta f(u), \tag{1}$ where $$f$$ is a function with increasing as well as decreasing branches as in the figure below, appears in different applications such as edge detection in digital images, models for population dynamics, or phase separation problems with supercooling and superheating. However, if the initial data for $$u$$ lie in the decreasing branch(es) of $$f$$, equation (1) is ill-posed within the classic framework of PDEs.

Nevertheless, solutions to (1) can be obtained by different regularization procedures. Moreover, these solutions often display complex dynamical features such as phase transitions, hysteresis, microstructure and coarsening, which put a comprehensive existence theory for arbitrary right hand side and initial data far out of reach.

### Detailed programme

1. Introduction:
• Gradient flow structure and some dynamic features of the forward-backward equation; different regularizations
2. The viscous approximation:
• Solutions to the approximate equation, boundedness and energy estimates
• Young-measure solutions to the limit equation
• Entropy inequality
• Two-phase problem
• Long-time behaviour
3. The Cahn-Hilliard equation (in 1D)
• Setup of approximate solutions and limit
• Convergence
• Two-phase problem

### Examination

The course will be completed by an oral exam on 30. and 31.07.15.

### Literature

Section 2 on the viscous approximation is based on the following papers.

• A. Novick-Cohen, B. Pego. Stable patterns in a viscous diffusion equation. Trans. Amer. Math. Soc. 324(1), 331–351, 1991.
• P. I. Plotnikov. Passing to the limit with respect to viscosity in an equation with variable parabolicity direction. Differential Equations 30(4), 614–622, 1994.
• L. C. Evans, M. Portilheiro. Irreversibility and hysteresis for a forward-backward diffusion equation. Math. Models Methods Appl. Sci. 14(11), 1599–1620, 2004.
• F. Smarrazzo, A. Tesei. Long-time behavior of solutions to a class of forward-backward parabolic equations. SIAM J. Math. Anal. 42(3), 1046–1093, 2010.

Section 3 on the Cahn-Hilliard equation is based on

• G. Bellettini, L. Bertini, M. Mariani, M. Novaga. Convergence of the one-dimensional Cahn-Hilliard equation. SIAM J. Math. Anal. 44(5), 3458–3480, 2012.

Some material on Young-measures can be found in Section 3 of

• S. Müller. Variational models for microstructure and phase transitions. Lecture Notes in Math. 1713, 85–210, Springer 1999.

## News

Prof. Dr. Lisa Sauermann has been honored with the von Kaven Award 2023 for her outstanding scientific achievements. (16.11.2023)

Prof. Dr. Angkana Rüland has been awarded a New Horizons in Mathematics Prize 2024 for her contributions to applied analysis (press release). (14.09.2023)

Prof. Dr. Angkana Rüland has been awarded the Calderon Prize that is awarded every two years by the Inverse Problems International Association. (06.09.2023)

Prof. Dr. Karl-Theodor Sturm has been elected into the Academia Europaea. (28.06.2022)

Florian Schweiger erhielt den Hausdorff-Gedächtnispreis 2021 der Fachgruppe Mathematik für die beste Dissertation. Er fertigte die Dissertation unter der Betreuung von Prof. Stefan Müller an. Unter anderen wurde Vanessa Ryborz mit einem Preis der Bonner Mathematischen Gesellschaft für ihre von Prof. Sergio Conti betreute Bachelorarbeit ausgezeichnet. (18.01.2022)

Contact

Managing Director: Prof. Dr. Juan J. L. Velázquez