# Functional Analysis

## PDEs with Memory Effects

Winter Term 2014/2015

Graduate Seminar on PDEs (S4B2)

Dr. M. Helmers, H. Pöttker

The seminar sessions will be on Fridays at 10:15 a.m. in room SR1.007.

### Synopsis

PDEs with memory effects appear in many different applications ranging from materials science to biological systems or models for phase transitions. In the simplest case, the memory effect is a sensitive dependence on the initial data; in more complicated cases, the solution at a time \(t>0\) or its properties depend on the history in the time interval \([0,t]\).

In the seminar, we are interested in the mathematical analysis of such phenomena and will concentrate on two, at first glance rather different, examples. The first, from biology, are equations for chemotaxis such as the Othmer-Stevens model \[\begin{align*} \partial_t u &= \operatorname{div} \big(\nabla u+\Phi(u,z)\nabla z\big), \\ \partial_t z &= \Psi(u,z), \end{align*}\] where \(u\) is the population density of certain bacteria and \(z\) is a vector of nutrients whose dynamics influence the movement of \(u\).

The second example stems from the theory of visco-elastic materials such as the equation \[\partial_{t t}^2 u - \partial_x \big( \Phi'(\partial_x u) + \mu \partial_{x t}^2 u\big) = f\] modelling the motion of a visco-elastic bar. Our goal in both cases is to study well-posedness as well as properties of solutions and the mechanisms of the memory effects. For this purpose, we will consider different approaches based on a wide range of available techniques, including explicit solutions, singular perturbation, compactness principles, semigroup theory, and time discretization with local minimization.

### Prerequisites

A sound knowledge of partial differential equations (classic and functional analytic methods, compare V2B2 and V3B1) is essential.

### Organization

The seminar sessions will be on Fridays at 10:15 a.m. in room SR1.007.

The tentative schedule is as follows:

Date | Topic | Speaker |
---|---|---|

14.11. | Oelschläger [1] | A. K. |

21.11. | Davis [2] | J. K. |

05.12. | Pego [3] | J. D. |

12.12. | Demoulini [4] | A. C. |

19.12. | Friesecke, Dolzmann [5] | M. V. |

09.01. | Ball, Sengül [6] | R. H. |

16.01. | Klein, Krejci [7] | T. V. |

### Literature

- K. Oelschläger. On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes.
*Probab. Th. Rel. Fields*82(4), 565–586, 1989. - B. Davis. Reinforced Random Walk.
*Probab. Th. Rel. Fields*84, 203-229, 1990. - R. Pego. Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability.
*Arch. Rational Mech. Anal.*97(4), 353–394, 1987. - S. Demoulini. Weak solutions for a class of nonlinear systems of viscoelasticity.
*Arch. Ration. Mech. Anal.*155(4), 299–334, 2000. - G. Friesecke, G. Dolzmann. Implicit time discretization and global existence for a quasilinear evolution with nonconvex energy.
*SIAM J. Math. Anal.*28(2), 363–380, 1997. - J. M. Ball, Y. Sengül. Quasistatic nonlinear viscoelasticity and gradient flows. arXiv preprint 1401.2715, 2014.
- O. Klein, P. Krejci. Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations.
*Nonlinear Anal. Real World Appl.*4(5), 755–785, 2003.