Partial Differential Equations

The group contributes to the following mathematical research topics of WIAS:

Analysis of Partial Differential Equations and Evolutionary Equations

Partial differential equations offer a powerful and versatile framework for the continuum description of phenomena in nature and technology with complex coupling and dependencies. At the Weierstrass Institute this research has three essential focuses: (a) Rigorous mathematical analysis of general evolution equations in terms of existence, uniqueness and regularity of different types of solutions, (b) Development of variational approaches using the toolbox of the calculus of variations, (c) Regularity results for solutions of elliptic and parabolic partial differential equations. [>> more]

Direct and inverse problems in thermomechanics

Thermomechanical models are the basis for the description of numerous technological processes. The consideration of phase transitions and of inelastic constitutive laws raise exciting new questions regarding the analysis of direct problems as well as the identification of materials parameters. [>> more]

Functional analysis and operator theory

At WIAS, functional analysis and operator theory are related, in particular, to problems of partial differential equations and evolutions equations as well as to analysis of multiscale, hybrid, and rate-independent models. [>> more]

Hysteresis operators and rate-independent systems

Time-dependent processes in physics, biology, and economics often exhibit a rate-independent input-output behavior. Quite often, such processes are accompanied by the occurrence of hysteresis phenomena induced by inherent memory effects. There are two methods to describe such processes at WIAS: rate independent systems and. hysteresis operators . [>> more]

Interacting stochastic particle systems

In the mathematical modeling of many processes and phenomena in the Sciences and Technology one employs systems with many random particles and interactions. In this context, we define the term "particle systems" very broadly and also include point processes with percolation properties and random graph structures as well as Gibbs interactions. It also includes random movements of these particles, such as those that occur in spatial models for communication. At WIAS, many macroscopic properties of these particle systems are investigated that arise from the microscopic rules, such as phase transitions (condensation, percolation, crystallization) and critical properties such as rescaling limits. [>> more]

Large deviations

The theory of large deviations, a branch of probability theory, provides tools for the description of the asymptotic decay rate of a small probability, as a certain parameter diverges or shrinks to zero. Examples are large times, low temperatures, large numbers of stochastic quantities, or an approximation parameter. This probabilistic theory is also indispensable in the treatment of a number of models in statistical physics, as it makes them accessible for analysis using variational techniques. Both theory and sophisticated applications in physics and chemistry are being investigated at WIAS. [>> more]

Modeling, analysis and numerics of phase field models

A diffuse phase field model is a mathematical model for describing microstructural phenomena and for predicting morphological evolution on the mesoscale. It is applied to a wide variety of material processes such as solidification, coarsening in alloys, crack propagation and martensitic transformations. [>> more]

Multi-scale modeling, asymptotic analysis, and hybrid models

To understand the interplay between different physical effects one often needs to consider models involving several length scales. The aim in this mathematical topic is the derivation of effective models for the efficient description of the processes. The understanding of the transfer between different scales relies on mathematical methods such as homogenization, asymptotic analysis, or Gamma convergence. The generated effective models are coupled partial differential equations combining volume and interfacial effects. [>> more]

Optimal Transport: Statistics, Numerics, and Partial Differential Equations

The theory of Optimal Transport has been immensely influential in connecting partial differential equations, geometry, and probability. On the one hand, research at WIAS is focused on applying methods and tools from Optimal Transportation Theory to problems in statistics, such as semi-supervised and unsupervised learning, clustering, text classification, as well as in image retrieval, clustering, segmentation, and classification by developing and analyzing new numerical algorithms and schemes. On the other hand, the theory of optimal transport is extended, e.g., towards unbalanced optimal transport and connections to evolutionary partial differential equations via gradient systems. [>> more]

Systems of partial differential equations: modeling, numerical analysis and simulation

The mathematical modelling of many scientific and technological problems leads to (initial) boundary value problems with systems of partial differential equations (PDEs). [>> more]

Variational methods

Many physical phenomena can be described by suitable functionals, whose critical points play the role of equilibrium solutions. Of particular interest are local and global minimizers: a soap bubble minimizes the surface area subject to a given volume and an elastic body minimizes the stored elastic energy subject to given boundary conditions. At WIAS, methids from the calcuus of variations are applied and further developed to solve problems in physics and technology such as continuum mechanics, quantum mechanics, and optimal control. [>> more]