# Model reduction in continuum thermodynamics: Modeling, analysis and computation (12w5029)

Arriving in Banff, Alberta Sunday, September 16 and departing Friday September 21, 2012

## Organizers

Eduard Feireisl (Academy of Sciences of the Czech Republic)

Josef Málek (Charles University)

## Objectives

The present project focuses on several aspects of model reduction in continuum fluid mechanics. The subject area of the workshop includes mathematical analysis and modeling of complete fluid systems, scale analysis and singular limits, and numerical methods applied to multiscale problems. The main objective of the workshop is to bring together experts in mathematical and numerical analysis as well as mathematical modeling to examine the recently emerging problems and to share ideas in a well focused environment. Given the diversity of the topics and the variety of reduced systems and their applications, these researchers would have probably never met together at any other meeting.

The principal topics to be discussed will be highlighted by means of key note lectures delivered by leading specialists in the respective field. These are:

1. Mathematical theory of complete fluid systems

Complete fluid systems play the role of primitive systems in the theory of model reduction. They are designed and believed to provide a complete description of the observed phenomena. Accordingly, the fluids considered are compressible, heat conducting, viscous, chemically reacting, or enjoying other properties as the case may be. Clearly, a mathematical description of these materials becomes rather involved, where the resulting system of equations reflects the basic physical principles of conservation of mass, balance of momentum and energy, among others.

The relevant mathematical theory can be developed either in the framework of classical description supposing that all fields are smooth functions of space and time, or using the more recent concept of "weak'' solutions, where the relevant physical principles are expressed by means of integral identities. New mathematical tools emerged quite recently to handle the problem of solvability of complete fluid systems, and the workshop should enhance further this development by exchanging ideas of purely theoretical as well as conceptual character.

2. Mathematical theory of incompressible fluids

The so-called incompressible fluids can already be viewed as an example of a model reduction obtained by means of the low Mach number (or incompressible) limit of a complete fluid system. Classical solvability of the underlying Navier-Stokes system represents an outstanding open problem of the theory of partial differential equations, also very popular as one of the "millenium problems''. Numerical experiments may to a certain extent indicate the limitations of the rigorous mathematical theory, and the newly emerging mathematical models may offer an attractive alternative to the classical systems.

3. Singular limits

As already pointed out, singular limits give rise to reduced models after performing a scale analysis and letting some characteristic numbers go to zero or become infinite. The incompressible Navier-Stokes system, the Euler equations of gas dynamics, the Oberbeck-Boussinesq and anelastic approximation may be viewed as singular limits of complete fluid systems. Singular limits are often performed formally by means of asymptotic expansion of all quantities with respect to a singular parameter, however, their rigorous justification is usually considerably more difficult. Recent development of the mathematical theory of complete fluid systems enables to perform rigorously certain singular limits, even in the case of the so-called ill-prepared data, where the primitive system is in a state that is "far away'' from the target stay.

4. Numerical analysis of multiscale problems.

Asymptotic analysis plays a crucial role in the design of efficient numerical methods for flows in a singular regime. These problems are characterized by multiple space and time scales, and by the fact that the standard numerical methods may either completely fail or become expensive. As the goal is to apply numerical methods to complete fluid systems, it is important to understand the qualitative changes of solutions in the singular limit regime. A typical example are rapid oscillations of acoustic waves in the low Mach number limit that can be eliminated by the method of acoustic filtering. Clearly, applying similar techniques requires a detailed mathematical analysis of the problem.

5. Mathematical modelling

A large variety of new materials represent a challenge in mathematical modeling. Liquid crystals, polymeric fluids, "smart'' materials require a substantial change of rheological laws as well as the underlying mathematical theory. Without understanding the physical structure of materials, it is impossible to develop a meaningful mathematical model. On the other hand, the mathematical properties of the rheologically more complex fluids may shed some light on the nowadays unsurmountable classical problems, the difficulty of which might be attributed to the fact that they are "incomplete'' so that the information provided is not sufficient for their well-posedness in the mathematical sense. Of course, these systems are more complex than the classical models of fluid mechanics and thermodynamics. Thus, the need to find appropriate approximate models is of high importance.

We believe that spreading the recently discovered ideas among a broad spectrum of specialists in the above mentioned fields can substantially enhance the development of the mathematical theory, effective numerical methods, and their rapid implementations in practical problems.

The principal topics to be discussed will be highlighted by means of key note lectures delivered by leading specialists in the respective field. These are:

1. Mathematical theory of complete fluid systems

Complete fluid systems play the role of primitive systems in the theory of model reduction. They are designed and believed to provide a complete description of the observed phenomena. Accordingly, the fluids considered are compressible, heat conducting, viscous, chemically reacting, or enjoying other properties as the case may be. Clearly, a mathematical description of these materials becomes rather involved, where the resulting system of equations reflects the basic physical principles of conservation of mass, balance of momentum and energy, among others.

The relevant mathematical theory can be developed either in the framework of classical description supposing that all fields are smooth functions of space and time, or using the more recent concept of "weak'' solutions, where the relevant physical principles are expressed by means of integral identities. New mathematical tools emerged quite recently to handle the problem of solvability of complete fluid systems, and the workshop should enhance further this development by exchanging ideas of purely theoretical as well as conceptual character.

2. Mathematical theory of incompressible fluids

The so-called incompressible fluids can already be viewed as an example of a model reduction obtained by means of the low Mach number (or incompressible) limit of a complete fluid system. Classical solvability of the underlying Navier-Stokes system represents an outstanding open problem of the theory of partial differential equations, also very popular as one of the "millenium problems''. Numerical experiments may to a certain extent indicate the limitations of the rigorous mathematical theory, and the newly emerging mathematical models may offer an attractive alternative to the classical systems.

3. Singular limits

As already pointed out, singular limits give rise to reduced models after performing a scale analysis and letting some characteristic numbers go to zero or become infinite. The incompressible Navier-Stokes system, the Euler equations of gas dynamics, the Oberbeck-Boussinesq and anelastic approximation may be viewed as singular limits of complete fluid systems. Singular limits are often performed formally by means of asymptotic expansion of all quantities with respect to a singular parameter, however, their rigorous justification is usually considerably more difficult. Recent development of the mathematical theory of complete fluid systems enables to perform rigorously certain singular limits, even in the case of the so-called ill-prepared data, where the primitive system is in a state that is "far away'' from the target stay.

4. Numerical analysis of multiscale problems.

Asymptotic analysis plays a crucial role in the design of efficient numerical methods for flows in a singular regime. These problems are characterized by multiple space and time scales, and by the fact that the standard numerical methods may either completely fail or become expensive. As the goal is to apply numerical methods to complete fluid systems, it is important to understand the qualitative changes of solutions in the singular limit regime. A typical example are rapid oscillations of acoustic waves in the low Mach number limit that can be eliminated by the method of acoustic filtering. Clearly, applying similar techniques requires a detailed mathematical analysis of the problem.

5. Mathematical modelling

A large variety of new materials represent a challenge in mathematical modeling. Liquid crystals, polymeric fluids, "smart'' materials require a substantial change of rheological laws as well as the underlying mathematical theory. Without understanding the physical structure of materials, it is impossible to develop a meaningful mathematical model. On the other hand, the mathematical properties of the rheologically more complex fluids may shed some light on the nowadays unsurmountable classical problems, the difficulty of which might be attributed to the fact that they are "incomplete'' so that the information provided is not sufficient for their well-posedness in the mathematical sense. Of course, these systems are more complex than the classical models of fluid mechanics and thermodynamics. Thus, the need to find appropriate approximate models is of high importance.

We believe that spreading the recently discovered ideas among a broad spectrum of specialists in the above mentioned fields can substantially enhance the development of the mathematical theory, effective numerical methods, and their rapid implementations in practical problems.