## Final Report Summary - MATHEF (Mathematical Thermodynamics of Fluids)

MATHEF is a five year research project focused on the mathematical theory of complete fluid systems developed on the basis of the fundamental laws of thermodynamics. The mathematical models studied are derived from the principles of continuum mechanics, where the properties of the fluid are described by means of observable physical fields (mass, momentum, temperature) represented by numerical functions of the time and the spatial position. Accordingly, the time evolution of the fluid is described by a system of non–linear partial differential equations.

The main objectives of the project are:

• Mathematical analysis of the models, identifying a proper concept of solution, and problems of whether or not a solution can be uniquely identified from the known data (well–posedness)

• Design of suitable numerical schemes to generate approximate solutions, studying how far the approximate solutions are from the exact ones (convergence of the scheme)

• Behavior of models in singular regimes, where, roughly speaking, their mathematical properties change qualitatively. Typical examples are flows of compressible fluids like air in the atmosphere when its speed is much lower than the speed of sound (low Mach number regime)

• Effects of perturbations of the system including those that are random (stochastic partial differential equations)

Solvability, dependence on the data

Rather surprisingly, we have shown that there are many physically well accepted models of inviscid fluids that admit infinitely many solutions for given data. Here solutions are understood in a generalized sense frequently used for this class of problems. These are solutions that develop uncontrollable oscillations even under certain restrictive conditions imposed by principles of thermodynamics (energy conservation, entropy dissipation). The way the existence of those solutions is shown seems highly non-constructive based on purely theoretical tools like Baire’s category argument. Still the hypothetical possibility that at least some of these solutions may be observable in real world problems opens a new perspective in fluid mechanics, replacing deterministic solutions by random ones that reflect the probability in which the system attains a given state. This idea has been further developed in the project along two rather different lines. First, we introduced the concept of a measure–valued solution, replacing the exact values of observable fields by probabilities. Second, we studied problems where the data as well as the external driving forces acting on the fluid are random.

Numerical analysis

Motivated by theoretical results, we have developed new numerical schemes that have been implemented and tested on a variety of real world problems including singular regimes. We performed a rigorous analysis of how far the numerical solutions are from the exact ones. We proposed a new approach to this problem based on measure–valued solutions. As soon as a numerical scheme produces a bounded family of approximate solutions (stability) and as soon as the latter solve the underlying system with an error that vanishes with the numerical parameter (consistency), it is relatively easy to show that the scheme generates a very general (measure–valued) solution of the exact system. On the other hand, we proved for a variety of problems that these generalized (measure–valued) solutions in fact coincide with the standard solutions as long as those exist (weak–strong uniqueness principle). This allows us to conclude that the numerical scheme always converges to the exact solution as long as the latter is available. Other rather surprising results may be shown via this approach. For instance, the fact that numerical solutions remain bounded uniformly with the vanishing numerical parameter guarantees convergence to the exact solution for certain schemes and models of viscous fluids. It is remarkable that existence of the exact solutions is not a priori assumed here.

Effect of stochastic perturbations

In many real world applications the data, including the initial state, the shape of the physical domain, and external forces, are known only with a certain probability. It is therefore of not only theoretical interest to extend the existing theory to problems with random terms. This is a rather delicate task for the system of differential equations that admit only generalized (weak) solutions. In particular, certain fields may not represent the standard stochastic processes required by the standard theory of stochastic differential equations. We extend the existing theory to the class of generalized solutions by developing a new theory of stochastic distributions. Here, each field is characterized by the probability that it belongs to a certain set in its natural range. Formal similarity with the concept of measure–valued solutions is immediate. Using this approach, we were able to establish results that are entirely new in the context of complete fluid systems. In particular, we have shown the existence of stochastically stationary solutions for the problem describing the motion of a compressible viscous fluid driven by a stochastic force of white noise type. We also extended a certain result on ill–posedness to problems driven by random forces. This might seem rather surprising and casts some doubt on the common belief that stochastic averaging may give rise to unique solvability of problems otherwise ill posed in the deterministic setting.

The main objectives of the project are:

• Mathematical analysis of the models, identifying a proper concept of solution, and problems of whether or not a solution can be uniquely identified from the known data (well–posedness)

• Design of suitable numerical schemes to generate approximate solutions, studying how far the approximate solutions are from the exact ones (convergence of the scheme)

• Behavior of models in singular regimes, where, roughly speaking, their mathematical properties change qualitatively. Typical examples are flows of compressible fluids like air in the atmosphere when its speed is much lower than the speed of sound (low Mach number regime)

• Effects of perturbations of the system including those that are random (stochastic partial differential equations)

Solvability, dependence on the data

Rather surprisingly, we have shown that there are many physically well accepted models of inviscid fluids that admit infinitely many solutions for given data. Here solutions are understood in a generalized sense frequently used for this class of problems. These are solutions that develop uncontrollable oscillations even under certain restrictive conditions imposed by principles of thermodynamics (energy conservation, entropy dissipation). The way the existence of those solutions is shown seems highly non-constructive based on purely theoretical tools like Baire’s category argument. Still the hypothetical possibility that at least some of these solutions may be observable in real world problems opens a new perspective in fluid mechanics, replacing deterministic solutions by random ones that reflect the probability in which the system attains a given state. This idea has been further developed in the project along two rather different lines. First, we introduced the concept of a measure–valued solution, replacing the exact values of observable fields by probabilities. Second, we studied problems where the data as well as the external driving forces acting on the fluid are random.

Numerical analysis

Motivated by theoretical results, we have developed new numerical schemes that have been implemented and tested on a variety of real world problems including singular regimes. We performed a rigorous analysis of how far the numerical solutions are from the exact ones. We proposed a new approach to this problem based on measure–valued solutions. As soon as a numerical scheme produces a bounded family of approximate solutions (stability) and as soon as the latter solve the underlying system with an error that vanishes with the numerical parameter (consistency), it is relatively easy to show that the scheme generates a very general (measure–valued) solution of the exact system. On the other hand, we proved for a variety of problems that these generalized (measure–valued) solutions in fact coincide with the standard solutions as long as those exist (weak–strong uniqueness principle). This allows us to conclude that the numerical scheme always converges to the exact solution as long as the latter is available. Other rather surprising results may be shown via this approach. For instance, the fact that numerical solutions remain bounded uniformly with the vanishing numerical parameter guarantees convergence to the exact solution for certain schemes and models of viscous fluids. It is remarkable that existence of the exact solutions is not a priori assumed here.

Effect of stochastic perturbations

In many real world applications the data, including the initial state, the shape of the physical domain, and external forces, are known only with a certain probability. It is therefore of not only theoretical interest to extend the existing theory to problems with random terms. This is a rather delicate task for the system of differential equations that admit only generalized (weak) solutions. In particular, certain fields may not represent the standard stochastic processes required by the standard theory of stochastic differential equations. We extend the existing theory to the class of generalized solutions by developing a new theory of stochastic distributions. Here, each field is characterized by the probability that it belongs to a certain set in its natural range. Formal similarity with the concept of measure–valued solutions is immediate. Using this approach, we were able to establish results that are entirely new in the context of complete fluid systems. In particular, we have shown the existence of stochastically stationary solutions for the problem describing the motion of a compressible viscous fluid driven by a stochastic force of white noise type. We also extended a certain result on ill–posedness to problems driven by random forces. This might seem rather surprising and casts some doubt on the common belief that stochastic averaging may give rise to unique solvability of problems otherwise ill posed in the deterministic setting.