Final Report Summary - FLUX (Towards regularity)
# Project
“Towards regularity” was a project devoted to the mathematical analysis of a wide spectrum of problems originating from fluid mechanics, aerodynamics, geophysics, phase transitions, image processing and meteorology. The project was divided into four distinct research topics:
- Compressible Navier-Stokes equations
- Crystal growth and image processing
- Regularity criteria
- Asymptotic analysis
## Compressible Navier-Stokes equtions
We studied quantitative and qualitative aspect of solutions to chosen particular models (Navier-Stokes and related systems). Our work in this field was aimed at answering the questions concerning existence and uniqueness of solutions, their stability, asymptotic behavior and structure of solutions, and finally their regularity. Using newly developed techniques we obtained numerous estimates for solutions and their existence in certain special cases. However, these techniques need further refinements for some models involving chemical reactions. We also began the analysis of the stability of special solutions to compressible Navier-Stokes equations in the framework of regular solutions. Additionally, we successfully adapted techniques from the theory of Schauder estimates for parabolic equations, thereby making the study of free boundary problems for viscous compressible fluids more feasible.
## Crystal growth and image processing
The crystal growth and image processing problem lead to: TV flows and its generalization; evolution of closed curves by singular mean curvature and Stefan type problems with Gibbs-Thomson relation involving singular mean curvature. We were interested in showing well-posedness of the systems and their asymptotic behavior. We developed new viscosity theory for formally second order parabolic singular problems. Due to the importance of the this work it may be called a **milestone**. We also investigated free boundary problems, which led us to show that Berg's effect is very rare. Therefore, we started investigating closely related free boundary problems involving fractional derivatives (c.f. fractional diffusion equation).
## Regularity criteria
The problem of regularity of weak solution to the non-stationary 3D Navier-Stokes equations is one of the most challenging problems of the modern PDE theory. We investigated local properties of weak solutions to the N-S system. We established several sufficient conditions of regularity of weak solutions to the Navier-Stokes equations in a neighborhood of a given point in space-time. We also showed the global existence of regular solutions being close to special regular global solutions like two-dimensional or axially-symmetric, with respect to non-homogenous and compressible Navier-Stokes equations. Some ideas that were explored during the project, have evolved into independent research projects.
## Asymptotic analysis
The issue of stability and long-time behavior of solutions to various evolutionary models was our priority. We proved the existence of special, global, regular solutions (e.g. self-similar, $2d$, axially symmetric, time-periodic) to Navier-Stokes equations and related models. We also worked on the decay of solutions, the correspondence between the geometry of domains and behavior of solutions, obtaining numerous results that can serve as starting points for further studies.
# Relevance
Most of the results that we obtained, can be implemented in direct applications, like simulating fluid flows or biological phenomena. The strategic objective of the program was to complement and develop a mathematical theory of partial differential equations stemming from real-life applications. We covered a wide scope of analysis of partial differential equations, enforcing cooperation of specialists from different fields in the theory of PDEs. We created opportunities to exchange full range of ideas and to obtain certain synergies. Most of the work was carried out by the partners in their home institutions but the secondments truly served the purpose of an intensive research. The exchange of knowledge was augmented through organization of project-related seminars, workshops and conferences.
We overcame certain fragmentation of research and eliminated artificial barriers by direct exchange of knowledge and ideas within our network. A combination of different techniques or even points of view on mathematics, which might have seemed unrelated at a first glance, resulted in several cases in surprising effects leading to the creation of new mathematical results and posting open questions.
# Impact
We strengthened bounds between the research centers and provided favorable conditions for mathematicians from different parts of the world to work together on the major up-to-date problems originating in the wide area of partial differential equations (PDE).
We hosted leading (world-famous) scientists in Europe, tightened cooperation between the partners across the world but also within ERA, granted knowledge transfer and gave promotion to the European science. We developed lasting collaboration with USA and Japan nodes and created new directions of cooperation.
# Further info
Website: www.impan.pl/~flux
“Towards regularity” was a project devoted to the mathematical analysis of a wide spectrum of problems originating from fluid mechanics, aerodynamics, geophysics, phase transitions, image processing and meteorology. The project was divided into four distinct research topics:
- Compressible Navier-Stokes equations
- Crystal growth and image processing
- Regularity criteria
- Asymptotic analysis
## Compressible Navier-Stokes equtions
We studied quantitative and qualitative aspect of solutions to chosen particular models (Navier-Stokes and related systems). Our work in this field was aimed at answering the questions concerning existence and uniqueness of solutions, their stability, asymptotic behavior and structure of solutions, and finally their regularity. Using newly developed techniques we obtained numerous estimates for solutions and their existence in certain special cases. However, these techniques need further refinements for some models involving chemical reactions. We also began the analysis of the stability of special solutions to compressible Navier-Stokes equations in the framework of regular solutions. Additionally, we successfully adapted techniques from the theory of Schauder estimates for parabolic equations, thereby making the study of free boundary problems for viscous compressible fluids more feasible.
## Crystal growth and image processing
The crystal growth and image processing problem lead to: TV flows and its generalization; evolution of closed curves by singular mean curvature and Stefan type problems with Gibbs-Thomson relation involving singular mean curvature. We were interested in showing well-posedness of the systems and their asymptotic behavior. We developed new viscosity theory for formally second order parabolic singular problems. Due to the importance of the this work it may be called a **milestone**. We also investigated free boundary problems, which led us to show that Berg's effect is very rare. Therefore, we started investigating closely related free boundary problems involving fractional derivatives (c.f. fractional diffusion equation).
## Regularity criteria
The problem of regularity of weak solution to the non-stationary 3D Navier-Stokes equations is one of the most challenging problems of the modern PDE theory. We investigated local properties of weak solutions to the N-S system. We established several sufficient conditions of regularity of weak solutions to the Navier-Stokes equations in a neighborhood of a given point in space-time. We also showed the global existence of regular solutions being close to special regular global solutions like two-dimensional or axially-symmetric, with respect to non-homogenous and compressible Navier-Stokes equations. Some ideas that were explored during the project, have evolved into independent research projects.
## Asymptotic analysis
The issue of stability and long-time behavior of solutions to various evolutionary models was our priority. We proved the existence of special, global, regular solutions (e.g. self-similar, $2d$, axially symmetric, time-periodic) to Navier-Stokes equations and related models. We also worked on the decay of solutions, the correspondence between the geometry of domains and behavior of solutions, obtaining numerous results that can serve as starting points for further studies.
# Relevance
Most of the results that we obtained, can be implemented in direct applications, like simulating fluid flows or biological phenomena. The strategic objective of the program was to complement and develop a mathematical theory of partial differential equations stemming from real-life applications. We covered a wide scope of analysis of partial differential equations, enforcing cooperation of specialists from different fields in the theory of PDEs. We created opportunities to exchange full range of ideas and to obtain certain synergies. Most of the work was carried out by the partners in their home institutions but the secondments truly served the purpose of an intensive research. The exchange of knowledge was augmented through organization of project-related seminars, workshops and conferences.
We overcame certain fragmentation of research and eliminated artificial barriers by direct exchange of knowledge and ideas within our network. A combination of different techniques or even points of view on mathematics, which might have seemed unrelated at a first glance, resulted in several cases in surprising effects leading to the creation of new mathematical results and posting open questions.
# Impact
We strengthened bounds between the research centers and provided favorable conditions for mathematicians from different parts of the world to work together on the major up-to-date problems originating in the wide area of partial differential equations (PDE).
We hosted leading (world-famous) scientists in Europe, tightened cooperation between the partners across the world but also within ERA, granted knowledge transfer and gave promotion to the European science. We developed lasting collaboration with USA and Japan nodes and created new directions of cooperation.
# Further info
Website: www.impan.pl/~flux