## Periodic Report Summary 1 - FLUX (Towards regularity)

1. Overview:

“Towards regularity” is a project devoted to the mathematical analysis of a wide spectrum of problems originating from different domains of real world applications. They include: fluid mechanics, aerodynamics, geophysics, phase transitions, image processing and meteorology. The language for rigorous formulation of models for these phenomena is the theory of partial differential equations.

The project is divided into four distinct research topics, which include:

-- Compressible Navier-Stokes equations

-- Crystal growth and image processing

-- Regularity criteria

-- Asymptotic analysis

1.1 Compressible Navier-Stokes equations

We plan to study quantitative and qualitative aspect of solutions to chosen particular models from this subject (most important are the Navier-Stokes system and Navier-Stokes-Fourier systems). Our work in this field will be aimed at answering the questions concerning existence and uniqueness of solutions, their stability (in particular for large velocity vectors), asymptotic behavior and structure of solutions, and finally their regularity.

1.2 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. Mathematically speaking, we are interested in showing well-posedness of the system and their asymptotic behavior. The central issue, however, is generation, evolution and persistence of facets.

1.3 Regularity criteria

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 are planning to investigate local properties of weak solutions to the NS system. Our main goal is to find out new sufficient conditions of regularity of weak solutions to the Navier-Stokes equations in a neighborhood of a given point in space-time. We are also going to show 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. Our other aim is to analyze the behavior of solutions near potential blow up points.

1.4 Asymptotic analysis

We focus on issues, where the prospective team members contributed significantly to the present state of art:

-- Long-time behavior of solutions and related problems, including stabilization of solutions to equilibria, existence of global attractors, time periodic and time almost-periodic solutions.

-- Scale analysis and singular limits, relations between different models of fluid dynamics.

-- Effect of domain geometries on the boundary behavior, results based on asymptotic analysis, scaling and study of boundary layers.

-- Stability of special regular solutions to Navier-Stokes and MHD equations.

2.Description of the performed work and achieved results

Since the start of the project we have had over 20 seconded researchers months during which we worked on the topics covering:

-- stability of solutions to compressible and incompressible Navier-Stokes equations and MHD systems in various domains under Dirichlet and slip boundary conditions,

-- decay of solutions to the generalized Navier-Stokes equations (with fractional diffusion),

-- regularity criteria for solutions to the Navier-Stokes equations,

-- analysis of variational solutions to singular mean curvature,

-- development of viscosity theory for second order parabolic singular problems.

We have obtained several results which among others include:

-- construction of various special solutions to Navier-Stokes equations and MHD systems. By special we mean 2d or close to 2d solutions without any smallness on the initial or external data,

-- significant progress in applying viscosity theory for formally second order parabolic singular problems,

-- better understanding of the Berg's effect,

-- Serrin type conditions for the Boussinesq system without any conditional assumptions on the temperature.

3. Impact:

It is well known that researchers in mathematics often specialize in similar techniques within a given institution. This results in a certain fragmentation of research and creates artificial barriers, limiting a possible general impact of results created by a particular school. Our goal is to overcome such a fragmentation by direct exchange of knowledge and ideas within our network. A combination of different techniques or even points of view on mathematics, which may seem unrelated at a first glance, can result in surprising effects leading to the creation of new mathematical tools. Their possible applications may easily exceed the scope of the project.

We expect the project to result in creating new directions of cooperation and tightening the existing bounds between the partner countries. Well-established cooperation between the European partners is the basis for the project. Scientific staff exchange at all the levels will lead to strengthening these bounds giving the prospects for even more efficient collaboration in the future.

“Towards regularity” is a project devoted to the mathematical analysis of a wide spectrum of problems originating from different domains of real world applications. They include: fluid mechanics, aerodynamics, geophysics, phase transitions, image processing and meteorology. The language for rigorous formulation of models for these phenomena is the theory of partial differential equations.

The project is divided into four distinct research topics, which include:

-- Compressible Navier-Stokes equations

-- Crystal growth and image processing

-- Regularity criteria

-- Asymptotic analysis

1.1 Compressible Navier-Stokes equations

We plan to study quantitative and qualitative aspect of solutions to chosen particular models from this subject (most important are the Navier-Stokes system and Navier-Stokes-Fourier systems). Our work in this field will be aimed at answering the questions concerning existence and uniqueness of solutions, their stability (in particular for large velocity vectors), asymptotic behavior and structure of solutions, and finally their regularity.

1.2 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. Mathematically speaking, we are interested in showing well-posedness of the system and their asymptotic behavior. The central issue, however, is generation, evolution and persistence of facets.

1.3 Regularity criteria

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 are planning to investigate local properties of weak solutions to the NS system. Our main goal is to find out new sufficient conditions of regularity of weak solutions to the Navier-Stokes equations in a neighborhood of a given point in space-time. We are also going to show 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. Our other aim is to analyze the behavior of solutions near potential blow up points.

1.4 Asymptotic analysis

We focus on issues, where the prospective team members contributed significantly to the present state of art:

-- Long-time behavior of solutions and related problems, including stabilization of solutions to equilibria, existence of global attractors, time periodic and time almost-periodic solutions.

-- Scale analysis and singular limits, relations between different models of fluid dynamics.

-- Effect of domain geometries on the boundary behavior, results based on asymptotic analysis, scaling and study of boundary layers.

-- Stability of special regular solutions to Navier-Stokes and MHD equations.

2.Description of the performed work and achieved results

Since the start of the project we have had over 20 seconded researchers months during which we worked on the topics covering:

-- stability of solutions to compressible and incompressible Navier-Stokes equations and MHD systems in various domains under Dirichlet and slip boundary conditions,

-- decay of solutions to the generalized Navier-Stokes equations (with fractional diffusion),

-- regularity criteria for solutions to the Navier-Stokes equations,

-- analysis of variational solutions to singular mean curvature,

-- development of viscosity theory for second order parabolic singular problems.

We have obtained several results which among others include:

-- construction of various special solutions to Navier-Stokes equations and MHD systems. By special we mean 2d or close to 2d solutions without any smallness on the initial or external data,

-- significant progress in applying viscosity theory for formally second order parabolic singular problems,

-- better understanding of the Berg's effect,

-- Serrin type conditions for the Boussinesq system without any conditional assumptions on the temperature.

3. Impact:

It is well known that researchers in mathematics often specialize in similar techniques within a given institution. This results in a certain fragmentation of research and creates artificial barriers, limiting a possible general impact of results created by a particular school. Our goal is to overcome such a fragmentation by direct exchange of knowledge and ideas within our network. A combination of different techniques or even points of view on mathematics, which may seem unrelated at a first glance, can result in surprising effects leading to the creation of new mathematical tools. Their possible applications may easily exceed the scope of the project.

We expect the project to result in creating new directions of cooperation and tightening the existing bounds between the partner countries. Well-established cooperation between the European partners is the basis for the project. Scientific staff exchange at all the levels will lead to strengthening these bounds giving the prospects for even more efficient collaboration in the future.

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Scientific Research**Numer rekordu**: 182648 /

**Ostatnia aktualizacja**: 2016-05-17