Mathematical methods for the study of equations of generalized viscous fluids

• Applicant: Privatdozent Dr. Jörg Wolf
• Subject Area: Mathematics
• Term: from 2014 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 256634185

In 1934 J. Leray has started the mathematical treatment of the Navier-Stokes equations by proving the existence of a weak solution in the whole space. Since this time these equations are one of the most studied partial differential equations with large contributions and many inspirations to various fields in Mathematical Analysis. Despite strong efforts of many mathematicians fundamental questions such as global existence and uniqueness of classical solutions are still open. One of the most important result in this direction is due to Caffarelli, Kohn and Nirenberg in 1984 by proving the regularity outside a possible set of singularities of dimension of maximal one. Since this time the regularity theory of weak solutions to the Navier-Stokes equations have been developed extensively. However, so far there is no local regularity theory which does not require the existence of a global pressure functions. In fact, such theory would be very helpful for the study of the local regularity of weak solutions to the equations of incompressible fluids with non-constant viscosity. The present research project aims to address this significant problem and will contribute largely to mathematical analysis of the equations of complex viscous fluids. The local pressure method which has been developed by the applicant will serve as one of the main tool for the research project. Recently, this method has been successfully used in order to prove the existence of a weak solution to the equations of non-Newtonian fluids. A major advantage of this method is the possibility to eliminate the pressure term in the local energy estimates which has not been possible before. On this basis we expect to proof the following main results:1. Extension of the Caffarelli-Kohn-Nirenberg theorem to the equations of non-Newtonian fluids.2. Local regularity criterion for solutions of equations to generalized viscous fluids together with an optimal bound of the Hausdorff dimension of the singular set.3. Existence and regularity of renormalized solutions to the equations of heat conducting fluids and turbulent flows due to Prandtl and Kolmogorov.By the present research project we wish to close an important gap in the regularity theory for models of complex viscous fluids. Furthermore the obtained results will give important innovations to other fields such as free boundary problems, Magneto hydrodynamic equations, Navier-Stokes-Fourier system compressible fluids.

• DFG Programme: Research Grants
• International Connection: Czech Republic
• Participating Persons: Professorin Dr. Maria Lukacova; Professor Dr. Joachim Naumann; Professorin Dr. Sarka Necasova