Research
Generalized Newtonian Fluids
Many fluids whose behavior cannot be adequately described by the Navier-Stokes equations can be modeled as so-called generalized Newtonian fluids. My current research focuses on investigating various aspects of the systems of equations with so-called p-structure resulting from this approach. A central problem is the improvement of the lower bound for the existence of weak solutions. Based on the various techniques developed in 59, 46, 25, 68, 19, 20 for the stationary and unsteady equations, the existence of weak solutions for p > 2N⁄N+2 could be proven in both cases. This bound is optimal. Further qualitative properties of the solutions were proven in 59, 56, 53, 41, 33, 25, b1. Analogous results for Herschel-Bulkley fluids and inhomogeneous generalized Newtonian fluids are contained in 32, 24, 17. Another central topic is the regularity theory for equations with p-structure. Satisfactory results for generalized Newtonian fluids are only known for periodic boundary conditions (cf. 53, 40, 33, 21). In the case of Dirichlet boundary conditions, fundamental questions remain unanswered that result from the divergence condition and the dependence of nonlinearity on the symmetric gradient. So far, only suboptimal results are known (18, 13). Many questions also remain unanswered in the field of numerical analysis of problems with p-structure. Initial results on convergence rates that depend on p, fully implicit space-time discretizations of flows of generalized Newtonian fluids are contained in 40 (cf. 37, 31 for time discretizations). A central problem here is to find a suitable measure for the error. Based on ideas from variational calculus, regularity theory, and numerical analysis, a “natural distance” has emerged that proves to be the appropriate measure of error (cf. 27, 29, 25, 21, 23, 10). In 29 and 23, optimal convergence rates independent of p were proven for the first time. As part of the DFG research group “Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis,” the ALBERTA software package was expanded in collaboration with K. Siebert to include a solver for flow problems of generalized Newtonian fluids (cf. 10 for test calculations in stationary cases). Recently, I have also been interested in the coupling of flows of generalized Newtonian fluids with elastic structures. Such “fluid-structure interaction” problems occur, among other things, in the modeling of blood flow in veins. A subproject in SFB/TR 71 “Geometric Partial Differential Equations” is dedicated to the analytical and numerical investigation of this problem.
Electrorheological Materials
Another focus of my current research, as during my research stays in Pittsburgh and Ferrara, is the modeling of electrorheological fluids (64, 65, 67, ü4, ü5, b2) and the mathematical treatment of the resulting systems of equations (43, 44, 38, 33, 28, 26, 25, 20, ü4, ü5, b2). Electrorheological fluids change their viscosity in response to an external electric field. This effect is reflected in the form of the stress tensor. The stationary or transient so-called p(.)-Laplace operator can be regarded as a mathematical prototype for this. Here, p is a given location- and time-dependent function. This opens up a wide field for the investigation of partial differential equations with p(.)-structure as well as for the investigation of function spaces with variable exponents and operators defined on them. Both areas are currently experiencing an enormous amount of publication activity. In addition to the works already listed, our own contributions to this problem area can be found in 39, 36, 35, 34, 68, b4.
Limit Values of Compressible Fluids
In 48, a consistent derivation of the Boussinesq approximation, the standard model for convection problems, was achieved for the first time. A mathematically rigorous justification of simplified subproblems is contained in 42, 30. In 11, 15, 66, 8, various incompressible models are justified as constitutive limit values for compressible fluids. This mathematically rigorous approach is motivated by studies from engineering literature. The works 22, 14 contain initial theoretical results for the Boussinesq approximation between coaxial cylinders.
Regularity of the Navier-Stokes Equations
Due to scaling properties of the Navier-Stokes equations, there are analogies between the N-dimensional instationary and the N+2-dimensional stationary equations. For N > 2, the regularity and uniqueness of weak solutions is an open problem in both cases. For the stationary equations, the existence of weak solutions that additionally have a one-sided maximum property for the Navier-Stokes equations has been proven. For N > 2, the regularity and uniqueness of weak solutions is an open problem in both cases. For the stationary equations, the existence of weak solutions that additionally possess a one-sided maximum property for the static pressure has been demonstrated (58, 57, 54, 52, 47). Due to this maximum property, it is possible to show local regularity of so-called weak “maximum solutions” (55, 54, 52, 47, 45, ü2, ü3). These ideas could be applied in 16 to the Rothe approximation in any dimension. Furthermore, for the first time in the theory of Navier-Stokes equations, it was possible to specify a sufficient regularity criterion that has a positive scaling dimension (45). A one-sided bound on the static pressure also plays a decisive role in solving an old problem posed by Leray. In 50, 49, it was shown that self-similar solutions of the unsteady Navier-Stokes equations lying in the space L∞ (I;L3( R3)) are necessarily trivial.
Modeling
Modeling new phenomena and innovative materials within the framework of continuum mechanics is another area that interests me. Collaboration with engineers and physicists yields new insights and inspiration for both sides. This results in physically sound equations whose mathematical analysis raises interesting and novel mathematical problems (cf. multipolar viscoelastic fluids 61, 63, ü1; electrorheological fluids 64, 65, 67, b2; limit values of compressible fluids 56, 48, 42, 11, 66, 8).