Flow equations for non-Newtonian fluids 1000-1M22RCN
The aim of this lecture is to familiarize students with the mathematical analysis of systems of nonlinear partial differential equations modeling the flow of non-Newtonian fluids. We will therefore learn mathematical methods in non-Newtonian fluid mechanics. Examples of motivations for considering such equations include blood flow, glacier movement, the dynamics of the Earth's mantle, the behavior of substances such as slime, Silly Putty, and quicksand.
The lecture will start with an outline of models and applications. We will then turn to their mathematical analysis. We will focus on the existence of solutions, but we will also discuss uniqueness and regularity.
We will present the development of the theory from the 1960s to recent results. This overview will familiarize us with techniques and methods useful in the study of nonlinear partial differential equations.
It is recommended that students first complete the lecture on the theory of partial differential equations and functional analysis.
In particular, we will address the following topics (depending on time available):
1. Introduction to the equations of non-Newtonian fluids. Derivation and motivation.
2. Existence of weak solutions. The method of monotonic operators. The Galerkin method, fixed-point theorems. Energy estimates. Bochner spaces. Integration by parts in Bochner spaces. Korn's inequality. The Aubin-Lions theorem. Weak continuum stability.
3. Definition of the Stokes operator and its properties. The method of higher energy estimates.
4. Definition of uniform integrability. Vitali's theorem.
5. Uniqueness of solutions. Regularity of solutions.
6. Maximum function and the Hardy-Littlewood theorem. The method of Lipschitzian truncation .
7. The concept of measure-valued solutions. The definition of Young measures. Reduction of Young measures to Dirac deltas.
8. Non-standard growth conditions. Orlicz spaces, Musielak-Orlicz spaces. Monotonicity methods for nonreflexive spaces.
Type of course
Prerequisites (description)
Course coordinators
Learning outcomes
Knowledge of selected analysis methods for nonlinear partial differential equations of fluid mechanics
Assessment criteria
The lecture will end with an oral examination. The student will be asked to presents a selected issue/issues, e.g. a scientific article or a part of a scientific article, or a monograph, or material that was covered in part of the lecture. The scope of the material to be presented should be agreed upon in advance with the lecturer.
Bibliography
1. Malek J., Necas J., Rokyta, M., Ruzicka M., Weak and Measure-valued Solutions to Evolutionary PDEs, Chapman & Hall 1996
2. Ladyzhenskaya, O.A. The boundary value problems of mathematical physics. Springer-Verlag, New York, 1985.
3. Frehse, J., Malek, J., Steinhauer, M. On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34 (2003), no. 5, 1064-1083
4. Chlebicka I., Gwiazda P., Świerczewska-Gwiazda A., Wróblewska-Kamińska A., Partial Differential Equations in Anisotropic Musielak-Orlicz Spaces, Springer, 2021
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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