An introduction to the mathematical theory of the Navier-Stokes

Garant předmětu:

Zdeněk Skalák

Lecturer:

Zdeněk Skalák

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The goal of the subject is to inform students about the basics the mathematical theory of the Navier-Stokes equations for the incompressible fluid. The content of the subject: The description of the Navier-Stokes equations, the introduction of the fundamental concepts, the definition of the fundamental function spaces, the description of the basic relations between the function spaces, the definition of the classical and weak solution, the expulsion of the pressure from the definition of the weak solution, Helmholtz decomposition, some elementary properties of the weak solution, the proof of the existence of the weak solutions by the Galerkin method in a general domain, the discussion of several different definitions of the weak solution, qualitative properties of the weak solution, energy inequality, strong energy inequality, the sufficient conditions for the energy equality, the problem of the uniqueness and regularity, the fundamental uniqueness theorem, the role of the initial conditions, brief discussion of the asymptotic behavior of the solution, brief discussion of large solutions, brief discussion of various proofs of the existence of the weak solution, mild solutions.

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Note:

Time-table for winter semester 2023/2024:

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Time-table for summer semester 2023/2024:

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The course is a part of the following study plans: