Partial Differential Equations II
Code  Completion  Credits  Range  Language 

2011090  ZK  3  2P+0C  Czech 
 Garant předmětu:
 Stanislav Kračmar
 Lecturer:
 Stanislav Kračmar
 Tutor:
 Stanislav Kračmar
 Supervisor:
 Department of Technical Mathematics
 Synopsis:

The course contains the notion of the weak formulation of the boundary problem for the linear elliptic equation of the 2nd order, the concept of the weak solution.
The essential topics are LaxMilgram's lemma, the theorem on the existence and uniqueness of a weak solution. Equivalence of the mentioned boundary problem and a problem of finding the minimum of a suitable quadratic functional.
Galerkin and Ritz's method of finding an approximate solution.
Fundamentals of vector field theory, div, rot operators and their properties, Gauss theorem, Stokes theorem, Weyl's vector field decomposition to the sum of the ∇φ field and the rot w field, Helmholz's decomposition, and related issues.
Euler, Stokes, Oseen and NavierStokes equations for an incompressible fluid. Classical and weak formulations of the boundary problem in the stationary case and mixed initialboundary problems in the nonstationary case for the NavierStokes equations.
 Requirements:
 Syllabus of lectures:

The course contains the notion of the weak formulation of the boundary problem for the linear elliptic equation of the 2nd order, the concept of the weak solution.
The essential topics are LaxMilgram's lemma, the theorem on the existence and uniqueness of a weak solution. Equivalence of the mentioned boundary problem and a problem of finding the minimum of a suitable quadratic functional.
Galerkin and Ritz's method of finding an approximate solution.
Fundamentals of vector field theory, div, rot operators and their properties, Gauss theorem, Stokes theorem, Weyl's vector field decomposition to the sum of the ∇φ field and the rot w field, Helmholz's decomposition, and related issues.
Euler, Stokes, Oseen and NavierStokes equations for an incompressible fluid. Classical and weak formulations of the boundary problem in the stationary case and mixed initialboundary problems in the nonstationary case for the NavierStokes equations.
 Syllabus of tutorials:
 Study Objective:
 Study materials:

• K. Rektorys: Variational Methods in Mathematics, Science and Engineering. Springer, Netherlands,1977.
• L. C. Evans: Partial differential equations. Graduate Studies in Mathematics, Vol 19, American Mathematical Society, Second Edition 2010.
• M. Chipot: Elliptic equations. An Introductory cource. Birkhauser Verlag, 2009.
• M. E. Taylor: Partial differential equations I. Applied Mathematical Sciences 115, Springer, 2011.
• R. Temam: NavierStokes equations. Theory and numerical analysis. North Holland, 1979, and later issues.
 Note:
 Timetable for winter semester 2023/2024:

06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon Tue Wed Thu Fri  Timetable for summer semester 2023/2024:
 Timetable is not available yet
 The course is a part of the following study plans: