Ebook: Lp-Theory for Incompressible Newtonian Flows: Energy Preserving Boundary Conditions, Weakly Singular Domains

This thesis is devoted to the study of the basic equations of fluid dynamics. First Matthias Köhne focuses on the derivation of a class of boundary conditions, which is based on energy estimates, and, thus, leads to physically relevant conditions. The derived class thereby contains many prominent artificial boundary conditions, which have proved to be suitable for direct numerical simulations involving artificial boundaries. The second part is devoted to the development of a complete Lp-theory for the resulting initial boundary value problems in bounded smooth domains, i.e. the Navier-Stokes equations complemented by one of the derived energy preserving boundary conditions. Finally, the third part of this thesis focuses on the corresponding theory for bounded, non-smooth domains, where the boundary of the domain is allowed to contain a finite number of edges, provided the smooth components of the boundary that meet at such an edge are locally orthogonal.

---

This thesis is devoted to the study of the basic equations of fluid dynamics. First Matthias K?hne focuses on the derivation of a class of boundary conditions, which is based on energy estimates, and, thus, leads to physically relevant conditions. The derived class thereby contains many prominent artificial boundary conditions, which have proved to be suitable for direct numerical simulations involving artificial boundaries. The second part is devoted to the development of a complete Lp-theory for the resulting initial boundary value problems in bounded smooth domains, i.e. the Navier-Stokes equations complemented by one of the derived energy preserving boundary conditions. Finally, the third part of this thesis focuses on the corresponding theory for bounded, non-smooth domains, where the boundary of the domain is allowed to contain a finite number of edges, provided the smooth components of the boundary that meet at such an edge are locally orthogonal.

Contents

· Navier-Stokes Equations

· Energy Preserving Boundary Condition

· Weakly Singular Domain

· Maximal Lp-Regularity

Target Groups

· Scientists, lecturers and graduate students in the fields of mathematical fluid dynamics and partial differential equations as well as experts in applied analysis.

The author

Matthias K?hne earned a doctorate of Mathematics under the supervision of Prof. Dr. Dieter Bothe at the Department of Mathematics at TU Darmstadt, where his research was supported by the cluster of excellence ''Center of Smart Interfaces'' and the international research training group ''Mathematical Fluid Dynamics''.

---

This thesis is devoted to the study of the basic equations of fluid dynamics. First Matthias K?hne focuses on the derivation of a class of boundary conditions, which is based on energy estimates, and, thus, leads to physically relevant conditions. The derived class thereby contains many prominent artificial boundary conditions, which have proved to be suitable for direct numerical simulations involving artificial boundaries. The second part is devoted to the development of a complete Lp-theory for the resulting initial boundary value problems in bounded smooth domains, i.e. the Navier-Stokes equations complemented by one of the derived energy preserving boundary conditions. Finally, the third part of this thesis focuses on the corresponding theory for bounded, non-smooth domains, where the boundary of the domain is allowed to contain a finite number of edges, provided the smooth components of the boundary that meet at such an edge are locally orthogonal.

Contents

· Navier-Stokes Equations

· Energy Preserving Boundary Condition

· Weakly Singular Domain

· Maximal Lp-Regularity

Target Groups

· Scientists, lecturers and graduate students in the fields of mathematical fluid dynamics and partial differential equations as well as experts in applied analysis.

The author

Matthias K?hne earned a doctorate of Mathematics under the supervision of Prof. Dr. Dieter Bothe at the Department of Mathematics at TU Darmstadt, where his research was supported by the cluster of excellence ''Center of Smart Interfaces'' and the international research training group ''Mathematical Fluid Dynamics''.

Content:
Front Matter....Pages 1-1
Front Matter....Pages 9-9
The Navier-Stokes Equations....Pages 11-19
Energy Preserving Boundary Conditions....Pages 21-32
Front Matter....Pages 33-33
L p -Theory for Incompressible Newtonian Flows....Pages 35-65
Tools and Methods....Pages 67-100
Maximal L p -Regularity in a Halfspace....Pages 101-114
Maximal L p -Regularity in a Bent Halfspace....Pages 115-126
Maximal L p -Regularity in a Bounded Smooth Domain....Pages 127-150
Front Matter....Pages 151-151
L p -Theory in Weakly Singular Domains....Pages 153-167
Back Matter....Pages 13-13

---

This thesis is devoted to the study of the basic equations of fluid dynamics. First Matthias K?hne focuses on the derivation of a class of boundary conditions, which is based on energy estimates, and, thus, leads to physically relevant conditions. The derived class thereby contains many prominent artificial boundary conditions, which have proved to be suitable for direct numerical simulations involving artificial boundaries. The second part is devoted to the development of a complete Lp-theory for the resulting initial boundary value problems in bounded smooth domains, i.e. the Navier-Stokes equations complemented by one of the derived energy preserving boundary conditions. Finally, the third part of this thesis focuses on the corresponding theory for bounded, non-smooth domains, where the boundary of the domain is allowed to contain a finite number of edges, provided the smooth components of the boundary that meet at such an edge are locally orthogonal.

Contents

· Navier-Stokes Equations

· Energy Preserving Boundary Condition

· Weakly Singular Domain

· Maximal Lp-Regularity

Target Groups

· Scientists, lecturers and graduate students in the fields of mathematical fluid dynamics and partial differential equations as well as experts in applied analysis.

The author

Matthias K?hne earned a doctorate of Mathematics under the supervision of Prof. Dr. Dieter Bothe at the Department of Mathematics at TU Darmstadt, where his research was supported by the cluster of excellence ''Center of Smart Interfaces'' and the international research training group ''Mathematical Fluid Dynamics''.

Content:
Front Matter....Pages 1-1
Front Matter....Pages 9-9
The Navier-Stokes Equations....Pages 11-19
Energy Preserving Boundary Conditions....Pages 21-32
Front Matter....Pages 33-33
L p -Theory for Incompressible Newtonian Flows....Pages 35-65
Tools and Methods....Pages 67-100
Maximal L p -Regularity in a Halfspace....Pages 101-114
Maximal L p -Regularity in a Bent Halfspace....Pages 115-126
Maximal L p -Regularity in a Bounded Smooth Domain....Pages 127-150
Front Matter....Pages 151-151
L p -Theory in Weakly Singular Domains....Pages 153-167
Back Matter....Pages 13-13
....