Fluids Under Control: The 2021 Prague-Sum Workshop Lectures

Preface
Contents
Existence and Regularity of Solutions for the Magnetohydrodynamic Flow with Navier-Type Boundary Conditions in 2-D
1 Introduction
1.1 Main Results
2 Basic Properties and Preliminary Results
2.1 Geometry of the Domain
2.2 Functional Spaces
2.3 Some Properties Related to the Curl Operator
2.4 An Important Identity Result in R2
2.5 Preliminary Results
3 Stokes Problem
3.1 Regularity of the Stokes Problem
3.2 Stokes Problem
4 Existence of the Solution for the MHD Problem
4.1 Regularity of the Solution
References
Well-Posedness and Optimal Control for 2-D Stochastic Second-Grade Fluids
1 Introduction
2 Functional Spaces
3 Well-Posedness of the State Equation
4 Control Problem
4.1 Stochastic Linearized State Equation and Gâteaux Differentiability of the Control-to-State Mapping
4.2 Stochastic Adjoint Equation
4.3 Duality Property for Y, Z, and (p,q)
4.4 Existence and Optimality Condition
References
Hopf Bifurcation for Navier–Stokes Flow Past a Rotating Obstacle
1 Introduction
2 Formulation of the Problem
3 Notation, Function Spaces, and Preliminary Considerations
4 On the Existence of an Analytic Steady-State Branch
5 Some Uniqueness Results
6 The Operator Q0 and Its Relevant Properties
7 On the Spectral Properties of the Linearized Operator
8 The Operator Q and Its Relevant Properties
9 A Time-Periodic Bifurcation Theorem
Appendix
References
Global Stabilization of a Rigid Body Moving in a Compressible Viscous Fluid
1 Introduction
1.1 Mathematical Description of the Model
2 Notations
3 Local Stabilization
3.1 Change of Variables
3.2 Linearized System
3.3 The Fluid-Structure Operator
3.4 Some Background on R-Sectorial Operators
3.5 R-Sectoriality of the Fluid-Structure Operator
3.6 Exponential Stability of the Fluid-Structure Operator
3.7 Maximal Lp-Lq Regularity for the Linearized Fluid-Structure System
3.8 Proof of Theorem 3.1 and Corollary 3.1
4 Global Stabilizability
References
Asymptotic Behavior of the Navier–Stokes Type Problem
1 Introduction
1.1 Discussion and Main Result
2 Notation and Preliminaries
3 Proof of Theorem 1.2
4 Proof of Theorem 1.3 and Corollary 1.1
5 Proof of Theorem 1.4
References
On Some Recent Results from the Theory of MHD Equations
1 Introduction
2 A Brief Survey of the Qualitative Theory of MHD and Hall's MHD Equations
3 Some Recent Results on the Role of Pressure in Theory of the MHD Equations
4 The L3–Local Regularity Criterion for the MHD Equations
References
Reduction of a 3D Mathematical Model of Flow Through an Axial Turbine to Two Spatial Dimensions and Restriction to One Period of the Infinite Periodical Domain
1 Introduction
2 Description of the Profile Cascade
3 Reduction of a 3D Mathematical Model of Flow Through an Axial Turbine to Two Spatial Dimensions
3.1 The 3D Axial Turbine Wheel
3.2 Equations of Motion
3.3 Transformation to the Rotating Frame
3.4 The Condition of Zero Velocity in the Radial Direction
3.5 A Simplification of the System (11)–(13)
3.6 A Remark on Eq.(10)
4 A Considered Mathematical Model
5 Restriction of the 2D Mathematical Model to One Spatial Period
5.1 The Framework of Classical Solutions
A Classical Solution in Domain O
A Classical Solution in One Spatial Period Ω
5.2 The Framework of Weak Solutions
Formal Derivation of a Weak Problem in Domain O and Definition of a Weak Solution
Remark (on the Boundary Condition (26))
A Weak Solution in One Spatial Period Ω
References
Steady Compressible Navier–Stokes–Fourier System with Slip Boundary Condition for the Velocity and Dirichlet Boundary Condition for the Temperature
1 Introduction
1.1 Constitutive Relations
1.2 Weak Formulation
1.3 Main Results
2 A Priori Estimates
3 Weak Compactness
4 Construction of the Solution
References
Classical Solution for the Compressible Flow with Free Surface in Three-Dimensional Exterior Domain
1 Introduction
1.1 Model
1.2 Main Result
2 Analysis of the Nonhomogeneous Boundary Conditions
2.1 Some Model Problem in ΩL
2.2 Some Nonlinear Estimates
2.3 Proof of Theorem 2.1
3 Estimates of the Lower Order Derivatives
3.1 Bound of D(η, u)
3.2 Proof of Theorem 3.1
4 Highest Order Derivatives Near the Boundary
5 Highest Order Derivatives Outside of BR+3
6 Highest Order Derivatives in ΩR+3
6.1 Solution of (78) in Eulerian Coordinates
6.2 Proof of (90)
7 Well-posedness Issue of (7)
7.1 Construction of the Local Solution
7.2 Argument on the Long Time Issue
8 Remark on the Property of the Maximal Regularity
8.1 Reduced Resolvent Problem in RN+
8.2 Complement of the Proof of (115)
9 Proof of (22)
10 Weighted Energy Estimate of Some Linear Problem
References
On Asymptotic Stability of Boussinesq Equations Without Heat Conduction
1 Introduction
2 Preliminaries
3 Global Existence of Solution
4 Nonlinear Asymptotic Stability
4.1 Linear Decay Estimates
4.2 Nonlinear Decay Estimates
5 Conclusion
References
Numerical Simulation of Fluid-Structure-Acoustic Interactions Models of Human Phonation Process
1 Introduction
2 Mathematical Model
2.1 Geometry Specification
2.2 Structural Model
2.3 Flow Model
2.4 FSI Coupled Problem
2.5 Acoustic Model
2.6 Glottis Closure Model
3 Numerical Approximation
3.1 Structural Model
3.2 Flow Model
3.3 Numerical Approximation of Acoustic Problem
3.4 Coupled Problem Solution
4 Numerical Results
4.1 Modal Analysis
4.2 Acoustic Resonant Frequencies of Vocal Tract
4.3 Phonation Onset and Flutter Instability
4.4 FSAI Simulations
5 Conclusion
References