Collected Papers in Honor of Yoshihiro Shibata

Contents
Preface
References
Global Wellposedness of the Primitive Equations with Nonlinear Equation of State in Critical Spaces
Abstract
1. Introduction
2. Preliminaries
3. Typical Ocean Densities
3.1. Linear Density
3.2. Equation of State by TEOS-10
3.3. Equation of State by McDongall–Jacket–Wright–Feistel
3.4. Equation of State by UNESCO-80
4. Main Result
5. Estimates for the Local Existence
6. A Priori Estimates
7. Proof of Theorem 4.1
7.1. Local Wellposedness
7.2. Global Wellposedness
Appendix A. Semilinear Evolution Equations and Maximal Lr-Regularity
References
On the Global Existence for the Compressible Euler–Riesz System
Abstract
Introduction
1. Main Results
2. A Local in Time Result for Non-decaying Data
2.1. A Priori Estimates
2.2. About the Proof of Existence
2.3. Uniqueness
3. A Global Existence Result
3.1. A Priori Estimates
3.2. Existence
3.3. The Proof of Uniqueness
3.4. Instability of Nontrivial Static Solutions in the Attractive Case
4. About Ideal Gases
4.1. Local Existence
4.2. Global Existence
4.3. Remark on Static Solutions
Appendix
Acknowledgements
References
Rotation Problem for a Two-Phase Drop
Abstract
1. Introduction
2. Linear Problem
3. The Nonlinear Problem
References
On the Stokes-Type Resolvent Problem Associated with Time-Periodic Flow Around a Rotating Obstacle
Abstract
1. Introduction
2. Notation
3. Main Results
4. The Resolvent Problem in the Whole Space
5. The Resolvent Problem in an Exterior Domain
6. The Time-Periodic Problem
References
Euler System with a Polytropic Equation of State as a Vanishing Viscosity Limit
Abstract
1. Introduction
2. Preliminary Material
2.1. Mathematical Theory of the Closed System
2.2. Transport Coefficients
2.3. Equation of State
2.4. Relative Energy
3. Main Results
3.1. Unconditional Convergence in the Absence of Boundary Layer
3.2. Conditional Result: Viscous Boundary Layer
4. Consistency of the Vanishing Dissipation/Radiation Approximation
4.1. Temperature for the Euler System
4.2. Consistency
4.2.1. Viscous Stress Consistency
4.2.2. Heat Flux Consistency
4.2.3. Radiation Entropy Convective Flux Consistency
5. Convergence
5.1. Velocity Regularization
5.2. Application of the Relative Energy Inequality
5.3. Integrals Controlled by the Consistency Estimates
5.4. Integrals Independent of the Boundary Layer
5.5. Boundary Layer
5.5.1. Viscous Stress
5.5.2. Convective Term
5.6. Strong Convergence
References
On the Hydrostatic Approximation of Compressible Anisotropic Navier–Stokes Equations–Rigorous Justification
Abstract
1. Introduction
2. Preliminaries
3. Main Result
3.1. Dissipative Weak Solutions of CNS
3.2. Strong Solution of CPE
3.3. Versatile Relative Entropy Inequality
3.4. Main Result
4. Convergence
4.1. Main Idea of Proof
4.2. Step 1
4.3. Step 2
4.4. Step 3
Acknowledgements
References
A Route to Chaos in Rayleigh–Bénard Heat Convection
Abstract
1. Introduction
2. linear Stability and Critical Rayleigh Number
3. Routes to Chaos
3.1. Roll Solutions on Bifurcation Branches in the Large
3.2. Time Evolution of Roll Solutions and the Secondary Hopf Bifurcation
3.3. Concluding Remark
Acknowledgements
References
Existence of Weak Solution to the Nonstationary Navier-Stokes Equations Approximated by Pressure Stabilization Method
Abstract
1. Introduction
2. Notations and Main Results
3. Preliminaries
4. Proof of Main Results
Acknowledgements
References
Resolvent Estimates for a Compressible Fluid Model of Korteweg Type and Their Application
Abstract
1. Introduction
2. Notation and Main Results
2.1. Notation
2.2. Main Results
3. Preliminaries
3.1. Some Inequalities
3.2. Compact Embeddings
3.3. Results of the Large Resolvent Parameter
3.4. Maximal Regularity
4. The Problem in Bounded Domains
4.1. Existence of Solutions
4.2. Uniqueness of Solutions
4.3. A Priori Estimates
4.4. Proof of Theorem 2.5
4.5. Proof of Theorem 2.6
5. The Whole Space Problem
5.1. Representation Formulas of Solutions
5.2. Estimates of P(ξ,λ) for γ=0.
5.3. Estimates of P(ξ,λ) for γ>0.
5.4. Proof of Theorem 5.1
6. The Problem in Exterior Domains
6.1. Construction of Parametrix
6.2. Uniqueness of Solutions
6.3. A Priori Estimates
6.4. An Auxiliary Problem
6.5. Proof of Theorem 2.1
7. Application to a Nonlinear Problem
7.1. Generation of an Analytic C0-Semigroup
7.2. Maximal Regularity with Exponential Stability
7.3. Estimates of Nonlinear Terms
7.4. Global Solvability of the Nonlinear Problem
References
Rate of the Enhanced Dissipation for the Two-jet Kolmogorov Type Flow on the Unit Sphere
Abstract
1. Introduction
2. Preliminaries
3. Analysis of the Linearized Operator
3.1. Settings and Basic Results
3.2. Verification of Assumption 4.6
3.3. Estimates for the Semigroup
4. Abstract Results
5. Appendix: Basic Formulas of Differential Geometry
Acknowledgements
References
Reacting Multi-component Fluids: Regular Solutions in Lorentz Spaces
Abstract
1. Introduction
2. Functional Spaces and the Main Result
3. Auxiliary Results and Linear Theory
4. A Priori Estimates
4.1. Velocity Bounds
4.2. Estimates for the Density
5. Existence
Acknowledgements
References
Global Well Posedness for a Q-tensor Model of Nematic Liquid Crystals
Abstract
1. Introduction
2. Maximal Lp–Lq Regularity
2.1. mathcalR-boundedness of Solution Operators
2.2. A Proof of Theorem 2.1
3. Decay Property of Solutions to the Linearized Problem
3.1. Decay Estimates for d
3.2. Decay Estimates for U and mathbbQ
3.2.1. Analysis of Low Frequency Parts
3.2.2. Analysis of High Frequency Parts
4. A Proof of Theorem 1.1
4.1. Analysis of Time Shifted Equations
4.2. Analysis of Compensation Equations
4.2.1. Estimates of Spatial Derivatives in Lp–Lq
4.2.2. Estimates of Time Derivatives in Lp–Lq
4.2.3. Estimates of the Lower Order Term in Linfty–Lq
4.3. Conclusion
References
Maximal Regularity for Compressible Two-Fluid System
Abstract
1. Introduction
1.1. Notation
1.2. Main Results
1.3. Discussion
2. Lagrangian Coordinates
3. Local Well-Posedness
3.1. Linearization Around the Initial Condition
3.2. Maximal Regularity
3.3. Preliminary Estimates
3.4. Estimate of the Right Hand Side of (3.3)
3.5. Contraction Argument—Proof of Theorem 1.1
4. Global Well-Posedness
4.1. Linearization Around the Constant State
4.2. Exponential Decay
4.3. Bounds for Nonlinearities
4.4. Proof of Theorem 1.2
Appendix
Acknowledgements
References
Steady Compressible Navier–Stokes–Fourier Equations with Dirichlet Boundary Condition for the Temperature
Abstract
1. Introduction
2. Formulation of the Problem: Main Result
3. Weak Compactness of Weak and Variational Entropy Ballistic Solutions
3.1. A Priori Estimates
3.2. Weak Compactness
4. Construction of the Solution
References
A Slightly Supercritical Condition of Regularity of Axisymmetric Solutions to the Navier–Stokes Equations
Abstract
1. Introduction
2. Auxiliary Facts
3. Proof of Proposition 1.4
4. Proof of Theorem 1.3
Acknowledgements
References
Spatial Pointwise Behavior of Time-Periodic Navier–Stokes Flow Induced by Oscillation of a Moving Obstacle
Abstract
1. Introduction
2. Results
2.1. Notation
2.2. Evolution Operator
2.3. Main Results
3. Proof of Theorem 2.1
3.1. Weak Form of the Integral Equation
3.2. Regularity in x
3.3. Regularity in t and the Pressure
4. Proof of Theorem 2.2
4.1. Reduction to the Whole Space Problem
4.2. Integral Equation for the Whole Space Problem
4.3. Reconstruction Procedure
References