Nonlinear Partial Differential Equations for Future Applications: Sendai, Japan, July 10–28 and October 2–6, 2017

Preface
Contents
An Introduction to Maximal Regularity for Parabolic Evolution Equations
	1 Introduction
	2 Maximal Regularity and Lp-Sobolev Spaces
		2.1 Linearization and Maximal Regularity
		2.2 Definition of Maximal Lp-Regularity
		2.3 Maximal Regularity for Non-autonomous Problems
	3 The Concept of mathcalR-Boundedness and the Theorem of Mikhlin
		3.1 mathcalR-Bounded Operator Families
		3.2 Fourier Multipliers and Mikhlin\'s Theorem
		3.3 mathcalR-sectorial Operators
	4 Lp-Sobolev Spaces
	5 Parabolic PDE Systems in the Whole Space
	6 Parabolic Boundary Value Problems
		6.1 The Shapiro-Lopatinksii Condition
		6.2 The Main Result on Parameter-Elliptic Boundary Value Problems
	7 Quasilinear Parabolic Evolution Equations
		7.1 Well-Posedness for Quasilinear Parabolic Evolution Equations
		7.2 Higher Regularity
	References
On Stability and Bifurcation in Parallel Flows of Compressible Navier-Stokes Equations
	1 Introduction
	2 Stability of Parallel Flows
	3 Outline of Proof of Theorem 2.1
		3.1 Notation
		3.2 Spectral Properties of the Linearized Semigroup
		3.3 Nonlinear Problem
	4 Instability and Bifurcation in Poiseuille Flows
		4.1 Notation
		4.2 Instability of Plane Poiseuille Flow
		4.3 Bifurcation of Wave Trains
	References
Uniform Regularity for a Compressible Gross-Pitaevskii-Navier-Stokes System
	1 Introduction
	2 Proof of Theorem 1.2
	References
Singular Limit Problem to the Keller-Segel System in Critical Spaces and Related Medical Problems—An Application of Maximal Regularity
	1 Introduction—The Singular Limit Problem
		1.1 Keller-Segel System in the Scaling Invariant Spaces
		1.2 The Chaplain-Anderson Model and the Fujie-Senba Equation
	2 Well-Posedness Issue in the Critical Setting
		2.1 Well-Posedness of the Full System
		2.2 Well-Posedness of the Keller-Segel System
		2.3 Two-Dimensional Critical Case for Keller-Segel System
		2.4 Singular Limit for the Keller-Segel System
		2.5 Formal Observation for the Singular Limit
	3 The Singular Limit Problem for the Chaplain-Anderson Systems
		3.1 The Well-Posedness
		3.2 Singular Limit Problem
	4 Preliminary Estimates
		4.1 Inequalities and Embeddings in Four Space Dimensions
		4.2 Heat Evolution on VMO
	5 Generalized Maximal Regularity
	6 Proof of Well-Posedness for Keller-Segel System
	7 Proof for the Singular Limit
	8 Proof for the Well-Posedness of Chaplain-Anderson and Fujie-Senba System
	9 Proof for the Singular Limit for Chaplain-Anderson Model
	References
HJB Equation, Dynamic Programming Principle, and Stochastic Optimal Control
	1 Introduction
	2 Stochastic Optimal Control Problem
		2.1 Strong Formulation of Optimal Control Problem
		2.2 Weak Formulation of Optimal Control Problem
		2.3 State Equation
	3 Dynamic Programming Principle and HJB Equation
		3.1 Verification Theorem, Necessary and Sufficient Conditions for Optimality
		3.2 Construction of Optimal Feedback Controls
		3.3 Uniqueness in Law
	4 Value Function and Proof of Dynamic Programming Principle
		4.1 Predictable Processes
		4.2 Canonical Reference Probability Space
		4.3 Independence of Value Function of Reference Probability Spaces
		4.4 Standard Reference Probability Spaces
		4.5 ``Conditioned\'\' Reference Probability Spaces
		4.6 Proof of the Dynamic Programming Principle
		4.7 Continuity of the Value Function in t
		4.8 Dynamic Programming Principle with Stopping Times
	5 Value Function Solves the HJB Equation
	References
Regularity of Solutions of Obstacle Problems –Old & New–
	1 Introduction
	2 A Linear Operator Case
	3 A Bellman Type Operator Case
		3.1 Bilateral Obstacles
		3.2 Unilateral Obstacles
	4 A Fully Nonlinear Operator Case
		4.1 Equi-Continuity
		4.2 C1,γ Estimates
	5 Appendix
	References
High-Energy Eigenfunctions of the Laplacian on the Torus and the Sphere with Nodal Sets of Complicated Topology
	1 Introduction
	2 An Inverse Localization Theorem on the Sphere
	3 An Inverse Localization Theorem on the Torus
	4 Proof of the Main Theorem
	5 Final Remark: Inverse Localization on the Sphere in Multiple Regions
	References