Nonlinear Partial Differential Equations for Future Applications: Sendai, Japan, July 10–28 and October 2–6, 2017 (Springer Proceedings in Mathematics & Statistics, 346)

This volume features selected, original, and peer-reviewed papers on topics from a series of workshops on Nonlinear Partial Differential Equations for Future Applications that were held in 2017 at Tohoku University in Japan. The contributions address an abstract maximal regularity with applications to parabolic equations, stability, and bifurcation for viscous compressible Navier–Stokes equations, new estimates for a compressible Gross–Pitaevskii–Navier–Stokes system, singular limits for the Keller–Segel system in critical spaces, the dynamic programming principle for stochastic optimal control, two kinds of regularity machineries for elliptic obstacle problems, and new insight on topology of nodal sets of high-energy eigenfunctions of the Laplacian. This book aims to exhibit various theories and methods that appear in the study of nonlinear partial differential equations. Preface 6 Contents 8 An Introduction to Maximal Regularity for Parabolic Evolution Equations 9 1 Introduction 9 2 Maximal Regularity and Lp-Sobolev Spaces 12 2.1 Linearization and Maximal Regularity 12 2.2 Definition of Maximal Lp-Regularity 14 2.3 Maximal Regularity for Non-autonomous Problems 18 3 The Concept of mathcalR-Boundedness and the Theorem of Mikhlin 19 3.1 mathcalR-Bounded Operator Families 19 3.2 Fourier Multipliers and Mikhlin's Theorem 30 3.3 mathcalR-sectorial Operators 35 4 Lp-Sobolev Spaces 37 5 Parabolic PDE Systems in the Whole Space 40 6 Parabolic Boundary Value Problems 50 6.1 The Shapiro-Lopatinksii Condition 50 6.2 The Main Result on Parameter-Elliptic Boundary Value Problems 54 7 Quasilinear Parabolic Evolution Equations 67 7.1 Well-Posedness for Quasilinear Parabolic Evolution Equations 67 7.2 Higher Regularity 72 References 77 On Stability and Bifurcation in Parallel Flows of Compressible Navier-Stokes Equations 79 1 Introduction 79 2 Stability of Parallel Flows 81 3 Outline of Proof of Theorem 2.1 85 3.1 Notation 85 3.2 Spectral Properties of the Linearized Semigroup 87 3.3 Nonlinear Problem 92 4 Instability and Bifurcation in Poiseuille Flows 94 4.1 Notation 95 4.2 Instability of Plane Poiseuille Flow 97 4.3 Bifurcation of Wave Trains 98 References 100 Uniform Regularity for a Compressible Gross-Pitaevskii-Navier-Stokes System 102 1 Introduction 102 2 Proof of Theorem 1.2 104 References 109 Singular Limit Problem to the Keller-Segel System in Critical Spaces and Related Medical Problems—An Application of Maximal Regularity 110 1 Introduction—The Singular Limit Problem 110 1.1 Keller-Segel System in the Scaling Invariant Spaces 111 1.2 The Chaplain-Anderson Model and the Fujie-Senba Equation 115 2 Well-Posedness Issue in the Critical Setting 119 2.1 Well-Posedness of the Full System 121 2.2 Well-Posedness of the Keller-Segel System 122 2.3 Two-Dimensional Critical Case for Keller-Segel System 124 2.4 Singular Limit for the Keller-Segel System 127 2.5 Formal Observation for the Singular Limit 130 3 The Singular Limit Problem for the Chaplain-Anderson Systems 131 3.1 The Well-Posedness 132 3.2 Singular Limit Problem 135 4 Preliminary Estimates 137 4.1 Inequalities and Embeddings in Four Space Dimensions 137 4.2 Heat Evolution on VMO 139 5 Generalized Maximal Regularity 140 6 Proof of Well-Posedness for Keller-Segel System 147 7 Proof for the Singular Limit 155 8 Proof for the Well-Posedness of Chaplain-Anderson and Fujie-Senba System 162 9 Proof for the Singular Limit for Chaplain-Anderson Model 174 References 186 HJB Equation, Dynamic Programming Principle, and Stochastic Optimal Control 190 1 Introduction 190 2 Stochastic Optimal Control Problem 191 2.1 Strong Formulation of Optimal Control Problem 193 2.2 Weak Formulation of Optimal Control Problem 193 2.3 State Equation 193 3 Dynamic Programming Principle and HJB Equation 194 3.1 Verification Theorem, Necessary and Sufficient Conditions for Optimality 196 3.2 Construction of Optimal Feedback Controls 197 3.3 Uniqueness in Law 198 4 Value Function and Proof of Dynamic Programming Principle 199 4.1 Predictable Processes 199 4.2 Canonical Reference Probability Space 200 4.3 Independence of Value Function of Reference Probability Spaces 200 4.4 Standard Reference Probability Spaces 200 4.5 ``Conditioned'' Reference Probability Spaces 201 4.6 Proof of the Dynamic Programming Principle 203 4.7 Continuity of the Value Function in t 206 4.8 Dynamic Programming Principle with Stopping Times 207 5 Value Function Solves the HJB Equation 208 References 210 Regularity of Solutions of Obstacle Problems –Old & New– 212 1 Introduction 212 2 A Linear Operator Case 216 3 A Bellman Type Operator Case 224 3.1 Bilateral Obstacles 224 3.2 Unilateral Obstacles 229 4 A Fully Nonlinear Operator Case 233 4.1 Equi-Continuity 234 4.2 C1,γ Estimates 238 5 Appendix 240 References 248 High-Energy Eigenfunctions of the Laplacian on the Torus and the Sphere with Nodal Sets of Complicated Topology 251 1 Introduction 251 2 An Inverse Localization Theorem on the Sphere 254 3 An Inverse Localization Theorem on the Torus 261 4 Proof of the Main Theorem 264 5 Final Remark: Inverse Localization on the Sphere in Multiple Regions 264 References 266