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

This volume presents selected papers from a workshop on Nonlinear Partial Differential Equations for Future Applications, covering topics such as maximal regularity, stability, bifurcation, and optimal control. The contributions aim to showcase theories and methods in the study of nonlinear partial differential equations.

Format: Paperback / softback
Length: 261 pages
Publication date: 17 April 2022
Publisher: Springer Verlag, Singapore

This volume presents a collection of selected, original, and peer-reviewed papers on diverse topics from a series of workshops on Nonlinear Partial Differential Equations for Future Applications, held in 2017 at Tohoku University in Japan. The contributions delve into various aspects, including abstract maximal regularity with applications to parabolic equations, stability, and bifurcation for viscous compressible Navier–Stokes equations, novel 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 distinct regularity machineries for elliptic obstacle problems, and a fresh perspective on the topology of nodal sets of high-energy eigenfunctions of the Laplacian. The primary objective of this book is to showcase a wide range of theories and methodologies employed in the study of nonlinear partial differential equations.

Abstract Maximal Regularity with Applications to Parabolic Equations
The contributions in this section explore the concept of abstract maximal regularity, which plays a crucial role in understanding the behavior of solutions to parabolic equations. The authors present new results and techniques for analyzing the stability and bifurcation of solutions to parabolic equations with various boundary conditions. They also discuss the application of abstract maximal regularity to the study of partial differential equations with non-smooth coefficients, including the heat equation and the Schrödinger equation.

Stability and Bifurcation of Viscous Compressible Navier–Stokes Equations
This section focuses on the stability and bifurcation analysis of viscous compressible Navier–Stokes equations. The authors present new results and methods for studying the behavior of solutions to these equations, including the study of blowup phenomena and the formation of singularities. They also discuss the application of stability and bifurcation theory to the design of control strategies for fluid flow problems.

New Estimates for a Compressible Gross–Pitaevskii–Navier–Stokes System
In this section, the authors present new estimates for a compressible Gross–Pitaevskii–Navier–Stokes system, which is a model for the study of fluid dynamics in astrophysics and cosmology. The estimates involve the use of novel mathematical techniques, such as singular perturbation analysis and homotopy methods, to obtain improved results for the system's solutions.

Singular Limits for the Keller–Segel System in Critical Spaces
The contributions in this section explore the singular limits of the Keller–Segel system in critical spaces. The Keller–Segel system is a model for the study of the dynamics of elastic membranes, and the singular limits play a crucial role in understanding the behavior of the system as it approaches certain critical points. The authors present new results and techniques for analyzing the singular limits and their implications for the behavior of the Keller–Segel system.

Dynamic Programming Principle for Stochastic Optimal Control
This section discusses the dynamic programming principle for stochastic optimal control. The dynamic programming principle is a powerful tool for solving optimal control problems in which the decision-maker must make decisions based on uncertain information. The authors present new results and methods for applying the dynamic programming principle to stochastic optimal control problems, including the study of optimal control strategies for systems with random inputs and outputs.

Regularity Machineries for Elliptic Obstacle Problems
The contributions in this section explore the use of regularity machineries for elliptic obstacle problems. Elliptic obstacle problems are a class of partial differential equations that arise in various applications, including fluid dynamics, solid mechanics, and biology. The regularity machineries involve the use of mathematical techniques, such as regularization and blowup analysis, to obtain solutions to these problems. The authors present new results and techniques for analyzing the behavior of solutions to elliptic obstacle problems and their applications to real-world problems.

New Insight on Topology of Nodal Sets of High-Energy Eigenfunctions of the Laplacian
The contributions in this section explore the topology of nodal sets of high-energy eigenfunctions of the Laplacian. The Laplacian is a fundamental operator in partial differential equations, and the high-energy eigenfunctions of the Laplacian play a crucial role in understanding the behavior of solutions to many physical problems. The authors present new results and techniques for analyzing the topology of nodal sets of high-energy eigenfunctions and their implications for the study of partial differential equations.

In conclusion, this volume presents a comprehensive collection of selected, original, and peer-reviewed papers on diverse topics related to Nonlinear Partial Differential Equations for Future Applications. The contributions cover a wide range of topics, including abstract maximal regularity, stability and bifurcation of 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, dynamic programming principle for stochastic optimal control, regularity machineries for elliptic obstacle problems, and new insight on topology of nodal sets of high-energy eigenfunctions of the Laplacian. The book aims to showcase various theories and methods that appear in the study of nonlinear partial differential equations and provide valuable insights to researchers and practitioners in the field.

Weight: 421g
Dimension: 235 x 155 (mm)
ISBN-13: 9789813348240
Edition number: 1st ed. 2021