Various Methods for the Analysis of PDEs

This book presents the proceedings of the Minisymposium "Various Methods for the Analysis of PDEs" held at the International Congress on Industrial and Applied Mathematics (ICIAM) 2023. This volume brings together a diverse group of researchers, practitioners, and experts who have shared their latest developments and innovations in the field of Partial Differential Equations (PDEs). The papers included in this volume reflect the high quality and breadth of research presented at the session. Covering a wide range of topics, this collection showcases the dynamic and interdisciplinary nature of the Analysis of PDEs. Each contribution has undergone a rigorous peer-review process to ensure the highest standards of academic excellence. Key topics include: Interpolation Inequalities: Novel contributions to the field, including stability results for the Sobolev inequality and the Gaussian logarithmic Sobolev inequality with explicit and dimensionally sharp constants. Strichartz Estimates: New estimates specifically for orthonormal families of initial data, extending traditional Strichartz estimates to provide deeper insights into the behavior of solutions to dispersive equations, including the wave equation, Klein-Gordon equation, and fractional Schrödinger equations. Asymptotic Behavior: Detailed analysis of the asymptotic behavior for the massive Maxwell-Klein-Gordon system under the Lorenz gauge condition in dimension (1+4), including scattering results. Time-Dependent Free Schrödinger Operator: A new characterization of this operator, highlighting its unique invariance under the Galilei group in Euclidean space-time. Lifespan Estimates: Analysis of the lifespan of solutions to the damped wave equation, with decay estimates for particular initial data in the case of nonlinearity with subcritical Fujita exponent. This book aims to provide readers with a profound and cohesive understanding of the current state of splitting optimization while inspiring future research and innovation in this dynamic field.