偏微分方程分析：2025–2026 年度综述

An annual survey of the math.AP arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 100 papers and organised into 36 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Existence and Regularity in Elliptic Boundary Value Problems (72.5%) — This cluster focuses on proving existence, uniqueness, and regularity of solutions to elliptic and parabolic PDEs with boundary conditions, including estimates in Sobolev spaces and applications to nonlinear Schrödinger and KdV-type equations.
• Global Well-Posedness for Fluid Dynamics and Coupled Systems (11.9%) — This cluster investigates global well-posedness, finite-time blow-up, and asymptotic behavior of solutions to incompressible and compressible fluid equations, including Navier-Stokes, Euler, and coupled multi-physics systems with surface interactions.
• Stabilization and Asymptotic Behavior of Dispersive and Heat Equations (8.2%) — This cluster studies stability, decay rates, and long-time asymptotic behavior of solutions to linear and semilinear wave, heat, and dispersive equations, including stochastic perturbations and inverse problems.
• Hyperbolic Conservation Laws and Geometric Flows (2.2%) — This cluster addresses scalar conservation laws, hyperbolic systems, and geometric flows, including curvature-driven dynamics, prescribed mean curvature problems, and numerical approximation schemes.
• Variational Methods and Optimal Transport in PDE Theory (2.1%) — This cluster develops variational formulations, optimal transport theory, and energy methods for PDEs, including gradient flows, control problems, and geometric measure theory applications.
• Nonlinear Schrödinger Equations and Dispersive Dynamics (2.0%) — This cluster focuses on well-posedness, blow-up, and long-time dynamics of nonlinear Schrödinger equations, including integrable systems, solitons, and Hamiltonian structures.
• Symmetry and Self-Similar Solutions in Nonlinear PDEs (0.7%) — This cluster examines symmetric solutions, self-similar profiles, and stability of blow-up phenomena for wave maps, reaction-diffusion systems, and related nonlinear PDEs with symmetry constraints.
• Analytic and Mathematical Modeling of Complex Physical Systems (0.4%) — This cluster addresses rigorous mathematical analysis of models arising in biology, physics, and engineering, including analyticity properties, well-posedness, and derivation of effective equations.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.