This volume contains the proceedings of the AMS Special Session on Spectral Calculus and Quasilinear Partial Differential Equations and the AMS Special Session on PDE Analysis on Fluid Flows, which were held in January 2017 in Atlanta, Georgia. These two sessions shared the underlying theme of the analysis aspect of evolutionary PDEs and mathematical physics. The articles address the latest trends and perspectives in the area of nonlinear dispersive equations and fluid flows. The topics mainly focus on using state-of-the-art methods and techniques to investigate problems of depth and richness arising in quantum mechanics, general relativity, and fluid dynamics.

Table of Contents:
N. Basharat, Y. Hu, and S. Zheng, Blowup rate for mass critical rotational nonlinear Schrodinger equations D. Cao, T. Mengesha, and T. Phan, Gradient estimates for weak solutions of linear elliptic systems with singular-degenerate coefficients R. Denlinger, Virial estimates for hard spheres Y. Du, G. Chen, and J. Liu, The almost global existence to classical solution for a 3-D wave equation of nematic liquid-crystals L. G. Farah, J. Holmer, and S. Roudenko, Instability of solitons-revisited, I: The critical generalized KdV equation L. G. Farah, J. Holmer, and S. Roudenko, Instability of solitons-revisited, II: The supercritical Zakharov-Kuznetsov equation C. Flores, S. Oh, and D. Smith, Stabilization of dispersion-generalized Benjamin-Ono D. Garrisi and V. Georgiev, Uniqueness of standing-waves for a non-linear Schrodinger equation with three pure-power combinations in dimension one S. Gustafson and D. Roxanas, Below-threshold solutions of a focusing energy-critical heat equation in $\mathbb{R}^4$ D. Li and X. Zhang, A regularity upgrade of pressure S. Miao, On large future-global-in-time solutions to energy-supercritical nonlinear wave equation J. Murphy, The nonlinear Schrodinger equation with an inverse-square potential Y. Shao and C. Wang, The harmonic map heat flow on conic manifolds K. Yamazaki, On the global regularity issue of the two-dimensional magnetohydrodynamics system with magnetic diffusion weaker than a Laplacian J. Zhang, S. Zheng, and S. Zhu, Orbital stability of standing waves for fractional Hartree equation with unbounded potentials.