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Mathematics has always played a key role for researches in fluid mechanics. The purpose of this handbook is to give an overview of items that are key to handling problems in fluid mechanics. Since the field of fluid mechanics is huge, it is almost impossible to cover many topics. In this handbook, we focus on mathematical analysis on viscous Newtonian fluid. The first part is devoted to mathematical analysis on incompressible fluids while part 2 is devoted to compressible fluids.
Contents Volume 1
Derivation of Equations for Incompressible and Compressible Fluids
Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids
Variational Modeling and Complex Fluids
Incompressible Fluids
The Stokes Equation in the Lp-Setting: Well-Posedness and Regularity Properties
Stokes Problems in Irregular Domains with Various Boundary Conditions
Leray’s Problem on Existence of Steady-State Solutions for the Navier-Stokes Flow
Stationary Navier-Stokes Flow in Exterior Domains and Landau Solutions
Steady-State Navier-Stokes Flow Around a Moving Body
Stokes Semigroups, Strong, Weak, and Very Weak Solutions for General Domains
Self-Similar Solutions to the Nonstationary Navier-Stokes Equations
Time-Periodic Solutions to The Navier-Stokes Equations
Large Time Behavior of the Navier-Stokes Flow
Critical Function Spaces for the Well-Posedness of the Navier-Stokes Initial Value Problem
Existence and Stability of Viscous Vortices
Models and Special Solutions of the Navier-Stokes Equations
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
Regularity Criteria for Navier-Stokes Solutions
Stable Self-Similar Profiles for Two 1D Models of the 3D Axisymmetric Euler Equations
Vorticity Direction and Regularity of Solutions to the Navier-Stokes Equations
Recent Advances Concerning Certain Class of Geophysical Flows
Equations for Polymeric Materials
Contents Volume 2
Modeling of Two-Phase Flows With and Without Phase Transitions
Equations for Viscoelastic Fluids
Modeling and Analysis of the Ericksen-Leslie Equations for Nematic Liquid Crystal Flows
Classical Well-Posedness of Free Boundary Problems in Viscous Incompressible Fluid Mechanics
Stability of Equilibrium Shapes in Some Free Boundary Problems Involving Fluids
Weak Solutions and Diffuse Interface Models for Incompressible Two-Phase Flows
Water Waves with or Without Surface Tension
Compressible Viscous Fluids
Concepts of Solutions in the Thermodynamics of Compressible Fluids
Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stability, and Longtime Behavior
Weak Solutions for the Compressible Navier-Stokes Equations with Density Dependent Viscosities
Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical Cases
Weak Solutions for the Compressible Navier-Stokes Equations in the Intermediate Regularity Class
Symmetric Solutions to the Viscous Gas Equations
Local and Global Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Energy Method
Fourier Analysis Methods for the Compressible Navier-Stokes Equations
Local and Global Existence of Strong Solutions for the Compressible Navier-Stokes Equations Near Equilibria via the Maximal Regularity
Local and Global Solvability of Free Boundary Problems for the Compressible Navier-Stokes Equations Near Equilibria
Contents Volume 3
Global Existence of Regular Solutions with Large Oscillations and Vacuum for Compressible Flows
Global Existence of Classical Solutions and Optimal Decay Rate for Compressible Flows via the Theory of Semigroups
Finite Time Blow-Up of Regular Solutions for Compressible Flows
Blow-Up Criteria of Strong Solutions and Conditional Regularity of Weak Solutions for the Compressible Navier-Stokes Equations
Well-Posedness and Asymptotic Behavior for Compressible Flows in One Dimension
Well-Posedness of the IBVPs for the 1D Viscous Gas Equations
Waves in Compressible Fluids: Viscous Shock, Rarefaction, and Contact Waves
Existence of Stationary Weak Solutions for Isentropic and Isothermal Compressible Flows
Existence of Stationary Weak Solutions for Compressible Heat Conducting Flows
Existence and Uniqueness of Strong Stationary Solutions for Compressible Flows
Low Mach Number Limits and Acoustic Waves
Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or Rotating Fluids
Scale Analysis of Compressible Flows from an Application Perspective
Weak and Strong Solutions of Equations of Compressible Magnetohydrodynamics
Multi-Fluid Models Including Compressible Fluids
Solutions for Models of Chemically Reacting Compressible Mixtures