The mathematical analysis of the incompressible Euler and Navier-Stokes equations : an introduction /
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| Author / Creator: | Bedrossian, Jacob, 1984- author. |
|---|---|
| Imprint: | Providence, Rhode Island : American Mathematical Society, [2022] ©2022 |
| Description: | 1 online resource ( xiii, 218 pages) : illustrations. |
| Language: | English |
| Series: | Graduate studies in mathematics, 1065-7339 ; 225 Graduate studies in mathematics ; v. 225. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/13144078 |
Table of Contents:
- Ideal incompressible fluids: the Euler equations
- Eulerian vs Lagrangian representations
- Incompressibility and Transport
- The incompressible homogeneous Euler equations
- Vorticity
- Symmetries and conservation laws
- Special explicit solutions
- Exercises
- Existence of solutions and continuation criteria for Euler
- Local existence of Hs solutions
- The Lipschitz continuation criterion
- The Beale-Kato-Majda theorem
- The global existence of strong solutions in 2D
- The Constantin-Fefferman-Majda criterion
- Exercises
- Incompressible viscous fluids: the Navier-Stokes equations
- Viscosity
- Non-dimensionalization
- Vorticity, symmetries, and balance laws
- Special explicit solutions
- Local existence of Hs solutions
- Strong solutions with initial datum in H1 : local and global
- Exercises
- Leray-Hopf weak solutions of Navier-Stokes
- Weak solutions
- Existence of weak solutions on the whole space via mollification
- The uniqueness of weak solutions in 2D
- Weak-strong uniqueness and the Prodi-Serrin class
- Partial regularity in time for Leray-Hopf weak solutions
- Existence of weak solutions on the periodic box via Galerkin
- Exercises
- Mild solutions of Navier-Stokes
- Mild formulation
- Scaling criticality
- Local-in-time well-posedness in H 1/2
- Local-in-time well-posedness in L3
- Local regularization
- Continuation of smooth solutions
- Exercises
- A survey of some advanced topics
- Local regularity and the Prodi-Serrin conditions
- Partial regularity of suitable weak solutions in 3D
- Bounded domains
- Stationary solutions of the Navier-Stokes equations
- Ruling out backward self-similar finite-time singularities
- Critical and supercritical well-posedness for Navier-Stokes
- Yudovich theory and 2D Euler with Lp vorticity
- Gradient growth in the 2D Euler equations
- The search for finite-time singularties in 3D Euler
- Hydrodynamic stability: Euler
- Hydrodynamic stability: Navier-Stokes
- The energy balance and Onsager's conjecture
- Appendix
- The contraction mapping principle
- Existence and uniqueness for ODEs
- Fourier transform
- Integral operators
- Sobolev Spaces
- Basic properties of the Poisson and heat equations
- Mollifiers
- Sobolev and Gagliardo-Nirenberg inequalities
- Compactness.
