An introduction to incompressible Navier-Stokes and Euler equations
In this course we will introduce some results on incompressible Navier-Stokes equations. The course will be divided in equal number of lectures into two parts.
For Part 1, we will select materials from the text book [MB] as described below.
Basic properties and concepts related to solutions of the Navier-Stokes (NS) equations and Euler equations. Several exact solutions with physical backgrounds (Chapter 1 of [MB]). More on the construction of exact solutions, explain how to construct exact solutions of 2.5D NS and Euler, and exact solutions of axisymmetric NS and Euler. Recover the velocity and the velocity gradients from the vorticity, and the vorticity-stream formulation (Chapter 2 of [MB]).
More on the construction of exact solutions, explain how to construct exact solutions of 2.5D NS and Euler, and exact solutions of axisymmetric NS and Euler. Recover the velocity and the velocity gradients from the vorticity, and the vorticity-stream formulation (Chapter 2 of [MB]).Energy methods to prove the local existence and uniqueness theory in Sobolev space setting for the NS and Euler, and the global existence in the 2D case (Sections 3.1-3.3 in Chapter 3 of [MB]).Local existence theory in the Holder space setting for the 3D Euler via the approach of particle trajectory. Beale-Kato-Majda regularity criterion for the 3D Euler. The global existence result for the 3D axisymmetric Euler without swirl (Sections 4.1-4.3 in Chapter 4 of [MB]).Some lower dimensional models that share the vortex stretching mechanism with the 3D Euler vorticity equation, in particular the 1D model of Constantin-Lax-Majda (Sections 5.1-5.2 in Chapter 5 of [MB]).The result of Yudovich on the global existence and uniqueness of weak solutions to the 2D Euler equation with bounded initial vorticity (L^1\cap L^\infty) (Section 8.2 in Chapter 8 of [MB]).
Part 2 will be on incompressible stationary Navier-Stokes equations.
We will survey classical and recent works on the following topics. In addition, we will outline proofs or give detailed proofs for some results. The topics are: Leray's problem on the existence of solutions of the nonhomogeneous boundary value problem for the incompressible stationary Navier-Stokes equations in bounded domains in two and dimensions; Leray's problem of steady Navier-Stokes flow past a body in the plane; Liouville theorems for three dimensional incompressible stationary Navier-Stokes equations; existence of regular solutions of the incompressible stationary Navier-Stokes equations in Euclidean space and flat torus in dimension less than or equal to 15; existence of regular solutions of the incompressible stationary Navier-Stokes equations with Dirichlet boundary data in dimension less than or equal to 6.
[MB] A.J. Majda and A. Bertozzi, Vorticity and incompressible flow. Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002.
Math 501 or Permission from instructor