### On the recently removed paper from the New York Journal of Mathematics

In the last week or so there has been some discussion on the internet about a paper (initially authored by Hill and Tabachnikov) that was initially accepted for publication in the Mathematical Intelligencer, but with the editor-in-chief of that …

### 254A, Notes 1: Local well-posedness of the Navier-Stokes equations

We now begin the rigorous theory of the incompressible Navier-Stokes equations

$\displaystyle \partial_t u + (u \cdot \nabla) u = \nu \Delta u - \nabla p \ \ \ \ \ (1)$

$\displaystyle \nabla \cdot u = 0,$

where ${\nu>0}$ is a given constant (the kinematic viscosity, or viscosity for short), ${u: I \times {\bf R}^d \rightarrow {\bf R}^d}$ is an unknown vector field (the velocity field), and ${p: I \times {\bf R}^d \rightarrow {\bf R}}$ is an unknown scalar field (the …

### 254A, Notes 2: Weak solutions of the Navier-Stokes equations

In the previous set of notes we developed a theory of “strong” solutions to the Navier-Stokes equations. This theory, based around viewing the Navier-Stokes equations as a perturbation of the linear heat equation, has many attractive features: solutions exist locally, …

### 254A, Notes 3: Local well-posedness for the Euler equations

We now turn to the local existence theory for the initial value problem for the incompressible Euler equations

$\displaystyle \partial_t u + (u \cdot \nabla) u = - \nabla p \ \ \ \ \ (1)$

$\displaystyle \nabla \cdot u = 0$

$\displaystyle u(0,x) = u_0(x).$

For sake of discussion we will just work in the non-periodic domain ${{\bf R}^d}$, ${d \geq 2}$, although the arguments here can be adapted without …