9 October 2018 | 8:44 pm

We now turn to the local existence theory for the initial value problem for the incompressible Euler equations For sake of discussion we will just work in the non-periodic domain , , although the arguments here can be adapted without much difficulty to the periodic setting. We will only work with solutions in which […]

2 October 2018 | 11:47 pm

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, are unique, depend continuously on the initial data, have a high degree of regularity, can […]

16 September 2018 | 5:31 pm

We now begin the rigorous theory of the incompressible Navier-Stokes equations where is a given constant (the kinematic viscosity, or viscosity for short), is an unknown vector field (the velocity field), and is an unknown scalar field (the pressure field). Here is a time interval, usually of the form or . We will either be […]