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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 much difficulty to the periodic setting. We will only work with solutions in which the pressure {p} is normalised in the usual fashion:

\displaystyle  p = - \Delta^{-1} \nabla \cdot \nabla \cdot (u \otimes u). \ \ \ \ \ (2)

Formally, the Euler equations (with normalised pressure) arise as the vanishing viscosity limit {\nu \rightarrow 0} of the Navier-Stokes equations

\displaystyle  \partial_t u + (u \cdot \nabla) u = - \nabla p + \nu \Delta u \ \ \ \ \ (3)

\displaystyle  \nabla \cdot u = 0

\displaystyle  p = - \Delta^{-1} \nabla \cdot \nabla \cdot (u \otimes u)

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

that was studied in previous notes. However, because most of the bounds established in previous notes, either on the lifespan {T_*} of the solution or on the size of the solution itself, depended on {\nu}, it is not immediate how to justify passing to the limit and obtain either a strong well-posedness theory or a weak solution theory for the limiting equation (1). (For instance, weak solutions to the Navier-Stokes equations (or the approximate solutions used to create such weak solutions) have {\nabla u} lying in {L^2_{t,loc} L^2_x} for {\nu>0}, but the bound on the norm is {O(\nu^{-1/2})} and so one could lose this regularity in the limit {\nu \rightarrow 0}, at which point it is not clear how to ensure that the nonlinear term {u_j u} still converges in the sense of distributions to what one expects.)

Nevertheless, by carefully using the energy method (which we will do loosely following an approach of Bertozzi and Majda), it is still possible to obtain local-in-time estimates on (high-regularity) solutions to (3) that are uniform in the limit {\nu \rightarrow 0}. Such a priori estimates can then be combined with a number of variants of these estimates obtain a satisfactory local well-posedness theory for the Euler equations. Among other things, we will be able to establish the Beale-Kato-Majda criterion – smooth solutions to the Euler (or Navier-Stokes) equations can be continued indefinitely unless the integral

\displaystyle  \int_0^{T_*} \| \omega(t) \|_{L^\infty_x( {\bf R}^d \rightarrow \wedge^2 {\bf R}^d )}\ dt

becomes infinite at the final time {T_*}, where {\omega := \nabla \wedge u} is the vorticity field. The vorticity has the important property that it is transported by the Euler flow, and in two spatial dimensions it can be used to establish global regularity for both the Euler and Navier-Stokes equations in these settings. (Unfortunately, in three and higher dimensions the phenomenon of vortex stretching has frustrated all attempts to date to use the vorticity transport property to establish global regularity of either equation in this setting.)

There is a rather different approach to establishing local well-posedness for the Euler equations, which relies on the vorticity-stream formulation of these equations. This will be discused in a later set of notes.

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