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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.

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 be continued in time as long as a sufficiently high regularity norm is under control, and tend to enjoy the same sort of conservation laws that classical solutions do. However, it is a major open problem as to whether these solutions can be extended to be (forward) global in time, because the norms that we know how to control globally in time do not have high enough regularity to be useful for continuing the solution. Also, the theory becomes degenerate in the inviscid limit ${\nu \rightarrow 0}$.

However, it is possible to construct “weak” solutions which lack many of the desirable features of strong solutions (notably, uniqueness, propagation of regularity, and conservation laws) but can often be constructed globally in time even when one us unable to do so for strong solutions. Broadly speaking, one usually constructs weak solutions by some sort of “compactness method”, which can generally be described as follows.

1. Construct a sequence of “approximate solutions” to the desired equation, for instance by developing a well-posedness theory for some “regularised” approximation to the original equation. (This theory often follows similar lines to those in the previous set of notes, for instance using such tools as the contraction mapping theorem to construct the approximate solutions.)
2. Establish some uniform bounds (over appropriate time intervals) on these approximate solutions, even in the limit as an approximation parameter is sent to zero. (Uniformity is key; non-uniform bounds are often easy to obtain if one puts enough “mollification”, “hyper-dissipation”, or “discretisation” in the approximating equation.)
3. Use some sort of “weak compactness” (e.g., the Banach-Alaoglu theorem, the Arzela-Ascoli theorem, or the Rellich compactness theorem) to extract a subsequence of approximate solutions that converge (in a topology weaker than that associated to the available uniform bounds) to a limit. (Note that there is no reason a priori to expect such limit points to be unique, or to have any regularity properties beyond that implied by the available uniform bounds..)
4. Show that this limit solves the original equation in a suitable weak sense.

The quality of these weak solutions is very much determined by the type of uniform bounds one can obtain on the approximate solution; the stronger these bounds are, the more properties one can obtain on these weak solutions. For instance, if the approximate solutions enjoy an energy identity leading to uniform energy bounds, then (by using tools such as Fatou’s lemma) one tends to obtain energy inequalities for the resulting weak solution; but if one somehow is able to obtain uniform bounds in a higher regularity norm than the energy then one can often recover the full energy identity. If the uniform bounds are at the regularity level needed to obtain well-posedness, then one generally expects to upgrade the weak solution to a strong solution. (This phenomenon is often formalised through weak-strong uniqueness theorems, which we will discuss later in these notes.) Thus we see that as far as attacking global regularity is concerned, both the theory of strong solutions and the theory of weak solutions encounter essentially the same obstacle, namely the inability to obtain uniform bounds on (exact or approximate) solutions at high regularities (and at arbitrary times).

For simplicity, we will focus our discussion in this notes on finite energy weak solutions on ${{\bf R}^d}$. There is a completely analogous theory for periodic weak solutions on ${{\bf R}^d}$ (or equivalently, weak solutions on the torus ${({\bf R}^d/{\bf Z}^d)}$ which we will leave to the interested reader.

In recent years, a completely different way to construct weak solutions to the Navier-Stokes or Euler equations has been developed that are not based on the above compactness methods, but instead based on techniques of convex integration. These will be discussed in a later set of notes.