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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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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 pressure field). Here {I} is a time interval, usually of the form {[0,T]} or {[0,T)}. We will either be interested in spatially decaying situations, in which {u(t,x)} decays to zero as {x \rightarrow \infty}, or {{\bf Z}^d}-periodic (or periodic for short) settings, in which one has {u(t, x+n) = u(t,x)} for all {n \in {\bf Z}^d}. (One can also require the pressure {p} to be periodic as well; this brings up a small subtlety in the uniqueness theory for these equations, which we will address later in this set of notes.) As is usual, we abuse notation by identifying a {{\bf Z}^d}-periodic function on {{\bf R}^d} with a function on the torus {{\bf R}^d/{\bf Z}^d}.

In order for the system (1) to even make sense, one requires some level of regularity on the unknown fields {u,p}; this turns out to be a relatively important technical issue that will require some attention later in this set of notes, and we will end up transforming (1) into other forms that are more suitable for lower regularity candidate solution. Our focus here will be on local existence of these solutions in a short time interval {[0,T]} or {[0,T)}, for some {T>0}. (One could in principle also consider solutions that extend to negative times, but it turns out that the equations are not time-reversible, and the forward evolution is significantly more natural to study than the backwards one.) The study of Euler equations, in which {\nu=0}, will be deferred to subsequent lecture notes.

As the unknown fields involve a time parameter {t}, and the first equation of (1) involves time derivatives of {u}, the system (1) should be viewed as describing an evolution for the velocity field {u}. (As we shall see later, the pressure {p} is not really an independent dynamical field, as it can essentially be expressed in terms of the velocity field without requiring any differentiation or integration in time.) As such, the natural question to study for this system is the initial value problem, in which an initial velocity field {u_0: {\bf R}^d \rightarrow {\bf R}^d} is specified, and one wishes to locate a solution {(u,p)} to the system (1) with initial condition

\displaystyle  	u(0,x) = u_0(x) 	\ \ \ \ \ (2)

for {x \in {\bf R}^d}. Of course, in order for this initial condition to be compatible with the second equation in (1), we need the compatibility condition

\displaystyle  	\nabla \cdot u_0 = 0 	\ \ \ \ \ (3)

and one should also impose some regularity, decay, and/or periodicity hypotheses on {u_0} in order to be compatible with corresponding level of regularity etc. on the solution {u}.

The fundamental questions in the local theory of an evolution equation are that of existence, uniqueness, and continuous dependence. In the context of the Navier-Stokes equations, these questions can be phrased (somewhat broadly) as follows:

  • (a) (Local existence) Given suitable initial data {u_0}, does there exist a solution {(u,p)} to the above initial value problem that exists for some time {T>0}? What can one say about the time {T} of existence? How regular is the solution?
  • (b) (Uniqueness) Is it possible to have two solutions {(u,p), (u',p')} of a certain regularity class to the same initial value problem on a common time interval {[0,T)}? To what extent does the answer to this question depend on the regularity assumed on one or both of the solutions? Does one need to normalise the solutions beforehand in order to obtain uniqueness?
  • (c) (Continuous dependence on data) If one perturbs the initial conditions {u_0} by a small amount, what happens to the solution {(u,p)} and on the time of existence {T}? (This question tends to only be sensible once one has a reasonable uniqueness theory.)

The answers to these questions tend to be more complicated than a simple “Yes” or “No”, for instance they can depend on the precise regularity hypotheses one wishes to impose on the data and on the solution, and even on exactly how one interprets the concept of a “solution”. However, once one settles on such a set of hypotheses, it generally happens that one either gets a “strong” theory (in which one has existence, uniqueness, and continuous dependence on the data), a “weak” theory (in which one has existence of somewhat low-quality solutions, but with only limited uniqueness results (or even some spectacular failures of uniqueness) and almost no continuous dependence on data), or no satsfactory theory whatsoever. In the former case, we say (roughly speaking) that the initial value problem is locally well-posed, and one can then try to build upon the theory to explore more interesting topics such as global existence and asymptotics, classifying potential blowup, rigorous justification of conservation laws, and so forth. With a weak local theory, it becomes much more difficult to address these latter sorts of questions, and there are serious analytic pitfalls that one could fall into if one tries too strenuously to treat weak solutions as if they were strong. (For instance, conservation laws that are rigorously justified for strong, high-regularity solutions may well fail for weak, low-regularity ones.) Also, even if one is primarily interested in solutions at one level of regularity, the well-posedness theory at another level of regularity can be very helpful; for instance, if one is interested in smooth solutions in {{\bf R}^d}, it turns out that the well-posedness theory at the critical regularity of {\dot H^{\frac{d}{2}-1}({\bf R}^d)} can be used to establish globally smooth solutions from small initial data. As such, it can become quite important to know what kind of local theory one can obtain for a given equation.

This set of notes will focus on the “strong” theory, in which a substantial amount of regularity is assumed in the initial data and solution, giving a satisfactory (albeit largely local-in-time) well-posedness theory. “Weak” solutions will be considered in later notes.

The Navier-Stokes equations are not the simplest of partial differential equations to study, in part because they are an amalgam of three more basic equations, which behave rather differently from each other (for instance the first equation is nonlinear, while the latter two are linear):

  • (a) Transport equations such as {\partial_t u + (u \cdot \nabla) u = 0}.
  • (b) Diffusion equations (or heat equations) such as {\partial_t u = \nu \Delta u}.
  • (c) Systems such as {v = F - \nabla p}, {\nabla \cdot v = 0}, which (for want of a better name) we will call Leray systems.

Accordingly, we will devote some time to getting some preliminary understanding of the linear diffusion and Leray systems before returning to the theory for the Navier-Stokes equation. Transport systems will be discussed further in subsequent notes; in this set of notes, we will instead focus on a more basic example of nonlinear equations, namely the first-order ordinary differential equation

\displaystyle  	\partial_t u = F(u) 	\ \ \ \ \ (4)

where {u: I \rightarrow V} takes values in some finite-dimensional (real or complex) vector space {V} on some time interval {I}, and {F: V \rightarrow V} is a given linear or nonlinear function. (Here, we use “interval” to denote a connected non-empty subset of {{\bf R}}; in particular, we allow intervals to be half-infinite or infinite, or to be open, closed, or half-open.) Fundamental results in this area include the Picard existence and uniqueness theorem, the Duhamel formula, and Grönwall’s inequality; they will serve as motivation for the approach to local well-posedness that we will adopt in this set of notes. (There are other ways to construct strong or weak solutions for Navier-Stokes and Euler equations, which we will discuss in later notes.)

A key role in our treatment here will be played by the fundamental theorem of calculus (in various forms and variations). Roughly speaking, this theorem, and its variants, allow us to recast differential equations (such as (1) or (4)) as integral equations. Such integral equations are less tractable algebraically than their differential counterparts (for instance, they are not ideal for verifying conservation laws), but are significantly more convenient for well-posedness theory, basically because integration tends to increase the regularity of a function, while differentiation reduces it. (Indeed, the problem of “losing derivatives”, or more precisely “losing regularity”, is a key obstacle that one often has to address when trying to establish well-posedness for PDE, particularly those that are quite nonlinear and with rough initial data, though for nonlinear parabolic equations such as Navier-Stokes the obstacle is not as serious as it is for some other PDE, due to the smoothing effects of the heat equation.)

One weakness of the methods deployed here are that the quantitative bounds produced deteriorate to the point of uselessness in the inviscid limit {\nu \rightarrow 0}, rendering these techniques unsuitable for analysing the Euler equations in which {\nu=0}. However, some of the methods developed in later notes have bounds that remain uniform in the {\nu \rightarrow 0} limit, allowing one to also treat the Euler equations.

In this and subsequent set of notes, we use the following asymptotic notation (a variant of Vinogradov notation that is commonly used in PDE and harmonic analysis). The statement {X \lesssim Y}, {Y \gtrsim X}, or {X = O(Y)} will be used to denote an estimate of the form {|X| \leq CY} (or equivalently {Y \geq C^{-1} |X|}) for some constant {C}, and {X \sim Y} will be used to denote the estimates {X \lesssim Y \lesssim X}. If the constant {C} depends on other parameters (such as the dimension {d}), this will be indicated by subscripts, thus for instance {X \lesssim_d Y} denotes the estimate {|X| \leq C_d Y} for some {C_d} depending on {d}.

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I have just uploaded to the arXiv the third installment of my “heatwave” project, entitled “Global regularity of wave maps V.  Large data local well-posedness in the energy class“. This (rather technical) paper establishes another of the key ingredients necessary to establish the global existence of smooth wave maps from 2+1-dimensional spacetime {\Bbb R}^{1+2} to hyperbolic space \mathbf{H} = \mathbf{H}^m.  Specifically, a large data local well-posedness result is established, constructing a local solution from any initial data with finite (but possibly quite large) energy, and furthermore that the solution depends continuously on the initial data in the energy topology.  (This topology was constructed in my previous paper.)  Once one has this result, the only remaining task is to show a “Palais-Smale property” for wave maps, in that if singularities form in the wave maps equation, then there exists a non-trivial minimal-energy blowup solution, whose orbit is almost periodic modulo the symmetries of the equation.  I anticipate this to the most difficult component of the whole project, and is the subject of the fourth (and hopefully final) installment of this series.

This local result is closely related to the small energy global regularity theory developed in recent years by myself, by Krieger, and by Tataru.  In particular, the complicated function spaces used in that paper (which ultimately originate from a precursor paper of Tataru).  The main new difficulties here are to extend the small energy theory to large energy (by localising time suitably), and to establish continuous dependence on the data (i.e. two solutions which are initially close in the energy topology, need to stay close in that topology).  The former difficulty is in principle manageable by exploiting finite speed of propagation (exploiting the fact (arising from the monotone convergence theorem) that large energy data becomes small energy data at sufficiently small spatial scales), but for technical reasons (having to do with my choice of gauge) I was not able to do this and had to deal with the large energy case directly (and in any case, a genuinely large energy theory is going to be needed to construct the minimal energy blowup solution in the next paper).  The latter difficulty is in principle resolvable by adapting the existence theory to differences of solutions, rather than to individual solutions, but the nonlinear choice of gauge adds a rather tedious amount of complexity to the task of making this rigorous.  (It may be that simpler gauges, such as the Coulomb gauge, may be usable here, at least in the case m=2 of the hyperbolic plane (cf. the work of Krieger), but such gauges cause additional analytic problems as they do not renormalise the nonlinearity as strongly as the caloric gauge.  The paper of Tataru establishes these goals, but assumes an isometric embedding of the target manifold into a Euclidean space, which is unfortunately not available for hyperbolic space targets.)

The main technical difficulty that had to be overcome in the paper was that there were two different time variables t, s (one for the wave maps equation and one for the heat flow), and three types of PDE (hyperbolic, parabolic, and ODE) that one has to solve forward in t, forward in s, and backwards in s respectively.  In order to close the argument in the large energy case, this necessitated a rather complicated iteration-type scheme, in which one solved for the caloric gauge, established parabolic regularity estimates for that gauge, propagated a “wave-tension field” by the heat flow, and then solved a wave maps type equation using that field as a forcing term.  The argument can eventually be closed using mostly “off-the-shelf” function space estimates from previous papers, but is remarkably lengthy, especially when analysing differences of two solutions.  (One drawback of using off-the-shelf estimates, though, is that one does not get particularly good control of the solution over extended periods of time; in particular, the spaces used here cannot detect the decay of the solution over extended periods of time (unlike, say, Strichartz spaces L^q_t L^r_x for q < \infty) and so will not be able to supply the long-time perturbation theory that will be needed in the next paper in this series.  I believe I know how to re-engineer these spaces to achieve this, though, and the details should follow in the forthcoming paper.)

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