We now begin the rigorous theory of the incompressible Navier-Stokes equations
where 
![{[0,T]}](https://s0.wp.com/latex.php?latex=%7B%5B0%2CT%5D%7D&bg=ffffff&fg=000000&s=0 "{[0,T]}")
In order for the system (1) to even make sense, one requires some level of regularity on the unknown fields 
As the unknown fields involve a time parameter 
for 
and one should also impose some regularity, decay, and/or periodicity hypotheses on 
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
of existence? How regular is the solution?
- (b) (Uniqueness) Is it possible to have two solutions
of a certain regularity class to the same initial value problem on a common time interval
- (c) (Continuous dependence on data) If one perturbs the initial conditions
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 
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
.
- (b) Diffusion equations (or heat equations) such as
.
- (c) Systems such as
,
, 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
where 
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 
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 
were bounded multiplicative functions, 
were fixed shifts, 
was a quantity going off to infinity, and 
was a generalised limit functional. Our main technical result asserted that these correlations were necessarily the uniform limit of periodic functions 
. Furthermore, if 
(weakly) pretended to be a Dirichlet character 
, then the 
whenever 
are integers with 
coprime to the periods of 
vanished completely. This was then used to verify several cases of the logarithmically averaged Elliott and Chowla conjectures.
is an additional parameter. Our main result was now as follows. If 
, then 
converged to zero on (doubly logarithmic) average as 
. If instead 
converged on (doubly logarithmic) average to a limit 
of 
(where “almost all” means “outside of a set of logarithmic density zero”), the non-logarithmic averages behave much like their logarithmic counterparts except for a possible additional twisting by an Archimedean character 
(which interacts with the Archimedean parameter 
). One consequence of this is that most of the recent results on the logarithmically averaged Chowla and Elliott conjectures can now be extended to their non-logarithmically averaged counterparts, so long as one excludes a set of exceptional scales 
either odd or equal to 
, so long as one excludes an exceptional set of scales.
is approximated by a nilsequence, but the nilsequence can vary with 
. The arguments then proceed largely as in the logarithmically averaged case. A key lemma to handle the dependence on the new parameter 
that was a quasimorphism in the sense that 
for all 
and some small 
, then there exists a real number 
for all small 
.
, continuing 
have been streamlined, made faster, and extended to larger heights than were previously possible. The best bound for 
now depends on the height to which one is willing to assume the Riemann hypothesis. Using the conservative verification up to height (slightly larger than) 
, which has been confirmed by independent work of 
. Using the verification up to height 
claimed by Gourdon-Demichel, this improves slightly to 
, and if one assumes the Riemann hypothesis up to height 
the bound improves to 
, contingent on a numerical computation that is still underway. (See the table below the fold for more data of this form.) This is broadly consistent with the expectation that the bound on 
for negative values of 
, but the writeup does not necessarily have to contain every single direction pursued in the project. If enough additional interesting findings are unearthed then one could always consider writing a second paper, for instance.
, the Euler equations).
as if they were exact identities (e.g., by differentiating both sides of the approximate identity), and we will not attempt to verify any regularity or convergence hypotheses in the expressions being manipulated. (The same holds for the exercises in this text, which also do not need to be justified at mathematical levels of rigour.) Of course, once we resume the mathematical portion of this course in subsequent notes, such issues will be an important focus of careful attention. This is a basic division of labour in mathematical modeling: non-rigorous heuristic reasoning is used to derive a mathematical model from physical (or other “real-life”) principles, but once a precise model is obtained, the analysis of that model should be completely rigorous if at all possible (even if this requires applying the model to regimes which do not correspond to the original physical motivation of that model). See the discussion by John Ball quoted at the end of 
be a finite 
. We say that 
of ![{[E:k]}](https://s0.wp.com/latex.php?latex=%7B%5BE%3Ak%5D%7D&bg=ffffff&fg=000000&s=0 "{[E:k]}")
of the extension. In that case, we call 
. The 
is an intermediate field betwen 
, and 
is a subgroup of 
is a subgroup of 
; namely 
and 
, then 
if and only if 
is a subgroup of 
. 
is isomorphic to the quotient group 
.
, and let 
be the degree 
Galois extension formed by adjoining a primitive 
root of unity (that is to say, 
). Then 
(the invertible elements of the ring 
). Amongst the intermediate fields, one has the cyclotomic fields of the form 
where 
divides 
and 
modulo 
be the field of rational functions of one indeterminate 
with complex coefficients, and let 
be the field formed by adjoining an 
to 
. Then 
corresponding to the field automorphism of 
to 
). The intermediate fields are of the form 
where 
and 
is a connected manifold and 
is a 
, we denote this relationship by 
. (Later on we will change our function notations slightly and write 
in place of the more traditional 
below; more on this below the fold.) If 
to be the group of 
which preserve the fibres of 
. We say that this covering map is a Galois cover if the cardinality of the group 
.
between 
, in such a way that the covering maps are compatible, in the sense that 
is the composition of 
and 
. This sort of compatibilty condition will be implicitly assumed whenever we chain together multiple instances of the 
notation. Two intermediate covers 
are equivalent if they cover each other, in a fashion compatible with all the other covering maps, thus 
and 
. We then have the analogous Galois correspondence:
is a subgroup of 
. 
and 
, then 
if and only if 
is a subgroup of 
. 
is isomorphic to the quotient group 
.
, and let 
be the 
. Then 
with covering map 
where 
have an algebraic structure (or a complex structure), then one can relate covering spaces to field extensions by considering the field of rational functions (or meromorphic functions) on the space. For instance, if 
and 
of rational functions on 
of rational functions. The covering 
relates the two coordinates 
by the relation 
, at which point one sees that the rational functions 
of the coordinate 
.) 
associated to these fields in such a fashion that (say) 
behaves like a “covering space” to 
with a group 
of deck transformations isomorphic to 
of the field 
of a commutative ring 
is defined as the set of 
that maps a commutative ring to its spectrum tends to act like an inverse of the operation that maps a space 
to a ring of functions on that space. For instance, if one considers the commutative ring ![{{\bf C}[z, z^{-1}]}](https://s0.wp.com/latex.php?latex=%7B%7B%5Cbf+C%7D%5Bz%2C+z%5E%7B-1%7D%5D%7D&bg=ffffff&fg=000000&s=0 "{{\bf C}[z, z^{-1}]}")
of regular functions on 
in ![{\{ f \in {\bf C}[z, z^{-1}]: f(z_0)=0\}}](https://s0.wp.com/latex.php?latex=%7B%5C%7B+f+%5Cin+%7B%5Cbf+C%7D%5Bz%2C+z%5E%7B-1%7D%5D%3A+f%28z_0%29%3D0%5C%7D%7D&bg=ffffff&fg=000000&s=0 "{\{ f \in {\bf C}[z, z^{-1}]: f(z_0)=0\}}")
, and one can check that these are the only such prime ideals (other than the zero ideal 
), giving an almost one-to-one correspondence between ![{\mathrm{Spec}( {\bf C}[z,z^{-1}] )}](https://s0.wp.com/latex.php?latex=%7B%5Cmathrm%7BSpec%7D%28+%7B%5Cbf+C%7D%5Bz%2Cz%5E%7B-1%7D%5D+%29%7D&bg=ffffff&fg=000000&s=0 "{\mathrm{Spec}( {\bf C}[z,z^{-1}] )}")
and 
that map from some other (affine) scheme 
to 
from 
is then a homomorphism 
of integral domains, such that 
is the composition of 
. Note that every prime ideal 
in the space 
defined here, namely the quotient map 
to the ring 
, which is an integral domain because 
defines a special point 
in 
.
AND 
) OR (NOT 
are some formulas). A typical such rule is 
be a measure-preserving system – a probability space 
equipped with a measure-preserving translation 
in this space as being “close” if 
for some 
for moderately large 
which is locally essentially constant in the sense that 
for 
-almost every 
such that 
for 
, then the averages 
converge in 
norm to the mean 
. (The mean ergodic theorem also applies to other 
spaces with 
, though it is usually proven first in the Hilbert space 
converges to 
for any measurable set 
. Then there exists a constant 
for 
.
at least 
, and partition 
, where 
is the level set
. Thus there exists 
such that 
, thus
converges to a limit 
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