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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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In the last week or so there has been some discussion on the internet about a paper (initially authored by Hill and Tabachnikov) that was initially accepted for publication in the Mathematical Intelligencer, but with the editor-in-chief of that journal later deciding against publication; the paper, in significantly revised form (and now authored solely by Hill), was then quickly accepted by one of the editors in the New York Journal of Mathematics, but then was removed from publication after objections from several members on the editorial board of NYJM that the paper had not been properly refereed or was within the scope of the journal; see this statement by Benson Farb, who at the time was on that board, for more details.  Some further discussion of this incident may be found on Tim Gowers’ blog; the most recent version of the paper, as well as a number of prior revisions, are still available on the arXiv here.

For whatever reason, some of the discussion online has focused on the role of Amie Wilkinson, a mathematician from the University of Chicago (and who, incidentally, was a recent speaker here at UCLA in our Distinguished Lecture Series), who wrote an email to the editor-in-chief of the Intelligencer raising some concerns about the content of the paper and suggesting that it be published alongside commentary from other experts in the field.  (This, by the way, is not uncommon practice when dealing with a potentially provocative publication in one field by authors coming from a different field; for instance, when Emmanuel Candès and I published a paper in the Annals of Statistics introducing what we called the “Dantzig selector”, the Annals solicited a number of articles discussing the selector from prominent statisticians, and then invited us to submit a rejoinder.)    It seems that the editors of the Intelligencer decided instead to reject the paper.  The paper then had a complicated interaction with NYJM, but, as stated by Wilkinson in her recent statement on this matter as well as by Farb, this was done without any involvement from Wilkinson.  (It is true that Farb happens to also be Wilkinson’s husband, but I see no reason to doubt their statements on this matter.)

I have not interacted much with the Intelligencer, but I have published a few papers with NYJM over the years; it is an early example of a quality “diamond open access” mathematics journal.  It seems that this incident may have uncovered some issues with their editorial procedure for reviewing and accepting papers, but I am hopeful that they can be addressed to avoid this sort of event occurring again.

 

 

Joni Teräväinen and I have just uploaded to the arXiv our paper “The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures“. This is a sequel to our previous paper that studied logarithmic correlations of the form

\displaystyle  f(a) := \lim^*_{x \rightarrow \infty} \frac{1}{\log \omega(x)} \sum_{x/\omega(x) \leq n \leq x} \frac{g_1(n+ah_1) \dots g_k(n+ah_k)}{n},

where {g_1,\dots,g_k} were bounded multiplicative functions, {h_1,\dots,h_k \rightarrow \infty} were fixed shifts, {1 \leq \omega(x) \leq x} was a quantity going off to infinity, and {\lim^*} was a generalised limit functional. Our main technical result asserted that these correlations were necessarily the uniform limit of periodic functions {f_i}. Furthermore, if {g_1 \dots g_k} (weakly) pretended to be a Dirichlet character {\chi}, then the {f_i} could be chosen to be {\chi}isotypic in the sense that {f_i(ab) = f_i(a) \chi(b)} whenever {a,b} are integers with {b} coprime to the periods of {\chi} and {f_i}; otherwise, if {g_1 \dots g_k} did not weakly pretend to be any Dirichlet character {\chi}, then {f} vanished completely. This was then used to verify several cases of the logarithmically averaged Elliott and Chowla conjectures.

The purpose of this paper was to investigate the extent to which the methods could be extended to non-logarithmically averaged settings. For our main technical result, we now considered the unweighted averages

\displaystyle  f_d(a) := \lim^*_{x \rightarrow \infty} \frac{1}{x/d} \sum_{n \leq x/d} g_1(n+ah_1) \dots g_k(n+ah_k),

where {d>1} is an additional parameter. Our main result was now as follows. If {g_1 \dots g_k} did not weakly pretend to be a twisted Dirichlet character {n \mapsto \chi(n) n^{it}}, then {f_d(a)} converged to zero on (doubly logarithmic) average as {d \rightarrow \infty}. If instead {g_1 \dots g_k} did pretend to be such a twisted Dirichlet character, then {f_d(a) d^{it}} converged on (doubly logarithmic) average to a limit {f(a)} of {\chi}-isotypic functions {f_i}. Thus, roughly speaking, one has the approximation

\displaystyle  \lim^*_{x \rightarrow \infty} \frac{1}{x/d} \sum_{n \leq x/d} g_1(n+ah_1) \dots g_k(n+ah_k) \approx f(a) d^{-it}

for most {d}.

Informally, this says that at almost all scales {x} (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 {d \mapsto d^{it}} (which interacts with the Archimedean parameter {d} in much the same way that the Dirichlet character {\chi} interacts with the non-Archimedean parameter {a}). 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 {x} of logarithmic density zero. For instance, the Chowla conjecture

\displaystyle  \lim_{x \rightarrow\infty} \frac{1}{x} \sum_{n \leq x} \lambda(n+h_1) \dots \lambda(n+h_k) = 0

is now established for {k} either odd or equal to {2}, so long as one excludes an exceptional set of scales.

In the logarithmically averaged setup, the main idea was to combine two very different pieces of information on {f(a)}. The first, coming from recent results in ergodic theory, was to show that {f(a)} was well approximated in some sense by a nilsequence. The second was to use the “entropy decrement argument” to obtain an approximate isotopy property of the form

\displaystyle  f(a) g_1 \dots g_k(p)\approx f(ap)

for “most” primes {p} and integers {a}. Combining the two facts, one eventually finds that only the almost periodic components of the nilsequence are relevant.

In the current situation, each {a \mapsto f_d(a)} is approximated by a nilsequence, but the nilsequence can vary with {d} (although there is some useful “Lipschitz continuity” of this nilsequence with respect to the {d} parameter). Meanwhile, the entropy decrement argument gives an approximation basically of the form

\displaystyle  f_{dp}(a) g_1 \dots g_k(p)\approx f_d(ap)

for “most” {d,p,a}. The arguments then proceed largely as in the logarithmically averaged case. A key lemma to handle the dependence on the new parameter {d} is the following cohomological statement: if one has a map {\alpha: (0,+\infty) \rightarrow S^1} that was a quasimorphism in the sense that {\alpha(xy) = \alpha(x) \alpha(y) + O(\varepsilon)} for all {x,y \in (0,+\infty)} and some small {\varepsilon}, then there exists a real number {t} such that {\alpha(x) = x^{it} + O(\varepsilon)} for all small {\varepsilon}. This is achieved by applying a standard “cocycle averaging argument” to the cocycle {(x,y) \mapsto \alpha(xy) \alpha(x)^{-1} \alpha(y)^{-1}}.

It would of course be desirable to not have the set of exceptional scales. We only know of one (implausible) scenario in which we can do this, namely when one has far fewer (in particular, subexponentially many) sign patterns for (say) the Liouville function than predicted by the Chowla conjecture. In this scenario (roughly analogous to the “Siegel zero” scenario in multiplicative number theory), the entropy of the Liouville sign patterns is so small that the entropy decrement argument becomes powerful enough to control all scales rather than almost all scales. On the other hand, this scenario seems to be self-defeating, in that it allows one to establish a large number of cases of the Chowla conjecture, and the full Chowla conjecture is inconsistent with having unusually few sign patterns. Still it hints that future work in this direction may need to split into “low entropy” and “high entropy” cases, in analogy to how many arguments in multiplicative number theory have to split into the “Siegel zero” and “no Siegel zero” cases.

This is the tenth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant {\Lambda}, continuing this post. Discussion of the project of a non-research nature can continue for now in the existing proposal thread. Progress will be summarised at this Polymath wiki page.

Most of the progress since the last thread has been on the numerical side, in which the various techniques to numerically establish zero-free regions to the equation H_t(x+iy)=0 have been streamlined, made faster, and extended to larger heights than were previously possible.  The best bound for \Lambda 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) 3 \times 10^{10}, which has been confirmed by independent work of Platt et al. and Gourdon-Demichel, the best bound remains at \Lambda \leq 0.22.  Using the verification up to height 2.5 \times 10^{12} claimed by Gourdon-Demichel, this improves slightly to \Lambda \leq 0.19, and if one assumes the Riemann hypothesis up to height 5 \times 10^{19} the bound improves to \Lambda \leq 0.11, 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 \Lambda should be inversely proportional to the logarithm of the height at which the Riemann hypothesis is verified.

As progress seems to have stabilised, it may be time to transition to the writing phase of the Polymath15 project.  (There are still some interesting research questions to pursue, such as numerically investigating the zeroes of H_t for negative values of t, 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.

Below the fold is the detailed progress report on the numerics by Rudolph Dwars and Kalpesh Muchhal.

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This coming fall quarter, I am teaching a class on topics in the mathematical theory of incompressible fluid equations, focusing particularly on the incompressible Euler and Navier-Stokes equations. These two equations are by no means the only equations used to model fluids, but I will focus on these two equations in this course to narrow the focus down to something manageable. I have not fully decided on the choice of topics to cover in this course, but I would probably begin with some core topics such as local well-posedness theory and blowup criteria, conservation laws, and construction of weak solutions, then move on to some topics such as boundary layers and the Prandtl equations, the Euler-Poincare-Arnold interpretation of the Euler equations as an infinite dimensional geodesic flow, and some discussion of the Onsager conjecture. I will probably also continue to more advanced and recent topics in the winter quarter.

In this initial set of notes, we begin by reviewing the physical derivation of the Euler and Navier-Stokes equations from the first principles of Newtonian mechanics, and specifically from Newton’s famous three laws of motion. Strictly speaking, this derivation is not needed for the mathematical analysis of these equations, which can be viewed if one wishes as an arbitrarily chosen system of partial differential equations without any physical motivation; however, I feel that the derivation sheds some insight and intuition on these equations, and is also worth knowing on purely intellectual grounds regardless of its mathematical consequences. I also find it instructive to actually see the journey from Newton’s law

\displaystyle  F = ma

to the seemingly rather different-looking law

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

\displaystyle  \nabla \cdot u = 0

for incompressible Navier-Stokes (or, if one drops the viscosity term {\nu \Delta u}, the Euler equations).

Our discussion in this set of notes is physical rather than mathematical, and so we will not be working at mathematical levels of rigour and precision. In particular we will be fairly casual about interchanging summations, limits, and integrals, we will manipulate approximate identities {X \approx Y} 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 these slides of Gero Friesecke for an expansion of these points.

Note: our treatment here will differ slightly from that presented in many fluid mechanics texts, in that it will emphasise first-principles derivations from many-particle systems, rather than relying on bulk laws of physics, such as the laws of thermodynamics, which we will not cover here. (However, the derivations from bulk laws tend to be more robust, in that they are not as reliant on assumptions about the particular interactions between particles. In particular, the physical hypotheses we assume in this post are probably quite a bit stronger than the minimal assumptions needed to justify the Euler or Navier-Stokes equations, which can hold even in situations in which one or more of the hypotheses assumed here break down.)

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