On the universality of the incompressible Euler equation on compact manifolds
Terence Tao
Discrete & Continuous Dynamical Systems - A 2018, 38(3): 1553-1565 doi: 10.3934/dcds.2018064
The incompressible Euler equations on a compact Riemannian manifold
take the form
$\partial_t u + \nabla_u u =- \mathrm{grad}_g p \\\mathrm{div}_g u =0.$
We show that any quadratic ODE
$\partial_t y =B(y,y)$
, where
$B \colon \mathbb{R}^n × \mathbb{R}^n \to \mathbb{R}^n$
is a symmetric bilinear map, can be linearly embedded into the incompressible Euler equations for some manifold
if and only if
obeys the cancellation condition
$\langle B(y,y), y \rangle =0$
for some positive definite inner product
. This allows one to construct explicit solutions to the Euler equations with various dynamical features, such as quasiperiodic solutions, or solutions that transition from one steady state to another, and provides evidence for the "Turing universality" of such Euler flows.
keywords: Euler equation quadratic ODE universality embedding Riemannian manifolds
Two remarks on the generalised Korteweg de-Vries equation
Terence Tao
Discrete & Continuous Dynamical Systems - A 2007, 18(1): 1-14 doi: 10.3934/dcds.2007.18.1
We make two observations concerning the generalised Korteweg de Vries equation $u_t + $uxxx$ = \mu ( |u|^{p-1} u )_x$. Firstly we give a scaling argument that shows, roughly speaking, that any quantitative scattering result for $L^2$-critical equation ($p=5$) automatically implies an analogous scattering result for the $L^2$-critical nonlinear Schrödinger equation $iu_t + $uxx$ = \mu |u|^4 u$. Secondly, in the defocusing case $\mu > 0$ we present a new dispersion estimate which asserts, roughly speaking, that energy moves to the left faster than the mass, and hence strongly localised soliton-like behaviour at a fixed scale cannot persist for arbitrarily long times.
keywords: Global well-posedness monotonicity formula.
Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm
M. Keel Tristan Roy Terence Tao
Discrete & Continuous Dynamical Systems - A 2011, 30(3): 573-621 doi: 10.3934/dcds.2011.30.573
We show that the Maxwell-Klein-Gordon equations in three dimensions are globally well-posed in $H^s_x$ in the Coulomb gauge for all $s > \sqrt{3}/2 \approx 0.866$. This extends previous work of Klainerman-Machedon [24] on finite energy data $s \geq 1$, and Eardley-Moncrief [11] for still smoother data. We use the method of almost conservation laws, sometimes called the "I-method", to construct an almost conserved quantity based on the Hamiltonian, but at the regularity of $H^s_x$ rather than $H^1_x$. One then uses Strichartz, null form, and commutator estimates to control the development of this quantity. The main technical difficulty (compared with other applications of the method of almost conservation laws) is at low frequencies, because of the poor control on the $L^2_x$ norm. In an appendix, we demonstrate the equations' relative lack of smoothing - a property that presents serious difficulties for studying rough solutions using other known methods.
keywords: $X^{s Global well-posedness Maxwell-Klein-Gordon equation b}$ spaces. Coulomb gauge I-method
An inverse theorem for the Gowers $U^{s+1}[N]$-norm
Ben Green Terence Tao Tamar Ziegler
Electronic Research Announcements 2011, 18(0): 69-90 doi: 10.3934/era.2011.18.69
This is an announcement of the proof of the inverse conjecture for the Gowers $U^{s+1}[N]$-norm for all $s \geq 3$; this is new for $s \geq 4$, the cases $s = 1,2,3$ having been previously established. More precisely we outline a proof that if $f : [N] \rightarrow [-1,1]$ is a function with ||$f$|| $U^{s+1}[N] \geq \delta$ then there is a bounded-complexity $s$-step nilsequence $F(g(n)\Gamma)$ which correlates with $f$, where the bounds on the complexity and correlation depend only on $s$ and $\delta$. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. In particular, one obtains an asymptotic formula for the number of $k$-term arithmetic progressions $p_1 < p_2 < ... < p_k \leq N$ of primes, for every $k \geq 3$.
keywords: Gowers norms nilsequences.
Linear approximate groups
Emmanuel Breuillard Ben Green Terence Tao
Electronic Research Announcements 2010, 17(0): 57-67 doi: 10.3934/era.2010.17.57
This is an informal announcement of results to be described and proved in detail in [3]. We give various results on the structure of approximate subgroups in linear groups such as $\SL_n(k)$. For example, generalizing a result of Helfgott (who handled the cases $n = 2$ and $3$), we show that any approximate subgroup of $\SL_n(\F_q)$ which generates the group must be either very small or else nearly all of $\SL_n(\F_q)$. The argument is valid for all Chevalley groups $G(\F_q)$. Extending work of Bourgain-Gamburd we also announce some applications to expanders, which will be proven in detail in [4].
keywords: expander graphs. Approximate groups growth

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