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Terence Tao

Publications and source records attributed to Terence Tao.

At least 289 records · Page 16Linked to original sources
arXiv2005

Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data

In any dimension $n \geq 3$, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear Schrödinger equation $i u_t + Δu = |u|^{\frac{4}{n-2}} u$ in $\R \times \R^n$ exist globally and scatter to free solutions; this generalizes the three and four dimensional results of Bourgain and Grillakis. Furthermore we have bounds on various spacetime norms of the solution which are of exponential type in the energy, which improves on the tower-type bounds of Bourgain. In higher dimensions $n \geq 6$ some new technical difficulties arise because of the very low power of the non-linearity.

math.AP↗
arXiv2005

Decoding by Linear Programming

This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector $f \in \R^n$ from corrupted measurements $y = A f + e$. Here, $A$ is an $m$ by $n$ (coding) matrix and $e$ is an arbitrary and unknown vector of errors. Is it possible to recover $f$ exactly from the data $y$? We prove that under suitable conditions on the coding matrix $A$, the input $f$ is the unique solution to the $\ell_1$-minimization problem ($\|x\|_{\ell_1} := \sum_i |x_i|$) $$ \min_{g \in \R^n} \| y - Ag \|_{\ell_1} $$ provided that the support of the vector of errors is not too large, $\|e\|_{\ell_0} := |\{i : e_i \neq 0\}| \le ρ\cdot m$ for some $ρ> 0$. In short, $f$ can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; $f$ is recovered exactly even in situations where a significant fraction of the output is corrupted.

math.MG↗
arXiv2005

The Bi-Carleson operator

We prove $L^p$ estimates for the Bi-Carleson operator, which is a natural hybrid of the Carleson maximal operator and the bilinear Hilbert transform. The methods used are essentially based on the treatment of the Walsh analogue of the operator in the prequel of this paper, but with additional technicalities due to the fact that in the Fourier model one cannot obtain perfect localization in both space and frequency.

math.CA↗
arXiv2004

Long-time decay estimates for the Schrödinger equation on manifolds

In this paper we develop a quantitative version of Enss' method to establish global-in-time decay estimates for solutions to Schrödinger equations on manifolds. To simplify the exposition we shall only consider Hamiltonians of the form $H := - {1/2} Δ_M$, where $Δ_M$ is the Laplace-Beltrami operator on a manifold $M$ which is a smooth compact perturbation of three-dimensional Euclidean space $\R^3$ which obeys the non-trapping condition. We establish a global-in-time local smoothing estimate for the Schrödinger equation $u_t = -iHu$. The main novelty here is the global-in-time aspect of the estimates, which forces a more detailed analysis on the low and medium frequencies of the evolution than in the local-in-time theory. In particular, to handle the medium frequencies we require the RAGE theorem (which reflects the fact that $H$ has no embedded eigenvalues), together with a quantitative version of Enss' method decomposing the solution asymptotically into incoming and outgoing components, while to handle the low frequencies we need a Poincare-type inequality (which reflects the fact that $H$ has no eigenfunctions or resonances at zero).

math.AP↗
arXiv2004

Multi-parameter paraproducts

We prove that the classical Coifman-Meyer theorem holds on any polydisc $\T^d$ of arbitrary dimension $d\geq 1$.

math.CA↗
arXiv2004

Geometric renormalization of large energy wave maps

There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the solution is large. In this report we present a different renormalization, based on the harmonic map heat flow, which works for large energy wave maps from two dimensions to hyperbolic spaces. We also observe an intriguing estimate of ``non-concentration'' type, which asserts roughly speaking that if the energy of a wave map concentrates at a point, then it becomes asymptotically self-similar.

math.AP↗
arXiv2004

Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal $f \in \C^N$ and a randomly chosen set of frequencies $Ω$ of mean size $τN$. Is it possible to reconstruct $f$ from the partial knowledge of its Fourier coefficients on the set $Ω$? A typical result of this paper is as follows: for each $M > 0$, suppose that $f$ obeys $$ # \{t, f(t) \neq 0 \} \le α(M) \cdot (\log N)^{-1} \cdot # Ω, $$ then with probability at least $1-O(N^{-M})$, $f$ can be reconstructed exactly as the solution to the $\ell_1$ minimization problem $$ \min_g \sum_{t = 0}^{N-1} |g(t)|, \quad \text{s.t.} \hat g(ω) = \hat f(ω) \text{for all} ω\in Ω. $$ In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for $α$ which depends on the desired probability of success; except for the logarithmic factor, the condition on the size of the support is sharp. The methodology extends to a variety of other setups and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one or two-dimensional) object from incomplete frequency samples--provided that the number of jumps (discontinuities) obeys the condition above--by minimizing other convex functionals such as the total-variation of $f$.

math.NA↗
arXiv2004

Sharp Strichartz estimates on non-trapping asymptotically conic manifolds

We obtain the Strichartz inequalities $$ \| u \|_{L^q_t L^r_x([0,1] \times M)} \leq C \| u(0) \|_{L^2(M)}$$ for any smooth $n$-dimensional Riemannian manifold $M$ which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$, and $2 < q, r \leq \infty$ are admissible Strichartz exponents ($\frac{2}{q} + \frac{n}{r} = \frac{n}{2}$). This corresponds with the estimates available for Euclidean space (except for the endpoint $(q,r) = (2, \frac{2n}{n-2})$ when $n > 2$). These estimates imply existence theorems for semi-linear Schrödinger equations on $M$, by adapting arguments from Cazenave and Weissler \cite{cwI} and Kato \cite{kato}. This result improves on our previous result in \cite{HTW}, which was an $L^4_{t,x}$ Strichartz estimate in three dimensions. It is closely related to the results of Staffilani-Tataru, Burq, Tataru, and Robbiano-Zuily, who consider the case of asymptotically flat manifolds.

math.AP↗
arXiv2004

Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations

In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schrödinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces,in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the $L^2$ topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity $H^{-3/4}$, and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval $[-T,T]$ and a locally Lipschitz continuous map taking the initial data in $H^{-3/4}$ to a distributional solution $u \in C^0 ([-T,T]; $H^{-3/4})$ which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV.

math.AP↗
arXiv2004

An uncertainty principle for cyclic groups of prime order

Let $G$ be a finite abelian group, and let $f: G \to \C$ be a complex function on $G$. The uncertainty principle asserts that the support $\supp(f) := \{x \in G: f(x) \neq 0\}$ is related to the support of the Fourier transform $\hat f: G \to \C$ by the formula $$ |\supp(f)| |\supp(\hat f)| \geq |G|$$ where $|X|$ denotes the cardinality of $X$. In this note we show that when $G$ is the cyclic group $\Z/p\Z$ of prime order $p$, then we may improve this to $$ |\supp(f)| + |\supp(\hat f)| \geq p+1$$ and show that this is absolutely sharp. As one consequence, we see that a sparse polynomial in $\Z/p\Z$ consisting of $k+1$ monomials can have at most $k$ zeroes. Another consequence is a short proof of the well-known Cauchy-Davenport inequality.

math.CA↗
arXiv2004

A Strichartz inequality for the Schroedinger equation on non-trapping asymptotically conic manifolds

We obtain an $L^4$ space-time Strichartz inequality for any smooth three-dimensional Riemannian manifold $(M,g)$ which is asymptotically conic at infinity and non-trapping, where $u$ is a solution to the Schrödinger equation $iu_t + {1/2} Δ_M u = 0$. The exponent $H^{1/4}(M)$ is sharp, by scaling considerations. In particular our result covers asymptotically flat non-trapping manifolds. Our argument is based on the interaction Morawetz inequality introduced by Colliander et al., interpreted here as a positive commutator inequality for the tensor product $U(t,z',z'') := u(t,z') u(t,z'')$ of the solution with itself. We also use smoothing estimates for Schrödinger solutions including a new one proved here with weight $r^{-1}$ at infinity and with the gradient term involving only one angular derivative.

math.AP↗
arXiv2004

Global well-posedness of the Benjamin-Ono equation in H^1(R)

We show that the Benjamin-Ono equation is globally well-posed in $H^s(\R)$ for $s \geq 1$. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in $H^s$ for any $s$. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.

math.AP↗
arXiv2004

A quantitative ergodic theory proof of Szemerédi's theorem

A famous theorem of Szemerédi asserts that given any density $0 < δ\leq 1$ and any integer $k \geq 3$, any set of integers with density $δ$ will contain infinitely many proper arithmetic progressions of length $k$. For general $k$ there are essentially four known proofs of this fact; Szemerédi's original combinatorial proof using the Szemerédi regularity lemma and van der Waerden's theorem, Furstenberg's proof using ergodic theory, Gowers' proof using Fourier analysis and the inverse theory of additive combinatorics, and Gowers' more recent proof using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring in particular the use of transfinite induction (and thus the axiom of choice), decomposing a general ergodic system as the weakly mixing extension of a transfinite tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is ``elementary'' in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.

math.CO↗
arXiv2004

On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation

We study the asymptotic behavior of large data radial solutions to the focusing Schrödinger equation $i u_t + Δu = -|u|^2 u$ in $\R^3$, assuming globally bounded $H^1(\R^3)$ norm (i.e. no blowup in the energy space). We show that as $t \to \pm \infty$, these solutions split into the sum of three terms: a radiation term that evolves according to the linear Schrödinger equation, a smooth function localized near the origin, and an error that goes to zero in the $\dot H^1(\R^3)$ norm. Furthermore, the smooth function near the origin is either zero (in which case one has scattering to a free solution), or has mass and energy bounded strictly away from zero, and obeys an asymptotic Pohozaev identity. These results are consistent with the conjecture of soliton resolution.

math.AP↗
arXiv2004

A singularity removal theorem for Yang-Mills fields in higher dimensions

In four and higher dimensions, we show that any stationary admissible Yang-Mills field can be gauge transformed to a smooth field if the $L^2$ norm of the curvature is sufficiently small. There are three main ingredients. The first is Price's monotonicity formula, which allows us to assert that the curvature is small not only in the $L^2$ norm, but also in the Morrey norm $M_2^{n/2}$. The second ingredient is a new inductive (averaged radial) gauge construction and truncation argument which allows us to approximate a singular gauge as a weak limit of smooth gauges with curvature small in the Morrey norm. The second ingredient is variant of Uhlenbeck's lemma, allowing one to place a smooth connection into the Coulomb gauge whenever the Morrey norm of the curvature is small; This variant was also proved independently by Meyer and Riviere. It follows easily from this variant that a $W^{1,2}$-connection can be placed in the Coulomb gauge if it can be approximated by smooth connections whose curvatures have small Morrey norm.

math.DG↗
arXiv2004

Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions

We show that in dimensions $n \geq 6$ that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm $\dot H^{n/2-1} \times \dot H^{n/2-2}$ of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge.

math.AP↗