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T. Tao

Publications and source records attributed to T. Tao.

16 recordsLinked to original sources
arXiv2025

Resonance density range governs two-plasmon decay saturation and enables hot-electron prediction in inertial confinement fusion

The saturation level of parametric instabilities critically determines their impact on fusion plasmas. We identify the resonance density range of two-plasmon decay as the critical parameter governing nonlinear saturation of ion density fluctuations and Langmuir waves, which drive hot-electron generation. Using this insight, we develop a predictive scaling model for the hot-electron energy fraction f_{hot} that depends only on the laser intensity I, with plasma conditions encoded via plasma ablation theory. The model can work for various experimental configurations-requiring only two (I, f_{hot}) data points to calibrate coefficients-and successfully reproduces results from prior OMEGA and OMEGA-EP experiments.

physics.plasm-ph↗
arXiv2022

Mitigating laser imprint with a foam overcoating

Foam has been suggested to reduce laser imprint because of its low density. In this paper, the two-dimensional radiation hydrodynamic code FLASH is applied to investigate and characterize the strength of laser imprint through analyzing areal density perturbation. There are two important factors for the mitigation of laser imprint besides the thermal smoothing of the conduction region (between the ablation front and the critical density surface) and the mass ablation of the ablation front. First, radiation ablation dynamically modulates density distribution not only to increase the frequency of the perturbed ablation front oscillation but also to decrease the amplitude of oscillation. Second, a larger length of the shocked compression region reduces the amplitude of the perturbed shock front oscillation. The smaller the perturbation of both ablation front and shock front, the smaller the areal density perturbation. Based on the above physical mechanisms, the optimal way of mitigating laser imprint with foam is that the dynamically modulated density distribution further reduces the amplitude of perturbation reaching the solid CH when the areal density perturbation of foam oscillates to the first minimum value. The optimal ranges of foam parameters to mitigate laser imprint are proposed with the aid of dimensional analysis: the foam thickness is about 2~3 times the perturbation wavelength, and the foam density is about 1/2~3/2 times the mass density corresponding to the critical density.

physics.plasm-ph↗
arXiv2018

Zero-viscosity limit of the Navier-Stokes equations with the Navier friction boundary condition

In this paper, we consider the zero-viscosity limit of the Navier-Stokes equations in a half space with the Navier friction boundary condition $$ (βu^{\varepsilon}-\varepsilon^γ\partial_y u^{\varepsilon})|_{y=0}=0, $$ where $β$ is a constant and $γ\in (0,1]$. In the case of $γ=1$, the convergence to the Euler equations and the Prandtl equation with the Robin boundary condition is justified for the analytic data. In the case of $γ\in (0,1)$, the convergence to the Euler equations and the linearized Prandtl equation is justified for the data in the Gevrey class $\frac 1 γ$.

math.AP↗
arXiv2009

Low-Energy Astrophysics: Stimulating the Reduction of Energy Consumption in the Next Decade

In this paper we address the consumption of energy by astronomers while performing their professional duties. Although we find that astronomy uses a negligible fraction of the US energy budget, the rate at which energy is consumed by an average astronomer is similar to that of a typical high-flying businessperson. We review some of the ways in which astronomers are already acting to reduce their energy consumption. In the coming decades, all citizens will have to reduce their energy consumption to conserve fossil fuel reserves and to help avert a potentially catastrophic change in the Earth's climate. The challenges are the same for astronomers as they are for everyone: decreasing the distances we travel and investing in energy-efficient infrastructure. The high profile of astronomy in the media, and the great public interest in our field, can play a role in promoting energy-awareness to the wider population. Our specific recommendations are therefore to 1) reduce travel when possible, through efficient meeting organization, and by investing in high-bandwidth video conference facilities and virtual-world software, 2) create energy-efficient observatories, computing centers and workplaces, powered by sustainable energy resources, and 3) actively publicize these pursuits.

astro-ph.IM↗
arXiv2008

Weakly turbulent solutions for the cubic defocusing nonlinear Schrödinger equation

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This weakly turbulent behavior is quantified by the growth of higher Sobolev norms: given any delta << 1, K >>1, s >1, we construct smooth initial data u_0 with ||u_0||_{{H}^s} < delta, so that the corresponding time evolution u satisfies ||u(T)||_{{H}^s} > K at some time T. This growth occurs despite the Hamiltonian's bound on ||u(t)||_{\dot{H}^1} and despite the conservation of the quantity ||u(t)||_{L^2}. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution's frequency support that simplifies the system of ODE's describing each Fourier mode's evolution. The second is a construction of solutions to these simpler systems of ODE's which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems.

math.AP↗
arXiv2008

On the permanent of random Bernoulli matrices

We show that the permanent of an $n \times n$ matrix with iid Bernoulli entries $\pm 1$ is of magnitude $n^{({1/2}+o(1))n}$ with probability $1-o(1)$. In particular, it is almost surely non-zero.

math.CO↗
arXiv2007

Resonant decompositions and the I-method for cubic nonlinear Schrodinger on R^2

The initial value problem for the cubic defocusing nonlinear Schrödinger equation $i \partial_t u + Δu = |u|^2 u$ on the plane is shown to be globally well-posed for initial data in $H^s (\R^2)$ provided $s>1/2$. The proof relies upon an almost conserved quantity constructed using multilinear correction terms. The main new difficulty is to control the contribution of resonant interactions to these correction terms. The resonant interactions are significant due to the multidimensional setting of the problem and some orthogonality issues which arise.

math.AP↗
arXiv2005

Polynomial upper bounds for the instability of the Nonlinear Schrödinger equation below the energy norm

We continue the study (initiated in \cite{ckstt:7}) of the orbital stability of the ground state cylinder for focussing non-linear Schrödinger equations in the $H^s(\R^n)$ norm for $1-\eps < s < 1$, for small $\eps$. In the $L^2$-subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the $H^1$-subcritical case then we cannot show this, but for defocussing equations we obtain global well-posedness and polynomial growth of $H^s$ norms for $s$ sufficiently close to 1.

math.AP↗
arXiv2004

Symplectic nonsqueezing of the KdV flow

We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle T. The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov. Unlike the work of Kuksin which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our results are similar to those obtained for the NLS flow by Bourgain. A major difficulty here though is the lack of any sort of smoothing estimate which would allow us to easily approximate the infinite dimensional KdV flow by a finite-dimensional Hamiltonian flow. To resolve this problem we invert the Miura transform and work on the level of the modified KdV (mKdV) equation, for which smoothing estimates can be established.

math.AP↗
arXiv2004

Collective Dynamics of Active Elements: Task Allocation and Pheromone Trailing

Collective behavior of active elements inspired by mass of biological organisms is addressed. Especially, two topics are focused on among amazing behaviors performed by colony of ants. First, task allocation phenomena are treated from the viewpoint of proportion regulation of population between different states. Using a dynamical model consisting of elements and external ``stock materials'', adaptability against various disturbances is numerically studied. In addition, a dynamical model for a colony ants interacting via two kind of pheromones is studied, in which simulated ants, as a mass, are shown to make an efficient foraging flexibly varying the foraging tactics according to feeding schedules. Finally, experiments are performed with robots moving in virtual pheromone fields simulated by CG and CCD camera feedback system. Trail formation processes are demonstrated by this multi-robot system.

q-bio.PE↗
arXiv2004

Global existence and scattering for rough solutions of a nonlinear Schroedinger equation on R^3

We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in $H^s(\rr^3)$ for $s > {4/5}$. The main new estimate in the argument is a Morawetz-type inequality for the solution $ϕ$. This estimate bounds $\|ϕ(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$, whereas the well-known Morawetz-type estimate of Lin-Strauss controls $\int_0^{\infty}\int_{\rr^3}\frac{(ϕ(x,t))^4}{|x|} dx dt

math.AP↗
arXiv2002

A refined global well-posedness result for Schrodinger equations with derivative

In this paper we prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>\frac12$ for data small in $L^{2}$. To understand the strength of this result one should recall that for $s<\frac12$ the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the ``I-method'' used by the same authors to obtain global well-posedness for $s>\frac23$. The same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in $H^{s}$, for $s>\frac12$.

math.AP↗
arXiv2001

Sharp Global well-posedness for KdV and modified KdV on $\R$ and $\T$

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based Sobolev spaces $H^s$ where local well-posedness is presently known, apart from the $H^{1/4} (\R)$ endpoint for mKdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura's transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

math.AP↗
arXiv2001

Global well-posedness for Schrödinger equations with derivative

We prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>2/3$ for small $L^{2}$ data. The result follows from an application of the ``I-method''. This method allows to define a modification of the energy norm $H^{1}$ that is ``almost conserved'' and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schrödinger equation on the line is globally well-posed for large data in $H^{s}$, for $s>2/3$ .

math.AP↗