Initialization and Extension (Theorems M6, M7, M8)

2020 ◽  
pp. 372-485
Author(s):  
Sergiu Klainerman ◽  
Jérémie Szeftel
Keyword(s):  
Iterative Procedure ◽  
Higher Order ◽  
Boundary Term ◽  
Decay Estimates ◽  
Extension Theorems ◽  
Prove Theorem

This chapter focuses on the proof for Theorem M6 concerning initialization, Theorem M7 concerning extension, and Theorem M8 concerning the improvement of higher order weighted energies. It first improves the bootstrap assumptions on decay estimates. The chapter then improves the bootstrap assumptions on energies and weighted energies for R and Γ‎ relying on an iterative procedure which recovers derivatives one by one. It also outlines the norms for measuring weighted energies for curvature components and Ricci coefficients. To prove Theorem M8, the chapter relies on Propositions 8.11, 8.12, and 8.13. Among these propositions, only the last two involve the dangerous boundary term.

Regge-Wheeler Type Equations

2020 ◽  
pp. 600-718
Author(s):  
Sergiu Klainerman ◽  
Jérémie Szeftel
Keyword(s):  
Wave Equations ◽  
Higher Order ◽  
Weighted Estimates ◽  
Prove Theorem ◽  
Type Wave ◽  
Derivatives Of

This chapter explores estimates for Regge-Wheeler type wave equations used in Theorem M1. It first proves basic Morawetz estimates for ψ‎. The chapter then proves rp-weighted estimates in the spirit of Dafermos and Rodnianski for ψ‎. In particular, it obtains as an immediate corollary the proof of Theorem 5.17 in the case s = 0 (i.e., without commutating the equation of ψ‎ with derivatives). It also uses a variation of the method of [5] to derive slightly stronger weighted estimates and prove Theorem 5.18 in the case s = 0. Finally, commuting the equation of ψ‎ with derivatives, the chapter completes the proof of Theorem 5.17 by controlling higher order derivatives of ψ‎.

Decay Estimates for α‎ and α‎ (Theorems M2, M3)

2020 ◽  
pp. 264-294
Author(s):  
Sergiu Klainerman ◽  
Jérémie Szeftel
Keyword(s):  
Parabolic Equation ◽  
Transport Equation ◽  
Decay Estimates ◽  
Prove Theorem

This chapter investigates the proof for Theorems M2 and M3. It relies on the decay of q to prove the decay estimates for α‎ and α‎. More precisely, the chapter relies on the results of Theorem M1 to prove Theorem M2 and M3. In Theorem M1, decay estimates are derived for q defined with respect to the global frame constructed in Proposition 3.26. To recover α‎ from q, the chapter derives a transport equation for α‎ where q is on the RHS. It then derives as Teukolsky–Starobinsky identity a parabolic equation for α‎. The chapter also improves bootstrap assumptions for α‎.

scholarly journals Decay estimates for higher-order elliptic operators

2020 ◽  
Vol 373 (4) ◽  
pp. 2805-2859
Author(s):  
Hongliang Feng ◽  
Avy Soffer ◽  
Zhao Wu ◽  
Xiaohua Yao
Keyword(s):  
Higher Order ◽  
Elliptic Operators ◽  
Decay Estimates

Decay of solutions of a higher order multidimensional nonlinear Korteweg–de Vries–Burgers system

1994 ◽  
Vol 124 (2) ◽  
pp. 263-271 ◽  
Author(s):  
Zhang Linghai
Keyword(s):  
Initial Value Problem ◽  
Higher Order ◽  
Decay Estimates ◽  
Initial Value ◽  
Decay Of Solutions ◽  
De Vries ◽  
Integral Estimation ◽  
Burgers System

We study decay estimates for the solutions to the initial value problem for a higher order multidimensional nonlinear Korteweg–de Vries–Burgers system. The method is integral estimation.

The Estimation of Parameters in Nonstationary Higher Order Continuous-Time Dynamic Models

1985 ◽  
Vol 1 (3) ◽  
pp. 369-385 ◽  
Author(s):  
A. R. Bergstrom
Keyword(s):  
Continuous Time ◽  
Iterative Procedure ◽  
Dynamic Models ◽  
Structural Parameters ◽  
Higher Order ◽  
Estimation Of Parameters ◽  
Initial State ◽  
Time Dynamic ◽  
Mixed Stock ◽  
Order Continuous

This paper is concerned with derivation of a new efficient algorithm for computing the exact Gaussian likelihood for structural parameters in nonstationary higher-order continuous-time dynamic models and with its application in the estimation of these parameters. The algorithm completely avoids the computation of the covariance matrix of the observations and is applicable to a system of any order with mixed stock and flow data. It is used as the basis for an iterative procedure in which the structural parameters and the initial state vector are estimated alternately.

The flow fields in and around a droplet moving axially within a tube

1970 ◽  
Vol 41 (4) ◽  
pp. 689-705 ◽  
Author(s):  
G. Hetsroni ◽  
S. Haber ◽  
E. Wacholder
Keyword(s):  
Flow Field ◽  
Viscous Fluid ◽  
Velocity Distribution ◽  
General Equation ◽  
Iterative Procedure ◽  
Settling Velocity ◽  
Circular Tube ◽  
Higher Order ◽  
Spherical Droplet ◽  
Special Cases

A solution is presented for the flow field in and around a single spherical droplet or bubble moving axially at an arbitrary radial location, within a long circular tube. In the tube there is viscous fluid flowing with a constant Poiseuillian velocity distribution far from the droplet.The settling velocity of the droplet or bubble is \begin{eqnarray*} U = \frac{2(\rho_i-\rho_e)ga^2}{9\mu_e}\frac{1+\alpha}{\frac{2}{3}+\alpha}\left[1-\frac{2+3\alpha}{3(1+\alpha)}\left(\frac{a}{R_0}\right)f\left(\frac{b}{R_0}\right)\right]+U_0\left[1-\left(\frac{b}{R_0}\right)^2\right.\\ \left. - \frac{2\alpha}{2+3\alpha}\left(\frac{a}{R_0}\right)^2\right] + O\left(\frac{a}{R_0}\right)^3. \end{eqnarray*} This is a general equation and it contains as special cases the familiar solutions of Stokes, Hadamard-Rybczynski, Brenner & Happel, Greenstein & Happel and Haberman & Sayre.The function describing the deviation of the interface from sphericity is solved and an iterative procedure is suggested for obtaining higher order solutions.

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