scholarly journals GLOBAL CHARACTERISTIC PROBLEM FOR EINSTEIN VACUUM EQUATIONS WITH SMALL INITIAL DATA: (I) THE INITIAL DATA CONSTRAINTS

2005 ◽  
Vol 02 (01) ◽  
pp. 201-277 ◽  
Author(s):  
GIULIO CACIOTTA ◽  
FRANCESCO NICOLÒ
Keyword(s):  
Initial Data ◽  
Existence Result ◽  
Subsequent Paper ◽  
Small Initial Data ◽  
Asymptotic Decay ◽  
Characteristic Problem ◽  
Einstein Vacuum Equations ◽  
Einstein Vacuum ◽  
Data Constraints ◽  
Global Existence Result

We show how to prescribe the initial data of a characteristic problem satisfying the constraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in detail the construction of the initial data and give a sketch of the existence result. This proof, which mimicks the analogous one for the non-characteristic problem in [19], will be the content of a subsequent paper.

scholarly journals CURVATURE BLOW UP ON A DENSE SUBSET OF THE SINGULARITY IN T3-GOWDY

2005 ◽  
Vol 02 (02) ◽  
pp. 547-564 ◽  
Author(s):  
HANS RINGSTRÖM
Keyword(s):  
Initial Data ◽  
Blow Up ◽  
Dense Subset ◽  
Cosmic Censorship ◽  
Einstein Vacuum Equations ◽  
Strong Cosmic Censorship ◽  
Einstein Vacuum ◽  
Original Argument ◽  
Gowdy Spacetimes ◽  
Generic Set

This paper is concerned with the Einstein vacuum equations under the additional assumption of T3-Gowdy symmetry. We prove that there is a generic set of initial data such that the corresponding solutions exhibit curvature blow up on a dense subset of the singularity. By generic, we mean a countable intersection of open sets (i.e. a Gδ set) which is also dense. Furthermore, the set of initial data is given the C∞ topology. This result was presented at a conference in Miami 2004. Recently, we have obtained a stronger result, but the argument to prove it is different and much longer. Therefore, we here wish to present the original argument. Finally, combining the results presented here with a paper by Chruściel and Lake, one obtains strong cosmic censorship for T3-Gowdy spacetimes.

Solutions with Singular Initial Data for a Model of Electrophoretic Separation

1990 ◽  
Vol 33 (1) ◽  
pp. 3-10 ◽  
Author(s):  
Joel D. Avrin
Keyword(s):  
Initial Data ◽  
Diffusion Coefficients ◽  
Existence Result ◽  
Strong Solutions ◽  
Approximate Solutions ◽  
Electrophoretic Separation ◽  
Small Diffusion ◽  
Singular Initial Data ◽  
Global Strong Solutions ◽  
Global Existence Result

AbstractUnique global strong solutions of a Cauchy problem arising in electrophoretic separation are constructed with arbitrary initial data in L1, thus generalizing an earlier global existence result. For small diffusion coefficients, the solutions can be viewed as approximate solutions for the corresponding zero-diffusion Riemann problem.

scholarly journals On the IAA version of the Doi–Edwards model versus the K-BKZ rheological model for polymer fluids: A global existence result for shear flows with small initial data

2016 ◽  
Vol 28 (1) ◽  
pp. 42-90 ◽  
Author(s):  
IONEL SORIN CIUPERCA ◽  
ARNAUD HEIBIG ◽  
LIVIU IULIAN PALADE
Keyword(s):  
Initial Data ◽  
Rheological Model ◽  
Shear Flows ◽  
Mathematical Framework ◽  
Smooth Solutions ◽  
Small Initial Data ◽  
Polymer Fluids ◽  
Edwards Model ◽  
Model Versus ◽  
Global Existence Result

This paper establishes the existence of smooth solutions for the Doi–Edwards rheological model of viscoelastic polymer fluids in shear flows. The problem turns out to be formally equivalent to a K-BKZ equation but with constitutive functions spanning beyond the usual mathematical framework. We prove, for small enough initial data, that the solution remains in the domain of hyperbolicity of the equation for all t≥0.

scholarly journals L2-CONCENTRATION PHENOMENON FOR ZAKHAROV SYSTEM BELOW ENERGY NORM

2009 ◽  
Vol 11 (01) ◽  
pp. 27-57 ◽  
Author(s):  
DAOYUAN FANG ◽  
SIJIA ZHONG
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Existence Result ◽  
Space Dimension ◽  
Blow Up ◽  
Energy Norm ◽  
Zakharov System ◽  
Concentration Result ◽  
Global Existence Result ◽  
Concentration Phenomenon

In this paper, we prove an L2-concentration result of Zakharov system in space dimension two, with radial initial data [Formula: see text], when blow up of the solution happens by I-method. In addition to that, we find a blow up character of this system. Furthermore, we improve the global existence result of Bourgain's to the above-mentioned spaces.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

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