scholarly journals GLOBAL ATTRACTOR FOR WEAKLY DAMPED, FORCED mKdV EQUATION BELOW ENERGY SPACE

2019 ◽  
pp. 1-33
Author(s):  
PRASHANT GOYAL
Keyword(s):  
Global Attractor ◽  
Mkdv Equation ◽  
Energy Space ◽  
Dimensional Torus ◽  
Resonant Frequencies ◽  
One Dimensional ◽  
Bilinear Estimates ◽  
The One ◽  
Close Investigation ◽  
Formula Method

We prove the existence of the global attractor in ${\dot{H}}^{s}$ , $s>11/12$ for the weakly damped and forced mKdV on the one-dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space has been established. We directly apply the $I$ -method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresponding to the KdV.

scholarly journals Fourth-order dispersive systems on the one-dimensional torus

2015 ◽  
Vol 6 (2) ◽  
pp. 237-263 ◽  
Author(s):  
Hiroyuki Chihara
Keyword(s):  
Fourth Order ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Dispersive Systems ◽  
The One

scholarly journals Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation

2014 ◽  
Vol 64 (1) ◽  
pp. 19-70 ◽  
Author(s):  
Fabrice Béthuel ◽  
Philippe Gravejat ◽  
Didier Smets
Keyword(s):  
Energy Space ◽  
Pitaevskii Equation ◽  
One Dimensional ◽  
The One

scholarly journals Well-posedness of the linear heat equation with a second order memory term

2020 ◽  
Vol 34 ◽  
pp. 03011
Author(s):  
Constantin Niţă ◽  
Laurenţiu Emanuel Temereancă
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Unique Solution ◽  
Source Term ◽  
Memory Term ◽  
Well Posedness ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Linear Heat ◽  
The One

In this article we prove that the heat equation with a memory term on the one-dimensional torus has a unique solution and we study the smoothness properties of this solution. These properties are related with some smoothness assumptions imposed to the initial data of the problem and to the source term.

scholarly journals Large semigroups of cellular automata

2011 ◽  
Vol 32 (6) ◽  
pp. 1991-2010 ◽  
Author(s):  
YAIR HARTMAN
Keyword(s):  
Cellular Automata ◽  
Field Structure ◽  
The Other ◽  
Entire Space ◽  
Dimensional Torus ◽  
Semigroup Action ◽  
Closed Set ◽  
One Dimensional ◽  
Shift Space ◽  
The One

AbstractIn this article, we consider semigroups of transformations of cellular automata which act on a fixed shift space. In particular, we are interested in two properties of these semigroups which relate to ‘largeness’: first, a semigroup has the ID (infinite is dense) property if the only infinite invariant closed set (with respect to the semigroup action) is the entire space; the second property is maximal commutativity (MC). We shall consider two examples of semigroups: one is spanned by cellular automata transformations that represent multiplications by integers on the one-dimensional torus, and the other one consists of all the cellular automata transformations which are linear (when the symbols set is the ring ℤ/sℤ). It will be shown that these two properties of these semigroups depend on the number of symbols s. The multiplication semigroup is ID and MC if and only if s is not a power of a prime. The linear semigroup over the mentioned ring is always MC but is ID if and only if s is prime. When the symbol set is endowed with a finite field structure (when possible), the linear semigroup is both ID and MC. In addition, we associate with each semigroup which acts on a one-sided shift space a semigroup acting on a two-sided shift space, and vice versa, in a way that preserves the ID and the MC properties.

Almost free action of the one-dimensional torus on a toric variety

2006 ◽  
Vol 197 (5) ◽  
pp. 681-703 ◽  
Author(s):  
D G Il'inskii
Keyword(s):  
Toric Variety ◽  
Free Action ◽  
Dimensional Torus ◽  
One Dimensional ◽  
The One

THE FUNDAMENTAL GROUP OF GALOIS COVER OF THE SURFACE 𝕋 × 𝕋

2008 ◽  
Vol 18 (08) ◽  
pp. 1259-1282 ◽  
Author(s):  
MEIRAV AMRAM ◽  
MINA TEICHER ◽  
UZI VISHNE
Keyword(s):  
Fundamental Group ◽  
Dimensional Torus ◽  
One Dimensional ◽  
Generic Projection ◽  
Galois Cover ◽  
The One ◽  
First Time

This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.

On absolutely nonmeasurable homomorphisms of commutative groups

2013 ◽  
Vol 50 (3) ◽  
pp. 287-295
Author(s):  
A. Kharazishvili
Keyword(s):  
Real Line ◽  
Quotient Group ◽  
Torsion Subgroup ◽  
Dimensional Torus ◽  
One Dimensional ◽  
The Real ◽  
The One

We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (G, +) for which the quotient group G/G0 is uncountable, where G0 denotes the torsion subgroup of G.

Existence of global attractor for the one-dimensional bipolar quantum drift-diffusion model

2017 ◽  
Vol 22 (4) ◽  
pp. 277-282 ◽  
Author(s):  
Yannan Liu ◽  
Wenlong Sun ◽  
Yeping Li
Keyword(s):  
Global Attractor ◽  
Diffusion Model ◽  
Drift Diffusion Model ◽  
Drift Diffusion ◽  
One Dimensional ◽  
The One

On Schauder bases in Hardy spaces

1983 ◽  
Vol 93 (3-4) ◽  
pp. 259-263 ◽  
Author(s):  
P. Oswald
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Schauder Basis ◽  
Dimensional Torus ◽  
Franklin System ◽  
One Dimensional ◽  
The Real ◽  
Real Hardy Space ◽  
Schauder Bases ◽  
The One

SynopsisIt is proved that in the case ½<p<l the periodic Franklin system forms a Schauder basis for the real Hardy space Hp(T) defined on the one-dimensional torus.In this note we prove the followingTheorem. The periodic Franklin system forms a Schauder basis in the real Hardyspace Hp(T) defined on the one-dimensional torus if ½<p< l.

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