scholarly journals Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3145
Author(s):  
Divyang G. Bhimani ◽  
Saikatul Haque
Keyword(s):  
Initial Data ◽  
Modulation Spaces ◽  
Well Posedness ◽  
Wiener Amalgam Spaces ◽  
General Initial Data ◽  
Amalgam Spaces ◽  
Loss Of Regularity ◽  
Bbm Equation

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

OPERATING FUNCTIONS ON MODULATION AND WIENER AMALGAM SPACES

2017 ◽  
Vol 230 ◽  
pp. 72-82 ◽  
Author(s):  
MASAHARU KOBAYASHI ◽  
ENJI SATO
Keyword(s):  
Open Problem ◽  
Composition Operators ◽  
Affirmative Answer ◽  
Modulation Spaces ◽  
Wiener Amalgam Spaces ◽  
Amalgam Spaces ◽  
Funct Anal

The goal of this paper is to characterize the operating functions on modulation spaces$M^{p,1}(\mathbb{R})$and Wiener amalgam spaces$W^{p,1}(\mathbb{R})$. This characterization gives an affirmative answer to the open problem proposed by Bhimani (Composition Operators on Wiener amalgam Spaces, arXiv: 1503.01606) and Bhimani and Ratnakumar (J. Funct. Anal.270(2016), pp. 621–648).

Singular convolution operators on Wiener amalgam spaces

2015 ◽  
Vol 13 (01) ◽  
pp. 1550003
Author(s):  
Meifang Cheng
Keyword(s):  
Modulation Spaces ◽  
Convolution Operators ◽  
Lebesgue Spaces ◽  
Wiener Amalgam Spaces ◽  
Amalgam Spaces

The purpose of this paper is to investigate some properties of the Wiener amalgam spaces. As applications, we study the boundedness properties of some singular convolution operators on such spaces. From our results, we will see that, besides modulation spaces, Wiener amalgam spaces are another good substitutions for Lebesgue spaces.

scholarly journals Well-Posedness of the Two-Dimensional Fractional Quasigeostrophic Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Yongqiang Xu
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Well Posedness ◽  
Two Dimensional ◽  
Small Initial Data ◽  
General Initial Data ◽  
Local Existence Of Solutions

This paper is concerned with the fractional quasigeostrophic equation with modified dissipativity. We prove the local existence of solutions in Sobolev spaces for the general initial data and the global existence for the small initial data when1/2≤α<1.

scholarly journals SHARPNESS OF SOME PROPERTIES OF WIENER AMALGAM AND MODULATION SPACES

2009 ◽  
Vol 80 (1) ◽  
pp. 105-116 ◽  
Author(s):  
ELENA CORDERO ◽  
FABIO NICOLA
Keyword(s):  
Modulation Spaces ◽  
Wiener Amalgam Spaces ◽  
Dilation Operator ◽  
Sharp Estimates ◽  
Amalgam Spaces ◽  
Scaling Arguments

AbstractWe prove sharp estimates for the dilation operator f(x)⟼f(λx), when acting on Wiener amalgam spaces W(Lp,Lq). Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces Mp,q, as well as the optimality of an estimate for the Schrödinger propagator on modulation spaces.

scholarly journals COMPOSITION OPERATORS ON WIENER AMALGAM SPACES

2019 ◽  
Vol 240 ◽  
pp. 257-274
Author(s):  
DIVYANG G. BHIMANI
Keyword(s):  
Composition Operator ◽  
Composition Operators ◽  
Complex Function ◽  
Affirmative Answer ◽  
Modulation Spaces ◽  
Wiener Amalgam Spaces ◽  
Real Analytic ◽  
Amalgam Spaces ◽  
Open Question ◽  
Funct Anal

For a complex function $F$ on $\mathbb{C}$, we study the associated composition operator $T_{F}(f):=F\circ f=F(f)$ on Wiener amalgam $W^{p,q}(\mathbb{R}^{d})\;(1\leqslant p<\infty ,1\leqslant q<2)$. We have shown $T_{F}$ maps $W^{p,1}(\mathbb{R}^{d})$ to $W^{p,q}(\mathbb{R}^{d})$ if and only if $F$ is real analytic on $\mathbb{R}^{2}$ and $F(0)=0$. Similar result is proved in the case of modulation spaces $M^{p,q}(\mathbb{R}^{d})$. In particular, this gives an affirmative answer to the open question proposed in Bhimani and Ratnakumar (J. Funct. Anal. 270(2) (2016), 621–648).

scholarly journals Klein-Gordon Equations on Modulation Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Guoping Zhao ◽  
Jiecheng Chen ◽  
Weichao Guo
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Stability Theory ◽  
Scattering Theory ◽  
Modulation Spaces ◽  
Well Posedness ◽  
Blowup Criterion ◽  
The Cauchy Problem ◽  
Klein Gordon

We consider the Cauchy problem for a family of Klein-Gordon equations with initial data in modulation spacesMp,1a. We develop the well-posedness, blowup criterion, stability of regularity, scattering theory, and stability theory.

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