scholarly journals Global Well-Posedness and Analyticity of the Primitive Equations of Geophysics in Variable Exponent Fourier–Besov Spaces

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 165
Author(s):  
Muhammad Zainul Abidin ◽  
Naeem Ullah ◽  
Omer Abdalrhman Omer
Keyword(s):  
Cauchy Problem ◽  
Besov Spaces ◽  
Variable Exponent ◽  
Three Dimensional ◽  
Primitive Equations ◽  
Well Posedness ◽  
Small Initial Data ◽  
Fourier Localization ◽  
Gevrey Class Regularity ◽  
The Cauchy Problem

We consider the Cauchy problem of the three-dimensional primitive equations of geophysics. By using the Littlewood–Paley decomposition theory and Fourier localization technique, we prove the global well-posedness for the Cauchy problem with the Prandtl number P=1 in variable exponent Fourier–Besov spaces for small initial data in these spaces. In addition, we prove the Gevrey class regularity of the solution. For the primitive equations of geophysics, our results can be considered as a symmetry in variable exponent Fourier–Besov spaces.

scholarly journals Global Well-Posedness for Fractional Navier-Stokes Equations in critical Fourier-Besov-Morrey Spaces

2017 ◽  
Vol 3 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Azzeddine El Baraka ◽  
Mohamed Toumlilin
Keyword(s):  
Cauchy Problem ◽  
Stokes Equations ◽  
Morrey Spaces ◽  
Navier Stokes ◽  
Well Posedness ◽  
Navier Stokes Equations ◽  
Small Initial Data ◽  
Localization Method ◽  
Fourier Localization ◽  
The Cauchy Problem

Abstract In this paper we study the Cauchy problem of the Fractional Navier-Stokes equations in critical Fourier-Besov-Morrey spaces FṄsp, λ,q(ℝ3) with . By making use of the Fourier localization method and the Littlewood-Paley theory as in [6] and [21], we get global well-posedness result with small initial data belonging to . The space FṄsp,λ,q(ℝ3) covers the classical spaces Ḃsq and FḂsp,q(ℝ3) (cf [7],[3], [19], [22]...). The result of this paper extends the works of [6] and [21].

scholarly journals Global Well-Posedness and Analyticity of Generalized Porous Medium Equation in Fourier-Besov-Morrey Spaces with Variable Exponent

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 498
Author(s):  
Muhammad Zainul Abidin ◽  
Jiecheng Chen
Keyword(s):  
Porous Medium ◽  
Variable Exponent ◽  
Porous Medium Equation ◽  
Morrey Spaces ◽  
Well Posedness ◽  
Small Initial Data ◽  
Localization Principle ◽  
Medium Equation ◽  
Fourier Localization ◽  
Gevrey Class Regularity

In this paper, we consider the generalized porous medium equation. For small initial data u0 belonging to the Fourier-Besov-Morrey spaces with variable exponent, we obtain the global well-posedness results of generalized porous medium equation by using the Fourier localization principle and the Littlewood-Paley decomposition technique. Furthermore, we also show Gevrey class regularity of the solution.

scholarly journals Dilation Properties for Weighted Modulation Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Elena Cordero ◽  
Kasso A. Okoudjou
Keyword(s):  
Cauchy Problem ◽  
Besov Spaces ◽  
Sharp Estimate ◽  
Modulation Spaces ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Unweighted Case ◽  
Vibrating Plate

We give a sharp estimate on the norm of the scaling operatorUλf(x)=f(λx)acting on the weighted modulation spacesMs,tp,q(ℝd). In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well-posedness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.

scholarly journals GLOBAL WELL-POSEDNESS FOR THE GENERALISED FOURTH-ORDER SCHRÖDINGER EQUATION

2012 ◽  
Vol 85 (3) ◽  
pp. 371-379 ◽  
Author(s):  
YUZHAO WANG
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Vries Equation ◽  
Well Posedness ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Image Position ◽  
Appl Math

AbstractWe study the Cauchy problem for the generalised fourth-order Schrödinger equation for data u0 in critical Sobolev spaces $\dot {H}^{1/2-3/2k}$. With small initial data we obtain global well-posedness results. Our proof relies heavily on the method developed by Kenig et al. [‘Well-posedness and scattering results for the generalised Korteweg–de Vries equation via the contraction principle’, Commun. Pure Appl. Math.46 (1993), 527–620].

Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4

2020 ◽  
Vol 32 (6) ◽  
pp. 1575-1598
Author(s):  
Zhaohui Huo ◽  
Yueling Jia
Keyword(s):  
Cauchy Problem ◽  
Sobolev Space ◽  
Initial Data ◽  
Well Posedness ◽  
Small Initial Data ◽  
Bilinear Estimates ◽  
The Cauchy Problem ◽  
Univariate Polynomial ◽  
Well Posed ◽  
Resonance Function

AbstractThe Cauchy problem of the 2D Zakharov–Kuznetsov equation {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space {H^{-1/4}}, and it is globally well-posed in {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

scholarly journals The Well Posedness for Nonhomogeneous Boussinesq Equations

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2110
Author(s):  
Yan Liu ◽  
Baiping Ouyang
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Positive Constant ◽  
Besov Spaces ◽  
Initial Velocity ◽  
Initial Temperature ◽  
Boussinesq Equations ◽  
Well Posedness ◽  
The Cauchy Problem ◽  
Critical Besov Spaces

This paper is devoted to studying the Cauchy problem for non-homogeneous Boussinesq equations. We built the results on the critical Besov spaces (θ,u)∈LT∞(B˙p,1N/p)×LT∞(B˙p,1N/p−1)⋂LT1(B˙p,1N/p+1) with 1<p<2N. We proved the global existence of the solution when the initial velocity is small with respect to the viscosity, as well as the initial temperature approaches a positive constant. Furthermore, we proved the uniqueness for 1<p≤N. Our results can been seen as a version of symmetry in Besov space for the Boussinesq equations.

scholarly journals Global solution to the incompressible Oldroyd-B model in hybrid Besov spaces

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3627-3639 ◽  
Author(s):  
Ruizhao Zia
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Global Solution ◽  
Besov Spaces ◽  
Coupling Constant ◽  
Unique Global Solution ◽  
Small Initial Data ◽  
Nonlinear Anal ◽  
The Cauchy Problem

This paper is dedicated to the Cauchy problem of the incompressible Oldroyd-B model with general coupling constant ? ?(0,1). It is shown that this set of equations admits a unique global solution in a certain hybrid Besov spaces for small initial data in ?Hs ??Bd/2 2,1 with - d/2 < s < d2-1. In particular, if d ? 3, and taking s=0, then ?H0 ? ?Bd/2 2,1 = B d/2 2,1. Since Bt2,? ? Bd/2 2,1 if t > d/2, this result extends the work by Chen and Miao [Nonlinear Anal.,68(2008), 1928-1939].

scholarly journals On the Cauchy Problem for the Two-Component Novikov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yongsheng Mi ◽  
Chunlai Mu ◽  
Weian Tao
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Transport Equation ◽  
Besov Spaces ◽  
Well Posedness ◽  
Two Component ◽  
The Cauchy Problem ◽  
Novikov Equation

We are concerned with the Cauchy problem of two-component Novikov equation, which was proposed by Geng and Xue (2009). We establish the local well-posedness in a range of the Besov spaces by using Littlewood-Paley decomposition and transport equation theory which is motivated by that in Danchin's cerebrated paper (2001). Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time, which extend some results of Himonas (2003) to more general equations.

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