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 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 On a Partial Boundary Value Condition of a Porous Medium Equation with Exponent Variable

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Huashui Zhan
Keyword(s):  
Porous Medium ◽  
Variable Exponent ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Medium Equation ◽  
Partial Boundary Value Condition ◽  
Initial Boundary Value

The initial-boundary value problem of a porous medium equation with a variable exponent is considered. Both the diffusion coefficient ax,t and the variable exponent px,t depend on the time variable t, and this makes the partial boundary value condition not be expressed as the usual Dirichlet boundary value condition. In other words, the partial boundary value condition matching up with the equation is based on a submanifold of ∂Ω×0,T. By this innovation, the stability of weak solutions is proved.

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].

Sign in / Sign up

Export Citation Format

Share Document