On the role of the Besov spaces for the solutions of the generalized burgers equation in homogeneous Sobolev spaces

Using the realizations, we study some convolution inequalities in the realized homogeneous Besov spaces ??Bs p,q(Rn) and the realized homogeneous Triebel-Lizorkin spaces ??Fs p,q(Rn). We also deduce for the homogeneous Sobolev spaces ?Wmp(Rn) in certain sense.

Download Full-text

Viscous Flow Around a Rigid Body Performing a Time-periodic Motion

Journal of Mathematical Fluid Mechanics ◽

10.1007/s00021-021-00556-4 ◽

2021 ◽

Vol 23 (1) ◽

Author(s):

Thomas Eiter ◽

Mads Kyed

Keyword(s):

Rigid Body ◽

Sobolev Spaces ◽

Periodic Motion ◽

Viscous Incompressible Fluid ◽

Body Motion ◽

Translational Velocity ◽

Ill Posed ◽

Homogeneous Sobolev Spaces ◽

The Mean ◽

Time Periodic

AbstractThe equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-posed in classical Sobolev spaces, a linear theory is developed in a framework of homogeneous Sobolev spaces.

Download Full-text

Bilinear forms on non-homogeneous Sobolev spaces

Forum Mathematicum ◽

10.1515/forum-2019-0311 ◽

2020 ◽

Vol 32 (4) ◽

pp. 995-1026

Author(s):

Carme Cascante ◽

Joaquín M. Ortega

Keyword(s):

Sobolev Spaces ◽

Bilinear Form ◽

Positive Measure ◽

Bilinear Forms ◽

Homogeneous Sobolev Spaces

AbstractIn this paper, we show that if {b\in L^{2}(\mathbb{R}^{n})}, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces {H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})}, {0<s<1}, by(f,g)\in H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})\to\int_{% \mathbb{R}^{n}}(\mathrm{Id}-\Delta)^{\frac{s}{2}}(fg)(\mathbf{x})b(\mathbf{x})% \mathop{}\!d\mathbf{x}is continuous if and only if the positive measure {\lvert b(\mathbf{x})\rvert^{2}\mathop{}\!d\mathbf{x}} is a trace measure for {H_{s}^{2}(\mathbb{R}^{n})}.

Download Full-text

1-Soliton solution of the generalized Burgers equation with generalized evolution

Applied Mathematics and Computation ◽

10.1016/j.amc.2011.05.031 ◽

2011 ◽

Vol 217 (24) ◽

pp. 10289-10294 ◽

Cited By ~ 8

Author(s):

Anjan Biswas ◽

Houria Triki ◽

T. Hayat ◽

Omar M. Aldossary

Keyword(s):

Soliton Solution ◽

Burgers Equation ◽

Generalized Burgers Equation

Download Full-text

SOLVING THE GENERALIZED BURGERS–KdV EQUATION BY A COMBINATION METHOD

Modern Physics Letters B ◽

10.1142/s0217984908016686 ◽

2008 ◽

Vol 22 (21) ◽

pp. 2021-2025 ◽

Cited By ~ 1

Author(s):

YUANXI XIE

Keyword(s):

Exact Solutions ◽

Burgers Equation ◽

Kdv Equation ◽

Combination Method ◽

Generalized Kdv Equation ◽

Generalized Burgers Equation

In view of the analysis on the characteristics of the generalized Burgers equation, generalized KdV equation and generalized Burgers–KdV equation, a combination method is presented to seek the explicit and exact solutions to the generalized Burgers–KdV equation by combining with those of the generalized Burgers equation and generalized KdV equation. As a result, many explicit and exact solutions for the generalized Burgers–KdV equation are successfully obtained by this technique.

Download Full-text