关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

This paper is devoted to the mathematical analysis of a magnetohydrodynamic viscous two-phase dusty fluid flow and heat transfer over permeable stretching or shrinking bodies. The wall boundary is subjected to a linear deformation as well as to a quadratic surface temperature. Such a highly nonlinear phenomenon, for the first time in the literature, is attacked to search for occurrence of exact solutions, whose numerical correspondences are already available for limited wall transpiration velocities. The obtained analytical solutions are found be in perfect line with the numerical computations. Besides this, exact solutions point to the existence of dual solutions for both permeable stretching and shrinking cases, which were not detected from the numerical studies up to date. The existence of such exact solutions and their parameter domain particularly depending on the wall suction or injection are successfully analyzed. The physical outcomes concerning the effects of suspended particles on the momentum a...

Magnetohydrodynamic two-phase dusty fluid flow and heat model over deforming isothermal surfaces

Regular solutions of the 2D Zakharov–Kuznetsov equation on a half-strip

The present discussion is about the unsteady two-dimensional flow of mixed convection and nonlinear thermal radiation in the presence of water-based carbon nanotubes over the vertically convected stretched sheet embedded in a Darcy’ Forchheimer porous media. Saffman’s proposed model is used for the suspension of fine dust particles in the nanofluid. A strong magnetic field (MHD) is applied normal to the flow which governs the Hall current effects. Khanafer Vafai Lightstone model estimated the effect of thermal conductivity and viscosity of the carbon nanotubes. Boundary layer approximation is utilized to built the nonlinear partial differential equations (PDEs). Similarity transformation is applied to convert these PDEs into the system of ordinary differential equations. Problem is solved numerically by bvp4c, using MATLAB software. It is observed through the analysis that the thermal field of nanofluid and the temperature boundary layer are much more higher than that of the dust phase, and these are further enhanced for the higher radiation parameter.

Hall current effect on unsteady rotational flow of carbon nanotubes with dust particles and nonlinear thermal radiation in Darcy–Forchheimer porous media

Energy decay for the modified Kawahara equation posed in a bounded domain

This paper studies nonhomogeneous initial-boundary value prob- lems for quasilinear one-dimensional odd-order equations posed on a bounded interval. For reasonable initial and boundary conditions we prove existence and uniqueness of global weak and regular solutions. Also we show the ex- ponential decay of the obtained solution with zero boundary conditions and right-hand side, and small initial data.

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Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval

Initial–boundary-value problems for the two-dimensional Zakharov–Kuznetsov equation posed on bounded rectangles and on a strip are considered. Spectral properties of a linearized operator and critical sizes of domains are studied. An exponential decay rate of regular solutions for the original nonlinear problems is proved.

Stabilization of Regular Solutions for the Zakharov-Kuznetsov Equation Posed on Bounded Rectangles and on a Strip

Initial-boundary value problems for the 2D Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip are considered. Spectral properties of a linearized operator and critical sizes of domains are studied. Exponential decay of regular solutions for the original nonlinear problems is proved.

Stabilization of regular solutions for the Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip

Well and ill-posedness of initial-boundary value problems for the KdV and Kawahara equations posed on a nite interval are discussed Non-existence of solutions to ill-posed problem for the KdV equation is proved as well as solvability, uniqueness, exponential decay and asymptotics of regular solutions to the Kawahara equation subject to reasonable boundary conditions

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Well and ill-posed problems for the KdV and Kawahara equations - doi: 10.5269/bspm.v26i1-2.7419

We study in a rectangle Q_T = (0, T)×(0, 1) global well-posedness of nonhomo-geneous initial-boundary value problems for general odd-order quasilinear partial differential equations. This class of equations includes well-known Korteweg–de Vries and Kawahara equations which model the dynamics of long small-amplitude waves in various media. Our study is motivated by physics and numerics and our main goal is to formulate a correct nonhomogeneous initial-boundary value problem under consideration in a bounded interval and to prove the existence and uniqueness of global in time weak and regular solutions in a large scale of Sobolev spaces as well as to study decay of solutions while t → ∞.

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Nikolai A. Larkin

Papers

Initial-boundary value problems for quasilinear dispersive equations posed on a bounded interval

Stabilization of Regular Solutions for the Zakharov-Kuznetsov Equation Posed on Bounded Rectangles and on a Strip

Stabilization of regular solutions for the Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip

Well and ill-posed problems for the KdV and Kawahara equations - doi: 10.5269/bspm.v26i1-2.7419

Odd-order quasilinear evolution equations posed on a bounded interval - 10.5269/bspm.v28i1.10816