A time-dependent perturbation solution from a steady state for Marangoni problem

Wang, Huichao; Wang, Quan; Liu, Ruikuan

doi:10.1080/00036811.2017.1317089

Cited by 4 publications

(4 citation statements)

References 19 publications

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“…It is also worth pointing out that Chen et al considered a class of nonlinear time fractional nonautonomous evolution equations with delay, and they give the definition of mild solutions for studied problem and established the new existence of mild solutions. Furthermore, by deleting the compactness condition on nonlocal function, they generalized and improved the results of the existence of mild solutions for a class of time fractional nonautonomous evolution equations with nonlocal conditions in Chen et al There are also many other interesting investigations about the existence of solutions for nonlinear evolution equations, see previous studies and references therein.…”

Section: Introductionmentioning

confidence: 95%

“…In addition, Ma et al established the T ‐weakly continuous operator theory for the abstract evolution equations. In particular, these two weakly continuous operator methods can effectively solve the existence of the solutions for the fully nonlinear elliptic problems, the degenerate quasi‐linear elliptic equations, and the nonlinear evolution equations, see other studies and so on.…”

Section: Introductionmentioning

confidence: 99%

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The existence of strong solution for a class of fully nonlinear equation

2019

Math Methods in App Sciences

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This article mainly investigates the existence of global strong solution of a class of fully nonlinear evolution equation and the strong solution of its steady-state equation. By using the T-compulsorily weakly continuous operator theory, the existence of the global strong solution of the fully nonlinear evolution equation is obtained. In addition, based on the acute angle principle, the W 2,p -strong solution for the corresponding stationary equation is also derived. KEYWORDS acute angle principle, fully nonlinear equation, strong solution, T-compulsorily weakly continuous operator theory MSC CLASSIFICATION 35A01; 35D35; 35D30; 47S99 Math Meth Appl Sci. 2020;43:1530-1542. wileyonlinelibrary.com/journal/mma

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

The existence of strong solution for a class of fully nonlinear equation

2019

Math Methods in App Sciences

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“…In this section, we present known results on the hexagonal pattern formation in surface tension driven convection also known as Marangoni convection to show that it fits into the framework we present in this study. The nondimensional form of the equations describing the Marangoni convection without gravity are, [8,38],…”

Section: Surface Tension Driven Convectionmentioning

confidence: 99%

“…From (38), using the notation (10), we obtain the below formula for the coefficients of the center manifold.…”

Section: Proofsmentioning

confidence: 99%

Dynamical Transition Theory of Hexagonal Pattern Formations

Şengül

2020

Preprint

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The main goal of this paper is to understand the formation of hexagonal patterns from the dynamical transition theory point of view. We consider the transitions from a steady state of an abstract nonlinear dissipative system. To shed light on the formation of mixed mode patterns such as the hexagonal pattern, we consider the case where the linearized operator of the system has two critical real eigenvalues, at a critical value λ c of a control parameter λ with associated eigenmodes having a roll and rectangular pattern. By using center manifold reduction, we obtain the reduced equations of the system near the critical transition value λ c . By a through analysis of these equations, we fully characterize all possible transition scenarios when the coefficients of the quadratic part of the reduced equations do not vanish. We consider three problems, two variants of the 2D Swift-Hohenberg equation and the 3D surface tension driven convection, to demonstrate that all the main theoretical results we obtain here are indeed realizable.

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Magneto-hydrodynamical model for plasma

Liu

Yang

2017

Z. Angew. Math. Phys.

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Basing on the Newton's second law and the Maxwell equations for the electromagnetic fields, we establish a new 3D incompressible magneto-hydrodynamics(MHD) equations for the motion of plasma under the standard Coulomb gauge. By using the Galerkin method, we prove a global weak solution for this 3D new model. keywords Maxwell equations, Plasma, Coulomb gauge, Magneto-hydrodynamics, Galerkin method. * Email:liuruikuan2008@163.com. Supported by NSFC(11401479) † Corresponding author:jiayan 1985@163.com; and obtained partial regularity theorems. Kang and Lee [10] and Wang and Zhang [22] gave other regularity criteria under the Ladyzhenskaya-Prodi-Serrin type conditions. Kang and Kim [11], [12] considered suitable weak solutions in the half space and gave a boundary regularity criteria.Recently, the Hall magneto-hydrodynamic (Hall-MHD) model was established by A. Marion, D. Pierre etc in [14]. The Hall-MHD model receives an increasing attentions from plasma physicists. It is believed to be the key for understanding the problem of magnetic reconnection. Indeed, space plasma observations provide strong evidence for the existence of frequent and fast changes in the topology of magnetic field lines, associating to important events such as solar flares [6]. For the existence of global weak and regular solutions of the (Hall-MHD) equations, we refer to [7], [8], [13]. For the generalized magnetohydrodynamics (MHD) and Navier-Stokes systems, we refer to [1], [6], [13], [17], [18], [19], [23], [24].However, the corresponding models in above papers are obtain by using certain approximations and assumptions. In this paper, basing on some basic physics principles, a new 3D incompressible magneto-hydrodynamic model for plasma shall be established without any assumptions. It is in particular that the model describes the un-static electronic field. Moreover, the existence of the global weak solution of this 3D model is obtained by the Galerkin method.

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A time-dependent perturbation solution from a steady state for Marangoni problem

Cited by 4 publications

References 19 publications

The existence of strong solution for a class of fully nonlinear equation

The existence of strong solution for a class of fully nonlinear equation

Dynamical Transition Theory of Hexagonal Pattern Formations

Magneto-hydrodynamical model for plasma

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