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30 September 2023, Volume 38 Issue 3

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Some Applications of Surface Curvatures in Theoretical Physics

YANG Yi-song

2023, 38(3):  221-253.  doi:10.13371/j.cnki.chin.q.j.m.2023.03.001

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In this survey article, we present two applications of surface curvatures in theoretical physics. The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy. In this formalism, the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics. We first show that there is an obstruction, arising from the spontaneous curvature, to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori. We then propose a scale-invariant anisotropic bending energy, which extends the Canham energy, and show that it possesses a unique toroidal energy minimizer, up to rescaling, in all parameter regime. Furthermore, we establish some genus-dependent topological lower and upper bounds, which are known to be lacking with the Helfrich energy, for the proposed energy. We also present the shape equation in our context, which extends the Helfrich shape equation. The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings. In this formalism, gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector. This setting provides a lucid exhibition of the interplay of the underlying geometry, matter energy, and topological characterization of the system. In both areas of applications, we encounter highly challenging nonlinear partial differential equation problems. We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

Some Applications of Group Actions in Complex Geometry

GUAN Daniel

2023, 38(3):  254-275.  doi:10.13371/j.cnki.chin.q.j.m.2023.03.002

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In this article, we give a further survey of some progress of the applications of group actions in the complex geometry after my earlier survey around 2020, mostly related to my own interests.


A New Existence Theorem for Global Attractors and its Application to a Non-Classical Diffusion Equation

QIN Yu-ming, CHEN Jia-le, JIANG Hui-te

2023, 38(3):  276-289.  doi:10.13371/j.cnki.chin.q.j.m.2023.03.003

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In this paper, we first survey existed theorems and propose all 46 related open problems of the existence of global attractors for autonomous dynamical systems, then establish a new existence theorem of global attractors which will be applied to a nonclassical diffusion equation for the norm-to-weak continuous, weakly compact semigroup on H01(Ω) and H2(Ω)∩H01(Ω) respectively. As an application of this new existence theorem of global attractors, we obtain the existence of the global attractors onH01(Ω) and H2(Ω)∩ H01(Ω) respectively for a nonclassical-diffusion equation.

Global Well-Posedness of the Initial-Boundary Value Problem on Incompressible MHD-Boussinesq Equations with Nonlinear Boundary Conditions

WANG Shu, SUN Rui

2023, 38(3):  290-310.  doi:10.13371/j.cnki.chin.q.j.m.2023.03.004

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The global well-posedness of another class of initial-boundary value problem on two/three-dimensional incompressible MHD-Boussinesq equations in the bounded domain with the smooth boundary is studied. The existence of a class of global weak solution to the initial boundary value problem for two/three-dimensional incompressible MHD-Boussinesq equation with the given pressure-velocity’s relation boundary condition for the fluid field, one generalized perfectly conducting boundary condition for the magnetic field and one density/temperature-velocity’s relation boundary condition for the density/temapture at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case for the smooth initial data is also proven.

On the Stokes Phenomenon and Representation Theory of Quantum Groups

XU Xiao-meng

2023, 38(3):  311-330.  doi:10.13371/j.cnki.chin.q.j.m.2023.03.005

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This is a survey paper that lists our research works in the study of Stokes phenomenon of meromorphic ordinary differential equations and its relation with representation theory of quantum groups