Periodic Solution Of Compressible Quantum Magnetohydrodynamic

In recent years,quantum hydrodynamic has become one of the hotpoints in the field of applied mathematics.In physics,quantum hydrodynamics is widely used in various fields,such as fluid model of nuclear and fluid model of superconductors,etc.In mathematics,quantum hydrodynamic model is a complex system coupling with hyperbolic,elliptical and nonlinear dispersion,which is a classic model of dispersion process^[57].This paper is concerned with the compressible quantum magnetohydrodynamic.When magnetic field is zero,the model becomes a compressible quantum Navier-Stokes model.When the Planck's constant is equal to zero,quantum effects disappear and the model will become a compressible MHD model.There have been a lot of conclusions about quantum Navier-Stokes model and MHD model,but the results of QMHD model are few.Investigating such kind of partial differential equation,can not only provide reasonable explanations of the aerospace,material science and other application fields from a mathematical perspective,but also the reasonably predict the relevant physical phenomena.From the theory of time-periodic solutions to non-linear partial differential equations whose origin may be put in 1830s.Up to now,very rich research methods and results are well known^[40].This paper mainly studies the periodic solution of compressible QMHD model,and the specific structure is as follows:In chapter 1,the physical background and research progress of the subject studied in this article are introduced through a large number of literatures.At the same time,the symbol descriptions and some results needed in this article are given;In chapter 2,the mathematical description of the model studied in this paper is carried out.After stating the main results of this paper,the existence and uniqueness of the periodic solution of the model studied in this paper are strictly proved.In chapter 3,we finally give the summary of the work in this paper.In this paper,the main highlights:First,from the structure of the equation,compared with the quantum Navier-Stokes model,the QMHD model has the influence of magnetic.And compared with the classical MHD model,the model studied in this paper has the quantum effect items.Second,in the method of the proof,we choose the regularization method to explore the existence of the periodic solution of the regularization problem through energy estimates and the topological degree theory,as the quantum effect term is a non-linear third-order differential operator.Through a limiting process,we establish the existence of periodic solutions of the QMHD model.Finally,we proved the uniqueness of the solution.