Global Well-posedness Of The Full Compressible Hall-MHD Equations

In physics,full compressible Hall-magnetohydrodynamic equations is an important model to describe the phenomenon of magnetic reconnection.The study of the well-posedness of the magnetohydrodynamic equations with Hall effect will help us to explain many astrophysics phenomenon.The work of this paper is to study the existence and uniqueness and the large time behavior of the global solution for full compressible Hall-magnetohydrodynamic equations in H3 space with the initial energy satisfying some smallness.Since when the Hall coefficient is equal to 0,full compressible Hall-magnetohydrodynamic equations degenerates into a full compressible magnetohydrodynamic equations,the full compressible magnetohydrodynamic equations with Hall effect can be regarded as a generalized form of the full compressible magnetohydrodynamic equations,the results in this part also applies to the full compressible magnetohydrodynamic equations.There are three main innovations in this article.First of all,from the perspective of the structure of the equation,the non-isoentropy equation is studied,which has more temperature terms than the isentropic equation,and processing this term will bring more mathematical difficulties.Secondly,in terms of the obtained results,in the study of the existence and uniqueness of the global solution,the required initial conditions are weaker than those required by the previous results.Finally,in terms of the method used,given the weak initial conditions and the Hall-effect term with a first-order derivative of the density term,it makes it more difficult to make low-order prior estimates of the global solution to the equations.To overcome this difficulty,the treatment of the density term will be different from the previous results.The structure of the article is as follows:The first chapter is the introduction,which mainly introduces the physical background,research significance and the mathematical description of the full compressible Hall-magnetohydrodynamic equations,and lists previous research results and related literature reviews.Finally,the research results of this paper are stated.The second chapter is the preliminary knowledge,which mainly gives the definitions of mathematical symbols in this paper,some commonly used inequalities and related lemmas needed later.The third chapter gives the proof of the existence and uniqueness of the global solution and the large time behavior.Under the conditions that the initial value belongs to the H3 space,the initial energy is suitably small,and the smallness depends on the H2 norm of the initial value of the density term,the velocity term and the magnetic field term,and the H1 norm of the temperature term.We give a lower-order a priori estimate of the time-independent solution and a higher-order estimate of the time-dependent solution by the energy method,then using local existence and uniqueness of solution and the continuity method to obtain the global existence and uniqueness of the H3 solution for the full compressible Hall-magnetohydrodynamic equations.Furthermore,we study the large time behavior of the global solution,and obtain that when the time tends to infinity,the solution obtained in the third chapter for the full compressible Hall-magnetohydrodynamic equations converges to the steady state solution.