Study On The Steady Euler Flows Through The Infinite Multidimensional Largely-open Nozzles

The steady Euler equation is the basic equation in aerodynamics,which mainly describes the motion of the fluid in the steady state.The thesis considers the infinite multi-dimensional largely-open nozzle,which includes the fully compression part and the fully rarefaction part.The compression-rarefaction is the essential difference between the compressible flows and the incompressible flows.For the infinite multidimensional largely-open nozzles problem,we study the well-posedness of the solution of the steady irrotational incompressible Euler equation,the well-posedness of the subsonic solution of the steady irrotational compressible Euler equation with the external force effects and the existence of the subsonic-sonic limit solution,and the low Mach number limits.Firstly,we consider the well-posedness of the solution of the steady incompressible Euler equation.The steady irrotational incompressible Euler equation can be transformed into an elliptic equation.Since there are two asymptotic behavers at the positive and negative infinity of the nozzle,the equivalent variational problem is obtained by constructing the corresponding special symmetric solutions at both directions,and the equation is transformed into the corresponding variational problem.The existence of the unique critical point is shown by the variational method.The regularity of the solution is proved by the standard elliptic technique.Finally,the uniqueness of the solution is proved by constructing the appropriate truncation function.Therefore,we prove the well-posedness of the solution of the steady incompressible Euler equation.Secondly,we study the well-posedness of the solution of the steady compressible Euler equation with the external force effects.First of all,due to the external forces,the appropriate truncation function is introduced based on the delicated phase space analysis,and the modified equation is a uniform elliptic equation.Then,we construct the variational problem by taking the solution of the steady incompressible Euler equation as the background profile,and prove the existence of the critical point.The effect of external force on the infinite asymptotic behavers is studied by using the scalling method.Finally,based on the regularity and uniqueness of the solution,the Bers technique shows that the solution of the modified equation is the subsonic solution of the steady compressible Euler equation when the flux is less than the critical one.Taking the subsonic solution as the approximate solution,the existence of the subsonicsonic limit solution is proved by the subsonic-sonic compactness framework.In conclusion,the well-posedness of the steady compressible Euler equation and the subsonic-sonic limit solution with the external force effects are eatablished.Finally,we consider the low Mach number limits.By introducing the energy functional related to the compressible-incompressible difference function,a series of uniform estimates of compressibility parameter are established.Thus,when the compressibility parameter tend to zero,the solutions of the steady compressible Euler equations converge smoothly and uniquely to the solutions of the steady incompressible Euler equations.And,the convergence rates of flow field including pressure is faster than that of unsteady flows.In this thesis,we extend the study of the well-posedness and low Mach number limit of the steady Euler equations of the infinite multi-dimensional largely-open nozzle problem,and introduce innovative research methods and techniques to further explore,which show the fine properties of the solutions of the steady Euler equations.The reseach results not only extend the mathematical theory of fluid dynamics,but also provide theoretical support for the further study and understanding the practical problems on the steady Euler flows.