The Well-posedness Of The Steady Incompressible Jet Flows Issuing From A Finitely Long Nozzle

This paper mainly involves with the well-posedness of irrotational,non-viscous,incompressible jet flows through a finitely long nozzle,that is,we study the existence of incompressible jet flows,and the existence and uniqueness of the symmetric jet flows.The main conclusion is that:Given two-dimensional finitely long nozzle and the horizontal velocity in the inlet of the nozzle,then we can show that there exists incompressible jet flows with one uniformly asymptotic direction,and the flow contains two free boundaries that initiate smoothly from the endpoints of the finite nozzle.Furthermore,we establish the corner regularity in the inlet of the nozzle,the asymptotic behavior of the jet in downstream,the positivity of the horizontal velocity of the flow,and the properties of the free boundaries and so on.In addition,the existence and uniqueness of incompressible symmetric jet flow issuing from a finitely long nozzle has also been studied,namely,we prove that there exists a unique symmetric jet flow if given two-dimensional finitely long nozzle and the horizontal velocity in the inlet of the nozzle,and then the free boundaries initiate smoothly from the endpoints of the finite nozzle.In chapter 1,we introduce the incompressible,irrotational,non-viscous Euler equations,the theory of the free boundary,and the background of the jet flows,meanwhile,we illustrate the main results of this paper.In chapter 2,based on the variational approach which has been developed by H.Alt,L.Caffarelli and A.Friedman,firstly,we convert the problem of two-dimensional irrotational,non-viscous,incompressible jet flows into that of the Laplace equation with Bernoulli-type free boundary.Furthermore,we adopt the variational method with two parameters,and prove the existence and uniqueness of the minimizer for the variational problem.Moreover,in view of the regularity theory of the elliptic equation and some properties of the free boundary,we obtain the existence and regularity of the involved free boundary problem,and then the main results are achieved.In chapter 3,we study the existence and uniqueness of the incompressible,irrotational,non-viscous symmetric jet flows,here we sketch of the variational approach with one parameter,and the regularity of the corner points in the inlet of the nozzle has been obtained with some compatibility conditions,and thus the well-posedness of the symmetric jet flows has been solved.In chapter 4,we state some prospects for the future research,and list some crucial and being studied problem.