Some Free Boundary Problems For Quasilinear Elliptic Equations

In this thesis,We study some free boundary problems for quasilinear elliptic equa-tions.Two kinds of fluid flow models are considered,including subsonic jet flow and subsonic-sonic flow in nozzles.The models we want to study are the ideal fluid con-trolled by the steady compressible Euler equations,and the fluid is inviscid,isenrtopic and irrotational.Such an ideal fluid satisfies the full potential equation,it is elliptic in subsonic region,hyperbolic in supersonic region,and degenerates in sonic state.For the subsonic jet flow,we study the jet flow of finite length divergent straight nozzle under the given external pressure.Here the boundary between the jet and the outside world is a free boundary.In mathematics,it can be described as a free boundary problem of quasilinear elliptic equations.At the boundary of the free boundary and the fixed boundary,the regularity of the solution is weak.For the subsonic-sonic flow in nozzles,we study two subsonic-sonic flows which the walls of nozzles are curved,where the walls of nozzles are the small pertubation of the straight solid walls.Here the sonic curve of the flow is a free boundary.First,we study the case where the inlet of the nozzles is a arc whose center is located at the vertex of the nozzles,and then we study the case where the inlet is disturbed more generally.In mathematics,these two models can be described as a free boundary problem of a quasilinear degenerate elliptic equation with non-local boundary conditions,and the equation degenerates exactly on the free boundary,the solution of the problem on the free boundary is singular.This thesis is mainly divided into two parts.The first part is the subsonic jet flow.This is a free boundary problem for a quasilinear elliptic equation which degenerates at the position of sonic curve.First,by means of some methods and techniques of the elliptic equation theory,we consider the fixed boundary problem of the corresponding regularized equation,and obtain the well-posedness result of the problem by a series of careful estimates.Then,for the free boundary problem,due to the unboundedness of the region and the fact that the problem satisfies the mixed boundary condition on the streamlines,the free boundary problem is solved by the well-posedness conclusion of the regularization equation and some exact estimates.It is proved that there is a critical angle which is not more than?,such that,when the angle of the nozzle is less than the critical angle,there is a unique subsonic jet flow,and there is no subsonic jet flow when the angle of the nozzles is equal to the critical angle.When the angle of the nozzles tends to the critical angle,either the length of the nozzles tend to 0 or the sonic point will appear at the inlet.It is important to note that the critical angle here does not depend on the location of the inlet of the nozzles.For the jet,the part away from the point of intersection with the nozzle is smooth and strictly concave relative to the fluid,almost tends to a horizontal line exponentially which parallel to the axis of symmetry at infinity.The second part is the subsonic-sonic flow in nozzles,including two cases:walls perturbation,inlet and walls perturbations together.The flows enter the nozzle along the normal direction of the inlet of the nozzle,satisfy the slip condition on the walls,reach the velocity of sonic at the outlet of the nozzle,and the flows out of the nozzle along its outer normal direction,but the position and shape of the sonic curve are unknown.So this problem is a quasilinear elliptic equation with free boundary,and the degeneracy occurs on the free boundary.In Chapter 2,we consider the case of walls perturbation.Because of the analysis of the compatibility and the related work,the wall of the nozzle are disturbed relative to the straight solid wall when it is far away from the intersection of the sonic curve and the walls of the nozzle,and is still straight solid wall near the intersection point.The problem studied here has a free boundary condition and a nonlocal boundary condition.For the existence of the subsonic-sonic flows in nozzles,there exists an open region.When the mass flux of the incoming flow is in this region,we use the Schauder fixed point theorem to prove it.First,for the given velocity at the inlet and the wall of the nozzle,the corresponding fixed boundary problem is established.By using the classical theory of the uniform elliptic equation and some estimates,the well-posedness results of the fixed boundary problem are obtained.Then we consider the free boundary problem,select the suitable space,and prove the existence of the solution of the free boundary problem by a series of energy estimates and the fixed-point theorem.For the uniqueness of the solution,the previous work usually transforms the free boundary condition into the fixed boundary condition and the non-local boundary condition into the general boundary condition by the appropriate coordinate transformation,then the uniqueness of the solution is obtained by the method of energy estimates.Due to the walls are not straight and the intersection point of the wall and the sonic curve is not fixed,the suitable coordinate transformation can not be found.Therefore,we prove uniqueness directly on the potential plane by energy estimates.The difficulties lie in the estimates of the velocity potential along the sonic curve and the nonlocal term at the upper wall.In the last chapter,we consider the simultaneous perturbation of the inlet and the wall of the nozzle.The difficulties lie in the fact that the equations satisfied by the fluid are not only degenerate on the sonic curve,but also the sonic curve is free.A new nonlocal boundary condition is generated by the inlet perturbation.By using similar techniques and methods,we prove that there exists a unique subsonic-sonic flow in the nozzle,which is a small perturbation of the corresponding symmetric flow and is smooth in the region without the sonic curve,it's singularity on the sonic curve.Compared with the previous proofs,the perturbation of the inlet changes the fixed point iteration scheme in the existence proof and makes the estimates of the inlet conditions more complicated in the uniqueness proof.