A Study On Some Free Boundary Value Problems In Fluid Mechanics And Materials Science

The present Ph.D. dissertation is concerned with two free boundary problems in fluid dynamics and material sciences. One is the free boundary problem of the expansion of gas into a vacuum, the other is the free boundary problem arising in the peeling phenomenon. Free boundary problems often appear in our daily life. Many problems in gas dynamics also involve free boundary problems.Firstly we investigate the problem of the expansion of gas from a wedge into a vacuum. Let θ be the half angle of the wedge. For a given9determined by the adiabatic exponent γ, we prove the global existence of the solution through the direct approach in the case0<θ<θ. Combining the previous result in [34], the global pseudo-steady solution is obtained by means of the direct approach for0<θ<π/2. Our analysis relies on the special symmetric structure of the characteristic form as well as characteristic decompositions.Next we consider a special Goursat problem, which comes from the inves-tigation of the stability of the problem of the expansion of gas into vacuum. It corresponds the interaction problem of two multi-dimensional rarefaction waves. We obtain the existence of the local solution to the problem.Finally we study a free boundary problem arising in peeling phenomenon, corresponding to the case of without initial interval. Under some compatibility conditions and the assumption of positive initial velocity, we prove the existence and uniqueness of the local classical solution. We obtain the global classical solution under two special kinds of assumptions.The whole contents are organized as follows.Chapter one is an introduction. We present the physical and mathematical background, research status, the main results of this thesis as well as the idea of proof and method.Chapter two is devoted to the problem of the expansion of gas from a wedge with small angle into a vacuum. We obtain the uniform a priori estimates of variables through direct approach by means of the special symmetric structure contained in the characteristic forms of the equations, therefore prove the existence of the global solution in the case0<θ≤θ. In chapter three, a special Goursat problem is considered, which describes the intersection of two multi-dimensional rarefaction waves. We introduce a transformation to get a new problem. For the new problem, we formulate an approximate solution of high order as the initial term of the iteration series, use the dyadic decomposition to deal with the degeneracy of the equation at t=0, and prove the convergence of the iteration series, therefore obtain the existence of the local solution. Then we pull back the solution to the former space to obtain the existence of the local solution to the problem we consider.In chapter four, we study a free boundary problem in peeling phenomenon. We introduce the physical model of peeling phenomenon and the derivation of the problem. We prove the existence and uniqueness of the local solution under some natural conditions and that of the global solution under two kinds of assumptions.