Some Piston Problems For The Relativistic Euler Equations

The paper is focused mainly on some piston problems for the Relativistic Eulerequations. Relativistic fluid dynamics is vastly applied in many diferent areas suchas plasma physics, astrophysics and nuclear physics. The piston problem is a specialphysical prototype. The main results in this Ph.Dissertation are given as follows:One-dimensional piston problem: When the piston is pushed forward relativelyto the gas in the tube, a shock is generated. Suppose that the total variations of theinitial data and the piston speed are small, we establish the global existence of shockfront solutions to the piston problem for the isentropic relativistic Euler equationsby a modified Glimm scheme, and prove that the non-relativistic limits of theentropy solutions is that of the corresponding problem for the classical isentropicEuler equations.Compared with known results, we consider not only a small perturbation ofthe piston speed, but also a small perturbation of a constant initial state for thegas in the tube. On this occasion, the position of the strong shock varies instead ofbeing fixed on the rightmost so that the interactions between the strong shocks andweak waves from the left and the right are more complicated. Thus we redefineapproaching waves, then make some local and uniform estimates (independentof large c1) between the waves. Except for considering the ingredient of theperturbation for the boundary, we introduce weighted strengths for the weak wavesto construct a new Glimm functional. In the aid of these estimates, we prove themonotone decreasing of the Glimm functional to obtain the convergence of theapproximate solutions. Therefore, we obtain the global existence of the entropysolutions. Since the approximate solutions are uniformly bounded, independent oflarge c, we establish the non-relativistic limits of the entropy solutions.Multidimensional piston problem: We consider such a physical phenomenonwhen the piston expands with the velocity depending only on t, the air outside is compressed, and a shock is generated. In this dissertation, we study the sphericallysymmetric piston problem for the relativistic Euler equations, consisting of conser-vation laws of momentum and energy. We not only establish the local existence ofthe strong shock solutions, but also prove that the non-relativistic limits are thatof the corresponding piston problem for the classical isentropic Euler equations influid dynamics.We present the existence and uniqueness of solutions (background solution)when the piston moves with constant speed. We construct a polynomial approxi-mate solution as the first term of iteration. We obtain a sequence of approximatesolutions by employing the Newton iteration, and prove the convergence of theapproximate solutions.It is worth stressing that, since the errors derived in the iteration process areindependent of c1, we obtain the non-relativistic limits of shock front solutionsby energy estimates.