相对论欧拉方程组一维活塞问题经典间断解的整体存在性

Global existence of classical discontinuous solution to piston problem of relativistic Euler equations

利用一阶拟线性方程组Cauchy问题及自由边值问题的经典解理论,通过引入Riemann不变量将方程组对角化,证明了当活塞的运动速度及气体的初始状态均为常数的小扰动时,相对论欧拉方程组的一维活塞问题的整体经典间断解存在唯一,且其解与未扰动情况下的解只相差小的扰动,激波速度与匀速情况下的激波速度也很接近,同样也不会出现真空.同时,还给出了解的一阶偏导数在t趋于无穷大时的衰减估计.

By the theory of classical solutions to the 1st order quasilinear hyperbolic system and the Riemann invariants diagnolizing the system, the paper proves the following theorem： when both the speed of piston and the initial state of air are small perturbations of constants, i） the piston problem of relativistic Euler equations admits a unique global classical discontinuous solution, ii） the difference between the perturbed and the unperturbed solutions is also small, iii） the shock speed of the solution is also close to that of the unperturbed problem, and iv） the vacuum state does not occur. Besides, the paper also gives a decay estimate of the first derivatives of the solution when t tends to infinity.