压差方程一维活塞问题激波解的整体存在性

Global existence of shock front solution to piston problem of pressure-gradient system

双曲守恒律方程组的活塞问题可被视为一阶拟线性双曲组的一种特殊的混合初边值问题,运用一阶拟线性双曲组经典间断解的结果,通过拼接子问题的经典解,以构造的方式证明了当活塞的运动速度及气体的初始状态均为常数的扰动时,相应的压差方程组一维活塞问题只包含一个激波的整体经典间断解存在唯一,而且证明了其解与未扰动情况下的解之间也只相差小的扰动,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.不仅如此,还给出了解的一阶偏导数在t趋于无穷时的衰减估计.

Piston problem of conservation laws can be regarded as a special kind of mixed initial-boundary value problem of first-order quasilinear hyperbolic sys- tems. Using the rusults of classical discontinuous solutions of first-order quasilinear hyperbolic systems, by piecing together classical solutions of subproblems, the pa- per proves by construction that when both the speed of the piston and the initial state of air are perturbations of constants, the corresponding piston problem of pressure-gradient system admits a unique global classical discontinuous solution which contains only one shock. Furthermore, the perturbation between the solu- tion and the unperturbed solution is also small~ the shock speed is also close to the shock speed of the unperturbed problem, and the vacuum state will still not appear. Besides, the paper also gives a decay estimate of the first derivative of the solution when t tends to infinity.