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
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摘要
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 1 st 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.
By the theory of classical solutions to the 1 st 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.
引文
[1]李大潜,秦铁虎.物理学与偏微分方程[M].北京:高等教育出版社,1997.
[2]Li Y C,Feng D M,Wang Z J.Global entropy solutions to the relativistic Euler equations for a class of large initial data[J].Z Angew Math Phys,2005,56:239-253.
[3]Xu Y L,Dou Y P.Global existence of shock front solutions in 1-dimensional piston problem in the relativistic Euler equations[J].Z Angew Math Phys,2008,59:244-263.
[4]Li T T,Zhao Y C.Global shock solutions to a class of piston problems for the system of one dimensional isentropic flow[J].Chin Ann Math Ser B,1991,12B(4):495-499.
[5]Li T T,Zhao Y C.Global perturbation of the Riemann problem for the system of one dimensional isentropic flow[M]//Lecture Notes in Mathematics.1306.Berlin/Heidelberg:SpringerVerlag,1988:131-140.
[6]李大潜,赵彦淳.拟线性双曲型方程组典型自由边界问题的整体经典解[J].中国科学:A辑,1990(1):8-20.
[7]Li T T,Yu W C.Boundary Value Problems for Quasilinear Hyperbolic System[M].Durham:Duke University,1985.
[2]Li Y C,Feng D M,Wang Z J.Global entropy solutions to the relativistic Euler equations for a class of large initial data[J].Z Angew Math Phys,2005,56:239-253.
[3]Xu Y L,Dou Y P.Global existence of shock front solutions in 1-dimensional piston problem in the relativistic Euler equations[J].Z Angew Math Phys,2008,59:244-263.
[4]Li T T,Zhao Y C.Global shock solutions to a class of piston problems for the system of one dimensional isentropic flow[J].Chin Ann Math Ser B,1991,12B(4):495-499.
[5]Li T T,Zhao Y C.Global perturbation of the Riemann problem for the system of one dimensional isentropic flow[M]//Lecture Notes in Mathematics.1306.Berlin/Heidelberg:SpringerVerlag,1988:131-140.
[6]李大潜,赵彦淳.拟线性双曲型方程组典型自由边界问题的整体经典解[J].中国科学:A辑,1990(1):8-20.
[7]Li T T,Yu W C.Boundary Value Problems for Quasilinear Hyperbolic System[M].Durham:Duke University,1985.