具有常外力项的零压欧拉方程组初值涉及狄拉克函数的黎曼问题
|
摘要
研究具有常外力项的一维零压欧拉方程组初值涉及狄拉克函数的黎曼问题.首先,研究对应的扰动初值问题;其次,基于一维非线性守恒律方程组弱解的稳定性理论,通过分析对应的扰动初值问题解的极限,最终得到了6类显示解.特别地,对于某些初值,捕捉到了在密度变量和内能变量上同时涉及狄拉克函数的狄拉克接触间断,这是一类重要的非线性现象.此外,这些结果清晰地刻画了常外力项对解结构的影响.
The Riemann problem with delta initial data for pressureless compressible Euler equations with a constant external force is considered. We first study the corresponding perturbed initial value problem. Then, under the stability theory of weak solutions to onedimensional nonlinear systems of conservation laws, six kinds of exact solutions are obtained by analyzing the limits of solutions to the corresponding perturbed initial value problem. In particular, for certain initial data, a delta contact discontinuity with Dirac delta function in both density and internal energy arises in the solutions, which is an important nonlinear phenomenon. Besides, the results show clearly the influence of the constant external force on the structures of solutions.
The Riemann problem with delta initial data for pressureless compressible Euler equations with a constant external force is considered. We first study the corresponding perturbed initial value problem. Then, under the stability theory of weak solutions to onedimensional nonlinear systems of conservation laws, six kinds of exact solutions are obtained by analyzing the limits of solutions to the corresponding perturbed initial value problem. In particular, for certain initial data, a delta contact discontinuity with Dirac delta function in both density and internal energy arises in the solutions, which is an important nonlinear phenomenon. Besides, the results show clearly the influence of the constant external force on the structures of solutions.
引文
[1] Lions P L. Mathematical topics in fluid mechanics:volume 1:incompressible models[M]. New York:Oxford University Press, 1996.
[2] Li J, Zhang T, Yang S L. Thes[M]. London:Longman, 1998.
[3] Kraiko A N. Discontinuity surfaces in medium without self-pressure[J]. Prikl Math Mech, 1979, 43:539-549.
[4] Kraiko A N, Sulaimanova S M. Two-phase flows of a gas-particle mixture near impermeable surfaces with the formation of sheets and filaments[J]. Prikl Math Mekh, 1983, 47:619-630.
[5] Nilsson B, Shelkovich V M. Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks[J]. Appl Anal, 2011, 90:1677-1689.
[6] Cheng H. Riemann problem for one-dimensional system of conservation laws of mass, momentum and energy in zero-pressure gas dynamics[J]. Differential Equations and Applications, 2012, 4:653-664.
[7]朱荔,盛万成.绝热流Chaplygin气体的黎曼问题[J].应用数学与计数数学学报,2010, 24:9-16.
[8] Pang Y. Delta shock wave with Dirac delta function in multiple components for the system of generalized Chaplygin gas dynamics[J]. Boundary Value Problems, 2016, 2016:1-20.
[9] Pang Y. Delta shock wave in the compressible Euler equations for a Chaplygin gas[J]. J Math Anal Appl, 2017, 448:245-261.
[10]庞一成,丁彦林,文永松.一维Chaplygin气体欧拉方程组中波的相互作用[J].数学的实践与认识,2017,47:261-271.
[11] Pang Y. Delta shock wave in a perfect fluid model with zero pressure[J]. Under review, 2018.
[12] E W, Sinai Y G, Rykov Y G. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in ashesion particle dynamics[J].Comm Math Phys, 1996, 177:349-380.
[13] Yang H, Sun W. The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws[J]. Nonlinear Analysis, 2007, 67:3041-3049.
[14] Wang Z, Zhang Q. The riemann problem with delta initial data for the one-dimensional chaplygin gas equations[J]. Acta Mathematica Scientia, 2012, 32:825-841.
[15]李建业,邵志强.零压相对论欧拉方程组含有狄拉克函数初値的黎曼问题[J].数学物理学报,2014, 34A:1083-1092.
[16] Wang L. The Riemann problem with Delta data for zero-pressure gas dynamics[J]. Chin Ann Math,2016, 37B:441-450.
[17] Shen C. The Riemann problem for the pressureless Euler system with the Coulomb-like friction term[J]. IMA Journal of Applied Mathematics, 2016, 81:76-99.
[18] Nedeljkov M, Oberguggenberger M. Interactions of delta shock waves in a strictly hyperbolic system of conservation laws[J]. J Math Anal Appl, 2008, 344:1143-1157.
[2] Li J, Zhang T, Yang S L. Thes[M]. London:Longman, 1998.
[3] Kraiko A N. Discontinuity surfaces in medium without self-pressure[J]. Prikl Math Mech, 1979, 43:539-549.
[4] Kraiko A N, Sulaimanova S M. Two-phase flows of a gas-particle mixture near impermeable surfaces with the formation of sheets and filaments[J]. Prikl Math Mekh, 1983, 47:619-630.
[5] Nilsson B, Shelkovich V M. Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks[J]. Appl Anal, 2011, 90:1677-1689.
[6] Cheng H. Riemann problem for one-dimensional system of conservation laws of mass, momentum and energy in zero-pressure gas dynamics[J]. Differential Equations and Applications, 2012, 4:653-664.
[7]朱荔,盛万成.绝热流Chaplygin气体的黎曼问题[J].应用数学与计数数学学报,2010, 24:9-16.
[8] Pang Y. Delta shock wave with Dirac delta function in multiple components for the system of generalized Chaplygin gas dynamics[J]. Boundary Value Problems, 2016, 2016:1-20.
[9] Pang Y. Delta shock wave in the compressible Euler equations for a Chaplygin gas[J]. J Math Anal Appl, 2017, 448:245-261.
[10]庞一成,丁彦林,文永松.一维Chaplygin气体欧拉方程组中波的相互作用[J].数学的实践与认识,2017,47:261-271.
[11] Pang Y. Delta shock wave in a perfect fluid model with zero pressure[J]. Under review, 2018.
[12] E W, Sinai Y G, Rykov Y G. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in ashesion particle dynamics[J].Comm Math Phys, 1996, 177:349-380.
[13] Yang H, Sun W. The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws[J]. Nonlinear Analysis, 2007, 67:3041-3049.
[14] Wang Z, Zhang Q. The riemann problem with delta initial data for the one-dimensional chaplygin gas equations[J]. Acta Mathematica Scientia, 2012, 32:825-841.
[15]李建业,邵志强.零压相对论欧拉方程组含有狄拉克函数初値的黎曼问题[J].数学物理学报,2014, 34A:1083-1092.
[16] Wang L. The Riemann problem with Delta data for zero-pressure gas dynamics[J]. Chin Ann Math,2016, 37B:441-450.
[17] Shen C. The Riemann problem for the pressureless Euler system with the Coulomb-like friction term[J]. IMA Journal of Applied Mathematics, 2016, 81:76-99.
[18] Nedeljkov M, Oberguggenberger M. Interactions of delta shock waves in a strictly hyperbolic system of conservation laws[J]. J Math Anal Appl, 2008, 344:1143-1157.