带群集耗散项的零压流方程的扰动黎曼问题

数学物理学报 ›› 2020, Vol. 40 ›› Issue (1): 49-62.

带群集耗散项的零压流方程的扰动黎曼问题

张庆玲,巴英

^{江汉大学数学与计算机科学学院数学系武汉 430056}

收稿日期:2018-05-15 出版日期:2020-01-26 发布日期:2020-04-08
作者简介:张庆玲
基金资助:
武汉市教育局教学研究重点项目(2017007)

The Perturbed Riemann Problem for the Pressureless Euler Equations with a Flocking Dissipation

Qingling Zhang,Ying Ba

^{Department of Mathematics, School of Mathematics and Computer Sciences, Jianghan University, Wuhan 430056}

Received:2018-05-15 Online:2020-01-26 Published:2020-04-08
Supported by:
武汉市教育局教学研究重点项目(2017007)

摘要/Abstract

摘要：

该文通过分析带群集耗散项的零压流方程初始波含狄拉克激波的波的相互作用，研究了其初值含三片常值和初值含狄拉克测度的两种扰动黎曼问题.当初值为三片常值时，通过广义Rankine-Hugoniot条件和广义熵条件，该文构造性地得到了整体解.进一步地，利用弱解的稳定性理论，通过分析初值为三片常值情形下解的结构并取极限，该文得到了初值含狄拉克测度的扰动黎曼问题的整体解.另外，在构造解的过程中，还引入了一种新的非经典解：狄拉克接触间断解.

关键词: 零压流, 黎曼问题, 狄拉克激波, 真空, 群集耗散, 波的相互作用

Abstract:

In this paper, the wave interactions involving the delta shock waves for the pressureless Euler equations with a flocking dissipation is considered and two kinds of perturbed Riemann problems are studied. The global existence of the generalzied solutions is obtained constructively by using the generalized Rankine-Hugoniot conditions and the entropy condition when the initial data are three piecewise constant states. By analyzing the global structure and the limit under the stability theory of weak solutions, the global soution for the delta inital data problem are constructively obtained. Moreover, a new kind of nonclassical wave -delta contact discontinuity has appeared in it.

Key words: Pressureless Euler equations, Riemann problem, Delta shock wave, Vacuum state, Flocking dissipation, Interaction

中图分类号:

O175.27

张庆玲,巴英. 带群集耗散项的零压流方程的扰动黎曼问题[J]. 数学物理学报, 2020, 40(1): 49-62.

Qingling Zhang,Ying Ba. The Perturbed Riemann Problem for the Pressureless Euler Equations with a Flocking Dissipation[J]. Acta mathematica scientia,Series A, 2020, 40(1): 49-62.

图/表 5

参考文献 27

1	Bouchut F. On Zero-pressure Gas Dynamics//Perthame B, et al. Advances in Kinetic Theory and Computing: Adv Math Appl Sci. Sinapore: World Scientific, 1994: 171-190
2	Brenier Y , Gangbo W , Savare G , Westdickenberg M . The sticky particles dynamicse with interactions. J Math Pures Appl, 2013, 99, 577- 617
3	Danilvo V G , Shelkovich V M . Dynamics of propagation and interaction of δ-shock waves in conservation law system. J Differential Equations, 2005, 221, 333- 381
4	Danilvo V G , Shelkovich V M . Delta-shock waves type solution of hyperbolic systems of conservation laws. Q Appl Math, 2005, 63, 401- 427
5	Ding Y , Huang F M . On a nonhomogeneous system of pressureless flow. Quart Appl Math, 2002, 62 (3): 509- 528
6	E W , Rykov Yu G , Sinai Ya G . Generalized varinational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm Math Phys, 1996, 177, 349- 380
7	Guo L , Li T , Pan L , Han X . The Riemann with delta inital data for the one-dimensional Chaplygin gas equations with a source term. Nonlinear Anal:Real Word Appl, 2018, 41, 588- 606
8	Ha S Y , Huang F , Wang Y . A global unique solvability of entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation. J Differential Equations, 2014, 2257, 1333- 1371
9	Ha S Y , Liu J G . A simple proof of the cucker-smale flocking dynamics and mean-field limit. Commun Math Sci, 2009, 7 (2): 297- 325
10	Huang F , Wang Z . Well posedness for pressureless flow. Comm Math Phys, 2001, 222, 117- 146
11	Jin C Y . Well posedness for pressureless Euler system with a flocking dissipation in Wasserstein space. Nonlinear Anal TMA, 2015, 128, 412- 422
12	Jin C Y . Existence and uniqueness of entropy solution to pressureless Euler system with a flocking dissipation. Acta Mathematica Scientia, 2016, 36B (5): 1262- 1284
13	Kalisch H , Mitrovic D . Singular solutions of a fully nonlinear 2×2 system of conservation laws. Proceedings of the Edinburgh Mathematical Society, 2012, 55, 711- 729
14	Kalisch H , Mitrovic D . Singular solutions for shallow water equations. IMA J Appl Math, 2012, 77, 340- 350
15	Li J, Zhang T, Yang S. The Two-dimensional Riemann Prolem in Gas Dynamics. Pitman Monogr Surv Pure Appl Math. Harlow: Longman Scientific and Technical, 1998
16	Nedeljkov M , Oberguggenberger M . Interactions of delta shock waves in a strictly hyperbolic system of conservation laws. J Math Anal Appl, 2008, 344, 1143- 1157
17	Shen C . The Riemann problem for the pressureless Euler system with the Coulomb-like friction term. IAM J Appl Math, 2016, 81, 76- 99
18	Shen C . The Riemann problem for the Chaplygin gas equations with a source term. Z Angew Math Mech, 1999, 96, 681- 695
19	Shen C , Sun M . Interactions of delta shocks for transport equations with split delta functions. J Math Anal Appl, 2009, 351, 747- 755
20	Sheng W, Zhang T. The Riemann Problem for Transportation Equations in Gas Dynamics. Providence, RI: Mem Amer Math Soc, 1999
21	Sun M . The exact Riemann solutions to the generalized Chaplygin gas equations with friction. Commun Nonlinear Sci Numer Simulat, 2016, 36, 342- 353
22	Wang L . The Riemann problem with delta data for zero-pressure gas dynamcis. Chinese Annals of Mathematics, 2016, 37 (3): 441- 450
23	Wang Z , Ding X . Uniqueness of generalized solution for the Cauchy problem of transportation equations. Acta Math Scientia, 1997, 17 (3): 341- 352
24	Wang Z , Huang F , Ding X . On the Cauchy problem of transportation equations. Acta Math Appl Sinica, 1997, 13 (2): 113- 122
25	Wang Z , Zhang Q . The Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations. Acta Mathematica Scientia, 2012, 32B (3): 825- 841
26	Yang H . Generalized plane delta-shock waves for n-dimensional zero-pressure gas dynamics. J Math Anal Appl, 2001, 260, 18- 35
27	Yang H , Sun W . The Riemann problem with delta initial data for a class of coupled hyperbolic system of conservation laws. Nonlinear Anal, 2007, 67, 3041- 3049

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