EVOLUTION AND INTERACTION OF $\delta$-WAVES IN THE ZERO-PRESSURE GAS DYNAMICS SYSTEM*

doi:10.1007/s10473-024-0510-7

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (5): 1801-1836.doi: 10.1007/s10473-024-0510-7

EVOLUTION AND INTERACTION OF $\delta$-WAVES IN THE ZERO-PRESSURE GAS DYNAMICS SYSTEM^*

Abhishek Das^1,†, K. T. Joseph²

1. Centre for Data Science, Institute of Technical Education and Research (ITER), Siksha 'O' Anusandhan (Deemed to be University), Bhubaneswar 751030, Odisha, India;
2. Centre for Applicable Mathematics, Tata Institute of Fundamental Research Sharada~Nagar, Bangalore 560065, Karnataka, India

收稿日期:2023-02-08 修回日期:2023-10-19 出版日期:2024-10-25 发布日期:2024-10-22
通讯作者: †Abhishek Das , E-mail,: abhishekdas@soa.ac.in; ad16.1992@gmail.com
作者简介:E-mail,: K. T. Joseph , ktj@tifrbng.res.inE-mail

EVOLUTION AND INTERACTION OF $\delta$-WAVES IN THE ZERO-PRESSURE GAS DYNAMICS SYSTEM^*

Abhishek Das^1,†, K. T. Joseph²

1. Centre for Data Science, Institute of Technical Education and Research (ITER), Siksha 'O' Anusandhan (Deemed to be University), Bhubaneswar 751030, Odisha, India;
2. Centre for Applicable Mathematics, Tata Institute of Fundamental Research Sharada~Nagar, Bangalore 560065, Karnataka, India

Received:2023-02-08 Revised:2023-10-19 Online:2024-10-25 Published:2024-10-22
Contact: †Abhishek Das , E-mail,: abhishekdas@soa.ac.in; ad16.1992@gmail.com
About author:E-mail,: K. T. Joseph , ktj@tifrbng.res.inE-mail

摘要/Abstract

摘要： Evolution and interaction of plane waves of the multidimensional zero-pressure gas dynamics system leads to the study of the corresponding one dimensional system. In this paper, we study the initial value problem for one dimensional zero-pressure gas dynamics system. Here the first equation is the Burgers equation and the second one is the continuity equation. We consider the solution with initial data in the space of bounded Borel measures. First we prove a general existence result in the algebra of generalized functions of Colombeau. Then we study in detail special solutions with $\delta$-measures as initial data. We study interaction of waves originating from initial data concentrated on two point sources and interaction with classical shock/rarefaction waves. This gives an understanding of plane-wave interactions in the multidimensional case. We use the vanishing viscosity method in our analysis as this gives the physical solution.

关键词: zero-pressure gas dynamics, delta-waves, interaction of waves

Abstract: Evolution and interaction of plane waves of the multidimensional zero-pressure gas dynamics system leads to the study of the corresponding one dimensional system. In this paper, we study the initial value problem for one dimensional zero-pressure gas dynamics system. Here the first equation is the Burgers equation and the second one is the continuity equation. We consider the solution with initial data in the space of bounded Borel measures. First we prove a general existence result in the algebra of generalized functions of Colombeau. Then we study in detail special solutions with $\delta$-measures as initial data. We study interaction of waves originating from initial data concentrated on two point sources and interaction with classical shock/rarefaction waves. This gives an understanding of plane-wave interactions in the multidimensional case. We use the vanishing viscosity method in our analysis as this gives the physical solution.

Key words: zero-pressure gas dynamics, delta-waves, interaction of waves

中图分类号:

35F25

Abhishek Das, K. T. Joseph. EVOLUTION AND INTERACTION OF $\delta$-WAVES IN THE ZERO-PRESSURE GAS DYNAMICS SYSTEM^*[J]. 数学物理学报(英文版), 2024, 44(5): 1801-1836.

Abhishek Das, K. T. Joseph. EVOLUTION AND INTERACTION OF $\delta$-WAVES IN THE ZERO-PRESSURE GAS DYNAMICS SYSTEM^*[J]. Acta mathematica scientia,Series B, 2024, 44(5): 1801-1836.

参考文献

[1] Albeverio S, Shelkovich V M.On the delta-shock front problem//Albeverio S, Shelkovich V M. Analytical Approaches to Multidimensional Balance Laws. New York: Nova Sci Publ, 2006: 45-88
[2] Bertsch M, Smarrazzo F, Terracina A, Alberto Tesei. Radon measure-valued solutions of first order scalar conservation laws. Adv Nonlinear Anal, 2020, 9(1): 65-107
[3] Biagioni H A, Oberguggenberger M. Generalized solutions to Burgers' equation. J Differential Equations, 1992, 97(2): 263-287
[4] Choudhury A P, Joseph K T, Sahoo M R. Spherically symmetric solutions of multidimensional zero-pressure gas dynamics system. J Hyperbolic Differ Equ, 2014, 11(2): 269-293
[5] Colombeau J F, Heibig A. Generalized solutions to Cauchy problems. Monatsh Math, 1994, 117: 33-49
[6] Colombeau J F.New Generalized Functions and Multiplication of Distributions. Amsterdam: North-Holland, 1984
[7] Colombeau J F, Heibig A, Oberguggenberger M. The Cauchy problem in a space of generalized functions, I-II. C R Acad Sci Paris Sér I Math, 1993, 317: 851-855; 1994, 319: 1179-1183
[8] Demengel F, Serre D. Nonvanishing singular parts of measure valued solutions for scalar hyperbolic equations. Comm Partial Differential Equations, 1991, 16(2-3): 221-254
[9] Gurbatov S N, Saichev A I. Probability distribution and spectra of potential hydrodynamic turbulence. Radiophysics and Quantum Electronics, 1984, 27(4): 303-313
[10] Gurbatov S N, Saichev A I.New approximation in the adhesion model for describing large-scale structure of the Universe. Cosmic Velocity Fields Proc 9th IAP Astrophysics Meeting, 1993
[11] Gurbatov S N, Saichev A I, S F Shandarin. Large-scale structure of the universe the zeldovich approximation and the adhesion model. Physics-Uspekhi, 2012, 55(3): 223-249
[12] Hopf E. The partial differential equation $u_t+uu_x=\mu u_{xx}$. Comm Pure Appl Math, 1950, 3: 201-230
[13] Joseph K T. Burgers' equation in the quarter plane, a formula for the weak limit. Comm Pure Appl Math, 1988, 41(2): 133-149
[14] Joseph K T. A Riemann problem whose viscosity solutions contain $\delta$-measures. Asymptotic Anal, 1993, 7(2): 105-120
[15] Joseph K T. One-dimensional adhesion model for large scale structures. Electron J Differential Equations, 2010, 2010(70): 1-15
[16] Kružkov S N. First order quasilinear equations with several independent variables. Mat Sb (N S ), 1970, 81(123): 228-255
[17] Lax P D. Hyperbolic systems of conservation laws II. Comm Pure Appl Math, 1957, 10: 537-566
[18] LeFloch P L. An existence and uniqueness result for two nonstrictly hyperbolic systems//Keyfitz B L, Shearer M (eds). Nonlinear Evolution Equations that Change Type. New York: Springer, 1990: 126-138
[19] Liu T P, Pierre M. Source-solutions and asymptotic behavior in conservation laws. J Differential Equations, 1984, 51(3): 419-441
[20] Vol'pert A I. The spaces BV and quasilinear equations. Mathematics of the USSR-Sbornik.1967, 2(2): 225-267
[21] Zeldovich Y B. Gravitational instability: An approximate theory for large density perturbations. Astron Astrophys, 1970, 5: 84-89

EVOLUTION AND INTERACTION OF $\delta$-WAVES IN THE ZERO-PRESSURE GAS DYNAMICS SYSTEM^*

EVOLUTION AND INTERACTION OF $\delta$-WAVES IN THE ZERO-PRESSURE GAS DYNAMICS SYSTEM^*

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 1

Metrics

本文评价

推荐阅读 10