THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS∗

THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗

THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗

THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗

THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗

摘要/Abstract

参考文献

Metrics

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

doi:10.1007/s10473-023-0416-9

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1717-1734.doi: 10.1007/s10473-023-0416-9

Fen HE¹, Zhen WANG^1,†, Tingting CHEN²

1. Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China;
2. School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

收稿日期:2022-01-05 修回日期:2022-06-07 发布日期:2023-08-08
通讯作者: †Zhen WANG, E-mail: zwang@whut.edu.cn
作者简介:Fen HE, E-mail: fenhe.zky@foxmail.com;Tingting CHEN, E-mail: chenting0617@163.com
基金资助:
*National Natural Science Foundation of China (11771442).

Fen HE¹, Zhen WANG^1,†, Tingting CHEN²

1. Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China;
2. School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China

Received:2022-01-05 Revised:2022-06-07 Published:2023-08-08
Contact: †Zhen WANG, E-mail: zwang@whut.edu.cn
About author:Fen HE, E-mail: fenhe.zky@foxmail.com;Tingting CHEN, E-mail: chenting0617@163.com
Supported by:
*National Natural Science Foundation of China (11771442).

摘要： In this paper, we study the shock waves for a mixed-type system from chemotaxis. We are concerned with the jump conditions for the left state which is located in the elliptical region and the right state in the hyperbolic region. Under the generalized entropy conditions, we find that there are different shock wave structures for different parameters. To guarantee the uniqueness of the solutions, we obtain the admissible shock waves which satisfy the generalized entropy condition in both parameters. Finally, we construct the Riemann solutions in some solvable regions.

关键词: mixed-type, shock waves, entropy condition, chemotaxis, conservation laws

Abstract: In this paper, we study the shock waves for a mixed-type system from chemotaxis. We are concerned with the jump conditions for the left state which is located in the elliptical region and the right state in the hyperbolic region. Under the generalized entropy conditions, we find that there are different shock wave structures for different parameters. To guarantee the uniqueness of the solutions, we obtain the admissible shock waves which satisfy the generalized entropy condition in both parameters. Finally, we construct the Riemann solutions in some solvable regions.

Key words: mixed-type, shock waves, entropy condition, chemotaxis, conservation laws

Fen HE, Zhen WANG, Tingting CHEN. THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1717-1734.

Fen HE, Zhen WANG, Tingting CHEN. THE SHOCK WAVES FOR A MIXED-TYPE SYSTEM FROM CHEMOTAXIS^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1717-1734.

[1] Carrillo J A, Li J, Wang Z A. Boundary spike-layer solutions of the singular Keller-Segel system: Existence and stability. Proc Lond Math Soc, 2021, 122: 42-68
[2] Chen G Q, Liu H. Formation of delta shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM Journal on Mathematical Analysis, 2003, 34: 925-938
[3] Delacruz R. Riemann problem for a

$2 \times 2$ hyperbolic system with linear damping. Acta Appl Math, 2020, 170: 631-647
[4] Fan J, Zhao K. Blow up criterion for a hyperbolic-parabolic system arising from chemotaxis. J Math Anal Appl, 2012, 394: 687-695
[5] Fontelos M A, Friedman A, Hu B. Mathematical analysis of a model for the initiation of angiogenesis. SIAM J Math Anal, 2002, 33: 1330-1355
[6] Goatin P, LeFloch P G. The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann Inst Henri Poincare Anal Non Lineaire, 2004, 21: 881-902
[7] Guo J, Xiao J X, Zhao H J, et al. Global solutions to a hyperbolic-parabolic coupled system with large initial data. Acta Mathematica Scientia, 2009, 29B: 629-641
[8] Horstmann D. From1970 until present: the Keller-Segel model in chemotaxis and its consequences. Jahresbericht der Deutschen Mathematiker-Vereinigung, 2003, 105: 103-165
[9] Hou Q Q, Wang Z A, Zhao K. Boundary layer problem on a hyperbolic system arising from chemotaxis. J Differential Equations, 2016, 261: 5035-5070
[10] Hsiao L, De Mottoni P. Existence and uniqueness of the Riemann problem for a nonlinear system of conservation laws of mixed type. Transactions of the American Mathematical Society, 1990, 332: 121-158
[11] Jin H Y, Li J, Wang Z A. Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity. J Differential Equations, 2013, 255: 193-219
[12] Keyfitz B L. Change of type in three-phase flow: A simple analogue. J Differential Equations, 1989, 80: 280-305
[13] Keyfitz B L. Admissibility conditions for shocks in conservation laws that change type. SIAM Journal on Mathematical Analysis, 1991, 22: 1284-1292
[14] Keyfitz B L, Kranzer H C. The Riemann problem for a class of hyperbolic conservation laws exhibiting a parabolic degeneracy. J Differential Equations, 1983, 47: 35-65
[15] Keller E F, Segel L A. A model for chemotaxis. J Theoretical Biology, 1971, 30: 225-234
[16] Keller E F, Segel L A. Traveling bands of chemotactic bacteria: a theoretical analysis. J Theoretical biology, 1971, 30: 235-248
[17] Keller E F, Segel L A. Initiation of slime mold aggregation viewed as an instability. J Theoretical Biology, 1970, 26: 399-415
[18] Lax P D. Hyperbolic systems of conservation laws. II. Comm Pure Appl Math, 1957, 10: 537-566
[19] Levine H A, Sleeman B D. A system of reaction diffusion equations arising in the theory of reinforced random walks. SIAM J Appl Math, 1997, 57: 683-730
[20] Li H C, Zhao K. Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis. J Differential Equations, 2015, 258: 302-338
[21] Li T, Pan R, Zhao K. Global dynamics of a hyperbolic-parabolic model arising from chemotaxis. SIAM J Appl Math, 2012, 72: 417-443
[22] Li J, Wang L, Zhang K. Asymptotic stability of a composite wave of two traveling waves to a hyperbolic-parabolic system modeling chemotaxis. Math Methods Appl Sci, 2013, 36: 1862-1877
[23] Li T, Wang Z A.Nonlinear stability of large amplitude viscous shock waves of a generalized hyperbolic-parabolic system arising in chemotaxis. Math Models Methods Appl Sci, 2010, 20: 1967-1998
[24] Li T, Wang Z A. Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis. J Differential Equations, 2011, 250: 1310-1333
[25] Li T, Wang Z A. Nonlinear stability of travelling waves to a hyperbolic-parabolic system modeling chemotaxis. SIAM J Appl Math, 2009, 70: 1522-1541
[26] Li T, Liu H, Wang L. Oscillatory traveling wave solutions to an attractive chemotaxis system. J Differential Equation, 2016, 261: 7080-7098
[27] Li T, Mathur N. Rienmann problem for a non-strictly hyperbolic system in chemotaxis. Discrete and Continuous Dynamical System Series B, 2022, 27(4): 2173-2187
[28] Li J Y, Li T, Wang Z A. Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity. Math Models Methods Appl Sci, 2014, 24: 2819-2849
[29] Mailybaev A A, Marchesin D. Lax shocks in mixed-type systems of conservation laws. Journal of Hyperbolic Differential Equations, 2008, 5: 295-315
[30] Marchesin D, Paes-Leme P J. A Riemann problem in gas dynamics with bifurcation//Witten M. Hyperbolic Partial Differential Equations. New York: Pergamon, 1986: 433-455
[31] Martinez V R, Wang Z, Zhao K. Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology. Indiana Univ Math J, 2018, 67: 1383-1424
[32] Othmer H G, Stevens A. Aggregation, blowup,collapse: the ABC's of taxis in reinforced random walks. SIAM J Appl Math, 1997, 57: 1044-1081
[33] Peng H, Wang Z A, Zhao K, et al. Boundary layers and stabilization of the singular Keller-Segel system. Kinet Relat Models, 2018, 11: 1085-1123
[34] Peng H, Wen H, Zhu C. Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis. Z Angew Math Phys, 2014, 65: 1167-1188
[35] Rascle M. The Riemann problem for a nonlinear non-strictly hyperbolic system arising in biology. Computers and Mathematics with Applications, 1985, 11: 223-238
[36] Smoller J.Shock Waves and Reaction-Diffusion Equations. New York: Springer-Verlag, 1994
[37] Wang Z A, Hillen T. Shock formation in a chemotaxis model. Mathematical Methods in the Applied Sciences, 2010, 31: 45-70
[38] Wang Z A, Xiang Z, Yu P. Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis. J Differential Equations, 2016, 260: 2225-2258

Viewed

Full text

	From	local

	Times	4
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	47

	From	Others

	Times	48
	Rate	100%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

Just accepted

Online first

Just accepted

Online first

[1]	Fei WU, Zejun WANG, Fangqi CHEN. THE GLOBAL EXISTENCE OF BV SOLUTIONS OF THE ISENTROPIC p-SYSTEM WITH LARGE INITIAL DATA^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1668-1674.
[2]	Minghua Yang, Jinyi Sun, Zunwei Fu, Zheng Wang. THE SINGULAR CONVERGENCE OF A CHEMOTAXIS-FLUID SYSTEM MODELING CORAL FERTILIZATION^*[J]. 数学物理学报(英文版), 2023, 43(2): 492-504.
[3]	Haiyang Jin, Kaiying Xu. BOUNDEDNESS OF A CHEMOTAXIS-CONVECTION MODEL DESCRIBING TUMOR-INDUCED ANGIOGENESIS^*[J]. 数学物理学报(英文版), 2023, 43(1): 156-168.
[4]	邱蜀燕, 穆春来, 易红. BOUNDEDNESS AND ASYMPTOTIC STABILITY IN A PREDATOR-PREY CHEMOTAXIS SYSTEM WITH INDIRECT PURSUIT-EVASION DYNAMICS[J]. 数学物理学报(英文版), 2022, 42(3): 1035-1057.
[5]	樊继山, 栗付才, Gen NAKAMURA. THE LOCAL WELL-POSEDNESS OF A CHEMOTAXIS-SHALLOW WATER SYSTEM WITH VACUUM[J]. 数学物理学报(英文版), 2021, 41(1): 231-240.
[6]	唐清泉, 辛巧, 穆春来. BOUNDEDNESS OF THE HIGHER-DIMENSIONAL QUASILINEAR CHEMOTAXIS SYSTEM WITH GENERALIZED LOGISTIC SOURCE[J]. 数学物理学报(英文版), 2020, 40(3): 713-722.
[7]	刘见礼, 薛杰. GLOBAL SIMPLE WAVE SOLUTIONS TO A KIND OF TWO DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS[J]. 数学物理学报(英文版), 2020, 40(1): 261-271.
[8]	张婷, 盛万成. GLOBAL SOLUTIONS OF THE PERTURBED RIEMANN PROBLEM FOR THE CHROMATOGRAPHY EQUATIONS[J]. 数学物理学报(英文版), 2019, 39(1): 57-82.
[9]	刘树君, 陈芳启, 王泽军. EXISTENCE OF GLOBAL L^∞ SOLUTIONS TO A GENERALIZED n×n HYPERBOLIC SYSTEM OF LEROUX TYPE[J]. 数学物理学报(英文版), 2018, 38(3): 889-897.
[10]	唐少君, 张澜. NONLINEAR STABILITY OF VISCOUS SHOCK WAVES FOR ONE-DIMENSIONAL NONISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH A CLASS OF LARGE INITIAL PERTURBATION[J]. 数学物理学报(英文版), 2018, 38(3): 973-1000.
[11]	徐田媛, 金春花, 季善明. DISCONTINUOUS TRAVELING WAVE ENTROPY SOLUTIONS OF A MODIFIED ALLEN-CAHN MODEL[J]. 数学物理学报(英文版), 2017, 37(6): 1740-1760.
[12]	陈化, 吕文斌, 吴少华. SOLVABILITY OF A PARABOLIC-HYPERBOLIC TYPE CHEMOTAXIS SYSTEM IN 1-DIMENSIONAL DOMAIN[J]. 数学物理学报(英文版), 2016, 36(5): 1285-1304.
[13]	何躏, 唐少君, 王涛. STABILITY OF VISCOUS SHOCK WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY[J]. 数学物理学报(英文版), 2016, 36(1): 34-48.
[14]	Corrado MASCIA. TWENTY-EIGHT YEARS WITH “HYPERBOLIC CONSERVATION LAWS WITH RELAXATION”[J]. 数学物理学报(英文版), 2015, 35(4): 807-831.
[15]	Debora AMADORI, Paolo BAITI, Andrea CORLI, Edda DAL SANTO. GLOBAL EXISTENCE OF SOLUTIONS FOR A MULTI-PHASE FLOW: A BUBBLE IN A LIQUID TUBE AND RELATED CASES[J]. 数学物理学报(英文版), 2015, 35(4): 832-854.