doi: 10.3934/dcdsb.2022034
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Stability of rarefaction wave for viscous vasculogenesis model

School of Mathematics, South China University of Technology, Guangzhou, 510641, China

*Corresponding author: Qingqing Liu

Received  July 2021 Revised  December 2021 Early access March 2022

Fund Project: The first author is supported by the National Natural Science Foundation of China (No. 12071153), Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515012360), Guangzhou Science and Technology Program (No. 202102021137) and the Fundamental Research Funds for the Central Universities (No. 2020ZYGXZR032)

In this paper, we are concerned with the large time behavior of solutions to the one-dimensional Cauchy problem on a hyperbolic-parabolic-elliptic model for vasculogenesis in the case when far field states of initial data are distinct. It turns out that the solutions exist for all time and tend to a weak rarefaction wave whose strength is not necessarily small under small perturbation. All the results are based on the assumption $ 2A-\frac{{\mu}a}{b}>0 $ which guarantees the dissipation of this model.

Citation: Qingqing Liu, Xiaoli Wu. Stability of rarefaction wave for viscous vasculogenesis model. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022034
References:
[1]

D. AmbrosiF. Bussolino and L. Preziosi, A review of vasculogenesis models, J. Theor. Med., 6 (2005), 1-19.  doi: 10.1080/1027366042000327098.

[2]

P.-H. Chavanis, Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology, C. R. Physique, 7 (2006), 318-330.  doi: 10.1016/j.crhy.2006.01.004.

[3]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2016.

[4]

C. M. Dafermos and R. H. Pan, Global $BV$ solutions for the $p$-system with frictional damping, SIAM J. Math. Anal., 41 (2009), 1190-1205.  doi: 10.1137/080735126.

[5]

M. Di Francesco and D. Donatelli, Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 79-100.  doi: 10.3934/dcdsb.2010.13.79.

[6]

C. Di Russo, Analysis and numerical approximations of hydrodynamical models of biological movements, Rend. Mat. Appl., 32 (2012), 117-367. 

[7]

C. Di Russo and A. Sepe, Existence and asymptotic behavior of solutions to a quasi-linear hyperbolic-parabolic model of vasculogenesis, SIAM J. Math. Anal., 45 (2013), 748-776.  doi: 10.1137/110858896.

[8]

R. J. DuanQ. Q. Liu and C. J. Zhu, Darcy's law and diffusion for a two-fluid Euler-Maxwell system with dissipation, Math. Models Methods Appl. Sci., 25 (2015), 2089-2151.  doi: 10.1142/S0218202515500530.

[9]

R. J. Duan and S. Q. Liu, Stability of rarefaction waves of the Navier-Stokes-Poisson system, J. Differential Equations, 258 (2015), 2495-2530.  doi: 10.1016/j.jde.2014.12.019.

[10]

R. J. DuanS. Q. LiuH. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84.  doi: 10.1007/s11425-015-5059-4.

[11]

R. J. Duan and X. F. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations, Commun. Pure Appl. Anal., 12 (2013), 985-1014.  doi: 10.3934/cpaa.2013.12.985.

[12]

F. FilbetP. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.  doi: 10.1007/s00285-004-0286-2.

[13]

A. GambaD. AmbrosiA. ConiglioA de CandiaS. Di TaliaE. GiraudoG. SeriniL. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamic in blood vessels formation, Phys. Rev. Lett., 90 (2003), 118101.  doi: 10.1103/PhysRevLett.90.118101.

[14]

I. GasserL. Hsiao and H. L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192 (2003), 326-359.  doi: 10.1016/S0022-0396(03)00122-0.

[15]

J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational. Mech. Anal., 95 (1986), 325-344.  doi: 10.1007/BF00276840.

[16]

F. M. HuangJ. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.  doi: 10.1007/s00205-009-0267-0.

[17]

F. M. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Indiana Univ. Math. J., 65 (2016), 1833-1875.  doi: 10.1512/iumj.2016.65.5914.

[18]

F. M. HuangZ. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014.

[19]

K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Methods Appl. Sci., 18 (2008), 1001-1025.  doi: 10.1142/S0218202508002930.

[20]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[21]

M. N. Jiang and C. J. Zhu, Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant, J. Differential Equations, 246 (2009), 50-77.  doi: 10.1016/j.jde.2008.03.033.

[22]

M. N. Jiang and C. J. Zhu, Convergence rates to nonlinear diffusion waves for p-system with nonlinear damping on quadrant, Discrete Contin. Dyn. Syst., 23 (2009), 887-918.  doi: 10.3934/dcds.2009.23.887.

[23]

Q. S. JiuY. Wang and Z. P. Xin, Vacuum behaviors around rarefaction waves to 1D compressible Navier-Stokes equations with density-dependent viscosity, SIAM J. Math. Anal., 45 (2013), 3194-3228.  doi: 10.1137/120879919.

[24]

S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math., 58 (2004), 211-250.  doi: 10.2206/kyushujm.58.211.

[25]

S. Kawashima and P. C. Zhu, Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid in the half space, Arch. Ration. Mech. Anal., 194 (2009), 105-132.  doi: 10.1007/s00205-008-0191-8.

[26]

Q. Q. LiuH. Y. Peng and Z.-A. Wang, Asymptotic stability of diffusion waves of a quasi-linear hyperbolic-parabolic model for vasculogenesis, SIAM J. Math. Anal., 54 (2022), 1313-1346.  doi: 10.1137/21M1418150.

[27]

Q. Q. LiuH. Y. Peng and Z.-A. Wang, Convergence to nonlinear diffusion waves for a hyperbolic-parabolic chemotaxis system modelling vasculogenesis, J. Differential Equations, 314 (2022), 251-286.  doi: 10.1016/j.jde.2022.01.021.

[28]

T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.  doi: 10.1007/BF01466726.

[29]

T. P. LiuT. YangS. H. Yu and H. J. Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Ration. Mech. Anal., 181 (2006), 333-371.  doi: 10.1007/s00205-005-0414-1.

[30]

T. P. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.  doi: 10.1007/s00220-003-1030-2.

[31]

T. LuoH. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a coupled compressible Navier-Stokes/Allen-Cahn system, Math. Methods Appl. Sci., 41 (2018), 4724-4736.  doi: 10.1002/mma.4925.

[32]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[33]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088.

[34]

A. Matsumura and K. Nishihara, Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335.  doi: 10.1007/BF02101095.

[35]

A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25.  doi: 10.1007/BF03167036.

[36]

M. Mei, Nonlinear diffusion waves for hyperbolic $p$-system with nonlinear damping, J. Differential Equations, 247 (2009), 1275-1296.  doi: 10.1016/j.jde.2009.04.004.

[37]

M. Mei, Best asymptotic profile for hyperbolic $p$-system with damping, SIAM J. Math. Anal., 42 (2010), 1-23.  doi: 10.1137/090756594.

[38]

K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations, 131 (1996), 171-188.  doi: 10.1006/jdeq.1996.0159.

[39]

K. Nishihara, Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. Differential Equations, 137 (1997), 384-395.  doi: 10.1006/jdeq.1997.3268.

[40]

K. Nishihara and T. Yang, Boundary effect on asymptotic behaviour of solutions to the $p$-system with linear damping, J. Differential Equations, 156 (1999), 439-458.  doi: 10.1006/jdeq.1998.3598.

[41]

R. H. Pan, Darcy's law as long-time limit of adiabatic porous media flow, J. Differential Equations, 220 (2006), 121-146.  doi: 10.1016/j.jde.2004.10.013.

[42]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[43]

H. Y. YinJ. S. Zhang and C. J. Zhu, Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system, Nonlinear Anal. Real World Appl., 31 (2016), 492-512.  doi: 10.1016/j.nonrwa.2016.01.020.

[44]

S. H. Yu, Nonlinear wave propagations over a Boltzmann shock profile, J. Amer. Math. Soc., 23 (2010), 1041-1118.  doi: 10.1090/S0894-0347-2010-00671-6.

[45]

C. J. Zhu and M. N. Jiang, $L^p$-decay rates to nonlinear diffusion waves for $p$-system with nonlinear damping, Sci. China Ser. A, 49 (2006), 721-739.  doi: 10.1007/s11425-006-0721-5.

show all references

References:
[1]

D. AmbrosiF. Bussolino and L. Preziosi, A review of vasculogenesis models, J. Theor. Med., 6 (2005), 1-19.  doi: 10.1080/1027366042000327098.

[2]

P.-H. Chavanis, Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology, C. R. Physique, 7 (2006), 318-330.  doi: 10.1016/j.crhy.2006.01.004.

[3]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer-Verlag, Berlin, 2016.

[4]

C. M. Dafermos and R. H. Pan, Global $BV$ solutions for the $p$-system with frictional damping, SIAM J. Math. Anal., 41 (2009), 1190-1205.  doi: 10.1137/080735126.

[5]

M. Di Francesco and D. Donatelli, Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 79-100.  doi: 10.3934/dcdsb.2010.13.79.

[6]

C. Di Russo, Analysis and numerical approximations of hydrodynamical models of biological movements, Rend. Mat. Appl., 32 (2012), 117-367. 

[7]

C. Di Russo and A. Sepe, Existence and asymptotic behavior of solutions to a quasi-linear hyperbolic-parabolic model of vasculogenesis, SIAM J. Math. Anal., 45 (2013), 748-776.  doi: 10.1137/110858896.

[8]

R. J. DuanQ. Q. Liu and C. J. Zhu, Darcy's law and diffusion for a two-fluid Euler-Maxwell system with dissipation, Math. Models Methods Appl. Sci., 25 (2015), 2089-2151.  doi: 10.1142/S0218202515500530.

[9]

R. J. Duan and S. Q. Liu, Stability of rarefaction waves of the Navier-Stokes-Poisson system, J. Differential Equations, 258 (2015), 2495-2530.  doi: 10.1016/j.jde.2014.12.019.

[10]

R. J. DuanS. Q. LiuH. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a two-fluid plasma model with diffusion, Sci. China Math., 59 (2016), 67-84.  doi: 10.1007/s11425-015-5059-4.

[11]

R. J. Duan and X. F. Yang, Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations, Commun. Pure Appl. Anal., 12 (2013), 985-1014.  doi: 10.3934/cpaa.2013.12.985.

[12]

F. FilbetP. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., 50 (2005), 189-207.  doi: 10.1007/s00285-004-0286-2.

[13]

A. GambaD. AmbrosiA. ConiglioA de CandiaS. Di TaliaE. GiraudoG. SeriniL. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamic in blood vessels formation, Phys. Rev. Lett., 90 (2003), 118101.  doi: 10.1103/PhysRevLett.90.118101.

[14]

I. GasserL. Hsiao and H. L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192 (2003), 326-359.  doi: 10.1016/S0022-0396(03)00122-0.

[15]

J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Rational. Mech. Anal., 95 (1986), 325-344.  doi: 10.1007/BF00276840.

[16]

F. M. HuangJ. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116.  doi: 10.1007/s00205-009-0267-0.

[17]

F. M. Huang and T. Wang, Stability of superposition of viscous contact wave and rarefaction waves for compressible Navier-Stokes system, Indiana Univ. Math. J., 65 (2016), 1833-1875.  doi: 10.1512/iumj.2016.65.5914.

[18]

F. M. HuangZ. P. Xin and T. Yang, Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.  doi: 10.1016/j.aim.2008.06.014.

[19]

K. Ide and S. Kawashima, Decay property of regularity-loss type and nonlinear effects for dissipative Timoshenko system, Math. Models Methods Appl. Sci., 18 (2008), 1001-1025.  doi: 10.1142/S0218202508002930.

[20]

W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.

[21]

M. N. Jiang and C. J. Zhu, Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant, J. Differential Equations, 246 (2009), 50-77.  doi: 10.1016/j.jde.2008.03.033.

[22]

M. N. Jiang and C. J. Zhu, Convergence rates to nonlinear diffusion waves for p-system with nonlinear damping on quadrant, Discrete Contin. Dyn. Syst., 23 (2009), 887-918.  doi: 10.3934/dcds.2009.23.887.

[23]

Q. S. JiuY. Wang and Z. P. Xin, Vacuum behaviors around rarefaction waves to 1D compressible Navier-Stokes equations with density-dependent viscosity, SIAM J. Math. Anal., 45 (2013), 3194-3228.  doi: 10.1137/120879919.

[24]

S. Kawashima and Y. Tanaka, Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math., 58 (2004), 211-250.  doi: 10.2206/kyushujm.58.211.

[25]

S. Kawashima and P. C. Zhu, Asymptotic stability of rarefaction wave for the Navier-Stokes equations for a compressible fluid in the half space, Arch. Ration. Mech. Anal., 194 (2009), 105-132.  doi: 10.1007/s00205-008-0191-8.

[26]

Q. Q. LiuH. Y. Peng and Z.-A. Wang, Asymptotic stability of diffusion waves of a quasi-linear hyperbolic-parabolic model for vasculogenesis, SIAM J. Math. Anal., 54 (2022), 1313-1346.  doi: 10.1137/21M1418150.

[27]

Q. Q. LiuH. Y. Peng and Z.-A. Wang, Convergence to nonlinear diffusion waves for a hyperbolic-parabolic chemotaxis system modelling vasculogenesis, J. Differential Equations, 314 (2022), 251-286.  doi: 10.1016/j.jde.2022.01.021.

[28]

T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.  doi: 10.1007/BF01466726.

[29]

T. P. LiuT. YangS. H. Yu and H. J. Zhao, Nonlinear stability of rarefaction waves for the Boltzmann equation, Arch. Ration. Mech. Anal., 181 (2006), 333-371.  doi: 10.1007/s00205-005-0414-1.

[30]

T. P. Liu and S. H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.  doi: 10.1007/s00220-003-1030-2.

[31]

T. LuoH. Y. Yin and C. J. Zhu, Stability of the rarefaction wave for a coupled compressible Navier-Stokes/Allen-Cahn system, Math. Methods Appl. Sci., 41 (2018), 4724-4736.  doi: 10.1002/mma.4925.

[32]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322.

[33]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13.  doi: 10.1007/BF03167088.

[34]

A. Matsumura and K. Nishihara, Global stability of the rarefaction wave of a one-dimensional model system for compressible viscous gas, Comm. Math. Phys., 144 (1992), 325-335.  doi: 10.1007/BF02101095.

[35]

A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25.  doi: 10.1007/BF03167036.

[36]

M. Mei, Nonlinear diffusion waves for hyperbolic $p$-system with nonlinear damping, J. Differential Equations, 247 (2009), 1275-1296.  doi: 10.1016/j.jde.2009.04.004.

[37]

M. Mei, Best asymptotic profile for hyperbolic $p$-system with damping, SIAM J. Math. Anal., 42 (2010), 1-23.  doi: 10.1137/090756594.

[38]

K. Nishihara, Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, J. Differential Equations, 131 (1996), 171-188.  doi: 10.1006/jdeq.1996.0159.

[39]

K. Nishihara, Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, J. Differential Equations, 137 (1997), 384-395.  doi: 10.1006/jdeq.1997.3268.

[40]

K. Nishihara and T. Yang, Boundary effect on asymptotic behaviour of solutions to the $p$-system with linear damping, J. Differential Equations, 156 (1999), 439-458.  doi: 10.1006/jdeq.1998.3598.

[41]

R. H. Pan, Darcy's law as long-time limit of adiabatic porous media flow, J. Differential Equations, 220 (2006), 121-146.  doi: 10.1016/j.jde.2004.10.013.

[42]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[43]

H. Y. YinJ. S. Zhang and C. J. Zhu, Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system, Nonlinear Anal. Real World Appl., 31 (2016), 492-512.  doi: 10.1016/j.nonrwa.2016.01.020.

[44]

S. H. Yu, Nonlinear wave propagations over a Boltzmann shock profile, J. Amer. Math. Soc., 23 (2010), 1041-1118.  doi: 10.1090/S0894-0347-2010-00671-6.

[45]

C. J. Zhu and M. N. Jiang, $L^p$-decay rates to nonlinear diffusion waves for $p$-system with nonlinear damping, Sci. China Ser. A, 49 (2006), 721-739.  doi: 10.1007/s11425-006-0721-5.

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