December  2021, 41(12): 5851-5869. doi: 10.3934/dcds.2021098

Sharp critical thresholds in a hyperbolic system with relaxation

Department of Mathematics, Iowa State University, Ames, IA 50011, USA

Received  December 2020 Revised  May 2021 Published  December 2021 Early access  July 2021

Fund Project: This research was supported by the National Science Foundation under Grant DMS1812666

We propose and study a one-dimensional $ 2\times 2 $ hyperbolic Eulerian system with local relaxation from critical threshold phenomena perspective. The system features dynamic transition between strictly and weakly hyperbolic. For different classes of relaxation we identify intrinsic critical thresholds for initial data that distinguish global regularity and finite time blowup. For relaxation independent of density, we estimate bounds on density in terms of velocity where the system is strictly hyperbolic.

Citation: Manas Bhatnagar, Hailiang Liu. Sharp critical thresholds in a hyperbolic system with relaxation. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5851-5869. doi: 10.3934/dcds.2021098
References:
[1]

M. Bhatnagar and H. Liu, Critical thresholds in one-dimensional damped Euler-Poisson systems, Math. Mod. Meth. Appl. Sci., 30(5) (2020), 891–916, 2020. doi: 10.1142/S0218202520500189.

[2]

M. Bhatnagar and H. Liu, Well-posedness and critical thresholds in a nonlocal Euler system with relaxation, Disc. Cont. Dyn. Sys., Online first ver., 2021. doi: 10.3934/dcds. 2021076.

[3]

J. A. CarrilloY. P. ChoiE. Tadmor and C. Tan, Critical thresholds in 1D Euler equations with non-local forces, Math. Mod. Meth. Appl. Sci., 26 (2016), 185-206.  doi: 10.1142/S0218202516500068.

[4]

J. A. CarrilloY. P. Choi and E. Zatorska, On the pressureless damped Euler-Poisson equations with quadratic confinement, Math. Mod. Meth. Appl. Sci., 26 (2016), 2311-2340.  doi: 10.1142/S0218202516500548.

[5]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3$^rd$ edition, 325, Springer-Verlag Berlin Heidelberg. doi: 10.1007/978-3-662-49451-6.

[6]

T. DoA. KiselevL. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, Arch. Rat. Mech. Anal., 228 (2017), 1-37.  doi: 10.1007/s00205-017-1184-2.

[7]

S. EngelbergH. Liu and E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana University Math. Journal, 50 (2001), 109-157.  doi: 10.1512/iumj.2001.50.2177.

[8]

S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Publishing Company, Amsterdam, 1962. doi: 10.1126/science. 140.3563.168.

[9]

S. M. He and E. Tadmor, Global regularity of two-dimensional flocking hydrodynamics, C. R. Math., 355 (2017), 795-805.  doi: 10.1016/j.crma.2017.05.008.

[10]

A. Kiselev and C. Tan, Global regularity for 1D Eulerian dynamics with singular interaction forces, SIAM J. Math. Anal., 50 (2018), 6208-6229.  doi: 10.1137/17M1141515.

[11]

P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, Journal of Math. Phys., 5 (1964), 611. doi: 10.1063/1.1704154.

[12]

T. Li and H. Liu, Critical thresholds in a relaxation model for traffic flows, Indiana Univ. Math. J., 57 (2008), 1409-1431.  doi: 10.1512/iumj.2008.57.3215.

[13]

T. Li and H. Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds, Disc. Cont. Dyn. Sys-Series A, 24 (2009), 511-521.  doi: 10.3934/dcds.2009.24.511.

[14]

T. Li and H. Liu, Critical thresholds in hyperbolic relaxation systems, J. Diff. Eqns., 247 (2009), 33-48.  doi: 10.1016/j.jde.2009.03.032.

[15]

H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math. Anal., 33(4) (2001), 930-945.  doi: 10.1137/S0036141001386908.

[16]

H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Commun. Math. Phys., 228 (2002), 435-466.  doi: 10.1007/s002200200667.

[17]

H. Liu and E. Tadmor, Critical thresholds in 2-D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), 1889-1910.  doi: 10.1137/S0036139902416986.

[18]

H. Liu and E. Tadmor., Rotation prevents finite-time breakdown, Physica D, 188 (2004), 262-276.  doi: 10.1016/j.physd.2003.07.006.

[19]

E. Tadmor and C. Tan., Critical thresholds in flocking hydrodynamics with non-local alignment, Phil. Trans. R. Soc. A., 372 (2014), 20130401.  doi: 10.1098/rsta.2013.0401.

[20]

E. Tadmor and D. Wei, On the global regularity of subcritical Euler-Poisson equations with pressure, J. Eur. Math. Soc., 10 (2008), 757-769.  doi: 10.4171/JEMS/129.

[21]

D. WeiE. Tadmor and H. Bae, Critical thresholds in multi-dimensional Euler-Poisson equations with radial symmetry, Commun. Math. Sci., 10 (2012), 75-86.  doi: 10.4310/CMS.2012.v10.n1.a4.

[22]

W. A. Yong, Intrinsic properties of conservation-dissipation formalism of irreversible thermodynamics, Phil. Trans. R. Soc. A., 378 (2020), 20190177.  doi: 10.1098/rsta.2019.0177.

show all references

References:
[1]

M. Bhatnagar and H. Liu, Critical thresholds in one-dimensional damped Euler-Poisson systems, Math. Mod. Meth. Appl. Sci., 30(5) (2020), 891–916, 2020. doi: 10.1142/S0218202520500189.

[2]

M. Bhatnagar and H. Liu, Well-posedness and critical thresholds in a nonlocal Euler system with relaxation, Disc. Cont. Dyn. Sys., Online first ver., 2021. doi: 10.3934/dcds. 2021076.

[3]

J. A. CarrilloY. P. ChoiE. Tadmor and C. Tan, Critical thresholds in 1D Euler equations with non-local forces, Math. Mod. Meth. Appl. Sci., 26 (2016), 185-206.  doi: 10.1142/S0218202516500068.

[4]

J. A. CarrilloY. P. Choi and E. Zatorska, On the pressureless damped Euler-Poisson equations with quadratic confinement, Math. Mod. Meth. Appl. Sci., 26 (2016), 2311-2340.  doi: 10.1142/S0218202516500548.

[5]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3$^rd$ edition, 325, Springer-Verlag Berlin Heidelberg. doi: 10.1007/978-3-662-49451-6.

[6]

T. DoA. KiselevL. Ryzhik and C. Tan, Global regularity for the fractional Euler alignment system, Arch. Rat. Mech. Anal., 228 (2017), 1-37.  doi: 10.1007/s00205-017-1184-2.

[7]

S. EngelbergH. Liu and E. Tadmor, Critical thresholds in Euler-Poisson equations, Indiana University Math. Journal, 50 (2001), 109-157.  doi: 10.1512/iumj.2001.50.2177.

[8]

S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Publishing Company, Amsterdam, 1962. doi: 10.1126/science. 140.3563.168.

[9]

S. M. He and E. Tadmor, Global regularity of two-dimensional flocking hydrodynamics, C. R. Math., 355 (2017), 795-805.  doi: 10.1016/j.crma.2017.05.008.

[10]

A. Kiselev and C. Tan, Global regularity for 1D Eulerian dynamics with singular interaction forces, SIAM J. Math. Anal., 50 (2018), 6208-6229.  doi: 10.1137/17M1141515.

[11]

P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, Journal of Math. Phys., 5 (1964), 611. doi: 10.1063/1.1704154.

[12]

T. Li and H. Liu, Critical thresholds in a relaxation model for traffic flows, Indiana Univ. Math. J., 57 (2008), 1409-1431.  doi: 10.1512/iumj.2008.57.3215.

[13]

T. Li and H. Liu, Critical thresholds in a relaxation system with resonance of characteristic speeds, Disc. Cont. Dyn. Sys-Series A, 24 (2009), 511-521.  doi: 10.3934/dcds.2009.24.511.

[14]

T. Li and H. Liu, Critical thresholds in hyperbolic relaxation systems, J. Diff. Eqns., 247 (2009), 33-48.  doi: 10.1016/j.jde.2009.03.032.

[15]

H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear conservation laws, SIAM J. Math. Anal., 33(4) (2001), 930-945.  doi: 10.1137/S0036141001386908.

[16]

H. Liu and E. Tadmor, Spectral dynamics of the velocity gradient field in restricted flows, Commun. Math. Phys., 228 (2002), 435-466.  doi: 10.1007/s002200200667.

[17]

H. Liu and E. Tadmor, Critical thresholds in 2-D restricted Euler-Poisson equations, SIAM J. Appl. Math., 63 (2003), 1889-1910.  doi: 10.1137/S0036139902416986.

[18]

H. Liu and E. Tadmor., Rotation prevents finite-time breakdown, Physica D, 188 (2004), 262-276.  doi: 10.1016/j.physd.2003.07.006.

[19]

E. Tadmor and C. Tan., Critical thresholds in flocking hydrodynamics with non-local alignment, Phil. Trans. R. Soc. A., 372 (2014), 20130401.  doi: 10.1098/rsta.2013.0401.

[20]

E. Tadmor and D. Wei, On the global regularity of subcritical Euler-Poisson equations with pressure, J. Eur. Math. Soc., 10 (2008), 757-769.  doi: 10.4171/JEMS/129.

[21]

D. WeiE. Tadmor and H. Bae, Critical thresholds in multi-dimensional Euler-Poisson equations with radial symmetry, Commun. Math. Sci., 10 (2012), 75-86.  doi: 10.4310/CMS.2012.v10.n1.a4.

[22]

W. A. Yong, Intrinsic properties of conservation-dissipation formalism of irreversible thermodynamics, Phil. Trans. R. Soc. A., 378 (2020), 20190177.  doi: 10.1098/rsta.2019.0177.

Figure 1.  Terminal roots of $ f(u)-u = 0 $
Figure 2.  Asymptotic invariant region
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