doi: 10.3934/dcds.2021098
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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 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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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