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November  2021, 41(11): 5271-5289. doi: 10.3934/dcds.2021076

Well-posedness and critical thresholds in a nonlocal Euler system with relaxation

Manas Bhatnagar and  Hailiang Liu ,

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

 

Received  November 2020 Revised  March 2021 Published  November 2021 Early access  April 2021

Fund Project: This research was supported in part by NSF grant DMS1812666

We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.

Keywords: Critical thresholds, global regularity, shock formation, hyperbolic systems, Euler systems.
Mathematics Subject Classification: Primary, 35L65; Secondary, 35B30.
Citation: Manas Bhatnagar, Hailiang Liu. Well-posedness and critical thresholds in a nonlocal Euler system with relaxation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5271-5289. doi: 10.3934/dcds.2021076
References:
[1]

G. R. Baker, X. Li and A. C. Morlet, Analytic structure of two 1D-transport equations with fluxes, Physica D, 91 (1996), 349-375.  doi: 10.1016/0167-2789(95)00271-5.  Google Scholar

[2]

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

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J. A. Carrillo, Y. P. Choi, E. 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

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J. A. Carrillo, Y. 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

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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

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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

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T. Do, A. Kiselev, L. 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

[8]

S. Engelberg, H. 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

[9]

L. C. F. Ferreira and J. C. Valencia-Guevara, Periodic solutions for a 1D-model with nonlocal velocity via mass transport, J. Diff. Equ., 260 (2016), 7093-7114.  doi: 10.1016/j.jde.2016.01.018.  Google Scholar

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D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, 224, Springer-Verlag Berlin Heidelberg, 2001.  Google Scholar

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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

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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

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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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, 53, Springer Science+Business Media, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[22]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[23]

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

[24]

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

[25]

D. Wei, E. 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

[26]

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]

G. R. Baker, X. Li and A. C. Morlet, Analytic structure of two 1D-transport equations with fluxes, Physica D, 91 (1996), 349-375.  doi: 10.1016/0167-2789(95)00271-5.  Google Scholar

[2]

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

[3]

J. A. Carrillo, Y. P. Choi, E. 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. Carrillo, Y. 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]

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

[7]

T. Do, A. Kiselev, L. 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

[8]

S. Engelberg, H. 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

[9]

L. C. F. Ferreira and J. C. Valencia-Guevara, Periodic solutions for a 1D-model with nonlocal velocity via mass transport, J. Diff. Equ., 260 (2016), 7093-7114.  doi: 10.1016/j.jde.2016.01.018.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, 224, Springer-Verlag Berlin Heidelberg, 2001.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, 53, Springer Science+Business Media, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[22]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[23]

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

[24]

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

[25]

D. Wei, E. 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

[26]

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

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