# American Institute of Mathematical Sciences

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

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

 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.

Citation: Manas Bhatnagar, Hailiang Liu. Well-posedness and critical thresholds in a nonlocal Euler system with relaxation. Discrete and 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. [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. [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. [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. [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] S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Publishing Company, Amsterdam, 1962. doi: 10.1126/science.140.3563.168. [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. [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. [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. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, 224, Springer-Verlag Berlin Heidelberg, 2001. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, 1990. doi: 10.1007/978-3-7091-6961-2. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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] S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Publishing Company, Amsterdam, 1962. doi: 10.1126/science.140.3563.168. [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. [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. [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. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, 224, Springer-Verlag Berlin Heidelberg, 2001. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [22] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, 1990. doi: 10.1007/978-3-7091-6961-2. [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. [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. [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. [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.
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