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Well-posedness and critical thresholds in a nonlocal Euler system with relaxation
Department of Mathematics, Iowa State University, Ames, IA 50011, USA |
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.
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. |
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. |
[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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