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Pointwise estimates of solutions for the multi-dimensional scalar conservation laws with relaxation
1. | Department of Mathematics, Shanghai Jiao Tong University, 800 Dong Chuan Road, 200240, Shanghai |
References:
[1] |
C. Arvanitis, Mesh redistribution strategies and finite element schemes for hyperbolic conservation laws, J. Sci. Comput., 34 (2008), 1-25.
doi: 10.1007/s10915-007-9155-7. |
[2] |
S. Balasubramanyam and S. V. Raghurama Rao, A grid-free upwind relaxation scheme for inviscid compressible flows, Internat. J. Numer. Methods Fluids, 51 (2006), 159-196.
doi: 10.1002/fld.1099. |
[3] |
S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases," 3rd edition, Cambridge Univ. Press, London, 1970. |
[4] |
G. Q. Chen, C. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830.
doi: doi:10.1002/cpa.3160470602. |
[5] |
D. Donatelli and C. Lattanzio, On the diffusive stress relaxation for multidimensional viscoelasticity, Commun. Pure Appl. Anal., 8 (2009), 645-654.
doi: 10.3934/cpaa.2009.8.645. |
[6] |
M. Di Francesco and D. Donatelli, Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 79-100. |
[7] |
L. C. Evans, Partial differential equations, in "Graduate Studies in Math.," 19, American Mathematical Society, Providence, RI, 1998. |
[8] |
H. Fan and J. Härterich, Conservation laws with a degenerate source: Traveling waves, large-time behavior and zero relaxation limit, Nonlinear Anal., 63 (2005), 1042-1069.
doi: 10.1016/j.na.2003.10.031. |
[9] |
D. Hoff and K. Zumbrum, Multi-dimensional diffusion wave for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. Journal, 44 (1995), 603-676.
doi: 10.1512/iumj.1995.44.2003. |
[10] |
S. Jin and Z. P. Xin, The relaxation schemes for system of conservative laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-277.
doi: 10.1002/cpa.3160480303. |
[11] |
S. Jin and Z. P. Xin, Numerical passage from systems of conservation laws to Hamilton-Jacobi equations, relaxation schemes, SIAM J. Numer. Anal., 35 (1998), 2385-2404.
doi: 10.1137/S0036142996314366. |
[12] |
B. Kwon and K. Zumbrun, Asymptotic behavior of multidimensional scalar relaxation shocks, J. Hyperbolic Differ. Equ., 6 (2009), 663-708. |
[13] |
D. L. Li, The Green's function of the Navier-Stokes equations for gas dynamics in $\mathbbR^3$, Commun. Math. Phys., 257 (2005), 579-619.
doi: 10.1007/s00220-005-1351-4. |
[14] |
M. B. Liu and Z. X. Cheng, Conservation laws. III. Relaxation limit, Rev. Colombiana Mat., 41 (2007), 107-115. |
[15] |
T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow, J. Differential Equations, 190 (2003), 131-149. |
[16] |
T.-P. Liu, Hyperbolic conservative laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175.
doi: 10.1007/BF01210707. |
[17] |
T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math, 50 (1997), 1113-1182.
doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D. |
[18] |
T.-P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimension, Comm. Math. Phys., 196 (1998), 145-173.
doi: 10.1007/s002200050418. |
[19] |
T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608.
doi: 10.1002/cpa.20011. |
[20] |
T.-P. Liu and S.-H. Yu, Green's function and large-time behavior of solutions of Boltzmann equation, 3-D waves, Bulletin. Inst. Math. Academia Sinica (N.S.), 1 (2006), 1-78. |
[21] |
T.-P. Liu and Y. Zeng, Large time behavior of solutions to general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997). |
[22] |
Y. Q. Liu and W. K. Wang, The pointwise estimates of solutions for dissipative wave equation in multi-dimensions, Discrete Contin. Dyn. Syst., 20 (2008), 1013-1028.
doi: 10.3934/dcds.2008.20.1013. |
[23] |
T. Luo and Z. P. Xin, Nonlinear stability of shock fronts for a relaxation system in several space dimensions, J. Differential Equations, 139 (1997), 365-408. |
[24] |
C. Mascia and K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J., 51 (2002), 773-904.
doi: 10.1512/iumj.2002.51.2212. |
[25] |
C. Mascia and K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., 37 (2005), 889-913.
doi: 10.1137/S0036141004435844. |
[26] |
C. Mascia and K. Zumbrun, Spectral stability of weak relaxation shock profiles, Comm. Partial Differential Equations, 34 (2009), 119-136. |
[27] |
R. Plaza and K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, Discrete Contin. Dyn. Syst., 10 (2004), 885-924.
doi: 10.3934/dcds.2004.10.885. |
[28] |
R. Kumar and M. K. Kadalbajoo, Efficient high-resolution relaxation schemes for hyperbolic systems of conservation laws, Internat. J. Numer. Methods Fluids, 55 (2007), 483-507.
doi: 10.1002/fld.1479. |
[29] |
Y.-J. Peng and S. Wang, Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters, Discrete Contin. Dyn. Syst., 23 (2009), 415-433.
doi: 10.3934/dcds.2009.23.415. |
[30] |
Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. |
[31] |
W. K. Wang and H. M. Xu, Pointwise estimate of solutions of isentropic Navier-Stokes equations in even multi-dimensions, Acta Math. Sci. Ser. B Engl. Ed., 21 (2001), 417-427. |
[32] |
W. K. Wang and T. Yang, The pointwise estimates of solutions of Euler equations with damping in multi-dimensions, J. Differential Equations, 173 (2001), 410-450. |
[33] |
W. K. Wang and X. F. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions, J. Hyperbolic Differ. Equ., 2 (2005), 673-695. |
[34] |
J. Xu and W.-A. Yong, Zero-relaxation limit of non-isentropic hydrodynamic models for semiconductors, Discrete Contin. Dyn. Syst., 25 (2009), 1319-1332.
doi: 10.3934/dcds.2009.25.1319. |
[35] |
W.-A. Yong and W. Jäger, On hyperbolic relaxation problems, in "Analysis and Numerics for Conservation Laws," Springer, Berlin, (2005), 495-520. |
[36] |
W.-A. Yong and K. Zumbrun, Existence of relaxation shock profiles for hyperbolic conservation laws, SIAM J. Appl. Math., 60 (2000), 1565-1575.
doi: 10.1137/S0036139999352705. |
[37] |
Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279.
doi: 10.1007/s002050050188. |
show all references
References:
[1] |
C. Arvanitis, Mesh redistribution strategies and finite element schemes for hyperbolic conservation laws, J. Sci. Comput., 34 (2008), 1-25.
doi: 10.1007/s10915-007-9155-7. |
[2] |
S. Balasubramanyam and S. V. Raghurama Rao, A grid-free upwind relaxation scheme for inviscid compressible flows, Internat. J. Numer. Methods Fluids, 51 (2006), 159-196.
doi: 10.1002/fld.1099. |
[3] |
S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases," 3rd edition, Cambridge Univ. Press, London, 1970. |
[4] |
G. Q. Chen, C. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math., 47 (1994), 787-830.
doi: doi:10.1002/cpa.3160470602. |
[5] |
D. Donatelli and C. Lattanzio, On the diffusive stress relaxation for multidimensional viscoelasticity, Commun. Pure Appl. Anal., 8 (2009), 645-654.
doi: 10.3934/cpaa.2009.8.645. |
[6] |
M. Di Francesco and D. Donatelli, Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 79-100. |
[7] |
L. C. Evans, Partial differential equations, in "Graduate Studies in Math.," 19, American Mathematical Society, Providence, RI, 1998. |
[8] |
H. Fan and J. Härterich, Conservation laws with a degenerate source: Traveling waves, large-time behavior and zero relaxation limit, Nonlinear Anal., 63 (2005), 1042-1069.
doi: 10.1016/j.na.2003.10.031. |
[9] |
D. Hoff and K. Zumbrum, Multi-dimensional diffusion wave for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. Journal, 44 (1995), 603-676.
doi: 10.1512/iumj.1995.44.2003. |
[10] |
S. Jin and Z. P. Xin, The relaxation schemes for system of conservative laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235-277.
doi: 10.1002/cpa.3160480303. |
[11] |
S. Jin and Z. P. Xin, Numerical passage from systems of conservation laws to Hamilton-Jacobi equations, relaxation schemes, SIAM J. Numer. Anal., 35 (1998), 2385-2404.
doi: 10.1137/S0036142996314366. |
[12] |
B. Kwon and K. Zumbrun, Asymptotic behavior of multidimensional scalar relaxation shocks, J. Hyperbolic Differ. Equ., 6 (2009), 663-708. |
[13] |
D. L. Li, The Green's function of the Navier-Stokes equations for gas dynamics in $\mathbbR^3$, Commun. Math. Phys., 257 (2005), 579-619.
doi: 10.1007/s00220-005-1351-4. |
[14] |
M. B. Liu and Z. X. Cheng, Conservation laws. III. Relaxation limit, Rev. Colombiana Mat., 41 (2007), 107-115. |
[15] |
T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow, J. Differential Equations, 190 (2003), 131-149. |
[16] |
T.-P. Liu, Hyperbolic conservative laws with relaxation, Comm. Math. Phys., 108 (1987), 153-175.
doi: 10.1007/BF01210707. |
[17] |
T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math, 50 (1997), 1113-1182.
doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D. |
[18] |
T.-P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimension, Comm. Math. Phys., 196 (1998), 145-173.
doi: 10.1007/s002200050418. |
[19] |
T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608.
doi: 10.1002/cpa.20011. |
[20] |
T.-P. Liu and S.-H. Yu, Green's function and large-time behavior of solutions of Boltzmann equation, 3-D waves, Bulletin. Inst. Math. Academia Sinica (N.S.), 1 (2006), 1-78. |
[21] |
T.-P. Liu and Y. Zeng, Large time behavior of solutions to general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997). |
[22] |
Y. Q. Liu and W. K. Wang, The pointwise estimates of solutions for dissipative wave equation in multi-dimensions, Discrete Contin. Dyn. Syst., 20 (2008), 1013-1028.
doi: 10.3934/dcds.2008.20.1013. |
[23] |
T. Luo and Z. P. Xin, Nonlinear stability of shock fronts for a relaxation system in several space dimensions, J. Differential Equations, 139 (1997), 365-408. |
[24] |
C. Mascia and K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J., 51 (2002), 773-904.
doi: 10.1512/iumj.2002.51.2212. |
[25] |
C. Mascia and K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., 37 (2005), 889-913.
doi: 10.1137/S0036141004435844. |
[26] |
C. Mascia and K. Zumbrun, Spectral stability of weak relaxation shock profiles, Comm. Partial Differential Equations, 34 (2009), 119-136. |
[27] |
R. Plaza and K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, Discrete Contin. Dyn. Syst., 10 (2004), 885-924.
doi: 10.3934/dcds.2004.10.885. |
[28] |
R. Kumar and M. K. Kadalbajoo, Efficient high-resolution relaxation schemes for hyperbolic systems of conservation laws, Internat. J. Numer. Methods Fluids, 55 (2007), 483-507.
doi: 10.1002/fld.1479. |
[29] |
Y.-J. Peng and S. Wang, Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters, Discrete Contin. Dyn. Syst., 23 (2009), 415-433.
doi: 10.3934/dcds.2009.23.415. |
[30] |
Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J., 14 (1985), 249-275. |
[31] |
W. K. Wang and H. M. Xu, Pointwise estimate of solutions of isentropic Navier-Stokes equations in even multi-dimensions, Acta Math. Sci. Ser. B Engl. Ed., 21 (2001), 417-427. |
[32] |
W. K. Wang and T. Yang, The pointwise estimates of solutions of Euler equations with damping in multi-dimensions, J. Differential Equations, 173 (2001), 410-450. |
[33] |
W. K. Wang and X. F. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimensions, J. Hyperbolic Differ. Equ., 2 (2005), 673-695. |
[34] |
J. Xu and W.-A. Yong, Zero-relaxation limit of non-isentropic hydrodynamic models for semiconductors, Discrete Contin. Dyn. Syst., 25 (2009), 1319-1332.
doi: 10.3934/dcds.2009.25.1319. |
[35] |
W.-A. Yong and W. Jäger, On hyperbolic relaxation problems, in "Analysis and Numerics for Conservation Laws," Springer, Berlin, (2005), 495-520. |
[36] |
W.-A. Yong and K. Zumbrun, Existence of relaxation shock profiles for hyperbolic conservation laws, SIAM J. Appl. Math., 60 (2000), 1565-1575.
doi: 10.1137/S0036139999352705. |
[37] |
Y. Zeng, Gas dynamics in thermal nonequilibrium and general hyperbolic systems with relaxation, Arch. Ration. Mech. Anal., 150 (1999), 225-279.
doi: 10.1007/s002050050188. |
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