2011, 30(4): 1107-1138. doi: 10.3934/dcds.2011.30.1107

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

Received  March 2010 Revised  August 2010 Published  May 2011

Our aim is to study the pointwise time-asymptotic behavior of solutions for the scalar conservation laws with relaxation in multi-dimensions. We construct the Green's function for the Cauchy problem of the relaxation system which satisfies the dissipative condition. Based on the estimate for the Green's function, we get the pointwise estimate for the solution. It is shown that the solution exhibits some weak Huygens principle where the characteristic 'cone' is the envelope of planes.
Citation: Shijin Deng, Weike Wang. Pointwise estimates of solutions for the multi-dimensional scalar conservation laws with relaxation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1107-1138. doi: 10.3934/dcds.2011.30.1107
References:
[1]

C. Arvanitis, Mesh redistribution strategies and finite element schemes for hyperbolic conservation laws,, J. Sci. Comput., 34 (2008), 1. 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. doi: 10.1002/fld.1099.

[3]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases,", 3rd edition, (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. 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. 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.

[7]

L. C. Evans, Partial differential equations,, Graduate studies in Math., 19 (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. 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. 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. 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. doi: 10.1137/S0036142996314366.

[12]

B. Kwon and K. Zumbrun, Asymptotic behavior of multidimensional scalar relaxation shocks,, J. Hyperbolic Differ. Equ., 6 (2009), 663.

[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. 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.

[15]

T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow,, J. Differential Equations, 190 (2003), 131.

[16]

T.-P. Liu, Hyperbolic conservative laws with relaxation,, Comm. Math. Phys., 108 (1987), 153. doi: 10.1007/BF01210707.

[17]

T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws,, Comm. Pure Appl. Math, 50 (1997), 1113. 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. 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. 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.

[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. 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.

[24]

C. Mascia and K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks,, Indiana Univ. Math. J., 51 (2002), 773. 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. doi: 10.1137/S0036141004435844.

[26]

C. Mascia and K. Zumbrun, Spectral stability of weak relaxation shock profiles,, Comm. Partial Differential Equations, 34 (2009), 119.

[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. 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. 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. 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.

[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.

[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.

[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.

[34]

J. Xu and W.-A. Yong, Zero-relaxation limit of non-isentropic hydrodynamic models for semiconductors,, Discrete Contin. Dyn. Syst., 25 (2009), 1319. doi: 10.3934/dcds.2009.25.1319.

[35]

W.-A. Yong and W. Jäger, On hyperbolic relaxation problems,, Analysis and Numerics for Conservation Laws, (2005), 495.

[36]

W.-A. Yong and K. Zumbrun, Existence of relaxation shock profiles for hyperbolic conservation laws,, SIAM J. Appl. Math., 60 (2000), 1565. 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. 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. 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. doi: 10.1002/fld.1099.

[3]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases,", 3rd edition, (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. 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. 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.

[7]

L. C. Evans, Partial differential equations,, Graduate studies in Math., 19 (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. 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. 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. 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. doi: 10.1137/S0036142996314366.

[12]

B. Kwon and K. Zumbrun, Asymptotic behavior of multidimensional scalar relaxation shocks,, J. Hyperbolic Differ. Equ., 6 (2009), 663.

[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. 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.

[15]

T. Li, Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow,, J. Differential Equations, 190 (2003), 131.

[16]

T.-P. Liu, Hyperbolic conservative laws with relaxation,, Comm. Math. Phys., 108 (1987), 153. doi: 10.1007/BF01210707.

[17]

T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws,, Comm. Pure Appl. Math, 50 (1997), 1113. 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. 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. 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.

[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. 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.

[24]

C. Mascia and K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks,, Indiana Univ. Math. J., 51 (2002), 773. 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. doi: 10.1137/S0036141004435844.

[26]

C. Mascia and K. Zumbrun, Spectral stability of weak relaxation shock profiles,, Comm. Partial Differential Equations, 34 (2009), 119.

[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. 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. 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. 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.

[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.

[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.

[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.

[34]

J. Xu and W.-A. Yong, Zero-relaxation limit of non-isentropic hydrodynamic models for semiconductors,, Discrete Contin. Dyn. Syst., 25 (2009), 1319. doi: 10.3934/dcds.2009.25.1319.

[35]

W.-A. Yong and W. Jäger, On hyperbolic relaxation problems,, Analysis and Numerics for Conservation Laws, (2005), 495.

[36]

W.-A. Yong and K. Zumbrun, Existence of relaxation shock profiles for hyperbolic conservation laws,, SIAM J. Appl. Math., 60 (2000), 1565. 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. doi: 10.1007/s002050050188.

[1]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[2]

Wen-Qing Xu. Boundary conditions for multi-dimensional hyperbolic relaxation problems. Conference Publications, 2003, 2003 (Special) : 916-925. doi: 10.3934/proc.2003.2003.916

[3]

Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983

[4]

Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961

[5]

Arno Berger. Multi-dimensional dynamical systems and Benford's Law. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 219-237. doi: 10.3934/dcds.2005.13.219

[6]

Xiaoling Sun, Xiaojin Zheng, Juan Sun. A Lagrangian dual and surrogate method for multi-dimensional quadratic knapsack problems. Journal of Industrial & Management Optimization, 2009, 5 (1) : 47-60. doi: 10.3934/jimo.2009.5.47

[7]

Hideo Kubo. On the pointwise decay estimate for the wave equation with compactly supported forcing term. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1469-1480. doi: 10.3934/cpaa.2015.14.1469

[8]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[9]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

[10]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307

[11]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883

[12]

Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92

[13]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

[14]

Franz Achleitner, Anton Arnold, Eric A. Carlen. On multi-dimensional hypocoercive BGK models. Kinetic & Related Models, 2018, 11 (4) : 953-1009. doi: 10.3934/krm.2018038

[15]

Anatoli F. Ivanov. On global dynamics in a multi-dimensional discrete map. Conference Publications, 2015, 2015 (special) : 652-659. doi: 10.3934/proc.2015.0652

[16]

Gerald Sommer, Di Zang. Parity symmetry in multi-dimensional signals. Communications on Pure & Applied Analysis, 2007, 6 (3) : 829-852. doi: 10.3934/cpaa.2007.6.829

[17]

Wenxiong Chen, Congming Li. A priori estimate for the Nirenberg problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 225-233. doi: 10.3934/dcdss.2008.1.225

[18]

Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks & Heterogeneous Media, 2015, 10 (4) : 749-785. doi: 10.3934/nhm.2015.10.749

[19]

L.R. Ritter, Akif Ibragimov, Jay R. Walton, Catherine J. McNeal. Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis. Conference Publications, 2009, 2009 (Special) : 630-639. doi: 10.3934/proc.2009.2009.630

[20]

Wen-Guei Hu, Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]