November  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. Google Scholar

[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. Google Scholar

[3]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases,", 3rd edition, (1970). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

L. C. Evans, Partial differential equations,, Graduate studies in Math., 19 (1998). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

M. B. Liu and Z. X. Cheng, Conservation laws. III. Relaxation limit,, Rev. Colombiana Mat., 41 (2007), 107. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Nonuniform Gases,", 3rd edition, (1970). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

L. C. Evans, Partial differential equations,, Graduate studies in Math., 19 (1998). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

M. B. Liu and Z. X. Cheng, Conservation laws. III. Relaxation limit,, Rev. Colombiana Mat., 41 (2007), 107. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

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

[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. Google Scholar

[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. Google Scholar

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