# American Institute of Mathematical Sciences

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.

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##### 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.
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