# American Institute of Mathematical Sciences

September  2019, 18(5): 2835-2854. doi: 10.3934/cpaa.2019127

## Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms

 1 School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai, China 2 School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China

* Corresponding author

Received  March 2019 Revised  March 2019 Published  April 2019

Fund Project: The first author is supported by the National Natural Science Foundation of China (No. 11801358 and No. 11571227). The second author is supported by the National Natural Science Foundation of China (No. 11771284 and No. 11831011)

We are concerned with the pointwise estimates of solutions to the scalar conservation law with a nonlocal dissipative term for arbitrary large initial data. Based on the Green's function method, time-frequency decomposition method as well as the classical energy estimates, pointwise estimates and the optimal decay rates are established in this paper. We emphasize that the decay rate is independent of the index s in the nonlocal dissipative term. This phenomenon is also coincident with the fact that the decay rate is determined by the low frequency part of the solution no matter the initial data is small or large.

Citation: Lijuan Wang, Weike Wang. Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2835-2854. doi: 10.3934/cpaa.2019127
##### References:
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Zhu, Energy method for multi-dimensional balance laws with non-local disspation, J. Math. Pures. Appl., 93 (2010), 572-598. doi: 10.1016/j.matpur.2009.10.007. Google Scholar [12] W. Gao and C. Zhu, Asymptotic decay toward the planar rarefaction waves for a model system of the radiating gas in two dimensions, Math. Models Methods Appl. Sci., 18 (2008), 511-541. doi: 10.1142/S0218202508002760. Google Scholar [13] J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840. Google Scholar [14] K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas, Quart. J. Mech. Appl. Math., 24 (1971), 155-168. Google Scholar [15] H. Dong, D. Du and D. Li, Finite time singularities and global well-posedness for fractal Burgers equations, Indiana Univ. Math. J., 58 (2009), 807-821. doi: 10.1512/iumj.2009.58.3505. Google Scholar [16] I. M. Held, R. T. Pierrehumbert, S. T. Garner and K. L. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech., 282 (1995), 1-20. doi: 10.1017/S0022112095000012. Google Scholar [17] E. Hopf, The partial differential equation $u_t+uu_x = \mu u_xx$, Commmu. Pure. Appl. Math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302. Google Scholar [18] D. Hoff and K. Zumbrum, Multi-dimensional diffusion wave for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003. Google Scholar [19] D. Hoff and K. Zumbrum, Pointwise decay estimates for multidimensional Navior-Stokes diffusion waves, Z. Angew Math. Phys., 48 (1997), 1-18. doi: 10.1007/s000330050049. Google Scholar [20] S. Kawashima and S. Nishibata, Cauchy problem for amodel system of the radiating gas: weak solutions with a jump and classical solutions, Math. Models Methods Appl. Sci., 9 (1999), 69-91. doi: 10.1142/S0218202599000063. Google Scholar [21] T. 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Rong, Decay of solutions to fractal parabolic conservation laws with large initial data, Commmu. Pure. Appl. Anal., 12 (2013), 973-984. doi: 10.3934/cpaa.2013.12.973. Google Scholar [27] Y. Liu and S. Kawashima, Asymptotic behavior of solutions to a model system of a radiating gas, Commmu. Pure. Appl. Anal., 10 (2011), 209-223. doi: 10.3934/cpaa.2011.10.209. Google Scholar [28] T-P. Liu and Y. Zeng, Large time behavior of solutions general quasilinear hyperbolic-parabolic systerms of conservation laws, A. M. S. memoirs, (1997), 599. doi: 10.1090/memo/0599. Google Scholar [29] T-P. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commu Math Phys, 196 (1998), 145-173. doi: 10.1007/s002200050418. Google Scholar [30] P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A., 40 (1989), 7193-7196. doi: 10.1103/PhysRevA.40.7193. Google Scholar [31] M. E. Schonbek, Decay of solution to parabolic conservation laws, Comm. Partial Differential Equations, 7 (1980), 449-473. doi: 10.1080/0360530800882145. Google Scholar [32] M. E. Schonbek, Uniform decay rates for parabolic conservation laws, Nonlinear Anal., 10 (1986), 943-956. doi: 10.1016/0362-546X(86)90080-5. Google Scholar [33] M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal., 35 (2003), 357-375. doi: 10.1137/S0036141002409362. Google Scholar [34] J. Smoller, Shock Waves and Reaction-diffusion Equations, Springer Science Business Media, 2012. Google Scholar [35] W. Wang and W. Wang, The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions, Nonlinear Anal. TMA, 71 (2009), 1180-1195. doi: 10.1016/j.na.2008.11.050. Google Scholar [36] W. Wang and W. Wang, Blow-up and global existence of solutions for a model system of the radiating gas, Nonlinear Analysis, 81 (2013), 12-30. doi: 10.1016/j.na.2012.12.010. Google Scholar [37] J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 56 (2001), 1-13. Google Scholar [38] L. Wang, W. Wang and X. Xu, Global existence of large solutions to conservation laws with nonlocal dissipation type terms (in Chinese), Sci. Sin. Math., 48 (2018), 589-608.Google Scholar [39] W. Wang and X. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimentions, J. Hyperbolic Differ. Equ., 3 (2005), 673-695. doi: 10.1142/S0219891605000580. Google Scholar [40] W. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differential Equations, 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937. Google Scholar [41] Z. Wu and W. Wang, Pointwise estimates of solution for non-isentropic Navier-Stokes-Poisson equations in multi-dimensions, Acta Math. Sci., 32 (2012), 1681-1702. doi: 10.1016/S0252-9602(12)60134-9. Google Scholar [42] W. Wang and Z. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions, J. Differential Equations, 248 (2010), 1617-1636. doi: 10.1016/j.jde.2010.01.003. Google Scholar [43] W. Wang and W. Wang, The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions, Nonlinear Anal., 71 (2009), 1180-1195. doi: 10.1016/j.na.2008.11.050. Google Scholar [44] Y. Wang and H. Zhao, Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating, Commmu. Pure. Appl. Anal., 17 (2018), 347-374. doi: 10.3934/cpaa.2018020. Google Scholar [45] C. Zhang, F. Li and J. Duan, Long-time behavior of a class of nonlocal partial differential equations, Discrete Contin. Dyn. Syst., Series B, 23 (2018), 749-763. doi: 10.3934/dcdsb.2018041. Google Scholar

show all references

##### References:
 [1] C. Chan, M. Czubak and L. Silvestre, Eventual regularization of the slightly supercritical fractional Burgers equation, Discrete Contin. Dyn. Syst., 27 (2010), 847-861. doi: 10.3934/dcds.2010.27.847. Google Scholar [2] C. Chan and M. Czubak, Regularity of solutions for the critical N-dimensional Burgers's equation, Ann. I. H. Poincaré-AN., 27 (2010), 471-501. doi: 10.1016/j.anihpc.2009.11.008. Google Scholar [3] L. A. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Annals of Mathematics, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903. Google Scholar [4] P. Constantin, A. J. Majda and E. Tabak, Formation of strong fronts in the 2-d quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533. Google Scholar [5] P. Constantin, A. Tarfulea and V. Vicol, Long time dynamics of forced critical SQG, Communications in Mathematical Physics, 335 (2015), 93-141. doi: 10.1007/s00220-014-2129-3. Google Scholar [6] P. Constantin and V. Vicol, Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Funct. Anal., 22 (2012), 1289-1321. doi: 10.1007/s00039-012-0172-9. Google Scholar [7] P. Constantin and J. Wu, Behavior of solutions of 2d quasi-geostrophic equations, SIAM J. Math. Anal., 30 (1999), 937-948. doi: 10.1137/S0036141098337333. Google Scholar [8] A. Cordoba and D. Cordoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1. Google Scholar [9] J. W. Cholewa and T. Dlotko, Fractional Navier-Stokes equations, Discrete Contin. Dyn. Syst., Series B, 23 (2018), 2967-2988. doi: 10.3934/dcdsb.2017149. Google Scholar [10] C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematicschen Wissenschaften, 325, 2005. doi: 10.1007/3-540-29089-3. Google Scholar [11] R. Duan, K. Fellner and C. Zhu, Energy method for multi-dimensional balance laws with non-local disspation, J. Math. Pures. Appl., 93 (2010), 572-598. doi: 10.1016/j.matpur.2009.10.007. Google Scholar [12] W. Gao and C. Zhu, Asymptotic decay toward the planar rarefaction waves for a model system of the radiating gas in two dimensions, Math. Models Methods Appl. Sci., 18 (2008), 511-541. doi: 10.1142/S0218202508002760. Google Scholar [13] J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840. Google Scholar [14] K. Hamer, Nonlinear effects on the propagation of sound waves in a radiating gas, Quart. J. Mech. Appl. Math., 24 (1971), 155-168. Google Scholar [15] H. Dong, D. Du and D. Li, Finite time singularities and global well-posedness for fractal Burgers equations, Indiana Univ. Math. J., 58 (2009), 807-821. doi: 10.1512/iumj.2009.58.3505. Google Scholar [16] I. M. Held, R. T. Pierrehumbert, S. T. Garner and K. L. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech., 282 (1995), 1-20. doi: 10.1017/S0022112095000012. Google Scholar [17] E. Hopf, The partial differential equation $u_t+uu_x = \mu u_xx$, Commmu. Pure. Appl. Math., 3 (1950), 201-230. doi: 10.1002/cpa.3160030302. Google Scholar [18] D. Hoff and K. Zumbrum, Multi-dimensional diffusion wave for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003. Google Scholar [19] D. Hoff and K. Zumbrum, Pointwise decay estimates for multidimensional Navior-Stokes diffusion waves, Z. Angew Math. Phys., 48 (1997), 1-18. doi: 10.1007/s000330050049. Google Scholar [20] S. Kawashima and S. Nishibata, Cauchy problem for amodel system of the radiating gas: weak solutions with a jump and classical solutions, Math. Models Methods Appl. Sci., 9 (1999), 69-91. doi: 10.1142/S0218202599000063. Google Scholar [21] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740. Google Scholar [22] A. Kiselev, F. Nazarov and R. Shterenberg, Blow-up and regularity for fractal Burgers equations, Dynamics of PDE, 5 (2008), 211-240. doi: 10.4310/DPDE.2008.v5.n3.a2. Google Scholar [23] A. Kiselev, F. Nazarov and A. Volberg, Global well-posedness for the critical 2d dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3. Google Scholar [24] D. B. Kotlow, Quasilinear parabolic equations and first order quasilinear conservation laws with bad Cauchy data, J. Math. Anal. Appl., 35 (1971), 563-576. doi: 10.1016/0022-247X(71)90204-6. Google Scholar [25] F. Li and W. Wang, The pointwise estimates of solutions to the parabolic consevation law in multi-dimensions, Nonlinear Differ. Equ. Appl., 21 (2014), 87-103. doi: 10.1007/s00030-013-0239-9. Google Scholar [26] F. Li and F. Rong, Decay of solutions to fractal parabolic conservation laws with large initial data, Commmu. Pure. Appl. Anal., 12 (2013), 973-984. doi: 10.3934/cpaa.2013.12.973. Google Scholar [27] Y. Liu and S. Kawashima, Asymptotic behavior of solutions to a model system of a radiating gas, Commmu. Pure. Appl. Anal., 10 (2011), 209-223. doi: 10.3934/cpaa.2011.10.209. Google Scholar [28] T-P. Liu and Y. Zeng, Large time behavior of solutions general quasilinear hyperbolic-parabolic systerms of conservation laws, A. M. S. memoirs, (1997), 599. doi: 10.1090/memo/0599. Google Scholar [29] T-P. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commu Math Phys, 196 (1998), 145-173. doi: 10.1007/s002200050418. Google Scholar [30] P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A., 40 (1989), 7193-7196. doi: 10.1103/PhysRevA.40.7193. Google Scholar [31] M. E. Schonbek, Decay of solution to parabolic conservation laws, Comm. Partial Differential Equations, 7 (1980), 449-473. doi: 10.1080/0360530800882145. Google Scholar [32] M. E. Schonbek, Uniform decay rates for parabolic conservation laws, Nonlinear Anal., 10 (1986), 943-956. doi: 10.1016/0362-546X(86)90080-5. Google Scholar [33] M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal., 35 (2003), 357-375. doi: 10.1137/S0036141002409362. Google Scholar [34] J. Smoller, Shock Waves and Reaction-diffusion Equations, Springer Science Business Media, 2012. Google Scholar [35] W. Wang and W. Wang, The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions, Nonlinear Anal. TMA, 71 (2009), 1180-1195. doi: 10.1016/j.na.2008.11.050. Google Scholar [36] W. Wang and W. Wang, Blow-up and global existence of solutions for a model system of the radiating gas, Nonlinear Analysis, 81 (2013), 12-30. doi: 10.1016/j.na.2012.12.010. Google Scholar [37] J. Wu, Dissipative quasi-geostrophic equations with $L^p$ data, Electron. J. Differential Equations, 56 (2001), 1-13. Google Scholar [38] L. Wang, W. Wang and X. Xu, Global existence of large solutions to conservation laws with nonlocal dissipation type terms (in Chinese), Sci. Sin. Math., 48 (2018), 589-608.Google Scholar [39] W. Wang and X. Yang, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in even space-dimentions, J. Hyperbolic Differ. Equ., 3 (2005), 673-695. doi: 10.1142/S0219891605000580. Google Scholar [40] W. Wang and T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differential Equations, 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937. Google Scholar [41] Z. Wu and W. Wang, Pointwise estimates of solution for non-isentropic Navier-Stokes-Poisson equations in multi-dimensions, Acta Math. Sci., 32 (2012), 1681-1702. doi: 10.1016/S0252-9602(12)60134-9. Google Scholar [42] W. Wang and Z. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions, J. Differential Equations, 248 (2010), 1617-1636. doi: 10.1016/j.jde.2010.01.003. Google Scholar [43] W. Wang and W. Wang, The pointwise estimates of solutions for a model system of the radiating gas in multi-dimensions, Nonlinear Anal., 71 (2009), 1180-1195. doi: 10.1016/j.na.2008.11.050. Google Scholar [44] Y. Wang and H. Zhao, Global existence and decay estimate of classical solutions to the compressible viscoelastic flows with self-gravitating, Commmu. Pure. Appl. Anal., 17 (2018), 347-374. doi: 10.3934/cpaa.2018020. Google Scholar [45] C. Zhang, F. Li and J. Duan, Long-time behavior of a class of nonlocal partial differential equations, Discrete Contin. Dyn. Syst., Series B, 23 (2018), 749-763. doi: 10.3934/dcdsb.2018041. Google Scholar
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