# American Institute of Mathematical Sciences

June  2022, 42(6): 2603-2636. doi: 10.3934/dcds.2021205

## Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

 1 School of Mathematical Sciences, Capital Normal University, Beijing 100048, China 2 Academy for Multidisciplinary Studies, Beijing 100048, China 3 School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSE and CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Houzhi Tang

Received  July 2021 Revised  November 2021 Published  June 2022 Early access  January 2022

In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate $(1+t)^{-\frac{1}{2}}$ in $L^\infty$ norm.

Citation: Hailiang Li, Houzhi Tang, Haitao Wang. Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2603-2636. doi: 10.3934/dcds.2021205
##### References:
 [1] S. J. Deng, Initial-boundary value problem for p-system with damping in half space, Nonlinear Anal., 143 (2016), 193-210.  doi: 10.1016/j.na.2016.05.009. [2] S. J. Deng and W. K. Wang, Half space problem for Euler equations with damping in 3-D, J. Differential Equations, 263 (2017), 7372-7411.  doi: 10.1016/j.jde.2017.08.013. [3] L. L. Du, Initial-boundary value problem of Euler equations with damping in $\mathbb{R}^{n}_{+}$, Nonlinear Anal., 176 (2018), 157-177.  doi: 10.1016/j.na.2018.06.014. [4] L. L. Du and H. T. Wang, Pointwise wave behavior of the Navier-Stokes equations in half space, Discrete Contin. Dyn. Syst., 38 (2018), 1349-1363.  doi: 10.3934/dcds.2018055. [5] D. Hoff and K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys., 48 (1997), 597-614.  doi: 10.1007/s000330050049. [6] D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003. [7] Y. Kagei and T. Kobayashi, On large time behavior of solutions to the compressible Navier-Stokes Equations in the half space in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 165 (2002), 89-159.  doi: 10.1007/s00205-002-0221-x. [8] Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal., 177 (2005), 231-330.  doi: 10.1007/s00205-005-0365-6. [9] Y. I. Kanel', On a model system of equations for one-dimensional gas motion, Diff. Eq., (in Russian), 4 (1968), 721–734. [10] S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194.  doi: 10.1017/S0308210500018308. [11] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, 1983. [12] S. Kawashima and T. Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. Math. Kyoto Univ., 21 (1981), 825-837.  doi: 10.1215/kjm/1250521915. [13] S. Kawashima and P. C. Zhu, Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space, J. Differential Equations, 244 (2008), 3151-3179.  doi: 10.1016/j.jde.2008.01.020. [14] A. V. Kazhikhov, Cauchy problem for viscous gas equations, Sibirsk. Mat. Zh., 23 (1982), 60-64. [15] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9. [16] K. Koike, Long-time behavior of a point mass in a one-dimensional viscous compressible fluid and pointwise estimates of solutions, J. Differential Equations, 271 (2021), 356-413.  doi: 10.1016/j.jde.2020.08.022. [17] D. L. Li, The Green's function of the Navier-Stokes equations for gas dynamics in $\mathbb{R}^3$, Comm. Math. Phys., 257 (2005), 579-619.  doi: 10.1007/s00220-005-1351-4. [18] H.-L. Li, A. Matsumura and G. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713.  doi: 10.1007/s00205-009-0255-4. [19] T. P. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173.  doi: 10.1007/s002200050418. [20] T. P. Liu and Y. N. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), viii+120 pp. doi: 10.1090/memo/0599. [21] A. Matsumura and T. Nishida, Initial boundary value problem for the equations of motion of compressible viscous and heat conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  doi: 10.1007/BF01214738. [22] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342. [23] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322. [24] G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418.  doi: 10.1016/0362-546X(85)90001-X. [25] W. K. Wang and Z. G. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions, J. Differ. Equ., 248 (2010), 1617-1636.  doi: 10.1016/j.jde.2010.01.003. [26] Z. G. Wu and W. K. Wang, Pointwise estimates for bipolar compressible Navier-Stokes-Poisson system in dimension three, Arch. Ration. Mech. Anal., 226 (2017), 587-638.  doi: 10.1007/s00205-017-1140-1. [27] Y. N. Zeng, L1 asymptotic behavior of compressible, isentropic, viscous 1-D flow, Comm. Pure Appl. Math., 47 (1994), 1053-1082.  doi: 10.1002/cpa.3160470804. [28] G. Zhang, H.-L. Li and C. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^3$, J. Differential Equations, 250 (2011), 866-891.  doi: 10.1016/j.jde.2010.07.035.

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##### References:
 [1] S. J. Deng, Initial-boundary value problem for p-system with damping in half space, Nonlinear Anal., 143 (2016), 193-210.  doi: 10.1016/j.na.2016.05.009. [2] S. J. Deng and W. K. Wang, Half space problem for Euler equations with damping in 3-D, J. Differential Equations, 263 (2017), 7372-7411.  doi: 10.1016/j.jde.2017.08.013. [3] L. L. Du, Initial-boundary value problem of Euler equations with damping in $\mathbb{R}^{n}_{+}$, Nonlinear Anal., 176 (2018), 157-177.  doi: 10.1016/j.na.2018.06.014. [4] L. L. Du and H. T. Wang, Pointwise wave behavior of the Navier-Stokes equations in half space, Discrete Contin. Dyn. Syst., 38 (2018), 1349-1363.  doi: 10.3934/dcds.2018055. [5] D. Hoff and K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys., 48 (1997), 597-614.  doi: 10.1007/s000330050049. [6] D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676.  doi: 10.1512/iumj.1995.44.2003. [7] Y. Kagei and T. Kobayashi, On large time behavior of solutions to the compressible Navier-Stokes Equations in the half space in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 165 (2002), 89-159.  doi: 10.1007/s00205-002-0221-x. [8] Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal., 177 (2005), 231-330.  doi: 10.1007/s00205-005-0365-6. [9] Y. I. Kanel', On a model system of equations for one-dimensional gas motion, Diff. Eq., (in Russian), 4 (1968), 721–734. [10] S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications, Proc. Roy. Soc. Edinburgh Sect. A, 106 (1987), 169-194.  doi: 10.1017/S0308210500018308. [11] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, 1983. [12] S. Kawashima and T. Nishida, Global solutions to the initial value problem for the equations of one-dimensional motion of viscous polytropic gases, J. Math. Kyoto Univ., 21 (1981), 825-837.  doi: 10.1215/kjm/1250521915. [13] S. Kawashima and P. C. Zhu, Asymptotic stability of nonlinear wave for the compressible Navier-Stokes equations in the half space, J. Differential Equations, 244 (2008), 3151-3179.  doi: 10.1016/j.jde.2008.01.020. [14] A. V. Kazhikhov, Cauchy problem for viscous gas equations, Sibirsk. Mat. Zh., 23 (1982), 60-64. [15] A. V. Kazhikhov and V. V. Shelukhin, Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  doi: 10.1016/0021-8928(77)90011-9. [16] K. Koike, Long-time behavior of a point mass in a one-dimensional viscous compressible fluid and pointwise estimates of solutions, J. Differential Equations, 271 (2021), 356-413.  doi: 10.1016/j.jde.2020.08.022. [17] D. L. Li, The Green's function of the Navier-Stokes equations for gas dynamics in $\mathbb{R}^3$, Comm. Math. Phys., 257 (2005), 579-619.  doi: 10.1007/s00220-005-1351-4. [18] H.-L. Li, A. Matsumura and G. Zhang, Optimal decay rate of the compressible Navier-Stokes-Poisson system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 196 (2010), 681-713.  doi: 10.1007/s00205-009-0255-4. [19] T. P. Liu and W. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Commun. Math. Phys., 196 (1998), 145-173.  doi: 10.1007/s002200050418. [20] T. P. Liu and Y. N. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Mem. Amer. Math. Soc., 125 (1997), viii+120 pp. doi: 10.1090/memo/0599. [21] A. Matsumura and T. Nishida, Initial boundary value problem for the equations of motion of compressible viscous and heat conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  doi: 10.1007/BF01214738. [22] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 337-342. [23] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  doi: 10.1215/kjm/1250522322. [24] G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418.  doi: 10.1016/0362-546X(85)90001-X. [25] W. K. Wang and Z. G. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions, J. Differ. Equ., 248 (2010), 1617-1636.  doi: 10.1016/j.jde.2010.01.003. [26] Z. G. Wu and W. K. Wang, Pointwise estimates for bipolar compressible Navier-Stokes-Poisson system in dimension three, Arch. Ration. Mech. Anal., 226 (2017), 587-638.  doi: 10.1007/s00205-017-1140-1. [27] Y. N. Zeng, L1 asymptotic behavior of compressible, isentropic, viscous 1-D flow, Comm. Pure Appl. Math., 47 (1994), 1053-1082.  doi: 10.1002/cpa.3160470804. [28] G. Zhang, H.-L. Li and C. Zhu, Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^3$, J. Differential Equations, 250 (2011), 866-891.  doi: 10.1016/j.jde.2010.07.035.
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