# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021205
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## 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 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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021205
##### References:
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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] Y. I. Kanel', On a model system of equations for one-dimensional gas motion, Diff. Eq., (in Russian), 4 (1968), 721–734.  Google Scholar [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.  Google Scholar [11] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, 1983. Google Scholar [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.  Google Scholar [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.  Google Scholar [14] A. V. Kazhikhov, Cauchy problem for viscous gas equations, Sibirsk. Mat. Zh., 23 (1982), 60-64.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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