# American Institute of Mathematical Sciences

March  2018, 38(3): 1349-1363. doi: 10.3934/dcds.2018055

## Pointwise wave behavior of the Navier-Stokes equations in half space

 1 Department of Applied Mathematics, Donghua University, Shanghai 201620, China 2 Institute of Natural Sciences and School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Haitao Wang, haitaowang.math@gmail.com.

Received  April 2017 Revised  September 2017 Published  December 2017

Fund Project: Du is supported by NSFC(Grant No. 11526049 and 11671075) and the Fundamental Research Funds for the Central Universities (No. 2232016D-22).

In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays with algebraic rate.

Citation: Linglong Du, Haitao Wang. Pointwise wave behavior of the Navier-Stokes equations in half space. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1349-1363. doi: 10.3934/dcds.2018055
##### References:
 [1] S. J. Deng, W. K. Wang and S.-H. Yu, Green's functions of wave equations in $R^n_+ × R_+$, Arch. Ration. Mech. Anal., 216 (2015), 881-903.  doi: 10.1007/s00205-014-0821-2. [2] S. J. Deng, Initial-boundary value problem for p-system with damping in half space, Nonlinear Analysis, 143 (2016), 193-210.  doi: 10.1016/j.na.2016.05.009. [3] S. J. Deng and S.-H. Yu, Green's function and pointwise convergence for compressible Navier-Stokes equations, Quart. Appl. Math., 75 (2017), 433-503.  doi: 10.1090/qam/1461. [4] L. L. Du, Characteristic half space problem for the Broadwell model, Netw. Heterog. Media, 9 (2014), 97-110.  doi: 10.3934/nhm.2014.9.97. [5] L. L. Du and Z. G. Wu, Solving the non-isentropic Navier-Stokes equations in Odd Space Dimensions: the Green Function Method, J. Math. Phys., 58 (2017), 101506, 38 pp. [6] C.-Y. Lan, H.-E. Lin and S.-H. Yu, The Green's function for the Broadwell model with a transonic boundary, J. Hyperbolic Differ. Equ., 5 (2008), 279-294.  doi: 10.1142/S0219891608001489. [7] T.-P. Liu and S.-H. Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math., 60 (2007), 295-356.  doi: 10.1002/cpa.20172. [8] 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. Mem. Amer. Math. Soc. , 125 (1997), ⅷ+120 pp. [9] T.-P. Liu and Y. N. Zeng, Compressible Navier-Stokes equations with zero heat conductivity, J. Differential Equations, 153 (1999), 225-291.  doi: 10.1006/jdeq.1998.3554. [10] 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. [11] Y. Kagei and T. Kobayashi, On large time behavior of solutions to the Compressible Navier-Stokes Equations in the half space in $R^3$, Arch. Ration. Mech. Anal., 165 (2002), 89-159.  doi: 10.1007/s00205-002-0221-x. [12] 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. [13] H. T. Wang and S.-H. Yu, Algebraic-complex scheme for Dirichlet-Neumann data for parabolic system, Arch. Ration. Mech. Anal., 211 (2014), 1013-1026.  doi: 10.1007/s00205-013-0699-4. [14] Y. Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous 1-D flow, Comm. Pure Appl. Math., 47 (1994), 1053-1082.  doi: 10.1002/cpa.3160470804.

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##### References:
 [1] S. J. Deng, W. K. Wang and S.-H. Yu, Green's functions of wave equations in $R^n_+ × R_+$, Arch. Ration. Mech. Anal., 216 (2015), 881-903.  doi: 10.1007/s00205-014-0821-2. [2] S. J. Deng, Initial-boundary value problem for p-system with damping in half space, Nonlinear Analysis, 143 (2016), 193-210.  doi: 10.1016/j.na.2016.05.009. [3] S. J. Deng and S.-H. Yu, Green's function and pointwise convergence for compressible Navier-Stokes equations, Quart. Appl. Math., 75 (2017), 433-503.  doi: 10.1090/qam/1461. [4] L. L. Du, Characteristic half space problem for the Broadwell model, Netw. Heterog. Media, 9 (2014), 97-110.  doi: 10.3934/nhm.2014.9.97. [5] L. L. Du and Z. G. Wu, Solving the non-isentropic Navier-Stokes equations in Odd Space Dimensions: the Green Function Method, J. Math. Phys., 58 (2017), 101506, 38 pp. [6] C.-Y. Lan, H.-E. Lin and S.-H. Yu, The Green's function for the Broadwell model with a transonic boundary, J. Hyperbolic Differ. Equ., 5 (2008), 279-294.  doi: 10.1142/S0219891608001489. [7] T.-P. Liu and S.-H. Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math., 60 (2007), 295-356.  doi: 10.1002/cpa.20172. [8] 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. Mem. Amer. Math. Soc. , 125 (1997), ⅷ+120 pp. [9] T.-P. Liu and Y. N. Zeng, Compressible Navier-Stokes equations with zero heat conductivity, J. Differential Equations, 153 (1999), 225-291.  doi: 10.1006/jdeq.1998.3554. [10] 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. [11] Y. Kagei and T. Kobayashi, On large time behavior of solutions to the Compressible Navier-Stokes Equations in the half space in $R^3$, Arch. Ration. Mech. Anal., 165 (2002), 89-159.  doi: 10.1007/s00205-002-0221-x. [12] 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. [13] H. T. Wang and S.-H. Yu, Algebraic-complex scheme for Dirichlet-Neumann data for parabolic system, Arch. Ration. Mech. Anal., 211 (2014), 1013-1026.  doi: 10.1007/s00205-013-0699-4. [14] Y. Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous 1-D flow, Comm. Pure Appl. Math., 47 (1994), 1053-1082.  doi: 10.1002/cpa.3160470804.
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