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 & Continuous Dynamical Systems - A, 2018, 38 (3) : 1349-1363. doi: 10.3934/dcds.2018055
References:
[1]

S. J. DengW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

C.-Y. LanH.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. J. DengW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

C.-Y. LanH.-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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[2]

Chang-Yeol Jung, Roger Temam. Interaction of boundary layers and corner singularities. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 315-339. doi: 10.3934/dcds.2009.23.315

[3]

Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $ \mathbb{R}^n_+ $. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033

[4]

Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282

[5]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004

[6]

Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021027

[7]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571

[8]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[9]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127

[10]

Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349

[11]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[12]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[13]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[14]

Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230

[15]

Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333

[16]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399

[17]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[18]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387

[19]

Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021011

[20]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (130)
  • HTML views (173)
  • Cited by (2)

Other articles
by authors

[Back to Top]