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. 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.

[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. 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.

[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.

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.

[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. 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.

[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.

[1]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001

[2]

Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2475-2487. doi: 10.3934/dcdss.2020139

[3]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks and Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007

[4]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791

[5]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307

[6]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861

[7]

Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075

[8]

Belkacem Said-Houari, Flávio A. Falcão Nascimento. Global existence and nonexistence for the viscoelastic wave equation with nonlinear boundary damping-source interaction. Communications on Pure and Applied Analysis, 2013, 12 (1) : 375-403. doi: 10.3934/cpaa.2013.12.375

[9]

Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control and Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015

[10]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure and Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[11]

Seick Kim, Longjuan Xu. Green's function for second order parabolic equations with singular lower order coefficients. Communications on Pure and Applied Analysis, 2022, 21 (1) : 1-21. doi: 10.3934/cpaa.2021164

[12]

David Hoff. Pointwise bounds for the Green's function for the Neumann-Laplace operator in $ \text{R}^3 $. Kinetic and Related Models, 2022, 15 (4) : 535-550. doi: 10.3934/krm.2021037

[13]

M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473

[14]

Vyacheslav A. Trofimov, Evgeny M. Trykin. A new way for decreasing of amplitude of wave reflected from artificial boundary condition for 1D nonlinear Schrödinger equation. Conference Publications, 2015, 2015 (special) : 1070-1078. doi: 10.3934/proc.2015.1070

[15]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497

[16]

Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253

[17]

Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367

[18]

Sergei Avdonin, Jeff Park, Luz de Teresa. The Kalman condition for the boundary controllability of coupled 1-d wave equations. Evolution Equations and Control Theory, 2020, 9 (1) : 255-273. doi: 10.3934/eect.2020005

[19]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179

[20]

Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure and Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (272)
  • HTML views (182)
  • Cited by (2)

Other articles
by authors

[Back to Top]