December  2017, 10(4): 1205-1233. doi: 10.3934/krm.2017046

Generalized Huygens' principle for a reduced gravity two and a half layer model in dimension three

1. 

Department of Applied Mathematics, Donghua University, Shanghai 201620, China

2. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Weike Wang

Received  September 2016 Revised  December 2016 Published  March 2017

Fund Project: The research of Z.G. Wu was sponsored by Natural Science Foundation of Shanghai (No. 16ZR1402100) and the Fundamental Research Funds for the Central Universities (No. 2232015D3-33). The research of W.K. Wang was supported by Natural Science Foundation of China (No. 11231006)

The Cauchy problem of the reduced gravity two and a half layer model in dimension three isconsidered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit two kinds of the generalized Huygens' waves. It is a significant different phenomenon from the Navier-Stokes system. Lastly, as a byproduct, we also extend $L^2(\mathbb{R}^3)$-decay rate to $L^p(\mathbb{R}^3)$-decay rate with $p>1$.

Citation: Zhigang Wu, Weike Wang. Generalized Huygens' principle for a reduced gravity two and a half layer model in dimension three. Kinetic & Related Models, 2017, 10 (4) : 1205-1233. doi: 10.3934/krm.2017046
References:
[1]

H. B. CuiL. Yao and Z. A. Yao, Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model, Communications on Pure Applied Analysis, 14 (2015), 981-1000. doi: 10.3934/cpaa.2015.14.981. Google Scholar

[2]

R. Duan and C.H. Zhou, On the compactness of the reduced-gravity two-and-a-half layer equations, J. Diff. Eqns., 252 (2012), 3506-3519. doi: 10.1016/j.jde.2011.12.012. Google Scholar

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R. J. DuanS. UkaiT. Yang and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Mathematical Models and Methods in Applied Sciences, 17 (2007), 737-758. doi: 10.1142/S021820250700208X. Google Scholar

[4]

R. J. DuanH. X. LiuS. Ukai and T. Yang, Optimal $L^p-L^q$ convergence rates for the compressible Navier-Stokes equations with potential force, J. Diff. Eqns., 238 (2007), 220-233. doi: 10.1142/S021820250700208X. Google Scholar

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D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana University Mathematics Journal, 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003. Google Scholar

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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

[7]

F. M. HuangJ. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116. doi: 10.1007/s00205-009-0267-0. Google Scholar

[8]

F. M. HuangA. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77. doi: 10.1007/s00205-005-0380-7. Google Scholar

[9]

Y. I. Kanel, Cauchy problem for the equations of gas dynamics with viscosity, Siberian Math. J., 20 (1979), 208-218. Google Scholar

[10]

S. Kawashima, System of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Manetohydrodynamics Ph. D thesis, Kyoto University, Kyoto, 1983.Google Scholar

[11]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358. 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]

T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbb{R}^3$, Commun. Math. Phys., 200 (1999), 621-659. Google Scholar

[14]

H. L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbb{R}^3$, Mathematical Methods in the Applied Sciences, 34 (2011), 670-682. doi: 10.1002/mma.1391. Google Scholar

[15]

T. P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math., 50 (1997), 1113-1182. doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D. Google Scholar

[16]

T. P. Liu and S. E. Noh, Wave propagation for the compressible Navier-Stokes equations, J. Hyperbolic Differ. Eqns., 12 (2015), 385-445. doi: 10.1142/S0219891615500113. Google Scholar

[17]

T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimension, Comm. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418. Google Scholar

[18]

T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465. doi: 10.1007/BF01466726. Google Scholar

[19]

T. P. Liu and S. H. Yu, The Green's function and large-time behavior of solutions for one dimensional Boltzmann equation, Comm. Pure. Appl. Math., 57 (2004), 1543-1608. doi: 10.1002/cpa.20011. Google Scholar

[20]

T. P. Liu and S. H. Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Commun. Pure Appl. Math., 60 (2007), 295-356. doi: 10.1002/cpa.20172. Google Scholar

[21]

T. P. Liu and S. H. Yu, Green's function of Boltzmann equation, 3-D waves, Bullet. Inst. of Math. Academia Sinica, 1 (2006), 1-78. Google Scholar

[22]

T. P. Liu and S. H. Yu, Solving Boltzmann equation, part Ⅰ: Green's function, Bullet. Inst. of Math. Academia Sinica, 6 (2011), 115-243. Google Scholar

[23]

T. P. Liu and S. H. Yu, Dirichlet-Neumann Kernel for hyperbolic-dissipative system in half space, Bullet. Inst. of Math. Academia Sinica, 7 (2012), 477-543. Google Scholar

[24]

T. P. Liu and Y. N. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Amer. Math. Soc., 7125 (1997), ⅷ+120 pp. doi: 10.1090/memo/0599. Google Scholar

[25]

T. P. Liu and Y. N. Zeng, Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc., 7125 (1997), ⅷ+120 pp. doi: 10.1090/memo/1105. Google Scholar

[26]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Jpn. Acad. Ser. A, 55 (1979), 337-342. doi: 10.3792/pjaa.55.337. Google Scholar

[27]

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

[28]

M. Okada and S. Kawashima, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ., 23 (1983), 55-71. doi: 10.1215/kjm/1250521610. Google Scholar

[29]

S. UkaiT. Yang and S. H. Yu, Nonlinear boundary layers of the Boltzmann equation. Ⅰ. Existence, Comm. Math. Phys., 236 (2003), 373-393. doi: 10.1007/s00220-003-0822-8. Google Scholar

[30]

G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation Cambridge University Press, 2006. doi: 10.1017/CBO9780511790447. Google Scholar

[31]

W. K. Wang and Z. G. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions, J. Diff. Eqns., 248 (2010), 1617-1636. doi: 10.1016/j.jde.2010.01.003. Google Scholar

[32]

W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler-Equations with damping in multi-dimensions, J. Diff. Eqns., 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937. Google Scholar

[33]

R. WeiY. Li and Z. A. Yao, Decay of the solution to a reduced gravity two and a half layer equations, Journal of Mathematical Analysis and Applications, 438 (2016), 946-961. doi: 10.1016/j.jmaa.2016.01.081. Google Scholar

[34]

Z. G. Wu and W. K. Wang, Pointwise estimates of solution for non-isentropic Navier-Stokes-Poisson equations in multi-dimensions, Acta Math. Sci., 32 (2012), 1681-1702. doi: 10.1016/S0252-9602(12)60134-9. Google Scholar

[35]

Z. G. Wu and W. K. Wang, Refined pointwise estimates for the Navier-Stokes-Poisson equations, Analysis and Applications, 14 (2016), 739-762. doi: 10.1142/S0219530515500153. Google Scholar

[36]

Z. G. Wu and W. K. Wang, Large time behavior and pointwise estimates for compressible Euler equations with damping, Science China Mathematics, 58 (2015), 1397-1414. doi: 10.1007/s11425-015-5013-5. Google Scholar

[37]

L. YaoZ. L. Li and W. J. Wang, Existence of spherically symmetric solutions for a reduced gravity two-and-a-half layer system, J. Diff. Eqns., 261 (2016), 1637-1668. doi: 10.1016/j.jde.2016.04.012. Google Scholar

[38]

S. H. Yu, Nonlinear wave propagation over a Boltzmann shock profile, J. Amer. Math. Soc., 23 (2010), 1040-1118. doi: 10.1090/S0894-0347-2010-00671-6. Google Scholar

[39]

Y. N. Zeng, $L^1$ asymptotic behavior of compressible isentropic viscous 1-D flow, Commun. Pure Appl. Math., 47 (1994), 1053-1082. doi: 10.1002/cpa.3160470804. Google Scholar

[40]

Y. N. Zeng, Thermal Non-equilibrium flows in three space dimensions, Arch. Rational Mech. Anal., 219 (2016), 27-87. doi: 10.1007/s00205-015-0892-8. Google Scholar

show all references

References:
[1]

H. B. CuiL. Yao and Z. A. Yao, Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model, Communications on Pure Applied Analysis, 14 (2015), 981-1000. doi: 10.3934/cpaa.2015.14.981. Google Scholar

[2]

R. Duan and C.H. Zhou, On the compactness of the reduced-gravity two-and-a-half layer equations, J. Diff. Eqns., 252 (2012), 3506-3519. doi: 10.1016/j.jde.2011.12.012. Google Scholar

[3]

R. J. DuanS. UkaiT. Yang and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential forces, Mathematical Models and Methods in Applied Sciences, 17 (2007), 737-758. doi: 10.1142/S021820250700208X. Google Scholar

[4]

R. J. DuanH. X. LiuS. Ukai and T. Yang, Optimal $L^p-L^q$ convergence rates for the compressible Navier-Stokes equations with potential force, J. Diff. Eqns., 238 (2007), 220-233. doi: 10.1142/S021820250700208X. Google Scholar

[5]

D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana University Mathematics Journal, 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003. Google Scholar

[6]

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

[7]

F. M. HuangJ. Li and A. Matsumura, Asymptotic stability of combination of viscous contact wave with rarefaction waves for one-dimensional compressible Navier-Stokes system, Arch. Ration. Mech. Anal., 197 (2010), 89-116. doi: 10.1007/s00205-009-0267-0. Google Scholar

[8]

F. M. HuangA. Matsumura and Z. P. Xin, Stability of contact discontinuities for the 1-D compressible Navier-Stokes equations, Arch. Ration. Mech. Anal., 179 (2006), 55-77. doi: 10.1007/s00205-005-0380-7. Google Scholar

[9]

Y. I. Kanel, Cauchy problem for the equations of gas dynamics with viscosity, Siberian Math. J., 20 (1979), 208-218. Google Scholar

[10]

S. Kawashima, System of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Manetohydrodynamics Ph. D thesis, Kyoto University, Kyoto, 1983.Google Scholar

[11]

S. Kawashima and A. Matsumura, Asymptotic stability of traveling wave solutions of systems for one-dimensional gas motion, Comm. Math. Phys., 101 (1985), 97-127. doi: 10.1007/BF01212358. 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]

T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in $\mathbb{R}^3$, Commun. Math. Phys., 200 (1999), 621-659. Google Scholar

[14]

H. L. Li and T. Zhang, Large time behavior of isentropic compressible Navier-Stokes system in $\mathbb{R}^3$, Mathematical Methods in the Applied Sciences, 34 (2011), 670-682. doi: 10.1002/mma.1391. Google Scholar

[15]

T. P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math., 50 (1997), 1113-1182. doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D. Google Scholar

[16]

T. P. Liu and S. E. Noh, Wave propagation for the compressible Navier-Stokes equations, J. Hyperbolic Differ. Eqns., 12 (2015), 385-445. doi: 10.1142/S0219891615500113. Google Scholar

[17]

T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimension, Comm. Math. Phys., 196 (1998), 145-173. doi: 10.1007/s002200050418. Google Scholar

[18]

T. P. Liu and Z. P. Xin, Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465. doi: 10.1007/BF01466726. Google Scholar

[19]

T. P. Liu and S. H. Yu, The Green's function and large-time behavior of solutions for one dimensional Boltzmann equation, Comm. Pure. Appl. Math., 57 (2004), 1543-1608. doi: 10.1002/cpa.20011. Google Scholar

[20]

T. P. Liu and S. H. Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Commun. Pure Appl. Math., 60 (2007), 295-356. doi: 10.1002/cpa.20172. Google Scholar

[21]

T. P. Liu and S. H. Yu, Green's function of Boltzmann equation, 3-D waves, Bullet. Inst. of Math. Academia Sinica, 1 (2006), 1-78. Google Scholar

[22]

T. P. Liu and S. H. Yu, Solving Boltzmann equation, part Ⅰ: Green's function, Bullet. Inst. of Math. Academia Sinica, 6 (2011), 115-243. Google Scholar

[23]

T. P. Liu and S. H. Yu, Dirichlet-Neumann Kernel for hyperbolic-dissipative system in half space, Bullet. Inst. of Math. Academia Sinica, 7 (2012), 477-543. Google Scholar

[24]

T. P. Liu and Y. N. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws, Amer. Math. Soc., 7125 (1997), ⅷ+120 pp. doi: 10.1090/memo/0599. Google Scholar

[25]

T. P. Liu and Y. N. Zeng, Shock waves in conservation laws with physical viscosity, Mem. Amer. Math. Soc., 7125 (1997), ⅷ+120 pp. doi: 10.1090/memo/1105. Google Scholar

[26]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Jpn. Acad. Ser. A, 55 (1979), 337-342. doi: 10.3792/pjaa.55.337. Google Scholar

[27]

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

[28]

M. Okada and S. Kawashima, On the equations of one-dimensional motion of compressible viscous fluids, J. Math. Kyoto Univ., 23 (1983), 55-71. doi: 10.1215/kjm/1250521610. Google Scholar

[29]

S. UkaiT. Yang and S. H. Yu, Nonlinear boundary layers of the Boltzmann equation. Ⅰ. Existence, Comm. Math. Phys., 236 (2003), 373-393. doi: 10.1007/s00220-003-0822-8. Google Scholar

[30]

G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation Cambridge University Press, 2006. doi: 10.1017/CBO9780511790447. Google Scholar

[31]

W. K. Wang and Z. G. Wu, Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions, J. Diff. Eqns., 248 (2010), 1617-1636. doi: 10.1016/j.jde.2010.01.003. Google Scholar

[32]

W. K. Wang and T. Yang, The pointwise estimates of solutions for Euler-Equations with damping in multi-dimensions, J. Diff. Eqns., 173 (2001), 410-450. doi: 10.1006/jdeq.2000.3937. Google Scholar

[33]

R. WeiY. Li and Z. A. Yao, Decay of the solution to a reduced gravity two and a half layer equations, Journal of Mathematical Analysis and Applications, 438 (2016), 946-961. doi: 10.1016/j.jmaa.2016.01.081. Google Scholar

[34]

Z. G. Wu and W. K. Wang, Pointwise estimates of solution for non-isentropic Navier-Stokes-Poisson equations in multi-dimensions, Acta Math. Sci., 32 (2012), 1681-1702. doi: 10.1016/S0252-9602(12)60134-9. Google Scholar

[35]

Z. G. Wu and W. K. Wang, Refined pointwise estimates for the Navier-Stokes-Poisson equations, Analysis and Applications, 14 (2016), 739-762. doi: 10.1142/S0219530515500153. Google Scholar

[36]

Z. G. Wu and W. K. Wang, Large time behavior and pointwise estimates for compressible Euler equations with damping, Science China Mathematics, 58 (2015), 1397-1414. doi: 10.1007/s11425-015-5013-5. Google Scholar

[37]

L. YaoZ. L. Li and W. J. Wang, Existence of spherically symmetric solutions for a reduced gravity two-and-a-half layer system, J. Diff. Eqns., 261 (2016), 1637-1668. doi: 10.1016/j.jde.2016.04.012. Google Scholar

[38]

S. H. Yu, Nonlinear wave propagation over a Boltzmann shock profile, J. Amer. Math. Soc., 23 (2010), 1040-1118. doi: 10.1090/S0894-0347-2010-00671-6. Google Scholar

[39]

Y. N. Zeng, $L^1$ asymptotic behavior of compressible isentropic viscous 1-D flow, Commun. Pure Appl. Math., 47 (1994), 1053-1082. doi: 10.1002/cpa.3160470804. Google Scholar

[40]

Y. N. Zeng, Thermal Non-equilibrium flows in three space dimensions, Arch. Rational Mech. Anal., 219 (2016), 27-87. doi: 10.1007/s00205-015-0892-8. Google Scholar

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