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

[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

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

[1]

Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267

[2]

Haibo Cui, Lei Yao, Zheng-An Yao. Global existence and optimal decay rates of solutions to a reduced gravity two and a half layer model. Communications on Pure & Applied Analysis, 2015, 14 (3) : 981-1000. doi: 10.3934/cpaa.2015.14.981

[3]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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 & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[9]

Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020139

[10]

Azniv Kasparian, Ivan Marinov. Duursma's reduced polynomial. Advances in Mathematics of Communications, 2017, 11 (4) : 647-669. doi: 10.3934/amc.2017048

[11]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487

[12]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098

[13]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767

[14]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657

[15]

Andrey Tsiganov. Poisson structures for two nonholonomic systems with partially reduced symmetries. Journal of Geometric Mechanics, 2014, 6 (3) : 417-440. doi: 10.3934/jgm.2014.6.417

[16]

Torsten Lindström. Discrete models and Fisher's maximum principle in ecology. Conference Publications, 2003, 2003 (Special) : 571-579. doi: 10.3934/proc.2003.2003.571

[17]

Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021

[18]

Arjuna Flenner, Gary A. Hewer, Charles S. Kenney. Two dimensional histogram analysis using the Helmholtz principle. Inverse Problems & Imaging, 2008, 2 (4) : 485-525. doi: 10.3934/ipi.2008.2.485

[19]

Claudio Giorgi, Diego Grandi, Vittorino Pata. On the Green-Naghdi Type III heat conduction model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2133-2143. doi: 10.3934/dcdsb.2014.19.2133

[20]

Hui Yin, Huijiang Zhao. Nonlinear stability of boundary layer solutions for generalized Benjamin-Bona-Mahony-Burgers equation in the half space. Kinetic & Related Models, 2009, 2 (3) : 521-550. doi: 10.3934/krm.2009.2.521

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (11)
  • HTML views (75)
  • Cited by (0)

Other articles
by authors

[Back to Top]