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Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space
Stability of the composite wave for the inflow problem on the micropolar fluid model
1. | School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China |
2. | School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China |
In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in half line $\mathbb{R}_{+}:=(0, \infty).$ Inspired by the relationship between the micropolar fluid model and Navier-Stokes system, we can prove that the composite wave consisting of the subsonic BL-solution, the contact wave, and the rarefaction wave for the inflow problem on micropolar fluid model is time-asymptotically stable. Meanwhile, we obtain the global existence of solutions based on the basic energy method.
References:
[1] |
M. T. Chen,
Global strong solutions for the viscous, micropolar, compressible flow, J. Partial Differ. Equ., 24 (2011), 158-164.
|
[2] |
M. T. Chen,
Blowup criterion for viscous compressible micropolar fluids with vacuum, Nonlinear Anal., Real World Appl., 13 (2012), 850-859.
doi: 10.1016/j.nonrwa.2011.08.021. |
[3] |
M. T. Chen, B. Huang and J. W. Zhang,
Blowup criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum, Nonlinear Anal., 79 (2013), 1-11.
doi: 10.1016/j.na.2012.10.013. |
[4] |
M. T. Chen, X. Y. Xu and J. W. Zhang,
Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum, Commun. Math. Sci., 13 (2015), 225-247.
doi: 10.4310/CMS.2015.v13.n1.a11. |
[5] |
H. B. Cui and H. Y. Yin,
Stationary solutions to the one-dimensional micropolar fluid model in a half line: existence, stability and convergence rate, J. Math. Anal. Appl., 449 (2017), 464-489.
doi: 10.1016/j.jmaa.2016.11.065. |
[6] |
A. C. Eringen,
Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
|
[7] |
J. Goodman,
Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344.
doi: 10.1007/BF00276840. |
[8] |
F. M. Huang and X. H. Qin,
Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation, J. Differential Equations, 246 (2009), 4077-4096.
doi: 10.1016/j.jde.2009.01.017. |
[9] |
F. M. Huang, J. 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. |
[10] |
F. M. Huang, A. Matsumura and X. D. Shi,
A gas-solid free boundary problem for compressible viscous gas, SIAM J. Math. Anal., 34 (2003), 1331-1355.
doi: 10.1137/S0036141002403730. |
[11] |
F. M. Huang, A. 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. |
[12] |
F. M. Huang, Z. P. Xin and T. Yang,
Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.
doi: 10.1016/j.aim.2008.06.014. |
[13] |
S. Kawashima, S. Nishibata and P. C. Zhu,
Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500.
doi: 10.1007/s00220-003-0909-2. |
[14] |
Q. Q. Liu and H. Y. Yin,
Stability of contact discontinuity for 1-D compressible viscous micropolar fluid model, Nonlinear Anal., 149 (2017), 41-55.
doi: 10.1016/j.na.2016.10.009. |
[15] |
Q. Q. Liu and P. X. Zhang,
Optimal time decay of the compressible micropolar fluids, J. Differential Equations, 260 (2016), 7634-7661.
doi: 10.1016/j.jde.2016.01.037. |
[16] |
T. P. Liu and Z. P. Xin,
Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.
|
[17] |
[0-8176-4008-8] G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Baston, Inc. , Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[18] |
A. Matsumura,
Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Methods Appl. Anal., 8 (2001), 645-666.
doi: 10.4310/MAA.2001.v8.n4.a14. |
[19] |
A. Matsumura and M. Mei,
Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146 (1999), 1-22.
doi: 10.1007/s002050050134. |
[20] |
A. Matsumura and K. Nishihara,
Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys., 222 (2001), 449-474.
doi: 10.1007/s002200100517. |
[21] |
N. Mujaković,
One-dimensional flow of a compressible viscous micropolar fluid: regularity of the solution, Rad. Mat., 10 (2001), 181-193.
|
[22] |
N. Mujaković,
Global in time estimates for one-dimensional compressible viscous micropolar fluid model, Glas. Mat. Ser.Ⅲ, 40 (2005), 103-120.
doi: 10.3336/gm.40.1.10. |
[23] |
N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: stabilization of the solution, Proceedings of the Conference on Applied Mathematics and Scientific Computing, Springer, Dordrecht, 2005,253-262.
doi: 10.1007/1-4020-3197-1_18. |
[24] |
N. Mujaković, Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: regularity of the solution, Bound. Value Probl. , Article ID 189748 (2008). |
[25] |
N. Mujaković, One-dimensional compressible viscous micropolar fluid model: stabilization of the solution for the Cauchy problem, Bound. Value Probl. , Article ID 796065 (2010). |
[26] |
T. Nakamura, S. Nishibata and T. Yuge,
Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111.
doi: 10.1016/j.jde.2007.06.016. |
[27] |
T. Nakamura and S. Nishibata,
Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas, J. Hyperbolic Differ. Equ., 8 (2011), 651-670.
doi: 10.1142/S0219891611002524. |
[28] |
B. Nowakowski,
Large time existence of strong solutions to micropolar equations in cylindrical domains, Nonlinear Anal., Real World Appl., 14 (2013), 635-660.
doi: 10.1016/j.nonrwa.2012.07.023. |
[29] |
Y. Qin, T. Wang and G. Hu,
The Cauchy problem for a 1D compressible viscous micropolar fluid model: analysis of the stabilization and the regularity, Nonlinear Anal., Real World Appl., 13 (2012), 1010-1029.
doi: 10.1016/j.nonrwa.2010.10.023. |
[30] |
X. H. Qin and Y. Wang,
Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 5 (2009), 2057-2087.
doi: 10.1137/09075425X. |
show all references
References:
[1] |
M. T. Chen,
Global strong solutions for the viscous, micropolar, compressible flow, J. Partial Differ. Equ., 24 (2011), 158-164.
|
[2] |
M. T. Chen,
Blowup criterion for viscous compressible micropolar fluids with vacuum, Nonlinear Anal., Real World Appl., 13 (2012), 850-859.
doi: 10.1016/j.nonrwa.2011.08.021. |
[3] |
M. T. Chen, B. Huang and J. W. Zhang,
Blowup criterion for the three-dimensional equations of compressible viscous micropolar fluids with vacuum, Nonlinear Anal., 79 (2013), 1-11.
doi: 10.1016/j.na.2012.10.013. |
[4] |
M. T. Chen, X. Y. Xu and J. W. Zhang,
Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum, Commun. Math. Sci., 13 (2015), 225-247.
doi: 10.4310/CMS.2015.v13.n1.a11. |
[5] |
H. B. Cui and H. Y. Yin,
Stationary solutions to the one-dimensional micropolar fluid model in a half line: existence, stability and convergence rate, J. Math. Anal. Appl., 449 (2017), 464-489.
doi: 10.1016/j.jmaa.2016.11.065. |
[6] |
A. C. Eringen,
Theory of micropolar fluids, J. Math. Mech., 16 (1966), 1-18.
|
[7] |
J. Goodman,
Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344.
doi: 10.1007/BF00276840. |
[8] |
F. M. Huang and X. H. Qin,
Stability of boundary layer and rarefaction wave to an outflow problem for compressible Navier-Stokes equations under large perturbation, J. Differential Equations, 246 (2009), 4077-4096.
doi: 10.1016/j.jde.2009.01.017. |
[9] |
F. M. Huang, J. 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. |
[10] |
F. M. Huang, A. Matsumura and X. D. Shi,
A gas-solid free boundary problem for compressible viscous gas, SIAM J. Math. Anal., 34 (2003), 1331-1355.
doi: 10.1137/S0036141002403730. |
[11] |
F. M. Huang, A. 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. |
[12] |
F. M. Huang, Z. P. Xin and T. Yang,
Contact discontinuity with general perturbations for gas motions, Adv. Math., 219 (2008), 1246-1297.
doi: 10.1016/j.aim.2008.06.014. |
[13] |
S. Kawashima, S. Nishibata and P. C. Zhu,
Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys., 240 (2003), 483-500.
doi: 10.1007/s00220-003-0909-2. |
[14] |
Q. Q. Liu and H. Y. Yin,
Stability of contact discontinuity for 1-D compressible viscous micropolar fluid model, Nonlinear Anal., 149 (2017), 41-55.
doi: 10.1016/j.na.2016.10.009. |
[15] |
Q. Q. Liu and P. X. Zhang,
Optimal time decay of the compressible micropolar fluids, J. Differential Equations, 260 (2016), 7634-7661.
doi: 10.1016/j.jde.2016.01.037. |
[16] |
T. P. Liu and Z. P. Xin,
Nonlinear stability of rarefaction waves for compressible Navier-Stokes equations, Comm. Math. Phys., 118 (1988), 451-465.
|
[17] |
[0-8176-4008-8] G. Lukaszewicz, Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Baston, Inc. , Boston, MA, 1999.
doi: 10.1007/978-1-4612-0641-5. |
[18] |
A. Matsumura,
Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Methods Appl. Anal., 8 (2001), 645-666.
doi: 10.4310/MAA.2001.v8.n4.a14. |
[19] |
A. Matsumura and M. Mei,
Convergence to travelling fronts of solutions of the p-system with viscosity in the presence of a boundary, Arch. Ration. Mech. Anal., 146 (1999), 1-22.
doi: 10.1007/s002050050134. |
[20] |
A. Matsumura and K. Nishihara,
Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Comm. Math. Phys., 222 (2001), 449-474.
doi: 10.1007/s002200100517. |
[21] |
N. Mujaković,
One-dimensional flow of a compressible viscous micropolar fluid: regularity of the solution, Rad. Mat., 10 (2001), 181-193.
|
[22] |
N. Mujaković,
Global in time estimates for one-dimensional compressible viscous micropolar fluid model, Glas. Mat. Ser.Ⅲ, 40 (2005), 103-120.
doi: 10.3336/gm.40.1.10. |
[23] |
N. Mujaković, One-dimensional flow of a compressible viscous micropolar fluid: stabilization of the solution, Proceedings of the Conference on Applied Mathematics and Scientific Computing, Springer, Dordrecht, 2005,253-262.
doi: 10.1007/1-4020-3197-1_18. |
[24] |
N. Mujaković, Nonhomogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: regularity of the solution, Bound. Value Probl. , Article ID 189748 (2008). |
[25] |
N. Mujaković, One-dimensional compressible viscous micropolar fluid model: stabilization of the solution for the Cauchy problem, Bound. Value Probl. , Article ID 796065 (2010). |
[26] |
T. Nakamura, S. Nishibata and T. Yuge,
Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line, J. Differential Equations, 241 (2007), 94-111.
doi: 10.1016/j.jde.2007.06.016. |
[27] |
T. Nakamura and S. Nishibata,
Stationary wave associated with an inflow problem in the half line for viscous heat-conductive gas, J. Hyperbolic Differ. Equ., 8 (2011), 651-670.
doi: 10.1142/S0219891611002524. |
[28] |
B. Nowakowski,
Large time existence of strong solutions to micropolar equations in cylindrical domains, Nonlinear Anal., Real World Appl., 14 (2013), 635-660.
doi: 10.1016/j.nonrwa.2012.07.023. |
[29] |
Y. Qin, T. Wang and G. Hu,
The Cauchy problem for a 1D compressible viscous micropolar fluid model: analysis of the stabilization and the regularity, Nonlinear Anal., Real World Appl., 13 (2012), 1010-1029.
doi: 10.1016/j.nonrwa.2010.10.023. |
[30] |
X. H. Qin and Y. Wang,
Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 5 (2009), 2057-2087.
doi: 10.1137/09075425X. |
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