-
Previous Article
Averaging of Hamilton-Jacobi equations along divergence-free vector fields
- DCDS Home
- This Issue
-
Next Article
On the asymptotic properties for stationary solutions to the Navier-Stokes equations
Asymptotic stability of viscous contact wave for the inflow problem of the one-dimensional radiative Euler equations
1. | School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan 430023, China |
2. | Hubei Key Laboratory of Mathematical Physics, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China |
3. | Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China |
This paper is devoted to the study of the inflow problem governed by the radiative Euler equations in the one-dimensional half space. We establish the unique global-in-time existence and the asymptotic stability of the viscous contact discontinuity solution. It is different from the case involved with the rarefaction wave for the inflow problem in our previous work [
References:
[1] |
A. M. Blokhin and Yu. L. Trakhinin,
Shock-wave stability for one model of radiation hydrodynamics, J. Appl. Mech. Tech. Phys., 37 (1996), 775-784.
doi: 10.1007/BF02369253. |
[2] |
C. Buet and B. Despres,
Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 85 (2004), 385-418.
doi: 10.1016/S0022-4073(03)00233-4. |
[3] |
J.-F. Coulombel, T. Goudon, P. Lafitte and C. Lin,
Analysis of large amplitude shock profiles for non-equilibrium radiative hydrodynamics: Formation of Zeldovich spikes, Shock Waves, 22 (2012), 181-197.
doi: 10.1007/s00193-012-0368-9. |
[4] |
B. Ducomet, E. Feireisl and S. Nečasova,
On a model in radiation hydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 797-812.
doi: 10.1016/j.anihpc.2011.06.002. |
[5] |
L. Fan, L. Ruan and W. Xiang,
Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1-25.
doi: 10.1016/j.anihpc.2018.03.008. |
[6] |
L. Fan, L. Ruan and W. Xiang,
Asymptotic stability of rarefaction wave for the inflow problem governed by the one-dimensional radiative Euler equations, SIAM J. Math. Anal., 51 (2019), 595-625.
doi: 10.1137/18M1203043. |
[7] |
W. Gao, L. Ruan and C. Zhu,
Decay rates to the planar rarefaction waves for a model system of the radiating gas in $n$-dimensions, J. Differential Equations, 244 (2008), 2614-2640.
doi: 10.1016/j.jde.2008.02.023. |
[8] |
P. Godillon-Lafitte and T. Goudon,
A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, Multiscale Model. Simul., 4 (2005), 1245-1279.
doi: 10.1137/040621041. |
[9] |
H. Hong,
Global stability of viscous contact wave for 1-D compressible Navier-Stokes equations, J. Differential Equations, 252 (2012), 3482-3505.
doi: 10.1016/j.jde.2011.11.015. |
[10] |
F. Huang and X. Li,
Convergence to the rarefaction wave for a model of radiating gas in one-dimension, Acta Math. Appl. Sin. Engl. Ser., 32 (2016), 239-256.
doi: 10.1007/s10255-016-0576-7. |
[11] |
F. 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. |
[12] |
F. Huang, A. Matsumura and Z. 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. |
[13] |
F. Huang, Z. 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. |
[14] |
S. Jiang, F. Li and F. Xie,
Nonrelativistic limit of the compressible Navier-Stokes-Fourier-P1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal., 47 (2015), 3726-3746.
doi: 10.1137/140987596. |
[15] |
S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics, Analysis of Systems of Conservation Laws (Aachen, 1997), Chapman Hall/CRC Monogr. Surv. Pure. Appl. Math., Chapman Hall/CRC, Boca Raton, FL, 99 1999, 87–127.. |
[16] |
S. Kawashima and S. Nishibata,
Shock waves for a model system of a radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117.
doi: 10.1137/S0036141097322169. |
[17] |
S. Kawashima and Y. Tanaka,
Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math., 58 (2004), 211-250.
doi: 10.2206/kyushujm.58.211. |
[18] |
C. Lattanzio, C. Mascia, T. Nguyen, R. Plaza and K. Zumbrun,
Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.
doi: 10.1137/09076026X. |
[19] |
C. Lattanzio, C. Mascia and D. Serre,
Shock waves for radiative hyperbolic-elliptic systems, Indiana Univ. Math. J., 56 (2007), 2601-2640.
doi: 10.1512/iumj.2007.56.3043. |
[20] |
C. Lin,
Asymptotic stability of rarefaction waves in radiative hydrodynamics, Commun. Math. Sci., 9 (2011), 207-223.
|
[21] |
C. Lin, J.-F. Coulombel and T. Goudon,
Shock profiles for non-equilibrium radiating gas, Phys. D, 218 (2006), 83-94.
doi: 10.1016/j.physd.2006.04.012. |
[22] |
C. Lin, J.-F. Coulombel and T. Goudon,
Asymptotic stability of shock profiles in radiative hydrodynamics, C. R. Math. Acad. Sci. Paris, 345 (2007), 625-628.
doi: 10.1016/j.crma.2007.10.029. |
[23] |
T.-P. Liu,
Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767-796.
doi: 10.1002/cpa.3160300605. |
[24] |
R. B. Lowrie, J. E. Morel and J. A. Hittinger,
The coupling of radiation and hydrodynamics, Astrophys. J., 521 (1999), 432-450.
doi: 10.1086/307515. |
[25] |
C. Mascia,
Small, medium and large shock waves for radiative Euler equations, Phys. D, 245 (2013), 46-56.
doi: 10.1016/j.physd.2012.11.008. |
[26] |
T. Nguyen, R. G. Plaza and K. Zumbrun,
Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Phys. D, 239 (2010), 428-453.
doi: 10.1016/j.physd.2010.01.011. |
[27] |
M. Nishikawa and S. Nishibata,
Convergence rates toward the travelling waves for a model system of the radiating gas, Math. Methods Appl. Sci., 30 (2007), 649-663.
doi: 10.1002/mma.800. |
[28] |
M. Ohnawa,
Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities, Kinet. Relat. Models, 5 (2012), 857-872.
doi: 10.3934/krm.2012.5.857. |
[29] |
M. Ohnawa,
$L^\infty$-stability of continuous shock waves in a radiating gas model, SIAM J. Math. Anal., 46 (2014), 2136-2159.
doi: 10.1137/130935252. |
[30] |
X. Qin and Y. Wang,
Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 41 (2009), 2057-2087.
doi: 10.1137/09075425X. |
[31] |
X. Qin and Y. Wang,
Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 43 (2011), 341-366.
doi: 10.1137/100793463. |
[32] |
C. Rohde, W. Wang and F. Xie,
Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: Superposition of rarefaction and contact waves, Commun. Pure Appl. Anal., 12 (2013), 2145-2171.
doi: 10.3934/cpaa.2013.12.2145. |
[33] |
L. Ruan and J. Zhang,
Asymptotic stability of rarefaction wave for hyperbolic-elliptic coupled system in radiating gas, Acta Math. Sci. Ser. B Engl. Ed., 27 (2007), 347-360.
doi: 10.1016/S0252-9602(07)60035-6. |
[34] |
L. Ruan and C. Zhu,
Asymptotic decay toward rarefaction wave for a hyperbolic-elliptic coupled system on half space, J. Partial Differential Equations, 21 (2008), 173-192.
|
[35] |
L. Ruan and C. Zhu,
Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas, J. Differential Equations, 249 (2010), 2076-2110.
doi: 10.1016/j.jde.2010.07.029. |
[36] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[37] |
W. G. Vincenti and C. H. Kruger Jr, Introduction to Physical Gas Dynamics, Wiley, New York, 1965.
doi: 10.1063/1.3047788. |
[38] |
J. Wang and F. Xie,
Singular limit to strong contact discontinuity for a 1D compressible radiation hydrodynamics model, SIAM J. Math. Anal., 43 (2011), 1189-1204.
doi: 10.1137/100792792. |
[39] |
J. Wang and F. Xie,
Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system, J. Differential Equations, 251 (2011), 1030-1055.
doi: 10.1016/j.jde.2011.03.011. |
[40] |
F. Xie,
Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1075-1100.
doi: 10.3934/dcdsb.2012.17.1075. |
show all references
References:
[1] |
A. M. Blokhin and Yu. L. Trakhinin,
Shock-wave stability for one model of radiation hydrodynamics, J. Appl. Mech. Tech. Phys., 37 (1996), 775-784.
doi: 10.1007/BF02369253. |
[2] |
C. Buet and B. Despres,
Asymptotic analysis of fluid models for the coupling of radiation and hydrodynamics, J. Quant. Spectrosc. Radiat. Transfer, 85 (2004), 385-418.
doi: 10.1016/S0022-4073(03)00233-4. |
[3] |
J.-F. Coulombel, T. Goudon, P. Lafitte and C. Lin,
Analysis of large amplitude shock profiles for non-equilibrium radiative hydrodynamics: Formation of Zeldovich spikes, Shock Waves, 22 (2012), 181-197.
doi: 10.1007/s00193-012-0368-9. |
[4] |
B. Ducomet, E. Feireisl and S. Nečasova,
On a model in radiation hydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 797-812.
doi: 10.1016/j.anihpc.2011.06.002. |
[5] |
L. Fan, L. Ruan and W. Xiang,
Asymptotic stability of a composite wave of two viscous shock waves for the one-dimensional radiative Euler equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1-25.
doi: 10.1016/j.anihpc.2018.03.008. |
[6] |
L. Fan, L. Ruan and W. Xiang,
Asymptotic stability of rarefaction wave for the inflow problem governed by the one-dimensional radiative Euler equations, SIAM J. Math. Anal., 51 (2019), 595-625.
doi: 10.1137/18M1203043. |
[7] |
W. Gao, L. Ruan and C. Zhu,
Decay rates to the planar rarefaction waves for a model system of the radiating gas in $n$-dimensions, J. Differential Equations, 244 (2008), 2614-2640.
doi: 10.1016/j.jde.2008.02.023. |
[8] |
P. Godillon-Lafitte and T. Goudon,
A coupled model for radiative transfer: Doppler effects, equilibrium and non equilibrium diffusion asymptotics, Multiscale Model. Simul., 4 (2005), 1245-1279.
doi: 10.1137/040621041. |
[9] |
H. Hong,
Global stability of viscous contact wave for 1-D compressible Navier-Stokes equations, J. Differential Equations, 252 (2012), 3482-3505.
doi: 10.1016/j.jde.2011.11.015. |
[10] |
F. Huang and X. Li,
Convergence to the rarefaction wave for a model of radiating gas in one-dimension, Acta Math. Appl. Sin. Engl. Ser., 32 (2016), 239-256.
doi: 10.1007/s10255-016-0576-7. |
[11] |
F. 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. |
[12] |
F. Huang, A. Matsumura and Z. 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. |
[13] |
F. Huang, Z. 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. |
[14] |
S. Jiang, F. Li and F. Xie,
Nonrelativistic limit of the compressible Navier-Stokes-Fourier-P1 approximation model arising in radiation hydrodynamics, SIAM J. Math. Anal., 47 (2015), 3726-3746.
doi: 10.1137/140987596. |
[15] |
S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics, Analysis of Systems of Conservation Laws (Aachen, 1997), Chapman Hall/CRC Monogr. Surv. Pure. Appl. Math., Chapman Hall/CRC, Boca Raton, FL, 99 1999, 87–127.. |
[16] |
S. Kawashima and S. Nishibata,
Shock waves for a model system of a radiating gas, SIAM J. Math. Anal., 30 (1999), 95-117.
doi: 10.1137/S0036141097322169. |
[17] |
S. Kawashima and Y. Tanaka,
Stability of rarefaction waves for a model system of a radiating gas, Kyushu J. Math., 58 (2004), 211-250.
doi: 10.2206/kyushujm.58.211. |
[18] |
C. Lattanzio, C. Mascia, T. Nguyen, R. Plaza and K. Zumbrun,
Stability of scalar radiative shock profiles, SIAM J. Math. Anal., 41 (2009/10), 2165-2206.
doi: 10.1137/09076026X. |
[19] |
C. Lattanzio, C. Mascia and D. Serre,
Shock waves for radiative hyperbolic-elliptic systems, Indiana Univ. Math. J., 56 (2007), 2601-2640.
doi: 10.1512/iumj.2007.56.3043. |
[20] |
C. Lin,
Asymptotic stability of rarefaction waves in radiative hydrodynamics, Commun. Math. Sci., 9 (2011), 207-223.
|
[21] |
C. Lin, J.-F. Coulombel and T. Goudon,
Shock profiles for non-equilibrium radiating gas, Phys. D, 218 (2006), 83-94.
doi: 10.1016/j.physd.2006.04.012. |
[22] |
C. Lin, J.-F. Coulombel and T. Goudon,
Asymptotic stability of shock profiles in radiative hydrodynamics, C. R. Math. Acad. Sci. Paris, 345 (2007), 625-628.
doi: 10.1016/j.crma.2007.10.029. |
[23] |
T.-P. Liu,
Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws, Comm. Pure Appl. Math., 30 (1977), 767-796.
doi: 10.1002/cpa.3160300605. |
[24] |
R. B. Lowrie, J. E. Morel and J. A. Hittinger,
The coupling of radiation and hydrodynamics, Astrophys. J., 521 (1999), 432-450.
doi: 10.1086/307515. |
[25] |
C. Mascia,
Small, medium and large shock waves for radiative Euler equations, Phys. D, 245 (2013), 46-56.
doi: 10.1016/j.physd.2012.11.008. |
[26] |
T. Nguyen, R. G. Plaza and K. Zumbrun,
Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Phys. D, 239 (2010), 428-453.
doi: 10.1016/j.physd.2010.01.011. |
[27] |
M. Nishikawa and S. Nishibata,
Convergence rates toward the travelling waves for a model system of the radiating gas, Math. Methods Appl. Sci., 30 (2007), 649-663.
doi: 10.1002/mma.800. |
[28] |
M. Ohnawa,
Convergence rates towards the traveling waves for a model system of radiating gas with discontinuities, Kinet. Relat. Models, 5 (2012), 857-872.
doi: 10.3934/krm.2012.5.857. |
[29] |
M. Ohnawa,
$L^\infty$-stability of continuous shock waves in a radiating gas model, SIAM J. Math. Anal., 46 (2014), 2136-2159.
doi: 10.1137/130935252. |
[30] |
X. Qin and Y. Wang,
Stability of wave patterns to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 41 (2009), 2057-2087.
doi: 10.1137/09075425X. |
[31] |
X. Qin and Y. Wang,
Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations, SIAM J. Math. Anal., 43 (2011), 341-366.
doi: 10.1137/100793463. |
[32] |
C. Rohde, W. Wang and F. Xie,
Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: Superposition of rarefaction and contact waves, Commun. Pure Appl. Anal., 12 (2013), 2145-2171.
doi: 10.3934/cpaa.2013.12.2145. |
[33] |
L. Ruan and J. Zhang,
Asymptotic stability of rarefaction wave for hyperbolic-elliptic coupled system in radiating gas, Acta Math. Sci. Ser. B Engl. Ed., 27 (2007), 347-360.
doi: 10.1016/S0252-9602(07)60035-6. |
[34] |
L. Ruan and C. Zhu,
Asymptotic decay toward rarefaction wave for a hyperbolic-elliptic coupled system on half space, J. Partial Differential Equations, 21 (2008), 173-192.
|
[35] |
L. Ruan and C. Zhu,
Asymptotic behavior of solutions to a hyperbolic-elliptic coupled system in multi-dimensional radiating gas, J. Differential Equations, 249 (2010), 2076-2110.
doi: 10.1016/j.jde.2010.07.029. |
[36] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-0873-0. |
[37] |
W. G. Vincenti and C. H. Kruger Jr, Introduction to Physical Gas Dynamics, Wiley, New York, 1965.
doi: 10.1063/1.3047788. |
[38] |
J. Wang and F. Xie,
Singular limit to strong contact discontinuity for a 1D compressible radiation hydrodynamics model, SIAM J. Math. Anal., 43 (2011), 1189-1204.
doi: 10.1137/100792792. |
[39] |
J. Wang and F. Xie,
Asymptotic stability of viscous contact wave for the 1D radiation hydrodynamics system, J. Differential Equations, 251 (2011), 1030-1055.
doi: 10.1016/j.jde.2011.03.011. |
[40] |
F. Xie,
Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1075-1100.
doi: 10.3934/dcdsb.2012.17.1075. |
[1] |
Yue-Jun Peng, Shu Wang. Asymptotic expansions in two-fluid compressible Euler-Maxwell equations with small parameters. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 415-433. doi: 10.3934/dcds.2009.23.415 |
[2] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020450 |
[3] |
Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316 |
[4] |
Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029 |
[5] |
Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 |
[6] |
Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605 |
[7] |
Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159 |
[8] |
Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315 |
[9] |
Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021017 |
[10] |
Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021003 |
[11] |
José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091 |
[12] |
Yueh-Cheng Kuo, Huey-Er Lin, Shih-Feng Shieh. Asymptotic dynamics of hermitian Riccati difference equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020365 |
[13] |
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 |
[14] |
Marcello D'Abbicco, Giovanni Girardi, Giséle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Equipartition of energy for nonautonomous damped wave equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 597-613. doi: 10.3934/dcdss.2020364 |
[15] |
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 |
[16] |
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 |
[17] |
Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 |
[18] |
Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227 |
[19] |
Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021006 |
[20] |
Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 |
2019 Impact Factor: 1.338
Tools
Metrics
Other articles
by authors
[Back to Top]