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On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications
Global solutions to a one-dimensional non-conservative two-phase model
1. | University of Stavanger (UiS), 4036 Stavanger |
2. | University of Stavanger, NO-4036 Stavanger, Norway |
3. | School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127 |
References:
[1] |
D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.
doi: 10.1007/s00220-003-0859-8. |
[2] |
D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.
doi: 10.1081/PDE-120020499. |
[3] |
D. Bresch and B. Desjardins, Stabilité de solutions faibles globales pour leséquations de Navier-Stokes compressible avec température, C. R. Math. Acad. Sci., 343 (2006), 219-224.
doi: 10.1016/j.crma.2006.05.016. |
[4] |
D. Bresch, B. Desjardins, J. M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Rational Mech. Anal., 196 (2010), 599-629.
doi: 10.1007/s00205-009-0261-6. |
[5] |
D. Bresch, X. D. Huang and J. Li, Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system, Comm. Math. Phys., 309 (2012), 737-755.
doi: 10.1007/s00220-011-1379-6. |
[6] |
F. Coquel, K. El Amine, E. Godlewki, B. Perthame and P. Rascle, A numerical method using upwind schemes for the resolution of two-phase flows, J. Comput. Phys., 136 (1997), 272-288.
doi: 10.1006/jcph.1997.5730. |
[7] |
Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.
doi: 10.1016/j.matpur.2003.11.004. |
[8] |
S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys., 192 (2003), 175-210.
doi: 10.1016/j.jcp.2003.07.001. |
[9] |
S. Evje and T. Flåtten, Weakly implicit numerical schemes for a two-fluid model, SIAM J. Sci. Comput., 26 (2005), 1449-1484.
doi: 10.1137/030600631. |
[10] |
S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM J. Math. Anal., 43 (2011), 1887-1922.
doi: 10.1137/100813932. |
[11] |
S. Evje and H. Y. Wen, Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operators, Discrete Contin. Dyn. Syst., 33 (2013), 4497-4530.
doi: 10.3934/dcds.2013.33.4497. |
[12] |
A. C. Fowler and P. E. Lisseter, Flooding and flow reversal in annular two-phase flow, SIAM J. Appl. Math., 52 (1992), 15-33.
doi: 10.1137/0152002. |
[13] |
Z. H. Guo, Q. S. Jiu and Z. P. Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427.
doi: 10.1137/070680333. |
[14] |
B.L. Keyfitz, R. Sanders, and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 541-563.
doi: 10.3934/dcdsb.2003.3.541. |
[15] |
H.-L. Li, J. Li and Z. P. Xin, Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations, Commun. Math. Phys., 281 (2008), 401-444.
doi: 10.1007/s00220-008-0495-4. |
[16] |
A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452.
doi: 10.1080/03605300600857079. |
[17] |
A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 39 (2008), 1344-1365.
doi: 10.1137/060658199. |
[18] |
A. Prosperetti and G. Tryggvason, Computational Methods for Multiphase Flow, Corrected paperback reprint of the 2007 original. Cambridge University Press, Cambridge, 2009. |
[19] |
M. D. Thanh, D. Kröner and N. T. Nam, Numerical approximation for a Baer-Nunziato model of two-phase flows, Appl. Numer. Math., 61 (2011), 702-721.
doi: 10.1016/j.apnum.2011.01.004. |
[20] |
L. Yao, H. L. Guo and Z. H. Guo, A note on viscous liquid-gas two-phase flow model with mass-dependent viscosity and vacuum, Nonlinear Anal. Real World Appl., 13 (2012), 2323-2342.
doi: 10.1016/j.nonrwa.2012.02.001. |
show all references
References:
[1] |
D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223.
doi: 10.1007/s00220-003-0859-8. |
[2] |
D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models Korteweg, lubrication, and shallow water systems, Comm. Partial Differential Equations, 28 (2003), 843-868.
doi: 10.1081/PDE-120020499. |
[3] |
D. Bresch and B. Desjardins, Stabilité de solutions faibles globales pour leséquations de Navier-Stokes compressible avec température, C. R. Math. Acad. Sci., 343 (2006), 219-224.
doi: 10.1016/j.crma.2006.05.016. |
[4] |
D. Bresch, B. Desjardins, J. M. Ghidaglia and E. Grenier, Global weak solutions to a generic two-fluid model, Arch. Rational Mech. Anal., 196 (2010), 599-629.
doi: 10.1007/s00205-009-0261-6. |
[5] |
D. Bresch, X. D. Huang and J. Li, Global weak solutions to one-dimensional non-conservative viscous compressible two-phase system, Comm. Math. Phys., 309 (2012), 737-755.
doi: 10.1007/s00220-011-1379-6. |
[6] |
F. Coquel, K. El Amine, E. Godlewki, B. Perthame and P. Rascle, A numerical method using upwind schemes for the resolution of two-phase flows, J. Comput. Phys., 136 (1997), 272-288.
doi: 10.1006/jcph.1997.5730. |
[7] |
Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.
doi: 10.1016/j.matpur.2003.11.004. |
[8] |
S. Evje and T. Flåtten, Hybrid flux-splitting schemes for a common two-fluid model, J. Comput. Phys., 192 (2003), 175-210.
doi: 10.1016/j.jcp.2003.07.001. |
[9] |
S. Evje and T. Flåtten, Weakly implicit numerical schemes for a two-fluid model, SIAM J. Sci. Comput., 26 (2005), 1449-1484.
doi: 10.1137/030600631. |
[10] |
S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM J. Math. Anal., 43 (2011), 1887-1922.
doi: 10.1137/100813932. |
[11] |
S. Evje and H. Y. Wen, Weak solutions of a gas-liquid drift-flux model with general slip law for wellbore operators, Discrete Contin. Dyn. Syst., 33 (2013), 4497-4530.
doi: 10.3934/dcds.2013.33.4497. |
[12] |
A. C. Fowler and P. E. Lisseter, Flooding and flow reversal in annular two-phase flow, SIAM J. Appl. Math., 52 (1992), 15-33.
doi: 10.1137/0152002. |
[13] |
Z. H. Guo, Q. S. Jiu and Z. P. Xin, Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients, SIAM J. Math. Anal., 39 (2008), 1402-1427.
doi: 10.1137/070680333. |
[14] |
B.L. Keyfitz, R. Sanders, and M. Sever, Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow, Discrete Contin. Dyn. Syst. Ser. B, 3 (2003), 541-563.
doi: 10.3934/dcdsb.2003.3.541. |
[15] |
H.-L. Li, J. Li and Z. P. Xin, Vanishing of Vacuum States and Blow-up Phenomena of the Compressible Navier-Stokes Equations, Commun. Math. Phys., 281 (2008), 401-444.
doi: 10.1007/s00220-008-0495-4. |
[16] |
A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Differential Equations, 32 (2007), 431-452.
doi: 10.1080/03605300600857079. |
[17] |
A. Mellet and A. Vasseur, Existence and uniqueness of global strong solutions for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 39 (2008), 1344-1365.
doi: 10.1137/060658199. |
[18] |
A. Prosperetti and G. Tryggvason, Computational Methods for Multiphase Flow, Corrected paperback reprint of the 2007 original. Cambridge University Press, Cambridge, 2009. |
[19] |
M. D. Thanh, D. Kröner and N. T. Nam, Numerical approximation for a Baer-Nunziato model of two-phase flows, Appl. Numer. Math., 61 (2011), 702-721.
doi: 10.1016/j.apnum.2011.01.004. |
[20] |
L. Yao, H. L. Guo and Z. H. Guo, A note on viscous liquid-gas two-phase flow model with mass-dependent viscosity and vacuum, Nonlinear Anal. Real World Appl., 13 (2012), 2323-2342.
doi: 10.1016/j.nonrwa.2012.02.001. |
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