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The algebraic representation for high order solution of Sasa-Satsuma equation
Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure
1. | Department of Mathematics, City University of Hong Kong, Hong Kong, China |
2. | School of Mathematical Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China |
3. | Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong |
References:
[1] |
R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784.
doi: 10.1090/S0894-0347-2014-00813-4. |
[2] |
J. W. Barrett and E. Süli, Reflections on Dubinskiĭs nonlinear compact embedding theorem, Publ. Inst. Math., 91 (2012), 95-110.
doi: 10.2298/PIM1205095B. |
[3] |
R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, Z. Angew. Math. Mech., 80 (2000), 733-744.
doi: 10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L. |
[4] |
W. E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.), 16 (2000), 207-218.
doi: 10.1007/s101140000034. |
[5] |
C.-J. Liu, Y.-G. Wang and T. Yang, A well-posedness Theory for the Prandtl equations in three space variables, arXiv:1405.5308v2, 2014. |
[6] |
C.-J. Liu, Y.-G. Wang and T. Yang, On the ill-posedness of the Prandtl equations in three space dimensions, Arch. Rational Mech. Anal., 220 (2016), 83-108.
doi: 10.1007/s00205-015-0927-1. |
[7] |
N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68 (2015), 1683-1741.
doi: 10.1002/cpa.21595. |
[8] |
F. K. Moore, Three-dimensional boundary layer theory, Adv. Appl. Mech., 4 (1956), 159-228.
doi: 10.1016/S0065-2156(08)70373-9. |
[9] |
O. A. Oleinik, On the properties of solutions of some elliptic boundary value problems, Matem. Sb., 30 (1952), 695-702. |
[10] |
O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, 1999. |
[11] |
L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner Reibung, in Verh. Int. Math. Kongr., Heidelberg, Germany 1904, Teubner, Germany 1905, 484-494. |
[12] |
M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space, I. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461; II. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491.
doi: 10.1007/s002200050305. |
[13] |
Z. P. Xin, Viscous boundary layers and their stability (I), J. Partial Differential Equations, 11 (1998), 97-124. |
[14] |
Z. P. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. in Math., 181 (2004), 88-133.
doi: 10.1016/S0001-8708(03)00046-X. |
show all references
References:
[1] |
R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang, Well-posedness of the Prandtl equation in Sobolev spaces, J. Amer. Math. Soc., 28 (2015), 745-784.
doi: 10.1090/S0894-0347-2014-00813-4. |
[2] |
J. W. Barrett and E. Süli, Reflections on Dubinskiĭs nonlinear compact embedding theorem, Publ. Inst. Math., 91 (2012), 95-110.
doi: 10.2298/PIM1205095B. |
[3] |
R. E. Caflisch and M. Sammartino, Existence and singularities for the Prandtl boundary layer equations, Z. Angew. Math. Mech., 80 (2000), 733-744.
doi: 10.1002/1521-4001(200011)80:11/12<733::AID-ZAMM733>3.0.CO;2-L. |
[4] |
W. E, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation, Acta Math. Sin. (Engl. Ser.), 16 (2000), 207-218.
doi: 10.1007/s101140000034. |
[5] |
C.-J. Liu, Y.-G. Wang and T. Yang, A well-posedness Theory for the Prandtl equations in three space variables, arXiv:1405.5308v2, 2014. |
[6] |
C.-J. Liu, Y.-G. Wang and T. Yang, On the ill-posedness of the Prandtl equations in three space dimensions, Arch. Rational Mech. Anal., 220 (2016), 83-108.
doi: 10.1007/s00205-015-0927-1. |
[7] |
N. Masmoudi and T. K. Wong, Local-in-time existence and uniqueness of solutions to the Prandtl equations by energy methods, Comm. Pure Appl. Math., 68 (2015), 1683-1741.
doi: 10.1002/cpa.21595. |
[8] |
F. K. Moore, Three-dimensional boundary layer theory, Adv. Appl. Mech., 4 (1956), 159-228.
doi: 10.1016/S0065-2156(08)70373-9. |
[9] |
O. A. Oleinik, On the properties of solutions of some elliptic boundary value problems, Matem. Sb., 30 (1952), 695-702. |
[10] |
O. A. Oleinik and V. N. Samokhin, Mathematical Models in Boundary Layer Theory, Chapman and Hall/CRC, 1999. |
[11] |
L. Prandtl, Über flüssigkeitsbewegungen bei sehr kleiner Reibung, in Verh. Int. Math. Kongr., Heidelberg, Germany 1904, Teubner, Germany 1905, 484-494. |
[12] |
M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equations on a half-space, I. Existence for Euler and Prandtl equations, Comm. Math. Phys., 192 (1998), 433-461; II. Construction of the Navier-Stokes solution, Comm. Math. Phys., 192 (1998), 463-491.
doi: 10.1007/s002200050305. |
[13] |
Z. P. Xin, Viscous boundary layers and their stability (I), J. Partial Differential Equations, 11 (1998), 97-124. |
[14] |
Z. P. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. in Math., 181 (2004), 88-133.
doi: 10.1016/S0001-8708(03)00046-X. |
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