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On the entropic property of the Ellipsoidal Statistical model with the prandtl number below 2/3
Global existence theorem for a model governing the motion of two cell populations
Department of Mathematics & Statistics, Mississippi State University, Mississippi State, MS 39762, USA |
This article is concerned with the existence of a weak solution to the initial boundary problem for a cross-diffusion system which arises in the study of two cell population growth. The mathematical challenge is due to the fact that the coefficient matrix is non-symmetric and degenerate in the sense that its determinant is $ 0 $. The existence assertion is established by exploring the fact that the total population density satisfies a porous media equation.
References:
| [1] |
M. Bertsch, M. E. Gurtin and D. Hilhorst, On interacting populations that disperse to avoid crowding: the case of equal dispersal velocities, Nonlinear Anal., 11 (1987), 493-499.
doi: 10.1016/0362-546X(87)90067-8. |
| [2] |
M. Bertsch, M. E. Gurtin, D. Hilhorst and L. A. Peletier,
On interacting populations that disperse to avoid crowding: Preservation of segregation, J. Math. Biology, 23 (1985), 1-13.
doi: 10.1007/BF00276555. |
| [3] |
F. Bubba, B. Perthame, C. Pouchol and M. Schmidtchen,
Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues, Arch. Rational Mech. Anal., 236 (2020), 735-766.
doi: 10.1007/s00205-019-01479-1. |
| [4] |
H. Byrne and M. A. J. Chaplain,
Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Mathematical and Computer Modelling, 24 (1996), 1-17.
doi: 10.1016/S0895-7177(96)00174-4. |
| [5] |
H. Byrne and D. Drasdo,
Individual-based and continuum models of growing cell populations: A comparison, Journal of mathematical biology, 58 (2009), 657-687.
doi: 10.1007/s00285-008-0212-0. |
| [6] |
J. A. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen,
Splitting schemes & segregation in reaction-(cross-)diffusion systems, SIAM J. Math. Anal., 50 (2018), 5695-5718.
doi: 10.1137/17M1158379. |
| [7] |
X. Chen, E. S. Daus and A. Jüngel,
Global existence analysis of cross-diffusion population systems for multiple species, Arch. Ration. Mech. Anal., 227 (2018), 715-747.
doi: 10.1007/s00205-017-1172-6. |
| [8] |
X. Chen and A. Jüngel, When do cross-diffusion systems have an entropy structure? arXiv: 1908.06873, [math.AP], 2019. |
| [9] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
| [10] |
L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS #74, American Mathematical Society, 1990. Third printing, 2002.
doi: 10.1090/cbms/074. |
| [11] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
| [12] |
M. E. Gurtin and A. C. Pipkin,
A note on interacting populations that disperse to avoid crowding, Quarterly Appl. Math., 42 (1984), 87-94.
doi: 10.1090/qam/736508. |
| [13] |
P. Gwiazda, B. Perthame and A. Świerczewska-Gwiazdak,
A two species hyperbolic-parabolic model of tissue growth, Comm. Partial Differential Equations, 44 (2019), 1605-1618.
doi: 10.1080/03605302.2019.1650064. |
| [14] |
A. Jüngel,
The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.
doi: 10.1088/0951-7715/28/6/1963. |
| [15] |
Q. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Tran. Math. Monographs, Vol. 23, AMS, Providence, RI, 1968. |
| [16] |
T. Lorenzi, A. Lorz and B. Perthame,
On interfaces between cell populations with different mobilities, Kinetic and Related Models, 10 (2017), 299-311.
doi: 10.3934/krm.2017012. |
| [17] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
| [18] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001.
doi: 10.1090/chel/343. |
show all references
References:
| [1] |
M. Bertsch, M. E. Gurtin and D. Hilhorst, On interacting populations that disperse to avoid crowding: the case of equal dispersal velocities, Nonlinear Anal., 11 (1987), 493-499.
doi: 10.1016/0362-546X(87)90067-8. |
| [2] |
M. Bertsch, M. E. Gurtin, D. Hilhorst and L. A. Peletier,
On interacting populations that disperse to avoid crowding: Preservation of segregation, J. Math. Biology, 23 (1985), 1-13.
doi: 10.1007/BF00276555. |
| [3] |
F. Bubba, B. Perthame, C. Pouchol and M. Schmidtchen,
Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues, Arch. Rational Mech. Anal., 236 (2020), 735-766.
doi: 10.1007/s00205-019-01479-1. |
| [4] |
H. Byrne and M. A. J. Chaplain,
Modelling the role of cell-cell adhesion in the growth and development of carcinomas, Mathematical and Computer Modelling, 24 (1996), 1-17.
doi: 10.1016/S0895-7177(96)00174-4. |
| [5] |
H. Byrne and D. Drasdo,
Individual-based and continuum models of growing cell populations: A comparison, Journal of mathematical biology, 58 (2009), 657-687.
doi: 10.1007/s00285-008-0212-0. |
| [6] |
J. A. Carrillo, S. Fagioli, F. Santambrogio and M. Schmidtchen,
Splitting schemes & segregation in reaction-(cross-)diffusion systems, SIAM J. Math. Anal., 50 (2018), 5695-5718.
doi: 10.1137/17M1158379. |
| [7] |
X. Chen, E. S. Daus and A. Jüngel,
Global existence analysis of cross-diffusion population systems for multiple species, Arch. Ration. Mech. Anal., 227 (2018), 715-747.
doi: 10.1007/s00205-017-1172-6. |
| [8] |
X. Chen and A. Jüngel, When do cross-diffusion systems have an entropy structure? arXiv: 1908.06873, [math.AP], 2019. |
| [9] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
| [10] |
L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS #74, American Mathematical Society, 1990. Third printing, 2002.
doi: 10.1090/cbms/074. |
| [11] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
| [12] |
M. E. Gurtin and A. C. Pipkin,
A note on interacting populations that disperse to avoid crowding, Quarterly Appl. Math., 42 (1984), 87-94.
doi: 10.1090/qam/736508. |
| [13] |
P. Gwiazda, B. Perthame and A. Świerczewska-Gwiazdak,
A two species hyperbolic-parabolic model of tissue growth, Comm. Partial Differential Equations, 44 (2019), 1605-1618.
doi: 10.1080/03605302.2019.1650064. |
| [14] |
A. Jüngel,
The boundedness-by-entropy method for cross-diffusion systems, Nonlinearity, 28 (2015), 1963-2001.
doi: 10.1088/0951-7715/28/6/1963. |
| [15] |
Q. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Tran. Math. Monographs, Vol. 23, AMS, Providence, RI, 1968. |
| [16] |
T. Lorenzi, A. Lorz and B. Perthame,
On interfaces between cell populations with different mobilities, Kinetic and Related Models, 10 (2017), 299-311.
doi: 10.3934/krm.2017012. |
| [17] |
J. Simon,
Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
| [18] |
R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001.
doi: 10.1090/chel/343. |
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