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Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain
Existence, uniqueness, stability and asymptotic behavior of solutions for a mathematical model of atherosclerosis
1. | Uni-CV, Cabo Verde and CEMAT, IST, Universidade de Lisboa, 1049-001 Lisbon, Portugal |
2. | Department of Mathematics and CEMAT/IST, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisboa |
3. | Department of Mathematics and CEMAT/IST, Faculty of Sciences and Technology, University of Algarve, Campus de Gambelas 8005-139 Faro, Portugal |
4. | Dept Math and CEMAT, IST, Universidade de Lisboa, 1049-001 Lisbon, Portugal |
References:
[1] |
N. F. Britton, Reaction-Diffusion Equations and their Applications to Biology, Academic Press Inc., London, 1986. |
[2] |
V. Calvez, A. Ebde, N. Meunier and A. Raoult, Mathematical and numerical modeling of the atherosclerotic plaque formation, ESAIM Proceedings, 28 (2009), 1-12.
doi: 10.1051/proc/2009036. |
[3] |
V. Calvez, J. Houot, N. Meunier, A. Raoult and G. Rusnakova, Mathematical and numerical modeling of early atherosclerotic lesions, ESAIM Proceedings, 30 (2010), 1-14.
doi: 10.1051/proc/2010002. |
[4] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin Heidelberg, 1979. |
[5] |
A. Friedman, Partial Differential Equations of Parabolic Type, R.E. Krieger Pub. Co., 1983. |
[6] |
H. Daniel, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin Heidelberg, 1981. |
[7] |
N. Filipovic, D. Nikolic, I. Saveljic, Z. Milosevic, T. Exarchos, G. Pelosi and O. Parodi, Computer simulation of three-dimensional plaque formation and progression in the coronary artery, Elsevier, Computers & Fluids, 88 (2013), 826-833.
doi: 10.1016/j.compfluid.2013.07.006. |
[8] |
W. Hao and A. Friedman, The LDL-HDL profile determines the risk of atherosclerosis- a mathematical model, PLoS ONE, 9 (2014), e90497.
doi: 10.1371/journal.pone.0090497. |
[9] |
N. El Khatib, S. Genieys and V. Volpert, Atherosclerosis initiation modeled as an inflammatory process, Math. Model Nat. Phenom., 2 (2007), 126-141.
doi: 10.1051/mmnp:2008022. |
[10] |
N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Mathematical modeling of atherosclerosis as an inflammatory disease, Phil. Trans. R. Soc. A, 367 (2009), 4877-4886.
doi: 10.1098/rsta.2009.0142. |
[11] |
N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Reaction-diffusion model of atherosclerosis development, J. Math. Biol., 65 (2012), 349-374.
doi: 10.1007/s00285-011-0461-1. |
[12] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Math. Society, 1996.
doi: 10.1090/gsm/012. |
[13] |
O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, American Math. Soc., 1968. |
[14] |
B. Liu and D. Tang, Computer simulations of atherosclerosis plaque growth in coronary arteries, Mol. Cell. Biomech., 7 (2010), 193-202. |
[15] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[16] |
C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Elsevier Science Ltd, Nonlinear Analysis, 26 (1996), 1889-1903.
doi: 10.1016/0362-546X(95)00058-4. |
[17] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[18] |
R. Ross, Atherosclerosis - an inflammatory disease, Massachussets Medical Soc., 340 (1999), 115-126. |
[19] |
F. Rothe, Global Solutions of Reaction-Diffusion System, Springer-Verlag, Berlin Heidelberg, 1984. |
[20] |
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic bounded value problems, Indiana University Math. Journal, 21 (1972), 979-1000. |
[21] |
T. Silva, A. Sequeira, R. Santos and J. Tiago, Mathematical modeling of atherosclerotic plaque formation coupled with a non-Newtonian model of blood flow, Hindawi Publishing Corporation Conf. Papers in Math., 2013 (2013), Article ID 405914, 14 pages.
doi: 10.1155/2013/405914. |
show all references
References:
[1] |
N. F. Britton, Reaction-Diffusion Equations and their Applications to Biology, Academic Press Inc., London, 1986. |
[2] |
V. Calvez, A. Ebde, N. Meunier and A. Raoult, Mathematical and numerical modeling of the atherosclerotic plaque formation, ESAIM Proceedings, 28 (2009), 1-12.
doi: 10.1051/proc/2009036. |
[3] |
V. Calvez, J. Houot, N. Meunier, A. Raoult and G. Rusnakova, Mathematical and numerical modeling of early atherosclerotic lesions, ESAIM Proceedings, 30 (2010), 1-14.
doi: 10.1051/proc/2010002. |
[4] |
P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer-Verlag, Berlin Heidelberg, 1979. |
[5] |
A. Friedman, Partial Differential Equations of Parabolic Type, R.E. Krieger Pub. Co., 1983. |
[6] |
H. Daniel, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin Heidelberg, 1981. |
[7] |
N. Filipovic, D. Nikolic, I. Saveljic, Z. Milosevic, T. Exarchos, G. Pelosi and O. Parodi, Computer simulation of three-dimensional plaque formation and progression in the coronary artery, Elsevier, Computers & Fluids, 88 (2013), 826-833.
doi: 10.1016/j.compfluid.2013.07.006. |
[8] |
W. Hao and A. Friedman, The LDL-HDL profile determines the risk of atherosclerosis- a mathematical model, PLoS ONE, 9 (2014), e90497.
doi: 10.1371/journal.pone.0090497. |
[9] |
N. El Khatib, S. Genieys and V. Volpert, Atherosclerosis initiation modeled as an inflammatory process, Math. Model Nat. Phenom., 2 (2007), 126-141.
doi: 10.1051/mmnp:2008022. |
[10] |
N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Mathematical modeling of atherosclerosis as an inflammatory disease, Phil. Trans. R. Soc. A, 367 (2009), 4877-4886.
doi: 10.1098/rsta.2009.0142. |
[11] |
N. El Khatib, S. Genieys, B. Kazmierczak and V. Volpert, Reaction-diffusion model of atherosclerosis development, J. Math. Biol., 65 (2012), 349-374.
doi: 10.1007/s00285-011-0461-1. |
[12] |
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces, American Math. Society, 1996.
doi: 10.1090/gsm/012. |
[13] |
O. Ladyzhenskaya, V. Solonnikov and N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, American Math. Soc., 1968. |
[14] |
B. Liu and D. Tang, Computer simulations of atherosclerosis plaque growth in coronary arteries, Mol. Cell. Biomech., 7 (2010), 193-202. |
[15] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[16] |
C. V. Pao, Quasisolutions and global attractor of reaction-diffusion systems, Elsevier Science Ltd, Nonlinear Analysis, 26 (1996), 1889-1903.
doi: 10.1016/0362-546X(95)00058-4. |
[17] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
[18] |
R. Ross, Atherosclerosis - an inflammatory disease, Massachussets Medical Soc., 340 (1999), 115-126. |
[19] |
F. Rothe, Global Solutions of Reaction-Diffusion System, Springer-Verlag, Berlin Heidelberg, 1984. |
[20] |
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic bounded value problems, Indiana University Math. Journal, 21 (1972), 979-1000. |
[21] |
T. Silva, A. Sequeira, R. Santos and J. Tiago, Mathematical modeling of atherosclerotic plaque formation coupled with a non-Newtonian model of blood flow, Hindawi Publishing Corporation Conf. Papers in Math., 2013 (2013), Article ID 405914, 14 pages.
doi: 10.1155/2013/405914. |
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