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July  2011, 10(4): 1079-1096. doi: 10.3934/cpaa.2011.10.1079

Nonlinear hyperbolic-elliptic systems in the bounded domain

N. V. Chemetov 1,

1. 

CMAF/Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal

Received  August 2010 Revised  October 2010 Published  April 2011

In the article we study a hyperbolic-elliptic system of PDE. The system can describe two different physical phenomena: 1st one is the motion of magnetic vortices in the II-type superconductor and 2nd one is the collective motion of cells. Motivated by real physics, we consider this system with boundary conditions, describing the flux of vortices (and cells, respectively) through the boundary of the domain. We prove the global solvability of this problem. To show the solvability result we use a "viscous" parabolic-elliptic system. Since the viscous solutions do not have a compactness property, we justify the limit transition on a vanishing viscosity, using a kinetic formulation of our problem. As the final result of all considerations we have solved a very important question related with a so-called "boundary layer problem", showing the strong convergence of the viscous solutions to the solution of our hyperbolic-elliptic system.
Keywords: solvability., flux through the boundary, Keller-Segel model, Nonlinear hyperbolic-elliptic system, mean-field vortex model for II-type superconductor.
Mathematics Subject Classification: Primary: 35D05, 35L60; Secondary: 78A25, 92C1.
Citation: N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1079-1096. doi: 10.3934/cpaa.2011.10.1079
References:
[1]

S. N. Antontsev and N. V. Chemetov, Flux of superconducting vortices through a domain, SIAM J. Math. Anal., 39 (2007), 263-280. doi: 10.1137/060655146.  Google Scholar

[2]

S. N. Antontsev and N. V. Chemetov, Superconducting Vortices: Chapman Full Model,, in, ().  doi: 10.1007/978-3-0346-0152-8_3.  Google Scholar

[3]

S. J. Chapman, A hierarchy of models for type-II superconductors, SIAM Review, 42 (2000), 555-598. doi: 10.1137/S0036144599371913.  Google Scholar

[4]

S. J. Chapman, A Mean-Field Model of Superconducting Vortices in Three Dimensions, SIAM J. Appl. Math., 55 (1995), 1259-1274. doi: 10.1137/S0036139994263665.  Google Scholar

[5]

G.-Q. Chen and H. Frid, On the theory of divergence-measure fields and its applications, Bol. Soc. Bras. Mat., 32 (2001), 401-433. doi: 10.1007/BF01233674.  Google Scholar

[6]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, 1999.  Google Scholar

[7]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.  Google Scholar

[8]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I) Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165; II) Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.  Google Scholar

[9]

C. De Lellis, Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio, Bourbaki Seminar, Preprint (2007), 1-26.  Google Scholar

[10]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, "Linear and Quasilinear Equations of Parabolic Type," American Mathematical Society, Providence RJ, 1968.  Google Scholar

[11]

O. A. Ladyzhenskaya and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York and London, 1968.  Google Scholar

[12]

J. L. Lions and E. Magenes, "Problèmes aux limites non Homogénes et Applications," Dunod, Paris, 1968.  Google Scholar

[13]

J. Malek, J. Necas, M. Rokyta and M. Ruzicka, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996.  Google Scholar

[14]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335. doi: 10.1090/S0002-9947-08-04656-4.  Google Scholar

[15]

P. I. Plotnikov, "Ultraparabolic Muskat Equations," Preprint No. 6, University Beira Interior, Covilhã (Portugal), 2000. Google Scholar

[16]

P. Plotnikov and S. Sazhenkov, Kinetic formulation for the Graetz-Nusselt ultra-parabolic equation, J. Math. Anal. Appl., 304 (2005), 703-724. doi: 10.1016/j.jmaa.2004.09.050.  Google Scholar

show all references

References:
[1]

S. N. Antontsev and N. V. Chemetov, Flux of superconducting vortices through a domain, SIAM J. Math. Anal., 39 (2007), 263-280. doi: 10.1137/060655146.  Google Scholar

[2]

S. N. Antontsev and N. V. Chemetov, Superconducting Vortices: Chapman Full Model,, in, ().  doi: 10.1007/978-3-0346-0152-8_3.  Google Scholar

[3]

S. J. Chapman, A hierarchy of models for type-II superconductors, SIAM Review, 42 (2000), 555-598. doi: 10.1137/S0036144599371913.  Google Scholar

[4]

S. J. Chapman, A Mean-Field Model of Superconducting Vortices in Three Dimensions, SIAM J. Appl. Math., 55 (1995), 1259-1274. doi: 10.1137/S0036139994263665.  Google Scholar

[5]

G.-Q. Chen and H. Frid, On the theory of divergence-measure fields and its applications, Bol. Soc. Bras. Mat., 32 (2001), 401-433. doi: 10.1007/BF01233674.  Google Scholar

[6]

L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, 1999.  Google Scholar

[7]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.  Google Scholar

[8]

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I) Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165; II) Jahresber. Deutsch. Math.-Verein., 106 (2004), 51-69.  Google Scholar

[9]

C. De Lellis, Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio, Bourbaki Seminar, Preprint (2007), 1-26.  Google Scholar

[10]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, "Linear and Quasilinear Equations of Parabolic Type," American Mathematical Society, Providence RJ, 1968.  Google Scholar

[11]

O. A. Ladyzhenskaya and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York and London, 1968.  Google Scholar

[12]

J. L. Lions and E. Magenes, "Problèmes aux limites non Homogénes et Applications," Dunod, Paris, 1968.  Google Scholar

[13]

J. Malek, J. Necas, M. Rokyta and M. Ruzicka, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996.  Google Scholar

[14]

B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model, Trans. Amer. Math. Soc., 361 (2009), 2319-2335. doi: 10.1090/S0002-9947-08-04656-4.  Google Scholar

[15]

P. I. Plotnikov, "Ultraparabolic Muskat Equations," Preprint No. 6, University Beira Interior, Covilhã (Portugal), 2000. Google Scholar

[16]

P. Plotnikov and S. Sazhenkov, Kinetic formulation for the Graetz-Nusselt ultra-parabolic equation, J. Math. Anal. Appl., 304 (2005), 703-724. doi: 10.1016/j.jmaa.2004.09.050.  Google Scholar

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