# American Institute of Mathematical Sciences

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

## Nonlinear hyperbolic-elliptic systems in the bounded domain

 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.
Citation: N. V. Chemetov. Nonlinear hyperbolic-elliptic systems in the bounded domain. Communications on Pure and 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. [2] S. N. Antontsev and N. V. Chemetov, Superconducting Vortices: Chapman Full Model,, in, ().  doi: 10.1007/978-3-0346-0152-8_3. [3] S. J. Chapman, A hierarchy of models for type-II superconductors, SIAM Review, 42 (2000), 555-598. doi: 10.1137/S0036144599371913. [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. [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. [6] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, 1999. [7] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983. [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. [9] C. De Lellis, Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio, Bourbaki Seminar, Preprint (2007), 1-26. [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. [11] O. A. Ladyzhenskaya and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York and London, 1968. [12] J. L. Lions and E. Magenes, "Problèmes aux limites non Homogénes et Applications," Dunod, Paris, 1968. [13] J. Malek, J. Necas, M. Rokyta and M. Ruzicka, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996. [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. [15] P. I. Plotnikov, "Ultraparabolic Muskat Equations," Preprint No. 6, University Beira Interior, Covilhã (Portugal), 2000. [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.

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##### 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. [2] S. N. Antontsev and N. V. Chemetov, Superconducting Vortices: Chapman Full Model,, in, ().  doi: 10.1007/978-3-0346-0152-8_3. [3] S. J. Chapman, A hierarchy of models for type-II superconductors, SIAM Review, 42 (2000), 555-598. doi: 10.1137/S0036144599371913. [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. [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. [6] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions," Studies in Advanced Mathematics, CRC Press, 1999. [7] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983. [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. [9] C. De Lellis, Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio, Bourbaki Seminar, Preprint (2007), 1-26. [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. [11] O. A. Ladyzhenskaya and N. N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Academic Press, New York and London, 1968. [12] J. L. Lions and E. Magenes, "Problèmes aux limites non Homogénes et Applications," Dunod, Paris, 1968. [13] J. Malek, J. Necas, M. Rokyta and M. Ruzicka, "Weak and Measure-valued Solutions to Evolutionary PDEs," Chapman & Hall, London, 1996. [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. [15] P. I. Plotnikov, "Ultraparabolic Muskat Equations," Preprint No. 6, University Beira Interior, Covilhã (Portugal), 2000. [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.
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