Hyperbolic equations of Von Karman type

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February 2016, 9(1): 125-137. doi: 10.3934/dcdss.2016.9.125

Hyperbolic equations of Von Karman type

Pascal Cherrier ^1, and Albert Milani ^2,

1.	Département de Mathématiques, UPMC (Paris VI), Paris, France
2.	Department of Mathematics, University of Wisconsin, Milwaukee, WI 53201, United States

Received September 2014 Revised February 2015 Published December 2015

Abstract
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We report some recent results on weak and semi-strong solutions to a coupled hyperbolic-elliptic system of Von Karman type on ${IR}^{2m}$, $m \in {IN}_{\geq 2}$.

Keywords: Hyperbolic equations, local solutions., von Karman equations, global solutions, weak solutions, strong solutions.

Mathematics Subject Classification: Primary: 32Q15, 35L30, 35K52, 53B35, 58J45; Secondary: 53C44, 53C8.

Citation: Pascal Cherrier, Albert Milani. Hyperbolic equations of Von Karman type. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 125-137. doi: 10.3934/dcdss.2016.9.125

References:

[1]	M. S. Berger, On von Karman's equations and the buckling of a thin elastic plate,, Comm. Pure App. Math., 20 (1967), 687. doi: 10.1002/cpa.3160200405. Google Scholar
[2]	P. Cherrier and A. Milani, Equations of von Karman type on compact Kähler manifolds,, Bull. Sc. Math., 116* (1992), 325. Google Scholar*
[3]	P. Cherrier and A. Milani, Parabolic equations of von Karman type on Kähler manifolds,, Bull. Sc. Math., 131 (2007), 375. doi: 10.1016/j.bulsci.2006.05.008. Google Scholar
[4]	P. Cherrier and A. Milani, Hyperbolic equations of von Karman type on Kähler manifolds,, Bull. Sc. Math., 136 (2012), 19. doi: 10.1016/j.bulsci.2011.07.017. Google Scholar
[5]	P. Cherrier and A. Milani, Linear and Quasi-Linear Evolution Equations in Hilbert Spaces,, GSM vol. 35, (2012). Google Scholar
[6]	P. Cherrier and A. Milani, Evolution Equations of von Karman Type,, To appear in the Lecture Notes of the Unione Matematica Italiana, (2015). doi: 10.1007/978-3-319-20997-5. Google Scholar
[7]	I. Chuesov and I. Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long-Time Dynamics,, Springer-Verlag, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar
[8]	A. Favini, M. A. Horn, I. Lasiecka, and D. Tataru, Global existence of solutions to a von Karman system with nonlinear boundary dissipation,, J. Diff. Int. Eqs., 9 (1996), 267. Google Scholar
[9]	A. Favini, M. A. Horn, I. Lasiecka, and D. Tataru, Addendum to the paper "Global existence of solutions to a von Karman system with nonlinear boundary dissipation'',, J. Diff. Int. Eqs., 10 (1997), 197. Google Scholar
[10]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,, Dunod - Gauthier-Villars, (1969). Google Scholar
[11]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary value Problems, Vol. I,, Springer Verlag, (1972). Google Scholar
[12]	A. Majda, Compressible Fluid Flows and Systems of Conservation Laws in Several Space Variables,, Springer-Verlag, (1984). doi: 10.1007/978-1-4612-1116-7. Google Scholar
[13]	Y. Shizuta, Strong continuity in time of the solution of the mixed problem for symmetric hyperbolic systems,, Nonlinear Anal., 30 (1997), 2517. doi: 10.1016/S0362-546X(97)00314-3. Google Scholar

show all references

References:

[1]	M. S. Berger, On von Karman's equations and the buckling of a thin elastic plate,, Comm. Pure App. Math., 20 (1967), 687. doi: 10.1002/cpa.3160200405. Google Scholar
[2]	P. Cherrier and A. Milani, Equations of von Karman type on compact Kähler manifolds,, Bull. Sc. Math., 116* (1992), 325. Google Scholar*
[3]	P. Cherrier and A. Milani, Parabolic equations of von Karman type on Kähler manifolds,, Bull. Sc. Math., 131 (2007), 375. doi: 10.1016/j.bulsci.2006.05.008. Google Scholar
[4]	P. Cherrier and A. Milani, Hyperbolic equations of von Karman type on Kähler manifolds,, Bull. Sc. Math., 136 (2012), 19. doi: 10.1016/j.bulsci.2011.07.017. Google Scholar
[5]	P. Cherrier and A. Milani, Linear and Quasi-Linear Evolution Equations in Hilbert Spaces,, GSM vol. 35, (2012). Google Scholar
[6]	P. Cherrier and A. Milani, Evolution Equations of von Karman Type,, To appear in the Lecture Notes of the Unione Matematica Italiana, (2015). doi: 10.1007/978-3-319-20997-5. Google Scholar
[7]	I. Chuesov and I. Lasiecka, Von Karman Evolution Equations: Well-Posedness and Long-Time Dynamics,, Springer-Verlag, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar
[8]	A. Favini, M. A. Horn, I. Lasiecka, and D. Tataru, Global existence of solutions to a von Karman system with nonlinear boundary dissipation,, J. Diff. Int. Eqs., 9 (1996), 267. Google Scholar
[9]	A. Favini, M. A. Horn, I. Lasiecka, and D. Tataru, Addendum to the paper "Global existence of solutions to a von Karman system with nonlinear boundary dissipation'',, J. Diff. Int. Eqs., 10 (1997), 197. Google Scholar
[10]	J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,, Dunod - Gauthier-Villars, (1969). Google Scholar
[11]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary value Problems, Vol. I,, Springer Verlag, (1972). Google Scholar
[12]	A. Majda, Compressible Fluid Flows and Systems of Conservation Laws in Several Space Variables,, Springer-Verlag, (1984). doi: 10.1007/978-1-4612-1116-7. Google Scholar
[13]	Y. Shizuta, Strong continuity in time of the solution of the mixed problem for symmetric hyperbolic systems,, Nonlinear Anal., 30 (1997), 2517. doi: 10.1016/S0362-546X(97)00314-3. Google Scholar

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