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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

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), 687-719. 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., 2e série, 116 (1992), 325-352.  Google Scholar

[3]

P. Cherrier and A. Milani, Parabolic equations of von Karman type on Kähler manifolds, Bull. Sc. Math., 131 (2007), 375-396. 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), 19-36. 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, American Mathematical Society, Providence, RI 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, {17}, 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, New York, 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), 267-294.  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), 197-200.  Google Scholar

[10]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod - Gauthier-Villars, Paris, 1969.  Google Scholar

[11]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary value Problems, Vol. I, Springer Verlag, New York, 1972.  Google Scholar

[12]

A. Majda, Compressible Fluid Flows and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 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., T.M.A., 30 (1997), 2517-2524. 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), 687-719. 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., 2e série, 116 (1992), 325-352.  Google Scholar

[3]

P. Cherrier and A. Milani, Parabolic equations of von Karman type on Kähler manifolds, Bull. Sc. Math., 131 (2007), 375-396. 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), 19-36. 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, American Mathematical Society, Providence, RI 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, {17}, 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, New York, 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), 267-294.  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), 197-200.  Google Scholar

[10]

J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod - Gauthier-Villars, Paris, 1969.  Google Scholar

[11]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary value Problems, Vol. I, Springer Verlag, New York, 1972.  Google Scholar

[12]

A. Majda, Compressible Fluid Flows and Systems of Conservation Laws in Several Space Variables, Springer-Verlag, New York, 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., T.M.A., 30 (1997), 2517-2524. doi: 10.1016/S0362-546X(97)00314-3.  Google Scholar

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