American Institute of Mathematical Sciences

Advanced
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. 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

[1]

Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1617-1633. doi: 10.3934/dcdsb.2016014
[2]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867
[3]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[4]

Yu-Zhu Wang, Yin-Xia Wang. Local existence of strong solutions to the three dimensional compressible MHD equations with partial viscosity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 851-866. doi: 10.3934/cpaa.2013.12.851
[5]

Honglv Ma, Jin Zhang, Chengkui Zhong. Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4721-4737. doi: 10.3934/dcdsb.2019027
[6]

Xiaoli Li, Dehua Wang. Global solutions to the incompressible magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 763-783. doi: 10.3934/cpaa.2012.11.763
[7]

Wanwan Wang, Hongxia Zhang, Huyuan Chen. Remarks on weak solutions of fractional elliptic equations. Communications on Pure & Applied Analysis, 2016, 15 (2) : 335-340. doi: 10.3934/cpaa.2016.15.335
[8]

Elder Jesús Villamizar-Roa, Henry Lamos-Díaz, Gilberto Arenas-Díaz. Very weak solutions for the magnetohydrodynamic type equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 957-972. doi: 10.3934/dcdsb.2008.10.957
[9]

Wendong Wang, Liqun Zhang. The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1261-1275. doi: 10.3934/dcds.2011.29.1261
[10]

Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293
[11]

Ondřej Kreml, Milan Pokorný. On the local strong solutions for the FENE dumbbell model. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 311-324. doi: 10.3934/dcdss.2010.3.311
[12]

Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic & Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041
[13]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613
[14]

Šárka Nečasová, Joerg Wolf. On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1539-1562. doi: 10.3934/dcds.2016.36.1539
[15]

Zhuan Ye. Remark on exponential decay-in-time of global strong solutions to 3D inhomogeneous incompressible micropolar equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6725-6743. doi: 10.3934/dcdsb.2019164
[16]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[17]

Cheng-Jie Liu, Ya-Guang Wang, Tong Yang. Global existence of weak solutions to the three-dimensional Prandtl equations with a special structure. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2011-2029. doi: 10.3934/dcdss.2016082
[18]

Daniel Pardo, José Valero, Ángel Giménez. Global attractors for weak solutions of the three-dimensional Navier-Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3569-3590. doi: 10.3934/dcdsb.2018279
[19]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1553-1561. doi: 10.3934/cpaa.2014.13.1553
[20]

Huajun Gong, Jinkai Li. Global existence of strong solutions to incompressible MHD. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1337-1345. doi: 10.3934/cpaa.2014.13.1337

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences