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Communications in Mathematical Physics

, Volume 30, Issue 1, pp 79–81 | Cite as

On global solutions for non-linear Hamiltonian evolution equations

  • J. Marsden
Article

Abstract

It is shown that partial differential equations of Hamiltonian type admit global solutions in time if (a) the initial data is near equilibrium (or the coupling constant is small) (b) the linear terms have positive energy and (c) the non-linear terms are smooth functions in the topology of the linearized energy norm. The non-linear terms need not have positive energy. The result is applied to non-linear wave equations in which the interaction energy is not necessarily positive.

Keywords

Neural Network Partial Differential Equation Initial Data Wave Equation Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Chadam, J. M.: Asymptotics for □u =m 2 u+G. — Global, I.: Existence and decay. Bull. Am. Math. Soc.76, 1032–5 (1970).Google Scholar
  2. 2.
    Chernoff, P., Marsden, J.: Hamiltonian systems and quantum mechanics (in preparation).Google Scholar
  3. 3.
    Fischer, A., Marsden, J.: The existence of complete spacetimes (in preparation).Google Scholar
  4. 4.
    Segal, I.: Non-linear semi-groups. Ann. Math.78, 339–364 (1963).Google Scholar

Copyright information

© Springer-Verlag 1973

Authors and Affiliations

  • J. Marsden
    • 1
  1. 1.Department of PhysicsUniversity of CaliforniaBerkeley

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