##### Department of Mathematics,

University of California San Diego

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### Special Recruitment Colloquium

## Chongchun Zeng

#### University of Virginia

## Wave equations with strong constraining potentials

##### Abstract:

In this talk, we consider a vector valued nonlinear wave equation of the unknown $u(t, x) \in R^n$. Suppose the energy density of the equation contains a nonlinear potential $V(u)/\epsilon^2$ which achieves its minimal value $0$ on a submanifold $M$ in $R^n$. As $\epsilon$ approaches $0$, i.e. as this potential approaches infinity, we are interested in the convergence of finite energy solutions. Through a multi-scale formal asymptotic expansion involving rapid oscillations, J. Keller and K. Rubinstein (1991) found that the singular limits of those solutions satisfy a hyperboic PDE system. We rigorously justified this convergence procedure and the local well-posedness of this system. In particular, when the initial data is well prepared, the limit system reduces to the wave map equation targeted on $M$. The comparison between the structures of the wave equation and the limit system and a more general picture of Hamiltonian PDEs with strong potentials, will also be briefly discussed.

Host: Hans Lindblad

### January 11, 2005

### 9:00 AM

### AP&M 6438

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