Nonlinear wave equations outside of obstacles
speaker: Christopher Sogge (Johns Hopkins University)abstract: Existence theorems for nonlinear wave equations in Minkowski space, $\mathbb{R}\times \mathbb{R}3$, are usually obtained by proving coupled $L2$ (energy) and $L\infty$ (decay) estimates for linear equations. Typically both estimates involve the invariant vector fields introduced by Klainerman. In the case of three spatial dimensions, solutions of linear wave equations decay only like $1t$, which is not integrable. Therefore, to get global existence for equations with quadratic nonlinearities, one needs additional structure, such as the null condition of Christodoulou and Klainerman, to have global solutions for small data.
The case Dirichlet wave equations outside of compact obstacles is more complicated. Here it is very difficult to prove $1t$ decay unless one makes strong assumptions, such as assuming that the obstacles are star-shaped. Also, one cannot use the invariant vector fields that generate hyperbolic rotations since they have large coefficients and do not preserve the boundary conditions. In recent joint work with Jason Metcalfe, we get around this by using local exponential decay estimates of Morawetz, Ralston & Strauss as well as ones of Ikawa to prove that there is global existence outside of a wide class of obstacles that includes nontrapping ones. We also only need to assume a null condition for interactions between same-speed waves.
We also present joint work with Jason Metcalfe and Ann Stewart for solutions of nonlinear wave equations in wave guides: $\mathbb{R}\times (\mathbb{R}n \times \Omega$, where $\Omega\subset \mathbb{R}d$ is a smooth, compact domain. We present results involving either Klein-Gordon or wave equations with either Dirichlet or Neumann boundary conditions. In the Dirichlet case we improve on earlier results of P. H. Lesky & R. Racke. They had earlier proven global existence for $n>5$, while we are able to obtain the natural results for $n\ge3$. In the Neumann case things are more complicated. An additional assumption must be placed on the nonlinear terms. Under this natural assumption we obtain the natural results that there is global existence for nonlinear wave equations when $n\ge4$ and for Klein-Gordon equations when $n\ge 3$. In all cases the proofs rely on orthogonality and the fact that the dispersive estimates obtained earlier for free space are uniform in the Klein-Gordon mass parameter. The role of orthogonality had been recognized earlier by Lesky and Racke.
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