Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. (English) Zbl 1067.58027

Let \(M\) be a closed \(d\)-dimensional Riemannian manifold. Let \(\Delta\) be the Laplace-Beltrami operator on \(M\). Fix numbers \(p\geq 2\), \(q < \infty\) such that \[ {2\over p} + {d \over q} = {d \over 2}. \] The authors show that for any finite time interval \(I\) the solution to the Schrödinger equation \[ i{\partial\over{\partial t}}v + \Delta v = 0, \text{ } v(0,x) = v_0(x), \] satisfies the Strichartz estimate \[ \| v\| _{L^p(I,L^q(M))} \leq C(I)\| v_0\| _{H^{1/p}(M)}. \] From this they deduce unique solvability of certain nonlinear Schrödinger equations. In the very special case that all geodesics of \(M\) are closed with a common period the estimate is improved to \[ \| v\| _{L^4(I\times M)} \leq C(I)\| v_0\| _{H^{s}(M)}, \] \(s>s_0(d)\), \(d\geq2\).


58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C22 Geodesics in global differential geometry
35Q55 NLS equations (nonlinear Schrödinger equations)
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