## 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 $\| v\| _{L^4(I\times M)} \leq C(I)\| v_0\| _{H^{s}(M)},$ $$s>s_0(d)$$, $$d\geq2$$.

### MSC:

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